Communications in Mathematical Physics - Volume 224

Commun. Math. Phys. 224, 1 – 2 (2001)

Communications in

Mathematical Physics

© Springer-Verlag 2001

Preface The present issue of CMP is dedicated to Joel L. Lebowitz, the Willliam Hill Professor of Mathematics and Physics of Rutgers University, in recognition of his outstanding contributions and scientific leadership in statistical physics and related areas of mathematical physics. In his research, Joel Lebowitz has addressed topics in statistical physics ranging over equilibrium and non-equilibrium phenomena. His works reflect the rare combination of a deep understanding of the relevant physics and the ability to see through the mathematical formalism in which physics is being expressed. He has reveled both in learning new physical phenomena and in shedding light on basic questions of physics through mathematically rigorous results. The subjects on which Joel has worked, with numerous collaborators, include: the theory of equilibrium fluids, rigorous approach to the liquidvapor transition, critical phenomena in Ising type models, the statistical mechanics of Coulomb systems, phase segregation studied in conjunction with pioneering implementations of Monte-Carlo simulations, ergodic theory in relation to fundamental issues of statistical mechanics, kinetic theory, and the structure of non-equilibrium steady states. Joel’s contributions have been widely recognized; in 1980 he was elected to the National Academy of Sciences and in 1992 he was awarded the Boltzmann Medal of the Union of Pure and Applied Physics.

2

M. Aizenman, H. Spohn

An inseparable aspect of the Lebowitz experience for the many whom he has touched has been the sense of his personal engagement and care. Having witnessed the holocaust as a young teenager, Joel emerged from the devastation and the inhumanely twisted reality that he was thrown into with a tenacious commitment to stand up for human rights and dignity. He has inspired many with the message that scientists should use the unique opportunities accorded to them to be at the forefront of that struggle. In 1999 he was awarded the Scientific Freedom and Responsibility Award of the American Association for the Advancement of Science for “. . . his tireless devotion to the rights of scientists in oppressive regimes throughout the world and his extraordinary creativity in finding ways to help these scientists survive their ordeal ”. In the late fifties Joel instituted a unique series of biannual meetings in Statistical Mechanics, which he has been running uninterrupted since then. These conferences have provided an invaluable forum for the presentation of recent results and for stimulating exchanges on both new emerging vistas and long outstanding fundamental issues. In a fitting reflection of the spirit of these meetings, Joel’s 70th birthday is being marked with two special issues: Physica A Vol. 279, Nos. 1–4, where the reader may also find a more complete biographical sketch, and this issue of CMP which presents rigorous results in fields related to Joel’s interests. Joel L. Lebowitz stands out in his never satisfied curiosity, the clarity of his thought, and his exceptional ability to reach out and stimulate others. He has inspired and guided numerous colleagues and students. This issue is dedicated with deep gratitude, with a sense of joy at having had the privilege to interact with him, and with best wishes for Joel’s continuing quest. Michael Aizenman Herbert Spohn

Commun. Math. Phys. 224, 3 – 16 (2001)

Communications in

Mathematical Physics

© Springer-Verlag 2001

Entropy Production in Quantum Spin Systems David Ruelle IHES, route de Chartes, 91440 Bures sur Yvette, France. E-mail: [email protected] Received: 7 June 2000 / Accepted: 5 November 2000

Abstract: We consider a quantum spin system consisting of a finite subsystem connected to infinite reservoirs at different temperatures. In this setup we define nonequilibrium steady states and prove that the rate of entropy production in such states is nonnegative. For several decades, Joel Lebowitz has been the soul of research in statistical mechanics. He now plays a central role in the development of new ideas which reshape our understanding of nonequilibrium. The present paper, dedicated to Joel on his 70th birthday, extends some of the new ideas to quantum systems. 1. Introduction Consider a physical situation where a “small” system S is connected to different “large” heat reservoirs Ra (a = 1, 2, . . . ) at different inverse temperatures βa . We want to define nonequilibrium steady states for the total system L = S + R1 + R2 + . . . , and verify that the rate of entropy production in such states is ≥ 0. The model which we discuss in this paper is that of a fairly realistic quantum spin system. In what follows we first describe the model and state our assumptions (A1), (A2), (A3). In this setup we introduce nonequilibrium steady states ρ as states which, in the distant past, described noninteracting reservoirs at different temperatures. Under suitable conditions we check that our definition does not depend on where we place the boundary between the small system and the reservoirs. Our definition of the entropy production eρ also does not depend on where the boundary between the small system and the reservoirs is placed. With this definition we prove eρ ≥ 0. By contrast with an earlier paper [4], we omit here assumptions of asymptotic abelianness in time which are difficult to verify, the definition of nonequilibrium steady states is more general, but we obtain less specific results.

4

D. Ruelle

2. Description of the Model (See [3, 1]) Let L be a countably infinite set. For each x ∈ L, let Hx be a finite dimensional complex Hilbert space, and write HX = ⊗x∈X Hx if X is a finite subset of L. We let AX be the C∗ -algebra of bounded operators on HX , and if Y ⊂ X we identify AY with a subalgebra of AX by the map AY  → AY ⊗1HX\Y ⊂ AX . We write L as a finite union L = ∪a≥0 Ra , where R0 = S is finite (small system) and the Ra with a > 0 are infinite (reservoirs). We can then define the quasilocal C∗ algebras Aa , A as the norm closures of

AX , AX X⊂Ra

X⊂L

repectively. Note that all these algebras have a common unit element 1. In this setup we assume that an interaction  : X → (X) is given such that (X) is a selfadjoint element of AX for every finite X ⊂ L. Also, for each reservoir, we prescribe an inverse temperature βa > 0 and a state σa on Aa . The assumptions (A1), (A2), (A3). Assumption A1. The interaction  satisfies  

 λ = enλ sup n≥0

x∈L Xx:cardX=n+1

(X) < ∞

for some λ > 0. The importance of this assumption is that it allows us to equip A with a one-parameter group (α t ) of automorphisms1 defining a time evolution. Introduce a linear operator δ : ∪X⊂L AX → A such that  δA = i [(Y ), A] if A ∈ AX . Y :Y ∩X=∅

If A ∈ AX , one checks that

δ m A ≤ A eλcardX m!(2λ−1  λ )m . The strongly continuous one-parameter group (α t ) of ∗-automorphisms of A is given by ∞ m  t m αt A = δ A m! m=0

if A ∈ ∪X⊂L AX and |t| < λ/2  λ . (More generally one could take A ∈ Aλ , where Aλ is defined in the Appendix.) Let  H = (X) X⊂

for finite  ⊂ L. Writing  → L if  eventually contains each finite X ⊂ L we have, assuming A ∈ A, lim eitH Ae−itH − α t A = 0 →L

uniformly for t in compact intervals of R. 1 See [1] Theorem 6.2.4 (or [3] Sect. 7.6).

Entropy Production in Quantum Spin Systems

5

Assumption A2. (X) = 0 if X ∩ S = ∅, X ∩ Ra  = ∅, X ∩ Rb  = ∅ for different a, b > 0. Note that the description of the interaction  is somewhat ambiguous because anything ascribed to (X) might also be ascribed to (Y ) for Y ⊃ X. Condition (A2) means that in our accounting, if a part of the interaction connects two different reservoirs, it must also involve the small system S. Assumption A3. If a > 0, let a be the restriction of the interaction  to subsets of Ra and write  a (X) = HRa ∩ . Ha = X⊂Ra ∩

Let also the interactions () be given such that

() λ ≤ K < ∞ and write



Ba =

(1)

() (X).

X⊂Ra ∩

We assume that, for a suitable sequence  → L, lim

Tr HRa ∩ (e−βa (Ha +Ba ) A) Tr HRa ∩ e−βa (Ha +Ba )

→L

= σa (A)

if A ∈ Aa : this defines a state σa on Aa , depending on the choice of (() ) and the sequence  → L. Furthermore we assume that for each finite X there is X such that () (Y ) = 0 if  ⊃ X and Y ⊂ X; therefore

[Ba , A] = 0

(2)

if  ⊃ X and A ∈ AX . In particular we can take all () = 0. Using (3) below, it is readily verified that σa is a βa -KMS state (see [2]) for the one-parameter group (α˘ at ) of automorphisms of Aa corresponding to the interaction a . [I do not know which of the βa -KMS states can be obtained in this manner]. Note that the assumptions (A1), (A2), (A3) can be explicitly verified in specific cases. From (A3) we obtain the following result. Lemma. lim eit (Ha +Ba ) Ae−it (Ha +Ba ) − α˘ at A = 0

→L

(3)

for a > 0, and 

lim eit (H +

→L

a>0

Ba )



Ae−it (H +

uniformly for t in compact intervals of R.

a>0

Ba )

− α t A = 0

(4)

6

D. Ruelle 



t it (H + a>0 Ba ) Ae−it (H + Proof. We  prove (4). Write α A = e i[H + a>0 Ba , A]. If A ∈ ∪X AX we see using (1) that t α A=

a>0

Ba )

and δ A =

∞ m  t m δ A m! 

m=0

converges uniformly in  for |t| < λ/2(  λ + K). Using also (2), it is shown in the m A → δ m A in A when  → L. Therefore Appendix that δ t A − α t A = 0 lim α

→L

when A ∈ ∪X AX , uniformly for |t| ≤ T < λ/2(  λ + K). But the condition A ∈ ∪X AX is removed by density, and the condition |t| ≤ T < λ/2(  λ + K) by use of the group property. The proof of (3) is similar.    The KMS state σ . The interaction a>0 βa a , evaluated at X is βa a (X) if X ⊂ Ra and 0 if X is not contained in one of the Ra . The corresponding one-parameter group (β t ) of automorphisms of A has, according to (A3), the KMS state2 σ = ⊗a≥0 σa where σ0 is the normalized trace on A0 = AS . In fact  Tr H (exp(− a βa (Ha + Ba ))A)  σ (A) = lim . (5) →L Tr H exp(− a βa (Ha + Ba )) Nonequilibrium steady states. We call nonequilibrium steady states (NESS) associated with σ the limits when T → ∞ of  1 T dt (α t )∗ σ T 0 using the w∗ -topology on the dual A∗ of A. With respect to this topology, the set % of NESS is compact, nonempty, and the elements of % are (α t )∗ -invariant states on A. This definition generalizes that given in [4] where, under stringent asymptotic abeliannes conditions, the existence of a single NESS was obtained. Dependence on the decomposition L = S + R1 + R2 + . . . 3 . Our definition of σ , and therefore of % depends on the choice of a decomposition of L into small system and reservoirs. If S is replaced by a finite set S  ⊃ S and the Ra by correspondingly  smaller sets Ra ⊂ Ra one checks that (A1), (A2),(A3) remain  valid. If a is the  restriction of  to subsets of Ra , the replacement of βa a by βa a changes (β t ) to a one-parameter group (β  t ) and σ to a state σ  . These changes are in fact bounded perturbations covered by Theorem 5.4.4 and Corollary 5.4.5 of [1]. The map σ → σ  (of KMS states for (β t ) to KMS states for (β  t )) is nonlinear (as can be guessed from (5)) and T T therefore we cannot expect that T1 0 dt (α t )∗ σ  has the same limit as T1 0 dt (α t )∗ σ 2 The state σ corresponds to the inverse temperature +1 rather than the inverse temperature −1 favored in the mathematical literature. 3 This section and the following Proposition are in the nature of a technical digression, and may be omitted by the reader essentially interested in the positivity of the entropy production.

Entropy Production in Quantum Spin Systems

7

in general, but the deviation is not really bad. The (central) decomposition of KMS states into extremal KMS states gives factor states. If σ is assumed to be a factor state, and T (α t ) is asymptotically abelian, one finds that lim T1 0 dt (α t )∗ σ does not depend on the decomposition L = S + R1 + R2 + . . . , as the following result indicates. Proposition. Using the above notation, assume that σ is a factor state, and that lim [α t A, B] = 0

t→∞

when A, B ∈ A. Then, when T → ∞,   1 T 1 T dt (α t )∗ σ  = lim dt (α t )∗ σ. lim T 0 T 0 Proof. Let us introduce the GNS representation (H, π, ') associated with σ so that if  1 T ρ = lim dt (α t )∗ σ, T 0 we have

 1 T dt (', π(α t A)'). T 0 By restricting T to a subsequence we may assume that in the weak operator topology  1 T lim dt π(α t A) = A¯ ∈ π(A) T 0 ρ(A) = lim

and by assumption we also have A¯ ∈ π(A) , hence A¯ ∈ π(A) ∩ π(A) = {λ1} since σ is a factor state. But we may write σ  (·) = (' , π(·)' ): this follows from the perturbation theory of [1] (see proof of Theorem 5.4.4). We have thus   1 T 1 T lim dt σ  (α t A) = lim dt (' , π(α t A)' ) T 0 T 0   1 T 1 T t = lim dt (', π(α A)') = lim dt σ (α t A) T 0 T 0 as announced.   Entropy production. For finite  ⊂ L we have defined  (X), H = X⊂

but HL , HRa do not make sense. We can however define [HL , HRa ] = lim [H , HRa ∩ ] = lim [H , Ha ]. →L

→L

We have indeed [H , Ha ] = [H − Ha , Ha ] = [H −

 b>0

Hb , Ha ]

8

D. Ruelle

and (A2) gives H −



Hb =

b>0





x∈S X:x∈X⊂

1 (X) card(X ∩ S)

[implying the existence of the limit lim→L (H − A]. Using (A1) we obtain



b>0 Hb )

= HL −



b>0 HRb

∈

[(X), Ha ] ≤ 2λ−1  λ (X) eλcardX , hence



[(X), Ha ] ≤ 2λ−1  λ eλ  λ ,

Xx

and [H , Ha ] has a limit [HL , HRa ] ∈ A when  → L with

[HL , HRa ] ≤ 2cardSλ−1 eλ  2λ . The operator i[HL , HRa ] may be interpreted as the rate of increase of the energy of the reservoir Ra or (since this energy is infinite) rather the rate of transfer of energy to Ra from the rest of the system. According to conventional wisdom we define the rate of entropy production in an (α t )∗ -invariant state ρ as eρ =



βa ρ(i[HL , HRa ])

a>0

(this definition does not require that ρ ∈ %). Remark. If we replace S by a finite set S  ⊃ S and the Ra by the correspondingly smaller sets Ra ⊂ Ra , we have noted earlier that (A1), (A2), (A3) remain satisfied. As a consequence of (A1) we have   i[HL , HRa − HRa ] = lim i[H , Ha − Ha ] = lim δ(Ha − Ha ) →L

→L

(where the operator δ has been defined just after (A3)), hence  ρ(i[HL , HRa − HRa ]) = lim ρ(δ(Ha − Ha )) = 0 →L

i.e. , the rate of entropy production is unchanged when S and the Ra are replaced by S  and the Ra . The reason why we do not have ρ(i[HL , HRa ]) = 0 is mathematically because HRa is “infinite” (HRa ∈ / A), and physically because our definition of ρ(i[HL , HRa ]) takes into account the flux of energy into Ra from S, but not the flux at infinity. Theorem. The entropy production in a NESS is nonnegative, i.e. , eρ ≥ 0 if ρ ∈ %.

Entropy Production in Quantum Spin Systems

9

We have seen that [HL , HRa ] = lim [H , Ha ] →L  Hb , Ha ]. = lim [H − →L

b>0



Therefore, using (A3) and [Hb + Bb , a>0 βa (Ha + Ba )] = 0, we find    βa [HL , HRa ] = lim [H − Hb , βa Ha ] →L

a>0

a>0

b>0

= lim [H − →L



Hb ,

→L



βa (Ha + Ba )]

a>0

b>0

= lim [H +



Bb ,



βa (Ha + Ba )]

a>0

b>0

in the sense of norm convergence. We also have, for some sequence of values of T tending to infinity and all A ∈ A,   1 T 1 T t ρ(A) = lim dt σ (α t A) = lim lim dt σ (α A), T →∞ T 0 T →∞ →L T 0 where, by (4),



t α A = eit (H +

a>0

Ba )



Ae−it (H +

a>0

Ba )

→ α t A in norm

when  → L, uniformly for t ∈ [0, T ]. Write  HB = H + Ba , G =



a>0



βa (Ha + Ba ) + log Tr H exp

a>0

 a>0



 βa (Ha + Ba ) .

a>0

Then the entropy production is eρ = ρ(i

−

i T →∞ →L T

βa [HL , HRa ]) = lim lim

 0

T

dt σ (eitHB [HB , G ] e−itHB )

and the convergence when  → L of the operator (eitHB [HB , G ]e−itHB ) is uniform for t ∈ [0, T ]. According to (A3) we may choose the  tending to L such that Tr H e−G (·) tends to σ (·) in the w ∗ -topology, hence  T i dt Tr H (e−G eitHB [HB , G ]e−itHB ) eρ = lim lim T →∞ →L T 0    1 T −G d itHB −itHB = lim lim (e dt Tr H e G e ) T →∞ →L T 0 dt 1 = lim lim Tr H (e−G eiT HB G e−iT HB ) − Tr H (e−G G ) T →∞ →L T and the theorem follows from the lemma below, applied with A = G , U = eiT HB and φ(s) = −e−s .

10

D. Ruelle

Lemma. Let A, U be a hermitian and a unitary n×n matrix respectively, and φ : R → R be an increasing function. Then tr(φ(A)U AU −1 ) ≤ tr(φ(A)A). Proof. As R. Seiler kindly pointed out to me, this lemma can be obtained readily from O. Klein’s inequality tr(f (B) − f (A) − (B − A)f  (A)) ≥ 0, where A, B are hermitian and f convex: take B = U AU −1 and φ = f  . Remark. We have



 

ρ(i[HL , HRa ]) = 0

a>0

because −



 

 ρ(i[HL , HRa ]) = lim ρ i H , H − Ha →L

a>0



=

d ρ αt dt

 X:X∩S=∅

a>0

 (X)

= 0, t=0

where we have used the fact that ρ is (α t )∗ -invariant. In particular, in the case of two reservoirs 0 ≤ eρ = (β1 − β2 )ρ(i[HL , HR1 ]) so that if the temperature β1−1 is less than β2−1 , i.e., β1 − β2 > 0, the flux of energy into R1 is ≥ 0: heat flows from the hot reservoir to the cold reservoir. 3. Proving Strict Positivity of eρ It is an obvious challenge to prove that eρ  = 0. A natural situation to discuss would correspond to Ra = Zν and a translationally invariant. But we need then ν ≥ 3 as discussed in [4]. Indeed, for ν < 3 one expects a nonequilibrium steady state to be in fact an equilibrium state at a temperature intermediate between the original temperatures of the reservoirs. Instead of a quantum spin system as described above, a gas of noninteracting fermions would probably be easier to treat first. 4. Complements and Relation with Recent Work of Jakši´c and Pillet After this paper was submitted for publication, two interesting contributions were posted to the mp arc archive: one by Jakši´c and Pillet4 and one by Maes et al.5 In this section and the next two, I am complying with the editor’s request to take into account remarks by the referees, and in particular to discuss the relations of my work with the two references mentioned above. 4 V. Jakši´c and C.-A. Pillet. “On entropy production in quantum statistical mechanics.” mp arc 00-309. 5 Chr. Maes, F. Redig, and M. Verschuere. “Entropy production for interacting particle systems.” mp arc

00-357.

Entropy Production in Quantum Spin Systems

11

Note that the definition of entropy production used above is based on the thermodynamic relation dQ = kT dS or, in the present case dS = a (kTa )−1 dQa . It can be considered a drawback that this definition does not relate directly to a microscopically defined entropy-like quantity, as is done in the papers of Jakši´c and Pillet, and Maes et al. We now discuss in detail the approach of Jakši´c and Pillet, and its relation with the present paper.6 We are given a C∗ -algebra A with identity, an element V = V ∗ ∈ A, time evolutions t (α˘ ), (α t ) (i.e. , strongly continuous one-parameter groups of ∗-automorphisms of A) such that  t1  tn−1   t

 α t (A) = α˘ t (A) + in dt1 dt2 . . . dtn α˘ tn (V ), . . . [α˘ t1 (V ), A] 0

0

n≥1

0

and an (α˘ t )-invariant state σ on A. Therefore (α t ) is a local perturbation by V of the “free” evolution given by (α˘ t ) and σ is an invariant state for the “free” evolution. We furthermore assume that (C1) There exists a time evolution (β t ) for which σ is a KMS state at inverse temperature +1. (C2) V is in the domain of the infinitesimal generator δβ of (β t ). [In fact Jakši´c and Pillet assume a temperature −1 in (C1); our choice of temperature +1 will bring a change of sign below in the definition of the entropy production. In the situation discussed earlier we have  V = (X), X∩S=∅

hence V λ ≤  λ cardS, and V ∈ Aλ . Note that Aλ is in the domain of the infinitesimal generator δβ of (β t ) (see the Appendix), hence (C2) holds. The advantage of the approach of Jakši´c and Pillet is that σ can be an arbitrary KMS state: the existence of “boundary terms” Ba as in (A3) is not required.] In this setup one introduces the observable −δβ (V ) and the entropy production in the state ρ is defined as ρ(−δβ (V )). [In our situation we have −δβ (V ) = −

 a>0

=



βa





i[(X), (Y )]

X⊂Ra Y :Y ∩S=∅

βa i[HL , HRa ]

a>0

so that ρ(−δβ (V )) = eρ is indeed the rate of entropy production in the state ρ.] 6 We have changed the notation of [2] to align it with the one used above.

12

D. Ruelle

Finite dimensional digression. For the purpose of motivation we discuss now the case where A would be the algebra of n × n matrices, and consider two states on A given by density matrices µ, ν. A relative entropy is then defined by Ent(µ|ν) = −tr(µ log µ − µ log ν) ≤ 0. If (α t ) is a one parameter group of ∗-automorphisms of A we have thus   d d Ent(µ ◦ α t |ν) = tr µ α t (log ν) . dt dt Suppose now that ν is preserved by the “free” evolution (α˘ t ), and that (α t ) is a perturbation of (α˘ t ), so that α t (A) = ei(H +V )t Ae−i(H +V )t ,

α˘ t (A) = eiH t Ae−iH t ,

then d t α (log ν) = α t (i[V , log ν]). dt Define now (β t ) by β t (A) = e−it log ν Aeit log ν so that ν is the corresponding KMS state (at inverse temperature +1). Then if δβ is the infinitesimal generator of (β t ) we have i[V , log ν] = δβ (V ), hence d t α (log ν) = α t (δβ (V )), dt

d Ent(µ ◦ α t |ν) = µ(α t (δβ (V ))). dt We obtain thus 

T

T

Ent(µ ◦ α |ν) − Ent(µ|ν) = 0

(µ ◦ α t )(δβ (V )) dt

or, taking µ = ν = σ , 0 ≤ −Ent(σ ◦ α T |σ ) =



T 0

(σ ◦ α t )(−δβ (V )) dt.

Entropy Production in Quantum Spin Systems

13

The infinite dimensional situation. If µ, ν are two faithful normal states on a von Neumann algebra M [in our case πσ (A) ], Araki has introduced a relative entropy Ent(µ|ν) in terms of a relative modular operator associated with µ, ν. We must refer the reader to [1] Definition 6.2.29 for details. Using this definition, Jakši´c and Pillet have worked out an infinite dimensional version of the finite dimensional calculation given above. They are able to prove the formula 

T 0

(σ ◦ α t )(−δβ (V )) dt = −Ent(σ ◦ α T |σ ) ≥ 0

which can be interpreted as an entropy balance, and gives in the limit ρ(−δβ (V )) ≥ 0 if ρ is a NESS. The proof is fairly technical. The approach of Jakši´c and Pillet has the interest of great generality. In particular σ can be an arbitrary KMS state.Also, instead of a spin lattice system one can consider fermions on a lattice. For a nonintertacting fermion model, Jakši´c and Pillet have announced a proof of strict positivity of the entropy production, as had been suggested above. Appendix: The Algebras Aλ The purpose of this Appendix is to complete the proof of (4) by establishing (10) below. On the way to this result we introduce “partial traces” π , and algebras Aλ which are of interest in their own right. For finite  ⊂ L, a map π : ∪X AX → A is defined by π A =

lim

Y →L\

tr HY A . dimHY

If the φi form an orthonormal basis of HY , and ψ  , ψ  ∈ H we have    tr HY A  1 ψ , = ψ (φi ⊗ ψ  , Aφi ⊗ ψ  ), dimHY dimHY i

hence π A ≤ A . The properties of the following lemma are then readily checked. Lemma. The map π extends to a unique linear norm-reducing map A → A . Furthermore if A ∈ A , π A = A π A∗ = (π A)∗ , π π = π π . Choose now some λ > 0. For A ∈ A , define     λ cardX

A λ = inf

AX e : AX = A . X⊂

X

14

D. Ruelle

By the inf by min. If  is replaced by a larger set  , and  compactness we may replace  Y AY = A with Y ⊂  , we have  

AY eλ cardY ≥

π AY eλ card(Y ∩) 

Y ⊂

Y

with Y π AY = π A = A. Therefore A λ does not depend on the choice of  provided A ∈ A . We have thus a norm . λ on ∪X AX , and we may define the Banach space Aλ by completion. Proposition. The inclusion map ∪X AX → A extends to a norm-reducing map ω : Aλ → A and ω is injective. Proof. ω is norm-reducing because A ≤ A λ for A ∈ ∪X AX . Note now that π : ∪X AX → A reduces the . λ -norm and extends thus to a linear norm-reducing map Aλ → Aλ , where Aλ is A equipped with the . λ -norm. Assume that A ∈ Aλ with A λ = a > 0. We may choose  and B ∈ A such that

A − B λ < a/3, hence B λ > 2a/3. Now ωA = 0 would imply π A = 0, hence a 2a < B λ = π (B − A) λ ≤ A − B λ < . 3 3 Therefore ω must be injective.   Corollary. Aλ is identified by ω to a dense ∗-subalgebra of A; Aλ is then a Banach algebra with respect to the norm . λ . Taking λ = 0 we may define A0 = A. With this definition, if λ < µ we have Aλ ⊃ Aµ , and the map Aµ → Aλ is norm-reducing.  Proof.If A, B ∈ A we may choose AX , BX ∈ AX such that A = X⊂ AX , B = X⊂ BX , and  

AX eλ cardX ,

B λ =

BX eλ cardX .

A λ = X⊂

Thus

AB λ ≤

 X

≤

X⊂

Y

 X

AX AY eλ card(X∪Y )

AX . AY eλ(cardX+cardY ) = A λ B λ .

Y

Therefore if A, B tend to limits A∞ , B∞ in Aλ , AB tends in Aλ to A∞ B∞ and A∞ B∞ λ ≤ A∞ λ B∞ λ . The rest is clear.   If  λ < ∞ and AX ∈ AX the formula  [(Y ), AX ] δAX = i Y :Y ∩X=∅

defines an element of Aλ . If λ > µ ≥ 0, and  λ < ∞, one also checks that δ m defines a map Aλ → Aµ such that

δA µ ≤ 2(λ − µ)−1 A λ  λ ,

δ m A µ ≤ A λ m!(2(λ − µ)−1  λ )m .

(6)

Entropy Production in Quantum Spin Systems

15

(The proof of (6) is basically the same as that of the standard case µ = 0).  + δ  , where We turn now to the proof of (10) below . We have δ = δ     δ A = i[ Ba , A] δ A = i[H , A], a>0

and (1) and (6) (for m = 1) yield

δA µ ≤ A λ .2(λ − µ)−1  λ , 

δ A µ ≤ A λ .2(λ − µ)−1  λ , 

δ A µ ≤ A λ .2(λ − µ)−1 K.

Given 4 > 0 and A ∈ Aλ we can find X such that A = A1 + A2 with A1 ∈ AX and

A2 λ < 4. Therefore  

(δ − δ )A µ ≤ (δ − δ )A1 µ + δA2 µ + δ A2 µ + δ A2 µ

(7)

= (δ − δ )A1 µ + 4.2(λ − µ)−1 (2  λ + K).

Taking  ⊃ X we also have

 δ A1 = 0

by (2), and  )A1 = i (δ − δ



[(Y ), A1 ]

Y :Y  ⊂,Y ∩X=∅

so that



(δ − δ )A1 µ ≤ A1 λ .2(λ − µ)−1  Xλ , (8)  where  Xλ = supx∈X Y x,Y ⊂X e(cardY −1)λ (Y ) . When  → L we have

 Xλ → 0 and (7), (8) yield

lim (δ − δ )A µ = 0.

(9)

→L

We can now prove that, if  λ < ∞ and A ∈ Aλ , m A = 0. lim δ m A − δ

(10)

→L

We have indeed m δ m A − δ A=

m−1  k=0

m−k−1 δ (δ − δ )δ k A

and, using (6),

6

δ k A 2λ/3 ≤ A λ .k! hence, by (9),

λ

 λ

k

,

lim (δ − δ )δ k A λ/3 = 0

→L

so that, using (6), m−k−1 m−k−1 (δ − δ )δ k A ≤ δ (δ − δ )δ k A 0

δ

≤ (δ − δ )δ k A λ/3 (m − k − 1)!

6

 λ

λ which tends to zero when  → L. This concludes the proof of (10).

m−k−1

,

16

D. Ruelle

References 1. Bratteli, O. and Robinson. D.W.: Operator algebras and quantum statistical mechanics I, II. New York: Springer, 1979–1981 (2nd ed. 1987–1997) 2. Haag, R., Hugenholtz, N.M. and Winnink, M.: On the equilibrium states in quantum statistical mechanics. Commun. Math. Phys. 5, 215–236 (1967) 3. Ruelle, D.: Statistical mechanics. Rigorous results. New York: Benjamin, 1969 4. Ruelle, D.: Natural nonequilibrium states in quantum statistical mechanics. J. Statist. Phys. 98, 57–75 (2000) Communicated by H. Spohn

Commun. Math. Phys. 224, 17 – 31 (2001)

Communications in

Mathematical Physics

A Rigorous Derivation of the Gross–Pitaevskii Energy Functional for a Two-dimensional Bose Gas
Elliott H. Lieb1 , Robert Seiringer2 , Jakob Yngvason2 1 Departments of Physics and Mathematics, Jadwin Hall, Princeton University, P. O. Box 708,

Princeton, NJ 08544, USA

2 Institut für Theoretische Physik, Universität Wien, Boltzmanngasse 5, 1090 Vienna, Austria

Received: 3 May 2000 / Accepted: 23 October 2000

Dedicated to Joel Lebowitz on the occasion of his 70th birthday Abstract: We consider the ground state properties of an inhomogeneous two-dimensional Bose gas with a repulsive, short range pair interaction and an external confining potential. In the limit when the particle number N is large but ρa ¯ 2 is small, where ρ¯ is the average particle density and a the scattering length, the ground state energy and density are rigorously shown to be given to leading order by a Gross–Pitaevskii (GP) energy functional with a coupling constant g ∼ 1/| ln(ρa ¯ 2 )|. In contrast to the 3D case the coupling constant depends on N through the mean density. The GP energy per particle depends only on Ng. In 2D this parameter is typically so large that the gradient term in the GP energy functional is negligible and the simpler description by a Thomas–Fermi type functional is adequate. 1. Introduction Motivated by recent experimental realizations of Bose–Einstein condensation the theory of dilute, inhomogeneous Bose gases is currently a subject of intensive studies. Most of this work is based on the assumption that the ground state properties are well described by the Gross–Pitaevskii (GP) energy functional (see the review article [1]). A rigorous derivation of this functional from the basic many-body Hamiltonian in an appropriate limit is not a simple matter, however, and has only been achieved recently for bosons with a short range, repulsive interaction in three spatial dimensions [2]. The present paper is concerned with the justification of the GP functional in two spatial dimensions. Several new issues arise. One is the form of the nonlinear interaction term in
the energy functional for the GP wave function . In three dimensions this term is 4πa ||4 , where a is the scattering length of the interaction potential. The rationale is the well known formula for the energy density of a homogeneous Bose gas, which,

© 2000 by the authors. Reproduction of this work, in its entirety, by any means, is permitted for noncommercial purposes.

18

E. H. Lieb, R. Seiringer, J. Yngvason

for dilute gases with particle density ρ, is 4π aρ 2 . This fact has been ‘known’ since the early 50’s but a rigorous proof is fairly recent [3]. In two dimensions the corresponding formula is 4πρ 2 | ln(ρa 2 )|−1 as proved in [4] by extension of the method of [3]. The formula was first stated by Schick [5];
other early references to this formula are [6–10]. It would seem natural to consider 4π ||4 | ln(||2 a 2 )|−1 as the interaction term in the GP functional, and this has indeed been suggested in [11, 12]. Such a term, however, is unnecessarily complicated for the purpose of leading order calculations. In fact, since the logarithm varies only slowly it turns out that one can use the same form as in the three dimensional case, but with an appropriate dimensionless coupling constant g replacing the scattering length, and still retain an exact theory (to leading order in ρ). It is often assumed that a justification of the GP functional depends on the existence of Bose Einstein condensation. Several remarks can be made about this: 1. We neither assume nor prove the existence of BE condensation, but we do demonstrate a kind of condensation over a distance that is fixed (i.e., non-thermodynamic) but whose length goes to infinity as the density goes to zero; 2. BE condensation does not exist in two dimensions when the temperature is positive, but it can, and most likely does, exist in the ground state; 3. In any event, when the density is low and the temperature is zero it appears to be likely that the system can be described for many purposes in terms of only a few macroscopic order parameters such as the density and phase – at least this is true for the dependence of the ground state energy and density upon an external potential. The functional we shall consider is E

GP

[] =

 

 |∇(x)|2 + V (x)|(x)|2 + 4πg|(x)|4 d2 x,

(1.1)

where V is the external confining potential and all integrals are over R2 . The choice of g is an issue on which there has not been unanimous opinion in the recent papers [12–18] on this subject. We shall prove that a right choice is g = | ln(ρa ¯ 2 )|−1 , where ρ¯ is a mean density that will be defined more precisely below. This mean density depends on the particle number N , which implies that the scaling properties of the GP functional are quite different in two and three dimensions. In the three-dimensional case the natural parameter is N a/aosc , with aosc being the length scale defined by the external confining potential.
If a/aosc is scaled like 1/N as N → ∞ this parameter is fixed and the gradient term |∇|2 in the GP functional is of the same order as the other terms. In two dimensions the corresponding parameter is N | ln(ρa ¯ 2 )|−1 . For a quadratic external 1/2 2 potential ρ¯ behaves like N /aosc and hence the parameter can only be kept fixed if a/aosc decreases exponentially with N . A slower decrease means that the parameter tends to infinity. This corresponds to the so-called Thomas Fermi (TF) limit where the gradient term has been dropped altogether and the functional is E TF [ρ] =

 

 V (x)ρ(x) + 4πgρ(x)2 d2 x,

(1.2)

defined for nonnegative functions ρ. Our main result, stated in Theorems 1.3 and 1.4 below, is that minimization of (1.2) reproduces correctly the ground state energy and density of the many-body Hamiltonian in the limit when N → ∞, ρa ¯ 2 → 0, but 2 −1 2 −1 N| ln(ρa ¯ )| → ∞. Only in the exceptional situation that N | ln(ρa ¯ )| stays bounded is there need for the full GP functional (1.1), cf. Theorems 1.1 and 1.2.

Derivation of Gross–Pitaevskii Energy Functional for 2D Bose Gas

19

We shall now describe the setting more precisely. The starting point is the Hamiltonian for N identical bosons in an external potential V and with pair interaction v, H

(N)

=

  v(xi − xj ), −∇i2 + V (xi ) +

N   i=1

(1.3)

i<j

acting on the totally symmetric wave functions in ⊗N L2 (R2 ). Units have been chosen so that h¯ = 2m = 1, where m is the particle mass. We assume that v is nonnegative and spherically symmetric with a finite scattering length a. (For the definition of scattering length in two dimensions see the appendix.) The external potential should be continuous and tend to ∞ as |x| → ∞. It is then possible and convenient to shift the energy scale so that minx V (x) = 0. For the TF limit theorem we shall require some additional properties of V to be specified later. The ground state energy ω of the one-particle operator −∇ 2 + V is a natural energy unit and gives rise to the length unit aosc ≡ ω−1/2 . In the sequel we shall be considering a limit where a/aosc tends to zero while N → ∞. Experimentally a/aosc can be changed in two ways: One can either vary aosc or a. The first alternative is usually simpler in practice but very recently a direct tuning of the scattering length itself has also been shown to be feasible [19]. Mathematically, both alternatives are equivalent, of course. −2 Vˆ (x/a ) and keeping Vˆ and v fixed. The The first corresponds to writing V (x) = aosc osc second corresponds to writing the interaction potential as v(x) = a −2 v(x/a), ˆ where vˆ has unit scattering length, and keeping V and vˆ fixed. This is equivalent to the first, since for given Vˆ and vˆ the ground state energy of (1.3), measured in units of ω, depends only on N and a/aosc . In the dilute limit when a is much smaller than the mean particle distance, the energy becomes independent of v. ˆ We shall measure all energies in terms of ω and lengths in terms of aosc and regard Vˆ and vˆ as fixed. The notation E QM (N, a) for the ground state energy of (1.3) is then justified. The quantum mechanical particle density is defined by  QM ρN,a (x) = N | (N) (x, x2 , . . . , xN )|2 d2 x2 . . . d2 xN , (1.4) where  (N) is a ground state for (1.3). The GP functional (1.1) has an obvious
domain of definition (cf. Eq. (2.1) in [2]). The infimum of E GP [] under the condition ||2 = N will be denoted by E GP (N, g). The infimum is obtained for a unique, positive function, denoted GP N,g , and the GP density GP (x) = GP (x)2 . is defined as ρN,g N,g of the TF functional (1.2) with the subsidiary condition
The ground state energy ρ = N is denoted E TF (N, g). The corresponding minimizer can be written explicitly; it is TF ρN,g (x) =

1 [µTF − V (x)]+ , 8πg

where [t]+ ≡ max{t, 0} and µTF is chosen so that the normalization condition N holds.

(1.5)


TF = ρN,g

20

E. H. Lieb, R. Seiringer, J. Yngvason

TF at coupling We now define the mean density ρ¯ as the average of the TF density ρN,1 TF , i.e., constant g = 1, weighted with N −1 ρN,1

1 N

ρ¯ =

 TF ρN,1 (x)2 d2 x.

(1.6)

It is clear that ρ¯ depends on N and when we wish to emphasize this we write ρ¯N . The definition (1.6) has the advantage that ρ¯ is easily computed; for instance, if V (x) ∼ |x|s for some s > 0, then
ρ¯N ∼ N s/(s+2) . It may appear more natural to define ρ¯ selfTF (x)2 d2 x with g = | ln(ρa consistently as ρ¯ = N1 ρN,g ¯ 2 )|−1 , which amounts to solving a nonlinear equation for ρ. ¯ Also, the TF density could be replaced by the GP density. However, since ρ¯ will only appear under a logarithm such sophisticated definitions are not needed for the leading order result we are after. The simple formula (1.6) is adequate for our purpose, but it should be kept in mind that the self-consistent definition may be relevant in computations beyond the leading order. With this notation we can now state the two dimensional analogue of Theorem I.1 in [2]. Theorem 1.1 (GP limit for the energy). If, for N → ∞, a 2 ρ¯N → 0 with N/| ln(a 2 ρ¯N )| fixed, then E QM (N, a)

lim

N→∞ E GP (N, 1/| ln(a 2 ρ¯N )|)

= 1.

(1.7)

The corresponding theorem for the density, cf. Theorem I.2 in [2], is Theorem 1.2 (GP limit for the density). If, for N → ∞, a 2 ρ¯N → 0 with γ ≡ N/| ln(a 2 ρ¯N )| fixed, then lim

N→∞

1 QM GP ρ (x) = ρ1,γ (x) N N,a

(1.8)

in the sense of weak convergence in L1 (R2 ). These theorems, however, are not particularly useful in the two dimensional case, because the hypothesis that N/| ln(a 2 ρ¯N )| stays bounded requires an exponential decrease of a with N . As remarked above, the TF limit, where N/| ln(a 2 ρ¯N )| → ∞, is much more relevant. Our treatment of this limit requires that V is asymptotically homogeneous and sufficiently regular in a sense made precise below. This condition can be relaxed, but it seems adequate for most practical applications and simplifies things considerably. Definition 1.1. We say that V is asymptotically homogeneous of order s > 0 if there is a function W with W (x)  = 0 for x  = 0 such that λ−s V (λx) − W (x) → 0 as λ → ∞ 1 + |W (x)|

(1.9)

and the convergence is uniform in x. The function W is clearly uniquely determined and homogeneous of order s, i.e., W (λx) = λs W (x) for all λ ≥ 0.

Derivation of Gross–Pitaevskii Energy Functional for 2D Bose Gas

21

Theorem 1.3 (TF limit for the energy). Suppose V is asymptotically homogeneous of order s > 0 and its scaling limit W is locally Hölder continuous, i.e., |W (x) − W (y)| ≤ (const.)|x − y|α for |x|, |y| = 1 for some fixed α > 0. If, for N → ∞, a 2 ρ¯N → 0 but N/| ln(a 2 ρ¯N )| → ∞, then E QM (N, a) = 1. (1.10) N→∞ E TF (N, 1/| ln(a 2 ρ¯N )|) To state the corresponding theorem for the density we need the minimizer of (1.2) with
g = 1, V replaced by W , and normalization ρ = 1. We shall denote this minimizer TF ; an explicit formula is by ρ˜1,1 lim

TF (x) = ρ˜1,1

1 TF [µ˜ − W (x)]+ , 8π

(1.11)

where µ˜ TF is determined by the normalization condition. Theorem 1.4 (TF limit for the density). Let V satisfy the same hypothesis as in Theorem 1.3. If, for N → ∞, a 2 ρ¯N → 0 but γ = N/| ln(a 2 ρ¯N )| → ∞, then γ 2/(s+2) QM 1/(s+2) TF x) = ρ˜1,1 (x) ρN,a (γ N→∞ N lim

(1.12)

in the sense of weak convergence in L1 (R2 ). Remark 1.1. For large N, ρ¯N behaves like (const.)N s/(s+2) . Moreover, prefactors are unimportant in the limit N → ∞, because ρ¯N stands under a logarithm. Hence Theorems 1.3 and 1.4 could also be stated with N s/(s+2) in place of ρ¯N . The proofs of these theorems follow from upper and lower bounds on the ground state energy E QM (N, a) that are derived in Sects. 3 and 4. For these bounds some properties of the minimizers of the functionals (1.1) and (1.2), discussed in the following section, are needed. 2. GP and TF Theory In this section we consider the functionals (1.1) and (1.2) with an arbitrary positive coupling constant g. Existence and uniqueness of minimizers is shown in the same way as in Theorem II.1 in [2]. The GP energy E GP (N, g) has the simple scaling property GP E GP (N, g) = N E GP (1, Ng). Likewise, N −1/2 GP N,g ≡ φγ depends only on γ ≡ Ng (2.1)
GP 2 and satisfies the normalization condition |φγ | = 1. The variational equation (GP equation) for the GP minimization problem, written in terms of φγGP , is −"φγGP + V φγGP + 8π γ (φγGP )3 = µGP (γ )φγGP ,

(2.2)

where the Lagrange multiplier (chemical potential) µGP (γ ) is determined by the subsidiary normalization condition. Multiplying (2.2) with φγGP and integrating we obtain  µGP (γ ) = E GP (1, γ ) + 4π γ φγGP (x)4 d2 x. (2.3) For the upper bound on the quantum mechanical energy in the next section we shall need a bound on the absolute value of the minimizer φγGP .

22

E. H. Lieb, R. Seiringer, J. Yngvason

Lemma 2.1 (Upper bound for the GP minimizer). φγGP 2∞ ≤

µGP (γ ) . 8π γ

(2.4)

Proof. φγGP is a continuous and positive function that satisfies the variational equation −"φγGP + U φγGP = µGP φγGP

(2.5)

with U = V + 8πγ (φγGP )2 . Let B = {x | φγGP (x)2 > µGP /(8π γ )}. Since V ≥ 0 we see that −"φγGP ≤ 0 on B, i.e., φγGP is subharmonic on B. Hence φγGP achieves its maximum  on the boundary of B, where φγGP (x)2 = µGP /(8π γ ), so B is empty.  The ground state energy E TF (N, g) of the TF functional (1.2) scales in the same way TF as E GP (N, g), i.e., E TF (N, g) = N E TF (1, Ng), and the corresponding minimizer ρN,g TF TF TF is equal to Nρ1,Ng . For short, we shall denote ρ1,γ by ργ . By (1.5) we have ργTF (x) =

1 [µTF (γ ) − V (x)]+ , 8π γ

with the chemical potential µTF (γ ) determined by the normalization condition 1. In the same way as in (2.3) we have  TF TF µ (γ ) = E (1, γ ) + 4π γ ργTF (x)2 d2 x.

(2.6)


ργTF = (2.7)

The chemical potential can also be computed from a variational principle: Lemma 2.2 (Variational principle for µTF ).  µTF (γ ) = Vρ + 8π γ ρ∞ . inf
ρ≥0, ρ=1

(2.8)

Proof. Obviously, the infimum is achieved for a multiple of a characteristic function for some measurable set R ⊂ R2 . If |R| denotes the Lebesgue measure of R, then    1 inf Vρ + 8πγ ρ = inf V + 8π γ (2.9) ∞
|R| R ρ=1 R     1 TF TF = inf V − µ (γ ) + 8π γ + µ (γ )|R| . |R| R R (2.10)
Now R (V − µTF (γ )) ≥ −8πγ , with equality for   x|V (x) < µTF (γ ) ⊆ R ⊆ x|V (x) ≤ µTF (γ ) . (2.11)   Corollary 2.1 (Properties of µTF (γ )). µTF (γ ) is a concave and monotonously increasing function of γ with µTF (0) = 0. Hence µTF (γ )/γ is decreasing in γ . Moreover, µTF (γ ) → ∞ and µTF (γ )/γ → 0 as γ → ∞.

Derivation of Gross–Pitaevskii Energy Functional for 2D Bose Gas

23

Proof. Immediate consequences of Lemma 2.2, using that minx V (x) = 0 and lim|x|→∞ V (x) = ∞.   Note that since E TF (1, γ ) ≥ 21 µTF (γ ) we also see that E TF (1, γ ) → ∞ with γ . In this limit the GP energy converges to the TF energy, provided the external potential satisfies a mild regularity and growth condition: Lemma 2.3 (TF limit of the GP energy). Suppose for some constants α > 0, L1 and L2 , |V (x) − V (y)| ≤ L1 |x − y|α eL2 |x−y| (1 + V (x)).

(2.12)

E GP (1, γ ) = 1. γ →∞ E TF (1, γ )

(2.13)

Then lim

Proof. It is clear that E TF (1, γ ) ≤ E GP (1, γ ). For the other direction, we use (j% ∗ ργTF )1/2 as a test function for E GP , where   1 1 j% (x) = exp − |x| . (2.14) 2π % 2 %
Note that j% = 1 and |∇j% | = % −1 j% . Therefore  

1 GP TF 2 TF TF 2 |∇j% ∗ ργ | + V (j% ∗ ργ ) + 4π γ (j% ∗ ργ ) E (1, γ ) ≤ 4j% ∗ ργTF    1 ≤ 2+ (j% ∗ V )ργTF + 4π γ (ργTF )2 , 4% (2.15) where we have used convexity for the last term. Moreover,    (j% ∗ V − V )ργTF = d2 xd2 yj% (x − y) (V (x) − V (y)) ργTF (x)   L1 −1 ≤ d2 xd2 y|x − y|α e(−% +L2 )|x−y| (1 + V (x))ργTF (x) 2π% 2   ≤ (const.) % α 1 + E TF (1, γ ) , (2.16) as long as % < L−1 2 . So we have E GP (1, γ ) ≤ (1 + (const.) % α )E TF (1, γ ) +

1 + (const.) % α . 4% 2

Optimizing over % gives as a final result   E GP (1, γ ) ≤ E TF (1, γ ) 1 + (const.)E TF (1, γ )−α/(α+2) .  

(2.17)

(2.18)

24

E. H. Lieb, R. Seiringer, J. Yngvason

Condition (2.12) is in particular fulfilled if V is homogeneous of some order s > 0 and locally Hölder continuous. In this case, E TF (1, γ ) = γ s/(s+2) E TF (1, 1)

(2.19)

TF γ 2/(s+2) ργTF (γ 1/(s+2) x) = ρ1,1 (x).

(2.20)

µTF (γ ) = γ s/(s+2) µTF (1).

(2.21)

and

By (2.7) we also have

If V is asymptotically homogeneous with a locally Hölder continuous limiting function W , we can prove corresponding formulas for the limit γ → ∞. This is the content of the next theorem, where we have included results on the GP → TF limit as well: Theorem 2.1 (Scaling limits). Suppose V satisfies the condition of Theorem 1.3. Let E˜ TF (1, 1) be the minimum of the TF functional (1.2) with g = 1 and N = 1 and V TF be the corresponding minimizer. Then replaced by W , and let ρ˜1,1 (i) limγ →∞ E GP (1, γ )/γ s/(s+2) = limγ →∞ E TF (1, γ )/γ s/(s+2) = E˜ TF (1, 1). GP (γ 1/(s+2) x) = ρ˜ TF (x), strongly in L2 (R2 ). (ii) limγ →∞ γ 2/(s+2) ρ1,γ 1,1 TF (x), uniformly in x. (iii) limγ →∞ γ 2/(s+2) ργTF (γ 1/(s+2) x) = ρ˜1,1 Proof. With the demanded properties of V , (2.13) holds. Using this and (1.9) one easily GP (γ 1/(s+2) x) is a minimizing sequence for the funcverifies (i). Moreover, γ 2/(s+2) ρ1,γ tional in question, so we can conclude as in Theorem II.2 in [2] that it converges to TF (x) strongly in L2 , proving (ii). (Remark: In Eq. (2.10) in [2] there is a misprint, ρ˜1,1 GP one should have ρ˜ GP on the left side.) To see (iii) let us define instead of ρ1,Na 1,Na   ρ γ (x) = γ 2/(s+2) ργTF γ 1/(s+2) x . (2.22) We can write ρ γ (x) =

 1 −s/(s+2) TF µ (γ ) − W (x) − %(γ , x) γ + 8π

(2.23)

with %(γ , x) = γ −s/(s+2) V (γ 1/(s+2) x) − W (x).

(2.24)

By assumption, |%(γ , x)| < δ(γ )(1 + W (x)) for some δ(γ ) with limγ →∞ δ(γ ) = 0.
Because ρ γ = 1 for all γ , we see from Eq. (2.23) that µTF (γ )γ −s/(s+2) converges to some c as γ → ∞. Moreover, we can conclude that the support of ρ γ is for large γ contained in some bounded set B independent of γ . Therefore   1 = lim ρ γ = (8π )−1 [c − W (x)]+ (2.25) γ →∞

by dominated convergence, so c is equal to the µ˜ TF of Eq. (1.11). Now  1 TF ρ γ (x) = µ˜ − W (x) − %¯ (γ , x) + 8π

(2.26)

Derivation of Gross–Pitaevskii Energy Functional for 2D Bose Gas

25

with %¯ (γ , x) = %(γ , x) + µ˜ TF − γ −s/(s+2) µTF (γ ).

(2.27)

¯ )(1 + W (x)) for some δ(γ ¯ ) with limγ →∞ δ(γ ¯ ) = 0. By Eqs. Again |¯% (γ , x)| < δ(γ (1.11) and (2.26) we thus have TF ¯ )  ργ − ρ˜1,1 ∞ < C δ(γ

(2.28)

with C = (8π)−1 supx∈B (1 + W (x)) < ∞.   The mean density for the TF theory is defined by  ρ¯γ ≡ N ργTF (x)2 d2 x.

(2.29)

For γ = N, i.e., g = 1 this is the same as (1.6). It satisfies Lemma 2.4 (Bounds on ρ¯γ ). For some constant C > 0, N

µTF (γ ) µTF (γ ) ≥ ρ¯γ ≥ CN . 8π γ γ

(2.30)

Proof. The upper bound is trivial. Because ρ γ , defined in (2.22), converges uniformly TF and µTF (γ )γ −s/(s+2) → µ ˜ TF as γ → ∞, we have the lower bound to ρ˜1,1 γ ρ¯γ ≥ 8πγ s/(s+2) µTF (γ )−1 N µTF (γ ) for some C > 0.



 TF 2 TF (ρ˜1,1 ) − 2ρ˜1,1 −ρ ∞

>C

(2.31)

 

Remark 2.1. With V asymptotically homogeneous of order s, µTF (γ )γ −s/(s+2) converges as γ → ∞, i.e. µTF (γ ) ∼ γ s/(s+2) for large γ . So the mean TF density for coupling constant g = 1, defined in (1.6), has the asymptotic behavior ρ¯ ∼ N s/(s+2) . 3. Upper Bound to the QM Energy As in the three dimensional case, cf. Eqs. (3.29) and (3.27) in [2], one has the upper bound

|∇φγGP |2 + V (φγGP )2 N J (φγGP )4 + 23 N 2 (φγGP 2∞ K)2 E QM (N, a) ≤ , (3.1) + N 1 − N φγGP 2∞ I (1 − N φγGP 2∞ I )2 where we have implicitly used that −"φγGP +V φγGP ≥ 0, which is justified by Lemma 2.1. The coefficients I , J and K are given by Eqs. (2.4)–(2.10) in [4]. They depend on the scattering length and a parameter b. We choose γ = N/| ln(a 2 ρ)| ¯ and b = ρ¯ −1/2 . (Recall that ρ¯ is short for ρ¯N .) With this choice we have (as long as a 2 ρ¯ < 1) J =

4π , | ln(a 2 ρ)| ¯

(3.2)

26

E. H. Lieb, R. Seiringer, J. Yngvason

and the error terms N φγGP 2∞ I ≤ (const.)

 µGP (γ )  1 + O(| ln(a 2 ρ)| ¯ −1 ) ρ¯

(3.3)

and K 2 N 2 φγGP 4∞ ≤ (const.)E GP (1, γ )

 µGP (γ )  ¯ −1 ) , 1 + O(| ln(a 2 ρ)| ρ¯

(3.4)

where we have used Lemma 2.1. So we have the upper bound E QM (N, a) E GP (N, 1/| ln(a 2 ρ)|) ¯

    ≤ 1 + O µGP (γ )/ρ¯ + O | ln(a 2 ρ)| ¯ −1 .

(3.5)

Now if γ is fixed as N → ∞, µGP (γ ) 1 1 ∼ ∼ . ρ¯ | ln(a 2 ρ)| ¯ N

(3.6)

If γ → ∞ with N we have instead, assuming that the external potential is asymptotically homogeneous of order s,  γ s/(s+2) µTF (γ ) µGP (γ ) ∼ TF , ∼ ρ¯ µ (N ) N

(3.7)

  E QM (N, a) 2 −s/(s+2) ≤ 1 + O | ln(a ρ)| ¯ E GP (N, 1/| ln(a 2 ρ)|) ¯

(3.8)

so in any case

holds as N → ∞ and a 2 ρ¯ → 0. 4. Lower Bound to the QM Energy Compared to the treatment of the 3D problem in [2] the new issue here is the TF case, i.e., γ = N/| ln(a 2 ρ)| ¯ → ∞, and we discuss this case first. The GP limit with γ fixed can be treated in complete analogy with the 3D case, cf. Remark 4.1 below. We introduce again the rescaled ρ γ as in (2.22) and also   v (x) = γ 2/(s+2) v γ 1/(s+2) x . (4.1) Note that the scattering length of v is a = a γ −1/(s+2) . Using V ≥ µTF (γ ) − 8π γργTF and (2.7) we see that  γ2 + γ −2/(s+2) Q E QM (N, a) ≥ E TF (N, γ /N ) + 4π N γ s/(s+2) ρ (4.2) TF − 8πN γ s/(s+2)  ργ − ρ˜1,1 ∞ ,

Derivation of Gross–Pitaevskii Energy Functional for 2D Bose Gas

with Q =
inf

 

||2 =1

|∇i | + 2

i



27

 v (xi − xj )||

2

TF − 8π γ ρ˜1,1 (xi )||2

.

(4.3)

j
Dividing space into boxes α of side length L with Neumann boundary conditions we get  Q≥ E hom (nα , L) − 8π γρα,max nα , (4.4) α TF in the box α, and E hom (n, L) is the where ρα,max denotes the maximal value of ρ˜1,1 energy of a homogeneous gas of n bosons in a box of side length L and Neumann boundary conditions. We can forget about the boxes where ρα,max = 0, because the energy of particles in these boxes is positive. We now want to use the lower bound on E hom given in [4], namely

E hom (n, L) ≥ 4π

  1 n2 1 − C| ln( a 2 n/L2 )|−1/5 . 2 2 2 L | ln( a n/L )|

(4.5)

a 2 n/L2 . Now if This bound holds for n > (const.)| ln( a 2 n/L2 )|1/5 and small enough the minimum in (4.4) is taken in some box α for some value nα , we have E hom (nα + 1, L) − E hom (nα , L) ≥ 8π γρα,max .

(4.6)

By a computation analogous to the upper bound (see [2]) one shows that E hom (n + 1, L) − E hom (n, L)    1 n 2 2 −1 . 1 + O | ln( a n/L )| ≤ 8π 2 L | ln( a 2 n/L2 )| Using Lemma 2.4 and the asymptotics of µTF (Remark 2.1) we see that  2/(s+2) 2  2/(s+2) a2n N a a2N s/(s+2) N 2 C ≤ 2 =N ≤ a ρ¯ 2 , L2 L γ L2 L γ

(4.7)

(4.8)

for some constant C, so (4.7) reads E hom (n + 1, L) − E hom (n, L)



 1 n 1 + | ln((γ /N )2/(s+2) L2 /C)| ≤ 8π 2 1+O . L | ln(a 2 ρ)| ¯ | ln(a 2 ρ)| ¯

(4.9)

So if L is fixed, our minimizing nα is at least ∼ ρα,max L2 N . If N is large enough and a 2 ρ¯ is small enough, we can thus use (4.5) in (4.4) to get     Q≥

 α

 n2  4π  α2 L

  Nρα,max  C     1 −   −2 . 2 ρ)|    | ln(a ¯ 2 2 a nα a N 1/5 | ln L2 | | ln L2 | 

1

(4.10)

28

E. H. Lieb, R. Seiringer, J. Yngvason

Lemma 4.1. For 0 < x, b < 1 we have

  b b2 1 x2 −2 x≥− 1+ . | ln x| | ln b| | ln b| (2| ln b|)2

(4.11)

1 −d Proof. Since ln x ≥ − de x for all d > 0 we have

x | ln b| 2x x 2 | ln b| −2 ≥ ≥ c(d)(bd ed| ln b|)−1/(1+d) edx 2+d − 2 2 b | ln x| b b b with c(d) = 2(2+d)/(1+d)



1 (2 + d)(2+d)/(1+d)

−

1 (2 + d)1/(1+d)

Choosing d = 1/| ln b| gives the desired result.



1 ≥ −1 − d 2 . 4

(4.12)

(4.13)

 

Note that the lemma above implies for k ≥ 1,   b b2 1 x2 k2 . −2 xk ≥ − 1+ | ln x| | ln b| | ln b| (2| ln b|)2

(4.14)

a 2 ρα,max we get the bound Applying this with x = a 2 nα /L2 and b = N  2 ρα,max L2 Q ≥ − 4πN γ 

  × 1+

α



1 4| ln( a 2 Nρ

α,max

)|2

 | ln( a 2 Nρα,max )|  C  2  1− 2 | ln(a ρ)| ¯ | ln aLN |1/5 2

−1     (4.15)

for (4.10). To estimate the error terms, note that as in (4.8),  2/(s+2) N 2 2 a N ∼ a ρ¯ , γ

(4.16)

TF  → 0 ¯ + O(ln | ln(a 2 ρ)|) ¯ for small a 2 ρ. ¯ Using  ργ − ρ˜1,1 so | ln( a 2 N )| = | ln(a 2 ρ)| ∞
TF 2
2 (Theorem 2.1 (iii)) and ρ γ → (ρ˜1,1 ) as γ → ∞ (which follows from the uniform convergence and boundedness of the supports) we get 

  E QM (N, a) 2 2 TF 2 lim inf TF ≥ 1 − (const.) ρα,max L − (ρ˜1,1 ) . (4.17) N→∞ E (N, 1/| ln(a 2 ρ)|) ¯ α

Since this holds for all choices of the boxes α with arbitrary small side length L, and by TF is continuous and has compact support, we can conclude the assumptions on V ρ˜1,1 lim inf N→∞

E QM (N, a) ≥1 E TF (N, 1/| ln(a 2 ρ)|) ¯

¯ → ∞. in the limit N → ∞, a 2 ρ¯ → 0 and N/| ln(a 2 ρ)|

(4.18)

Derivation of Gross–Pitaevskii Energy Functional for 2D Bose Gas

29

Remark 4.1 (The GP case). In the derivation of the lower bound we have assumed that γ → ∞ with N , i.e. N  | ln(a 2 ρ)|, ¯ which seems natural because otherwise the scattering length would have to decrease exponentially with N . However, for fixed γ one can use the methods of [2] (with slight modifications: One uses the 2D bounds on the homogeneous gas and Lemma 4.1) to compute a lower bound in terms of the GP energy. The result is lim inf N→∞

E QM (N, a) ≥1 E GP (N, 1/| ln(a 2 ρ)|) ¯

(4.19)

¯ fixed. in the limit N → ∞, a 2 ρ¯ → 0 with γ = N/| ln(a 2 ρ)| 5. The Limit Theorems We have now all the estimates needed for Theorems 1.1–1.4. The upper bound (3.8) and the lower bound (4.19) prove Theorem 1.1. The energy limit Theorem 1.3 for the TF case follows from (3.8), Theorem 2.1 (i) and (4.18). The convergence of the energies implies the convergence of the densities in the usual way by variation of the external potential. Replacing V (x) by V (x) + δγ s/(s+2) Y (γ −1/(s+2) x) for some positive Y ∈ C0∞ and redoing the upper and lower bounds we see that Theorem 1.3 and Theorem 2.1 (i) hold with W replaced by W + δY . Differentiating with respect to δ at δ = 0 yields γ 2/(s+2) QM 1/(s+2) TF x) = ρ˜1,1 (x) ρN,a (γ N→∞ N lim

(5.1)

in the sense of distributions. Since the functions all have norm 1, we can conclude that there is even weak L1 -convergence. Remark 5.1 (The 3D case). In [2] the analogues of Theorems 1.1 and 1.2 were shown for the three-dimensional Bose gas. Using the methods developed here one can extend these results to analogues of Theorems 1.3 and 1.4. In 3D the coupling constant is g = a, so γ = N a. Moreover, the relevant mean 3D density is ρ¯γ ∼ N (N a)−3/(s+3) . A. Appendix: Scattering Length in Two Dimensions Due to the logarithmic behavior of the Green function of the two dimensional Laplacian the definition of the scattering length is slightly more delicate in two dimensions than in three. For a nonnegative potential v(x), depending only on |x| and with finite range R0 , it is naturally defined by the following variational principle: Theorem A.1. Let R > R0 and consider the functional    1 2 2 ER [φ] = |∇φ(x)| + v(x)|φ(x)| d2 x. 2 |x|≤R

(A.1)

30

E. H. Lieb, R. Seiringer, J. Yngvason


Then, in the subclass of functions such that (|φ|2 + |∇φ|2 ) < ∞ and φ(x) = 1 for |x| = R, there is a unique function φ0 that minimizes ER [φ]. This function is nonnegative and rotationally symmetric, and satisfies the equation 1 −"φ0 (x) + v(x)φ0 (x) = 0 2

(A.2)

for |x| ≤ R in the sense of distributions, with boundary condition φ0 (x) = 1 for |x| = R. For R0 < |x| < R, φ0 (x) = ln(|x|/a)/ ln(R/a)

(A.3)

for a unique number a called the scattering length. For the proof see [4], where generalizations to other dimensions and potentials with a negative part are also discussed. Note that the factor 21 in (A.1) and (A.2) is due to the reduced mass of the two body problem. If v has infinite range it is easy to extend
∞the definition of the scattering length for nonnegative v under the assumption that |x|≥R1 v(x)d2 x < ∞ for some R1 . In fact, one may then simply cut off the potential at some point R0 > R1 (i.e., set v(x) = 0 for |x| > R0 ) and consider the limit of the scattering lengths of the cut off potentials as R0 → ∞. See [4] for details. References 1. Dalfovo, F., Giorgini, S., Pitaevskii, L.P. and Stringari, S.: Theory of Bose–Einstein condensation in trapped gases. Rev. Mod. Phys. 71, 463–512 (1999) 2. Lieb, E.H., Seiringer, R. and Yngvason, J.: Bosons in a Trap: A Rigorous Derivation of the Gross– Pitaevskii Energy Functional. Phys. Rev. A 61, 043602-1–043602-13 (2000); arXiv: math-ph/9908027, mp_arc 99-312. See also: Proceedings of ‘Quantum Theory and Symmetries’ (Goslar, 18-22 July 1999), edited by H.-D. Doebner, V.K. Dobrev, J.-D. Hennig and W. Luecke, Singapore: World Scientific, 2000; arXiv math-ph/9911026, mp_arc 99–439 3. Lieb, E.H. andYngvason, J.: Ground State Energy of the Low Density Bose Gas. Phys. Rev. Lett. 80, 2504– 2507 (1998); arXiv math-ph/9712138, mp_arc 97-631. A more leisurely presentation is in Differential Equations and Mathematical Physics, Proceedings of 1999 conference at the Univ. of Alabama, R. Weikard and G. Weinstein, eds., Cambridge, MA: International Press, 2000, pp. 295–306 4. Lieb, E.H. and Yngvason, J.: Ground State Energy of a Dilute Two-dimensional Bose Gas. J. Stat. Phys. 103, 509–526 (2001); arXiv: math-ph/0002014 5. Schick, M.: Two-Dimensional System of Hard Core Bosons. Phys. Rev. A 3, 1067–1073 (1971) 6. Hines, D.F., Frankel, N.E. and Mitchell, D.J.: Hard disc Bose gas. Physics Letters 68A, 12–14 (1978) 7. Popov, V.N.: On the theory of the superfluidity of two- and one-dimensional Bose systems. Theor. and Math. Phys. 11, 565–573 (1977) 8. Fisher, D.S. and Hohenberg, P.C.: Dilute Bose gas in two dimensions. Phys. Rev. B 37, 4936–4943 (1988) 9. Kolomeisky, E.B. and Straley, J.P.: Renormalization group analysis of the ground state properties of dilute Bose systems in d spatial dimensions. Phys. Rev. B 46, 11749–11756 (1992) 10. Ovchinnikov, A.A.: On the description of a two-dimensional Bose gas at low densities. J. Phys. Condens. Matter 5, 8665–8676 (1993). See also JETP Letters 57, 477 (1993); Mod. Phys. Lett. 7, 1029 (1993) 11. Shevchenko, S.I.: On the theory of a Bose gas in a nonuniform field. Sov. J. Low Temp. Phys. 18, 223–230 (1992) 12. Kolomeisky, E.B., Newman, T.J., Straley, J.P. and Qi, X.: Low-dimensional Bose liquids: Beyond the Gross–Pitaevskii approximation. Phys. Rev. Lett. 85, 1146–1149 (2000); arXiv: cond-mat/0002282 13. Kim, S., Won, C., Oh, S.D. and Jhe, W.: Bose–Einstein condensation in a two-dimensional trap. arXiv: cond-mat/0003342 (2000) 14. Kim, S., Won, C., Oh, S.D. and Jhe, W.: Two-dimensional condensation of dilute Bose atoms in harmonic trap. J. Korean Phys. Soc. 37, 665 (2000); arXiv: cond-mat/9904087 15. Garcia-Ripoll, J.J. and Perez-Garcia, V.M.:Anomalous rotational properties of Bose–Einstein condensates in asymmetric traps. Phys. Rev. A 64, 013602 (2001); arXiv: cond-mat/0003451 (2000)

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31

16. Gonzalez, A. and Perez, A.: Ground-state properties of bosons in three- and two-dimensional traps. Int. J. Mod. Phys. B 12, 2129–38 (1998) 17. Heinrichs, S. and Mullin, W.J.: Quantum-Monte-Carlo Calculations for Bosons in a Two-Dimensional Harmonic Trap. J. Low Temp. Phys. 113, 231–6 (1998) 18. Bayindir, M. and Tanatar, B.: Bose–Einstein condensation in a two-dimensional, trapped, interacting gas. Phys. Rev. A 58, 3134–7 (1998) 19. Cornish, S.L., Claussen, N.R., Roberts, J.L., Cornell, E.A. and Wieman, C.E.: Stable 85 Rb Bose–Einstein Condensates with Widely Tunable Interactions. Phys. Rev. Lett. 85, 1795–98 (2000) Communicated by H. Spohn

Commun. Math. Phys. 224, 33 – 63 (2001)

Communications in

Mathematical Physics

© Springer-Verlag 2001

Quantum Lattice Models at Intermediate Temperature J. Fröhlich1 , L. Rey-Bellet2 , D. Ueltschi3, 1 Institut für Theoretische Physik, ETH Hönggerberg, 8093 Zürich, Switzerland.

E-mail: [email protected]

2 Department of Mathematics, University of Virginia, Charlottesville, VA 22903, USA.

E-mail: [email protected]

3 Department of Physics, Princeton University, Jadwin Hall, Princeton, NJ 08544, USA.

E-mail: [email protected] Received: 6 December 2000 / Accepted: 18 July 2001

Dedicated to Joel Lebowitz on the occasion of his seventieth birthday Abstract: We analyze the free energy and construct the Gibbs-KMS states for a class of quantum lattice systems, at low temperature and when the interactions are almost diagonal, in a suitable basis. The models we study may have continuous symmetries, our results, however, apply to intermediate temperatures where discrete symmetries are broken but continuous symmetries are not. Our results are based on quantum Pirogov– Sinai theory and a combination of high and low temperature expansions.

1. Introduction In this paper we study the low temperature phase diagram for a class of quantum lattice systems. Starting with [PS, Sin], Pirogov–Sinai theory has evolved [KP, Zah, BKL, BS, BI, BK] into a very powerful tool to study the pure phases, their coexistence and the firstorder phase transitions in classical spin systems at low temperature. In recent years a large part of the Pirogov–Sinai theory has been extended to quantum systems [Pir, BKU, DFF, DFFR, KU], quantum spin systems as well as fermionic and bosonic lattice gases, and applied to a variety of models [FR, DFF2, GKU] to describe insulating phases associated with discrete symmetry breaking. Here we formulate the Pirogov–Sinai theory in terms of tangent functionals to the free energy. This allows us to discuss the completeness of the phase diagram avoiding the difficulties associated with boundary conditions. We reformulate results of [BKU, DFF, DFFR, KU] in this framework, and extend the theory to a class of models where discrete symmetries are broken at intermediate temperatures. This applies in particular to some systems with continuous symmetries. For this, we consider the restricted ensembles introduced in [BKL] that are very useful to analyze phases which are associated to a family of configurations rather than to a single configuration.  Supported by the US National Science Foundation, grant PHY 9820650

34

J. Fröhlich, L. Rey-Bellet, D. Ueltschi

The models that we consider have Hamiltonians, for finite volumes , of the form H = V + T , where V is a classical Hamiltonian (i.e. diagonal in a suitable basis) and T is a (usually small) quantum perturbation. In typical situations the suitable basis is the basis of occupation numbers of position operators. Electronic systems provide a large class of interesting models. The classical interaction V describes the many-body short range and classical interaction between the spin- 21 fermions as well as external fields and chemical potentials:

V = Jx,σ nx,σ + Jxy,σ σ  nx,σ ny,σ  + · · · . x∈ σ ∈{↑,↓}

x,y∈ σ,σ  ∈{↑,↓}

A typical quantum perturbation T is the kinetic energy
† T = txy,σ (cxσ cyσ + h.c.), <x,y>⊂ σ ∈{↑,↓}

† where cxσ and cxσ are the creation and annihilation operators and < x, y > denotes pairs of nearest neighbors. Often, in such systems, the behavior at low temperatures arises from a subtle interplay between the (classical) potential energy and the kinetic energy. In this paper two such mechanisms are considered and combined, each of which we now illustrate with an example.

Example 1 (Hubbard Model). In this case the (classical) interaction is only on-site:
V = U nx↑ nx↓ − µ(nx↑ + nx↓ ). x∈

For suitable values of U and µ, the ground states of V have an infinite degeneracy (in the thermodynamic limit): each site is occupied by a single particle of arbitrary spin. However the kinetic energy lifts this degeneracy and induces an effective antiferromagnetic interaction between nearest neighbors. The perturbative methods of [DFFR, DFF2] shows that, in this parameter range, this system is equivalent, in the sense of statistical mechanics, to the Heisenberg antiferromagnet, up to controlled error terms. If the hopping coefficients are asymmetric (e.g. txy,↑ txy,↓ ) then quantum Pirogov–Sinai implies the coexistence of two antiferromagnetic phases at low enough temperatures [DFFR, KU, DFF2]. Rigorous results for the Hubbard model are reviewed in [Lieb]. Example 2 (Extended Hubbard Model). This variant of the Hubbard model includes a nearest neighbor interaction: 

V = (nx↑ + nx↓ )(ny↑ + ny↓ ). U nx↑ nx↓ − µ(nx↑ + nx↓ ) + W x∈

<x,y>⊂

If the interaction between nearest neighbors is repulsive then for suitable values of U , W and µ the ground states of V are chessboard configurations where empty sites alternate with sites occupied with one particle of arbitrary spin. The degeneracy of the

Quantum Lattice Models at Intermediate Temperature

35

ground states is infinite in the thermodynamic limit but we have a spatial ordering of the particles. Using a restricted ensemble we associate a pure phase to this spatial ordering by neglecting the spin degrees of freedom. The methods of this paper imply the existence of only two pure phases in the intermediate temperature range βt 1

and

βW 1.

The temperature is so low that the spatial ordering of the particles survives but so high that the spins are in a disordered phase. The continuous symmetry (if txy,↑ = txy,↓ ) is not broken in this parameter regime. These two models illustrate some of the mechanisms arising from the competition between classical and quantum effects, where the system remains insulating and no continuous symmetry is broken. Our main result, Theorem 4.4, provides tools to describe the phase diagram of such models, in particular the coexistence of several phases and the associated first-order phase transitions. The main technical ingredient in this paper is a combined low-temperature and hightemperature expansion for suitable contour models obtained using the perturbation theory developed in [DFFR]. This paper is organized as follows. In Sect. 2 we describe the general formalism of quantum lattice systems and the perturbation theory of [DFFR]. Section 3 is devoted to the Pirogov–Sinai theory. In Sect. 4 we state the results of Pirogov–Sinai theory for quantum systems. The extended Hubbard model is discussed in Sect. 5 as an illustration. In Sect. 6 we prove our main result by studying a contour model and deriving the required bounds on the contours. 2. General Framework of Quantum Lattice Models 2.1. Basic set-up. We consider a quantum mechanical system on a ν-dimensional lattice Zν , as considered, e.g., in [Rue, Isr, BR, Sim]. We will need a slight modification of the usual formalism in order to treat fermionic lattice gases [DFFR] and to accommodate the fact that fermionic creation and annihilation operators do not commute but anticommute. A quantum lattice system is defined by the following data: (i) Hilbert space. For convenience we choose a total ordering (denoted by the symbol ) of the sites in Zν . We choose the spiral order, depicted in Fig.1 for ν = 2, and an analogous ordering for ν ≥ 3. This ordering has the property that, for any finite set A, the set A := {z ∈ Zν , z  A} of lattice sites which are smaller than A, or belong to A, is finite. To each lattice site a ∈ Zν is associated a finite-dimensional Hilbert space Ha and, for any finite subset A = {a1 ≺ · · · ≺ an } ⊂ Zν , the corresponding Hilbert space HA is given by the ordered tensor product HA = Ha1 ⊗ · · · ⊗ Han .

(2.1)

We further require that there be a Hilbert space isomorphism φa : Ha −→ H, for all a ∈ Zν . (ii) Field and observable algebras. For any finite subset A ⊂ Zν an operator algebra FA , the field algebra, is given. The algebra FA is isomorphic to the algebra B(HA ) of bounded operators on HA , but in general FA  = B(HA ), rather FA ⊂ B(HA ). The algebra FA is a ∗-algebra equipped with a C ∗ -norm obtained from the operator norm on B(HA ). If A ⊂ B and a ≺ b, for all a ∈ A and all b ∈ B \ A, then there is a natural

36

J. Fröhlich, L. Rey-Bellet, D. Ueltschi

✛ ❄

✻ 5

t

4

t

3

6

1

2

t

t

t t

(0, 0)

7

t

✲

Fig. 1. Spiral order in Z2

embedding of FA into FB : An operator K ∈ FA corresponds to the operator K ⊗ 1HB\A in FB . In the following we denote by K both operators. For the infinite system the field algebra is the C ∗ algebra given by F=



FA

norm

,

(2.2)

AZν

(the limit being taken through a sequence of increasing subsets of Zν , where increasing refers to the (spiral) ordering defined above). The algebras FA contain the observable algebras OA which have the same embedding properties as the field algebras and, moreover, satisfy the following commutativity condition: If A ∩ B = ∅, then for any K ∈ FA , L ∈ OB we have [K, L] = 0.

(2.3)

For the infinite system the observable algebra O is given by O=



OA

norm

.

(2.4)

AZν

The group of space translations Zν acts as a ∗-automorphism group {τa }a∈Zν on the algebras F and O, with FX+a = τa (FX ),

OX+a = τa (OX ),

(2.5)

for any X ⊂ Zν and a ∈ Zν . (iii) Interactions, dynamics and free energy. An interaction H = {HA } is given: This is a map from the finite sets A ⊂ Zν to self-adjoint operators HA in the observable algebra OA . We assume the interaction to be translation invariant or periodic, i.e., there is a lattice # ⊆ Zν , with dim# = ν, such that τa HA = HA+a , for all a ∈ # and all A ⊂ Zν . We will consider finite range or exponentially decaying interactions. The norm of an interaction is defined as
H r = sup HA  er|A| , (2.6) a∈Zν Aa

for some r > 0. Here |A| denotes the cardinality of the smallest connected subset of Zν which contains A. We shall denote by Br = {H : H r < ∞} the corresponding Banach space of interactions.

Quantum Lattice Models at Intermediate Temperature

37

 For a finite box , we denote H the finite-volume Hamiltonian given by H = A⊂ HA . Here, we consider only periodic boundary conditions, i.e. is the ν-dimensional torus (Z/LZ)ν , L being the size of . In the sequel we will consider infinite volume limits; the notation limZν will stand for limL→∞ . If H ∈ Br , the interaction H determines a one-parameter group of ∗-automorphisms, {αt }t∈R on F. These automorphisms are constructed as the limit (in the strong topology) of the automorphisms αt given by for K ∈ FA , A ⊂  by αt (K) = eitH K e−itH .

(2.7)

The proof is standard (see e.g. [BR]). Note that one makes crucial use of the commutativity condition (2.3). For an interaction H and at inverse temperature β the partition function is defined as β

Z = Tr e−βH ;

(2.8)

the free energy f (H ) is then f (H ) = −

1 1 β lim ν log Z . β Z ||

(2.9)

Existence of the limit is a well-known result, see [Isr, Sim]. Notice that f (H ) is a concave function of the interaction H . (iv) KMS states and tangent functionals. A state w on O is a positive normalized linear functional on O. A state w is periodic if w ◦ τa = w, for all a in a lattice # ⊂ Zν and invariant if # = Zν . A KMS state at inverse temperature β is a state wβ which satisfies the KMS condition wβ (Kαt (L)) = wβ (αt−iβ (K)L).

(2.10)

For finite systems with periodic boundary conditions it is easy to check that the Gibbs state given by wβ ( · ) = (Tr e−βH )−1 Tr( e−βH · )

(2.11)

satisfies the KMS condition. The set of KMS states is convex, and w is called extremal if it cannot be written as a linear combination of KMS states. The state w is clustering if lim w(Kτa (L)) = w(K)w(τa L),

a→∞

(2.12)

for all K, L ∈ O. Note that a state w is extremal if it is clustering. The state w is exponentially clustering if, for any local observables K ∈ OA , L ∈ OB we have the property w(Kτa (L)) − w(K)w(τa L)
CK,L e−|a|/ξ

(2.13)

with ξ > 0; here CK,L depends on K and L only. If we consider the free energy as a function of the interaction, KMS states at inverse temperature β are in one-to-one correspondence with tangent functionals to the free energy. The free energy f is a concave function of the interaction H and a linear functional α on Br is said to be tangent to f at H if for all interaction K ∈ Br we have f (H + K)
f (H ) + α(K).

(2.14)

38

J. Fröhlich, L. Rey-Bellet, D. Ueltschi

To an invariant state w we associate a tangent functional α defined by α(K) = w(AK ),

(2.15)

 where AK = X0 |X|−1 KX (and similarly for periodic states). The results of Israel and Araki [Isr,Ara] show that if α is a tangent functional at H , then the invariant state w defined in (2.15) is a KMS state at temperature β and, conversely, for any KMS state at temperature β there is a unique tangent functional α. The identification of KMS states with tangent functionals will be very useful to describe the phase diagrams arising from Pirogov–Sinai theory. Example. As an illustration of the general formalism we consider spin 1/2 fermions, as in the examples treated in this paper. The Hilbert space Ha is isomorphic to C4 . We † and caσ the creation and annihilation operators of a particle at site a with spin denote caσ σ ∈ {↑, ↓}. One can construct an explicit representation of the creation and annihilation † ∈ / B(Ha ). operators as operators in B(Ha ), see e.g. Sect. 4.2 in [DFFR], but caσ , caσ † , a ∈ A, The algebras FA ⊂ B(HA ) are chosen to be the algebras generated by caσ , caσ σ ∈ {↑, ↓}. The observable algebras OA are chosen as the algebras generated by pairs of creation or annihilation operators. It is easy to check that the elements FA and OA satisfy the commutativity condition (2.3). Classical interactions. A particular class of interactions consists of the classical interactions. Let {ej }j ∈I be an orthonormal basis of H. Then, for A ⊂ Zν , EA = {⊗a∈A ejaa }, with ejaa = φa−1 ej ,

(2.16)

is an orthonormal basis of HA . We denote by C(EA ) the abelian subalgebra of OA consisting of all operators which are diagonal in the basis EA . An interaction V is called classical, if there exists a basis {ej }j ∈I of H such that VA ∈ C(EA ), for all A ⊂ Zν .

(2.17)

The set .A of configurations in A is defined as the set of all assignments {ja }{a∈A} of an element ja ∈ I to each a. A configuration  ωA is an element in .A . There is a one-to-one correspondence between basis vectors a∈A ejaa of HA and configurations on A:  a∈A

ejaa ←→ ωA ≡ {ja }a∈A .

(2.18)

In the sequel we shall use the notation eωA to denote the basis vector defined by the configuration ωA via the correspondence (2.18). Since a classical interaction V only depends on the numbers 0A (ωA ) = %eωA |VA |eωA &

(2.19)

we may view 0A as a (real-valued) function on the set of configurations. Similarly the algebra C(EA ) may be viewed as the ∗-algebra of complex-valued functions on the set of configurations .A .

Quantum Lattice Models at Intermediate Temperature

39

2.2. Perturbation theory for interactions. The interactions we will study have the form H = V +λT , where V is a classical interaction, T is a perturbation and λ a small parameter. A typical situation is the following: the classical part of the interaction has infinitely many ground states, i.e. the number of ground states of the finite-volume Hamiltonian H diverges as || → ∞, but the perturbation T lifts this degeneracy (completely or partially). This is usually easy to check this using standard perturbation theory for the finite-volume Hamiltonian V + λT . Standard perturbation theory however does not work in the thermodynamic limit, the norm of the error growing with || and other methods are required. Such methods have been developed in [DFFR] and applied in [FR, DFF2] (see also [KU] for an alternative approach). ˜ which is equivalent to H and which can be The idea is to construct an interaction H cast in the form ˜ = V˜ (λ) + T˜ (λ), H

(2.20)

where now the degeneracy of the ground states of V˜ is lifted and T˜ (λ) is suitably small with respect to V˜ (λ). ˜ are equivalent if there exists a ∗-automorphism Recall that two interactions H and H of the algebra O of local observables such that H˜ A = γ (HA ),

(2.21)

˜ ∈ Br˜ . A convenient way of for all A. In particular, if H ∈ Br , there exists r˜ such that H constructing equivalent interactions is with a family of unitary transformations U . Let SA , A ⊂ Zν , be a family of antiselfadjoint operators, periodic or translation invariant,  with SA ∈ OA and Sr < ∞ for some r > 0. We set S = A⊂ SA and then U = exp(S ) is unitary. It is shown in [DFFR] that if Sr is small enough then the ˜ ∈ Br˜ for unitary equivalent Hamiltonians H˜  = U H U−1 define an interaction H ˜ is equivalent to H . some r˜ > 0 and H We consider now an interaction of the form H = V + λT which satisfy the following conditions: (P1) The interaction V is classical and of finite range. Moreover, we assume that V is given by a translation-invariant m-potential. This last condition means that we can assume (if necessary by passing to a physically equivalent interaction) that there exists at least one configuration ω minimizing all 00X , i.e., 00X (ω) = min 00X (ω ),  ω

(2.22)

for all X. For any m-potential, the set of all configurations for which Eq. (2.22) holds is the set of ground states of 00 . (P2) The perturbation interaction T is in some space Banach space Br for some r > 0. Since, by condition (P1), the ground states can be determined locally, there is a corresponding decomposition of the Hilbert space HA for all A: high

low ⊕ HA , HA = HA

(2.23)

low is the subspace spanned by the ground states of V . We can decompose any where HA low and Hhigh : operator KA ∈ B(HA ) according to their action on HA A

KA = KAll + KAhh + KAlh ,

(2.24)

40

J. Fröhlich, L. Rey-Bellet, D. Ueltschi

with low low ⊂ HA KAll HA high KAhh HA low KAlh HA

⊂ ⊂

high HA high HA

high

KAll HA

= 0,

low KAhh HA = 0, high

KAlh HA

low ⊂ HA .

Accordingly we decompose any interaction T : T = T ll + T hh + T lh ,

(2.25)

The following theorem shows that, for any integer n ≥ 1, it is possible to construct an interaction H (n) equivalent to H with the property that H (n) is block diagonal up to order n. Note that this is a constructive result and an algorithm is given in [DFFR] which (n) allows one to construct the unitary transformations U and the interactions H (n) . Theorem 2.1. Consider an interaction of the form H = V + λT ,

(2.26)

where V satisfies Condition (P1) and T satisfies Condition (P2). For any integer n ≥ 1 there is rn > 0 and λn > 0 such that for |λ| < λn there is an interaction H (n) = V + T (n) ∈ Brn , equivalent to H , with T (n)lh rn = O(λn+1 ).

(2.27)

This theorem is useful to analyze the low temperature behavior of quantum spin systems when the ground states of V have infinite degeneracy and T lifts this degeneracy (totally or partially). Consider for example the typical case where the degeneracy is lifted in second order perturbation theory. In that case we may take n = 1 and we have T (1)lh = O(λ2 ):


(1)ll (1)hh (1)lh H (1) = V + λj T j + λj T j + λj T j . (2.28) j ≥1

j ≥1

j ≥2

We then decompose H (1) = V˜ + T˜ into a new “classical part” V˜ given by V˜ = V +

2
j =1

(1)ll

λj T j

,

(2.29)

and T˜ contains all remaining terms. The new perturbation satisfies the bounds T˜ = hh lh O(λ3 ), T˜ = O(λ), and T˜ = O(λ2 ). If V˜ is a classical interaction with a sufficiently regular zero-temperature phase diagram, then Pirogov–Sinai techniques can be applied to study the phase diagrams of V˜ + T˜ for sufficiently small λ (see below). Note that this perturbation scheme is not only useful to analyze the low-temperature behavior of the model. The new “classical part” V˜ does not need to be classical at all. For example, see [DFFR, DFF2], if one applies this perturbation scheme to the Hubbard model at half-filling, V˜ is given by the Heisenberg model and this gives a rigorous proof of the equivalence of both models up to controlled error terms. ll

Quantum Lattice Models at Intermediate Temperature

41

3. Phase Diagrams, Contour Models, and Pirogov–Sinai Theory A phase diagram in Thermodynamics is a partition of a space of physical parameters in domains corresponding to phases; the free energy varies very smoothly inside a domain. However, first derivatives or of higher order may have discontinuities when crossing the boundary between two domains, and in this case one talks of phase transitions. The first proof of a phase transition was proposed by Peierls for the Ising model [Pei]. It was extended by Pirogov and Sinai [PS, Sin] to situations where different phases are not related by a symmetry. Important extensions and simplifications of the Pirogov– Sinai theory include Kotecký and Preiss [KP], Zahradník [Zah], Bricmont et al. [BKL] and [BS], Borgs and Imbrie [BI], Borgs and Kotecký [BK, BK2]. An exposition of the Pirogov–Sinai theory can be found in [EFS]. Another extension of the Peierls argument was done in Fröhlich and Lieb [FL] using reflection positivity [FSS, DLS]. 3.1. Phase diagrams. We consider the Banach space Br of periodic interactions, with the norm defined in (2.6). Here r is any positive number, but further assumptions (bounds for the weights of the contours, see below) can be verified in given models only if r is large enough. To a given interaction H ∈ Br and temperature β we associate the set of all translation invariant (or periodic) KMS states or, equivalently [Ara, Isr], the set of all tangent functionals to the free energy f (H ). The set of periodic KMS states forms a simplex, so that it is enough to describe the extremal states, or the corresponding tangent functionals. We denote the set of extremal states by E β (H ). In order to define a phase diagram we consider a smooth (p−1)-dimensional manifold on the Banach space Br of periodic interactions; it is described by an application u ) → H u , from a connected open set U ⊂ Rp−1 into Br . For m = 1, 2, 3, . . . , we introduce E (m) = {H ∈ Br : |E β (H )| = m}; accordingly, we partition the set U as U=

∞

∪ U (m) ,

m=1

(3.1)

where u ∈ U (m) iff H u ∈ E (m) . The decomposition (3.1) is called the phase diagram of H u. The phase diagram of H u , u ∈ U ⊂ Rp−1 , is said to satisfy the Gibbs phase rule if the following conditions hold. Here, we call “boundary” of U (i) the set (U¯ (i) \ U (i) ) ∩ U, with U¯ (i) the closure of U (i) . (i) U = U (1) ∪ · · · ∪ U (p) . (ii) (a) U (1) consists of p connected components, each of which is a (p−1)-dimensional manifold. The boundary of U (1) is U (2) ∪ · · · ∪ U (p) .  (b) U (2) consists of p2 connected components, each of which is a (p − 2)-dimensional manifold.The boundary of U (2) is U (3) ∪ · · · ∪ U (p) . p (q) (c) U consists of q connected components, each of which is a (p − q)-dimensional manifold. The boundary of U (q) is U (q+1) ∪ · · · ∪ U (p) . (d) U (p) consists of a single point u0 . In other words, the phase diagram of H u satisfies the Gibbs phase rule iff it is homeomorphic to a connected, open neighborhood U  of the boundary of the positive octant of Rp , in such a way that u0 is mapped onto the origin, U (p−1) is mapped onto the union of axis ∪i {ai > 0, aj = 0, j  = i}, and so on...

42

J. Fröhlich, L. Rey-Bellet, D. Ueltschi

Connected components of U (1) are the one-phase region, or pure phase region, U (2) is the region of coexistence of two phases, . . . , U (p) is the point of coexistence of all p phases. We will call a phase diagram which satisfies the Gibbs phase rule regular if the free energy is a real analytic function of u in each one-phase region, and if all connected components of the manifold U (j ) are smooth (C 1 ). 3.2. Contour models. A contour A is a pair (A, α), where A ⊂ Zν is a finite connected set and is the support of A; to describe α, let us introduce the closed unit cell C(x) ⊂ Rν centered at x, i.e. C(x) = {y ∈ Rν : |y − x|∞
21 }. The boundary B(A) of A ⊂ Zν is the union of plaquettes B(A) = {C(x) ∩ C(y) : x ∈ A, y ∈ / A}.

(3.2)

The boundary B(A) decomposes into connected components; each connected component b is given a label αb ∈ {1, . . . , p}, and α = (αb ). Let  ⊂ Zν finite, with periodic boundary conditions.A set of contours {A1 , . . . , Ak } is admissible iff • Ai ⊂ , and dist (Ai , Aj )  1 if i  = j . • Labels αj are matching in the following sense. Let W =  \ ∪kj =1 Aj ; then each connected component of W must have the same label on its boundaries. For j ∈ {1, . . . , p}, let Wj be the union of all connected components of W with labels j on their boundaries. β,u For each j ∈ {1, . . . , p}, we give ourselves a complex function gj (“free energy of a restricted ensemble”), that is real analytic in u ∈ U. We suppose that the limit β → ∞ β,u of gj exists, and we write β,u

eiu = lim Re gi , β→∞

1
i
p,

(3.3)

e0u = min eiu .

(3.4)

i

We consider the partition function (2.8) for an interaction H u = V u + T , where the periodic interaction T is a perturbation of V u . We assume that the partition function can be rewritten as β,u,T

Z

=




k

{A1 ,...,Ak } j =1

w β,u,T (Aj )

p

β,u

e−βgi

|Wi |

,

(3.5)

i=1

where the sum is over admissible sets of contours in .1 The weight w β,u,T (A) of a contour A is a complex function of β, u, and T , that behaves nicely for β large and T in a neighborhood of 0. Precisely, we assume that there exists a set W ⊂ R+ × Br , that is open and connected, and whose closure contains (∞, 0); furthermore, we suppose that for all u ∈ U and all (β, T ) ∈ W, and all contours A, β,u −βgi || 1 The sum includes the case k = 0, and the corresponding term is p . It is however j =1 e

irrelevant, since it does not contribute to the infinite-volume free energy (3.6).

Quantum Lattice Models at Intermediate Temperature

43

• w β,u,T is periodic with period =, i.e. we have w β,u,T (τa A) = w β,u,T (A) for all a ∈ (=Z)ν and all A. Here τa is the translation operator. u • |w β,u,T (A)|
e−βe0 |A| e−τ |A| for a large enough constant τ (depending on ν, p, and =). Furthermore, |

u ∂ β,u,T w (A)|
β|A|C e−βe0 |A| e−τ |A| ∂ui

and |

u ∂ β,u,T +ηK (A)|
β|A|CKr e−βe0 |A| e−τ |A| w ∂η

for a uniform constant C. • limβ→∞ limT →0 w β,u,T (A) = 0. This means that the weights represent the correction to the situation (β = ∞, T = 0). • wβ,u,T (A) is real analytic in u; for all K ∈ Br , wβ,u,T +ηK (A) is real analytic in η in a neighborhood of 0 (the neighborhood depends on K). Finally, the free energy is f β,u,T = −

1 1 β,u,T lim ν log Z . β Z ||

(3.6)

We also assume the following properties for f β,u,T : • f β,u,T is real, and concave as a function of T ;  • whenever H u + T = H u + T  , we have 



f β,u,T = f β,u ,T .

(3.7)

Although these properties seem difficult to verify in the context of a contour model, they are usually clear in the original physical model.

3.3. The Pirogov–Sinai theory. The results of the Pirogov–Sinai theory are usually presented in terms of existence of many Gibbs states for a given interaction. However, it is more convenient to think of the Pirogov–Sinai theory as to express the free energy in a suitable form for the description of first-order phase transitions: the free energy is given as the minimum of C 1 functions (“metastable free energies”), that intersect themselves by making angles, hence a first-order phase transition when varying parameters so as to cross an intersection. The free energy at zero temperature is given by (3.4); in typical situations this is the minimum over energies of some important configurations (the “potential ground states”). The Pirogov–Sinai theory shows that in contour models, this structure extends at low temperatures. In the quantum situation one is also interested in adding a perturbation to a “nice” model; the metastable free energies then depend not only on β, but also on the quantum perturbation. We claim that the Pirogov–Sinai theory allows to construct metastable free energies that satisfy the following properties.

44

J. Fröhlich, L. Rey-Bellet, D. Ueltschi

Properties of the metastable free energies. We consider a contour model that satisfies the β,u,T for (β, T , u) ∈ structure described in Sect. 3.2. Then there exist p real functions fi W × U, such that β,u,T

(a) f β,u,T = mini fi ; β,u,T (b) limβ→∞ limT →0 fi = eiu , and limβ→∞ limT →0

∂ ∂uj

β,u,T

fi

∂ u ∂uj ei ; β,u,T +ηK fi

=

(c) for all K ∈ Br , there exists a neighborhood NK of 0 such that is C 1 as ∂ β,u,T +ηK a function of (u, η) in U × NK , and | ∂η fi |
CKr for a constant C depending on ν, p, = only;  β,u,T β,u,T β,u,T (d) fi is a real analytic function of u in M{i} = u : fi < fj ∀ j = i . Notice that the point (d) implies that the free energy f β,u,T is a real analytic function of u in ∪i M{i} (which is the region of uniqueness, as will be seen below). The proof of these properties involves the full artillery of the Pirogov–Sinai theory. The item (c) is not really standard and may appear as superfluous technicalities, but it plays a role when establishing the properties of the phase diagram, see Theorem 3.1 below. Since the present paper is only aimed at studying a special class of quantum models, we content ourselves with an outline of the proof, so as to make it plausible for readers who have knowledge of the details of the Pirogov–Sinai theory. A review of the Pirogov–Sinai theory is expected to appear shortly and will contain a detailed proof of these properties. Sketch of the proof of these properties. We heavily rely on [BKU], which itself follows [PS, Sin, Zah, BI, BK, BK2]. Our metastable free energies are defined as the real part of the metastable free energies of [BKU], which are complex in general. The first step consists in defining the metastable free energies. This can be done by introducing truncated contour activities and truncated partition functions following the inductive procedure of [BKU], Eqs. (5.6)–(5.12). One obtains metastable free energies (n) fj (that depend on β, u, T ). One can then prove the claims of Lemma A.1 i), iii), iv), β,u,T

(n)

= limn→∞ fj . v), vi) of [BKU]. We then set fj At this point we have well-defined metastable free energies depending on β, u and T (that is, they are functionals on the Banach space of interactions), and the free energy of the system is given by the minimum of the metastable free energies, as stated in item β,u,T β,u,T = eiu , and that fi is real analytic in (a). It is also clear that limβ→∞ limT →0 fi u on M{i} . What remains to be done is to check differentiable properties. β,u,T +ηK For given T and K, we consider fj as a function of (u, η). This is a mild complication of the situation in [BKU], since the metastable free energies here depend on p parameters instead of p − 1. One then gets the items ii) and vii) of Lemma A.1 – the partial derivatives with respect to η of the truncated contour activities and of the partition function with given external label satisfying the claims of the lemma with a constant C0 Kr instead of C0 . Finally, the metastable free energies are given as convergent series of clusters of contours, the weights of those obeying suitable bounds. This leads to item (c). / 0 We show now that these metastable free energies allow for a complete characterization of tangent functionals, under the extra assumption that the situation at zero temperature and without perturbation satisfies the Gibbs phase rule in a strong sense.

Quantum Lattice Models at Intermediate Temperature

45

The stronger condition for the Gibbs phase rule is that, for some u0 ∈ U, we have that all “potential ground state energies” are equal, eiu0 = eju0 for all i, j , and that the matrix of derivatives

∂   (3.8) eiu − epu 1
i,j
p−1 ∂uj has an inverse that is uniformly bounded. Actually, energies eiu may not be differentiable; β,u in this case, we consider the same matrix with Re gi instead of eiu , and we suppose that it has an inverse for all β large enough, the inverse matrix being uniformly bounded with respect to u ∈ U, and β  const. Theorem 3.1 (Stability of the phase diagram). Assume that there exist metastable free β,u,T energies fi , 1
i
p, that satisfy all points (a)–(d) of the properties above. We assume in addition that the strong version of the Gibbs phase rule, described above, is satisfied. Then for β large enough and T r small enough (depending on p and on the bound of the inverse of the matrix of derivatives (3.8)), there exists U  ⊂ U such that the phase diagram for H u + T , u ∈ U  , at inverse temperature β, satisfies the Gibbs phase rule and is regular. Theorem 3.1 states that there exists u0 ∈ U  such that the set of tangent functionals to  the free energy at H u0 + T is a simplex with p extremal points. More generally, we have the decomposition U  = U  (1) ∪ · · · ∪ U  (p) such that for u ∈ U  (q) , the set of tangent functionals at H u + T is a q-dimensional simplex. This “completeness” of the phase diagram was addressed in [Zah] and [BW]. The approach was however different and involved studying the Gibbs states, which is more intricate and does not easily extend to the quantum case. It is simpler to look at tangent functionals, and then to use existing results on their equivalence with DLR or KMS states. Notice that the Pirogov–Sinai theory also provides various extra information, such as the fact that the limit of U  (q) , as T → 0 and β → ∞, is equal to U (q) . Also, the extremal equilibrium states can be shown to be exponentially clustering. We do not claim these properties here however, because doing so would require extra assumptions and technicalities in the description of the abstract contour model. Proof of Theorem 3.1. Items (b) and (c) of the properties of metastable free energies β,u ,T

β,u0 ,T

(with η = 0) imply that there exists u0 such that fi 0 = fj the matrix of derivatives

∂   β,u,T fi − fpβ,u,T 1
i,j
p−1 ∂uj

for all i, j , and that (3.9)

has a bounded inverse, uniformly in u in a neighborhood U  of u0 . Let us define β,u,T

Mi = {u ∈ U  : fi and, for Q ⊂ {1, . . . , p}, MQ =

 i∈Q

β,u,T

= min fj j

Mi \

 i ∈Q /

Mi

},

(3.10)

(3.11)

46

J. Fröhlich, L. Rey-Bellet, D. Ueltschi

(notice that M{i}  Mi ). By the implicit function theorem, each MQ is described by a C 1 function from an open subset of Rp−|Q| into U  . If we set U (q) = ∪|Q|=q MQ the phase diagram satisfies the Gibbs phase rule, provided there are exactly |Q| tangent functionals at H u + T for each u ∈ MQ . β,u,T Each metastable free energy fj , j ∈ Q, defines a tangent functional αj : for all β,u,T +ηK

∂ fj |η=0 . Notice that item (c) ensures boundedness K ∈ Br , we set αj (K) = ∂η 2 of the tangent functional. We show now that these tangent functionals are linearly independent, and that any other tangent functional is a linear combination of these ones. We examine the manifold where q phases coexist; without loss of generality, we can choose u˜ ∈ MQ with Q = {1, . . . , q}. The determinant of (3.9) can be written as a linear combination of determinants of

∂   β,u,T ˜ ˜ fi − fqβ,u,T , 1
i,j
q−1 ∂ukj

(3.12)

with k1 , . . . , kq−1 being q − 1 different indices. Since the determinant of (3.9) differs from 0, at least one of the determinants in the previous equation differs from 0. Without loss of generality we can assume that

∂   β,u,T ˜ ˜ fi − fqβ,u,T 1
i,j
q−1 ∂uj

(3.13)

is not singular. p−1 Our analysis is local, so we can take u˜ = 0 and H u = H 0 + j =1 uj K j . Then β,u,T

|u=0 , and non-singularity of (3.13) shows that (3.7) implies that αj (K i ) = ∂u∂ i fj αj , 1
j
q, are linearly independent. Furthermore, it also implies that for all tangent functionals α  the system of equations for ξ = (ξ1 , . . . , ξq ), α  (K i ) =

q


ξj αj (K i ),

i = 1, . . . , q − 1,

(3.14)

j =1

has a unique solution with gj (u, η) =

β,u,T +ηK fj ,



j ξj

= 1 . Now we consider any K ∈ Br ; we define

1
j
q, and 

 g1 (u, η) − gq (u, η)   .. g(u, η) =  . . gq−1 (u, η) − gq (u, η)

(3.15)

∂ g(0, 0) is an isomorphism, and g(u, η) is a map of class C 1 We have g(0, 0) = 0, ∂u by item (c) of the properties metastable free energies. By the implicit function theorem 2 One may wonder whether the functional α is linear. It is actually, because α can be obtained as the j j limit of linear functionals that are tangent to the free energy, uniquely defined for all points of M{j } – a region of parameters where the concave free energy has a unique tangent functional.

Quantum Lattice Models at Intermediate Temperature

47

there exists a map u(η) such that g(u(η), η) = 0. We introduce the interactions R(η) = K +

q−1 1
uj (η)K j , η

(3.16)

j =1

uj (0)K j .

(3.17)

= · · · = fqβ,0,T +ηR(η) .

(3.18)

R = lim R(η) = K + η→0

q−1
j =1

Then using (3.7) we have β,0,T +ηR(η)

f β,0,T +ηR(η) = f1

Differentiating with respect to η, we obtain (recall that α  is tangent to f β,0,T +ηR(η) at η = 0) α  (R) = α1 (R) = · · · = αq (R). (3.19)  Then obviously α  (R) = j ξj αj (R), and it follows by linearity of the tangent functionals that 

α (K) =

q


ξj αj (K).

(3.20)

j =1

0 /

4. Results of the Quantum Pirogov–Sinai Theory We summarize in this section the results obtained in [BKU, DFF, DFFR, KU], and in the present paper. All results concern the situation where the interaction has the form H = V + T , where V is a classical interaction satisfying the standard Pirogov–Sinai framework, and T is a small perturbation. The temperature will be assumed to be small. The results however split into four classes, according to whether we use the perturbation methods of [DFFR] (Sect. 2.2), and whether we include high temperature expansions to analyze phases at intermediate temperatures. In this section, we implicitly assume all properties of the metastable free energies, see Subsect. 3.3, to be valid – without these properties the statements below would not include completeness, i.e. we could not ascertain to have identified all the periodic Gibbs states of the systems. 4.1. Quantum perturbation of classical model with finitely many ground states. In this case the classical interaction V has finitely many ground states and the phase diagram of V + T is, at low temperatures and for sufficiently small T a small deformation of the zero temperature phase diagram of V . The extension of the Pirogov–Sinai theory to this class of quantum systems goes back to [Pir] and was proved in [BKU, DFF]. ν (a) Structure. We denote by . = {1, . . . , M}Z the space of classical configurations; the dimension ν of the physical space is always supposed to be bigger or equal to 2. The interaction has the form H = V +T , where V is a block interaction and is diagonal with

48

J. Fröhlich, L. Rey-Bellet, D. Ueltschi

respect to the basis of classical configurations: if A = U (x) ≡ {y : |y − x|∞
R} for some x ∈ Zν , VA |e&ω = 0x (ωU (x) ) |e&ω ,

(4.1)

and VA = 0 if there is no x with U (x) = A. The function 0x depends on µ ∈ U ⊂ Rp−1 , and we assume that its derivatives ∂µ∂ j 0x (ωU (x) ) are bounded uniformly in x, µ, ω, j .

A finite set G = {g (1) , . . . , g (p) } ⊂ . of periodic configurations is given, that contains all ground states of V for all µ (see below the precise assumption). We write GA = {gA : g ∈ G}. We suppose that 0x (gU (x) ) is independent of x, for all g ∈ G, and µ this value is denoted by eg (this is the mean energy of the configuration g). (b) Assumptions. (A1) A gap separates the excitations: for all ωU (x) ∈ / GU (x) , 0x (ωU (x) ) − min 0x (gU (x) )  D g∈G

(uniformly in µ). (A2) The zero temperature phase diagram is (linearly) regular: there is µ0 ∈ U such µ µ µ that eg 0 = eg 0 for all g, g  ∈ G, and the inverse of the matrix of derivatives MG , see (3.8), is uniformly bounded. (c) Properties of Gibbs states. Theorem 4.1. Assume (A1) and (A2) hold true. There exist β0 , c < ∞ (depending on ν, R, p, M and on the periods of {g (j ) } and H only) such that if βD  β0 and T c /D
1, the phase diagram of the quantum model satisfies the Gibbs phase rule and is regular in a neighborhood U  ⊂ U of µ0 . In the single phase region, i.e. if µ ∈ Mβ ({g}), the KMS state w β,µ,T (·) is close to the ground state g: for all K ∈ OA , limβ→∞,T r →0 w β,µ,T (K) = %eg |K|eg &. The condition T c /D
1 means that T is a perturbation with respect to V ; c plays the role of the perturbative parameter: from Definition (2.6) of the norm  · c , TA  must be very small if c is very large. The proof of this theorem follows from [BKU, DFF]. 4.2. Models with infinite degeneracy. Consider a model whose classical part has infinitely many ground states, and a perturbation which lifts this degeneracy completely. The pertubation methods of [DFFR] (see Sect. 2.2) permits one in certain cases to analyze this by constructing an equivalent interaction with a new classical part which has finitely many ground states. In this case the new perturbation has a slightly more complicated form than in Sect. 4.1 and the following theorem deals with this situation. This situation was considered in [DFFR] (for a different approach see [KU]). ν (a) Structure. The space of classical configurations is again . = {1, . . . , M}Z . We consider two sets G, D ⊂ ., with D ⊂ G finite, D = {d (1) , . . . , d (p) } is a finite set of periodic configurations; G may be infinite and will represent the configurations of low energy. For A ⊂ Zν , the Hilbert space HA has the following decomposition HA = low ⊕ Hhigh , where Hlow is the subspace spanned by the low energy configurations HA A A gA ∈ GA . The interaction has the form H = V + T , where V is a classical block

Quantum Lattice Models at Intermediate Temperature

49

interaction with uniformly bounded derivatives ∂µ∂ 0x (ωU (x) ), and T is a perturbation j that is submitted to some restrictions, see the assumptions below. (b) Assumptions. / GU (x) , (B1) A gap separates high and low energies: for all ωU (x) ∈ 0x (ωU (x) ) − max 0x (gU (x) )  D0 . g∈G

(B2) Gap with the ground states: we assume that 0x (dU (x) ) is independent of x for d ∈ D, and for all ωU (x) ∈ / DU (x) , 0x (ωU (x) ) − min 0x (dU (x) )  D d∈D

(and we assume that D
D0 ). (B3) The perturbation may be decomposed T = K + K  + K  ; for all A, low KA HA = 0,

KA HA

⊂ HA ;

KA HA

low ⊂ HA

high

low KA HA ⊂ HA , high

high

high

(there is no assumption on K).3 µ (B4) The zero temperature phase diagram is (linearly) regular, i.e. all energies ed are µ equal for some µ0 ∈ U, and the matrix MD [see (3.8)] has a uniformly bounded inverse. (c) Properties of Gibbs states. Theorem 4.2. Assume (B1)–(B4) hold true. There exist β0 , c < ∞ (depending on ν, R, p, M and on the periods of {d (j ) } and H only) such that if βD  β0 , Kc /D
1, K  c /D0
1, K  c /D0
1 the phase diagram of the quantum model satisfies the Gibbs phase rule and is regular in U  ⊂ U, U   µ0 . In the single phase region, i.e. if µ ∈ Mβ ({d}), the KMS state w β,µ,T (·) is close to the ground state d: for all K ∈ OA , limβ→∞,T r →0 w β,µ,T (K) = %ed |K|ed &. The proof of this theorem is given in [DFFR]. A somewhat different method yielding similar results has been developed later in [KU]. 4.3. Combined high and low temperature expansions. Here we consider models whose classical part V has partially ordered ground states, typically described by periodic configurations of holes and particles but still with infinite degeneracy due to, e.g., degeneracy of the spin at each site. Together with the quantum perturbation the system may have a continuous symmetry. We will suppose that the temperature is low and, in addition, that βT c is actually small (i.e. the temperature is large compared to T ) and we will prove that in this case one phase corresponds to each periodic configuration of holes and particles and that in this phase the spin degrees of freedom are in a disordered phase. This situation has many similarities with that of [BKL], and could be called “a theory of restricted ensembles in quantum lattice systems”. 3 Motivation comes from (2.25). It is however slightly more general, and it is just what is required in the proof of Theorem 4.2.

50

J. Fröhlich, L. Rey-Bellet, D. Ueltschi ν

(a) Structure. As before, let . = {1, . . . , M}Z . Intermediate temperature phases will be characterized by “motives” giving partial information on the underlying configurations. In order to describe this, we consider a partition of {1, . . . , M}: N 

{1, . . . , M} =

Ij

with Ii ∩ Ij = ∅.

(4.2)

j =1

We denote N = {1, . . . , N} (and N ≡ NZν ). For n ∈ N , we write .n = {ω ∈ . : ωx ∈ Inx ∀x}. Let G = {g (1) , . . . , g (p) } ⊂ N be a finite set of periodic configurations; this is the set of motives and a pure phase will be associated with each of these configurations. We write .G = ∪g∈G .g . The interaction has the form H = V +T , where V is a classical block interaction with uniformly bounded derivatives w.r.t. µ, and T is a perturbation. We introduce restricted partition functions for each g ∈ G: let 
g Z = e−β x,U (x)⊂ 0x (ωU (x) ) (4.3) ω ∈.g,

and hβ,µ =− g

1 1 g lim log Z . β Zν ||

(4.4)

β,µ

µ

The ground energies are eg = limβ→∞ hg , g ∈ G. (b) Assumptions. (C1) For all configurations ωU (x) ∈ / .G,U (x) , we have  0x (ωU (x) ) − min 0x (ωU (x) )  D.  ω ∈.G

Moreover, we assume that min

ωU (x) ∈.g,U (x)

0x (ωU (x) ) = eµ (g)

independently of x, for all g ∈ G. (C2) We need a condition that ensures that no phase transition takes place in a restricted ensemble .g ; in other words, spatial correlations should decay quickly enough. The following condition is stronger, and amounts to saying that there is no correlation between different sites. For all g ∈ G, we suppose that there exists an on-site interaction 0g such that for all x: g

0x (ωU (x) ) = 0x (ωx ) for all ω ∈ .g . µ µ (C3) The zero temperature phase diagram is regular with eg 0 = eg 0 , g, g  ∈ G, for µ some µ0 ∈ U, and the matrix MG , see (3.8), has a uniformly bounded inverse.4 µ

β,µ

4 If {e } are not C 1 , we consider the matrix of derivatives of h g g is bounded uniformly w.r.t. µ and large β.

for β large; it must have an inverse that

Quantum Lattice Models at Intermediate Temperature

51

(c) Gibbs states at intermediate temperature. Theorem 4.3. Assume (C1)–(C3) hold true. There exist β0 , c < ∞ (depending on ν, R, p, M and on the periods of {g (j ) } and H only) such that if β0
βD < ∞ and βT c
1, the phase diagram satisfies the Gibbs phase rule and is regular in U  ⊂ U, U   µ0 . In the single phase region, i.e. if µ ∈ Mβ ({g}), the KMS state w β,µ,T (·) is close β,µ,T (K) = (Tr(P ))−1 Tr(KP ), to the motive g: for all K ∈ OA , lim β→∞,T r →0 w A A where PA is the projection given by ωA ∈.g,A |eωA &%eωA | . Remark. It follows from our assumptions that T is small compared to V ; more precisely, T c /D
1/β0 . This theorem is actually a consequence of Theorem 4.4 below, see the remark after Theorem 4.4. 4.4. Infinite degeneracy, high and low temperature expansions. Here we consider systems where phases result from subtle interplay between potential and kinetic energy, combining the effect described in Sects. 4.2 and 4.3. The quantum perturbation lifts partially the degeneracy of the classical interaction, leading at intermediate temperatures, to spatially ordered phases. Hereafter we describe the general framework in a rather abstract way; it will be illustrated in Sect. 5, and the reader may gain better understanding by working out a concrete application. ν (a) Structure. The space of classical configurations is . = {1, . . . , M}Z ; we consider a partition like in (4.2) and define similarly N and .n . We consider a (possibly infinite) set G ⊂ N that represents low energy configurations; the Hilbert spaces decompose in low ⊕ Hhigh , where Hlow is the subspace spanned by the the following way: HA = HA A A low-energy configurations gA ∈ GA . The interaction has the form H = V + T ; V is a block interaction with uniformly bounded derivatives ∂µ∂ 0x (ωU (x) ); the perturbation j

T decomposes further T = K + K  + K  ; we shall require different assumptions on K, K  , K  , motivated by the perturbation theory of Sect. 2.2. We suppose that a finite set D = {d (1) , . . . , d (p) } ⊂ G is given, that corresponds to possible ground states. For each d ∈ D, we define the corresponding restricted partition function 
d Z = e−β x,U (x)⊂ 0x (ωU (x) ) (4.5) ω ∈.d,

and the corresponding restricted free energy β,µ

hd µ

=−

1 1 d , lim ν log Z β Z ||

β,µ

and ed = limβ→∞ hd . (b) Assumptions. (D1) A gap separates high and low energies: for all ωU (x) ∈ / .G,U (x) ,  0x (ωU (x) ) − max 0x (ωU (x) )  D0 .  ω ∈.G

(4.6)

52

J. Fröhlich, L. Rey-Bellet, D. Ueltschi

(D2) Gap with the ground states: for all ωU (x) ∈ / .D,U (x) ,  0x (ωU (x) ) − min 0(ωU (x) )  D.  ω ∈.D

(D3) For all d ∈ D, there exists an on-site interaction 0d such that for all ω ∈ .d and all x, 0x (ωU (x) ) = 0dx (ωx ). Moreover, we suppose that µ

min 0dx (ωx ) = ed

ωx ∈Idx

independently of x. (D4) The quantum perturbation T = K + K  + K  has the same properties as in (B3), with respect to the decomposition into low and high energy states. µ µ (D5) There is µ0 ∈ U such that ed 0 = ed 0 , d, d  ∈ D, and the matrix of derivaµ tives (3.8) has a uniformly bounded inverse (see the footnote of (C3) if ed is not differentiable). (c) Properties of Gibbs states. Theorem 4.4. Assume (D1)–(D5) hold true. There exist β0 , c < ∞ (depending on ν, R, p, M and on the periods of {d (j ) } and H only) such that if β0
βD < ∞, βKc
1, K  c /D0
1, K  c /D0
1, and βK  2c /D0
1, the phase diagram satisfies the Gibbs phase rule and is regular in an open set U  ⊂ U that contains µ0 . In the single phase region, i.e. if µ ∈ Mβ ({d}), the KMS state w β,µ,T (·) is close β,µ,T (K) = (Tr(P ))−1 Tr(KP ), to the motive d: for all K ∈ OA , lim β→∞,T r →0 w A A where PA is the projection given by ωA ∈.d,A |eωA &%eωA | . This theorem follows from the contour representation obtained in Sect. 6, together with the Pirogov–Sinai theory. Remarks. 1. Theorem 4.3 is an immediate consequence of Theorem 4.4. Indeed, we clearly recover the setting of Sect. 4.3 by choosing G = . (i.e. all configurations have low energy), and K  = K  = 0. 2. These two theorems also generalize results of [Uel]: they can be applied to the Hubbard model

† H = −t (cxσ cyσ + h.c.) + U nx↑ nx↓ , (4.7) <x,y> σ =↑,↓

x

to show that the high temperature phase extends to 



(β, t, U ) : βt small

and

 (β, t, U ) : βt 2 /U small

(standard high temperature expansions apply when both βt and βU are small).

Quantum Lattice Models at Intermediate Temperature

53

5. Example: Extended Hubbard Model This is a Hubbard model where particles interact among each other when their distance is smaller than or equal to 1. Explicitly,



† H = −t (cxσ cyσ + h.c.) + U nx↑ nx↓ + W nx ny − µ nx . <x,y>⊂ σ =↑,↓

x∈

<x,y>⊂

x∈

(5.1) † , cxσ are creation, annihilation, operators of a fermion of spin σ at site x; Here, cx,σ † cxσ is the number of < x, y > stands for a set of nearest neighbor sites; nxσ = cxσ particles of spin σ at x (it has eigenvalues 0 and 1); nx = nx↑ + nx↓ is the total number of particles at x. The coefficient t represents the hopping, and will be taken to be small compared to the nearest-neighbor repulsion W ; µ is the chemical potential. The classical limit t → 0 was studied in [J¸ed, BJK]. The stability of the chessboard phase M(0,2) (see below) with small t is a straightforward application of [DFF]; a later study devoted to it is [BK3]. Weνstart by analyzing the classical interactions. The configuration space is . = {0, ↑, ↓, 2}Z and the corresponding classical interaction can be written as (taking R = 21 )

0x (ωU (x) ) =

U
W δωy ,2 + ν−1 2ν 2 y∈U (x)

Here we introduced qy ∈ {0, 1, 2}:   0 qy = 1  2




qy q z −

⊂U (x)

µ
qy . 2ν

(5.2)

y∈U (x)

if ωy = 0 if ωy =↑ or ωy =↓ if ωy = 2.

(5.3)

The interaction can also be written as a sum over pairs of n.n. sites; this simplifies the analysis of the zero temperature phase diagram, and the search for symmetries (see below). This pair interaction is given by 0<x,y> (qx , qy ) =

U µ (δqx ,2 + δqy ,2 ) + W qx qy − (qx + qy ). 2ν 2ν

(5.4)

This model has a hole-particle symmetry. Introducing the unitary operator U such that † † U −1 = cxσ and U cxσ U −1 = cxσ , we see that U T U −1 = T .As for the potential, U cxσ the effect of the symmetry can be exhibited by considering classical configurations; defining qx = 2 − qx , and µ = U + 4νW − µ, we easily check that 

  µ 0µ <x,y> (qx , qy ) = 0<x,y> (qx , qy ) + C,

(5.5)

where C = −U/ν − 4W + 2µ/ν does not depend on (qx , qy ). As a result, the phase diagrams (U, µ) are symmetric along the line µ= for any temperature.

U + 2νW, 2

(5.6)

54

J. Fröhlich, L. Rey-Bellet, D. Ueltschi µ ν|W |

µ νW

M1

M2

2

U ν|W |

4

M(0,2)

-2

M(1,2)

M2 4

M1

2 M(0,1)

M0 2

U νW

M0

(a) (b) Fig. 2. Zero temperature phase diagrams of the extended Hubbard model, (a) when W < 0 and (b) when W > 0. The dashed line represents the hole-particle symmetry, see (5.6)

The zero temperature phase diagrams with t = 0 are depicted in Fig. 2, in both cases W < 0 and W > 0. In the case W < 0, it decomposes into three domains M0 , M1 , and M2 ; M0 and M2 have a unique translation invariant ground state with respectively 0 and 2 particles at each site. In M1 , any configurations with one particle per site is a ground state; there is degeneracy 2|| since each particle has spin ↑ or ↓. The situation W > 0 presents a richer structure with six domains. Domains M0 , M1 and M2 have the same features as with attractive n.n. interactions. In between domains M(0,2) , M(1,2) and M(0,1) now appear. M(0,2) consists in two ground states, the two 1 chessboard configurations with alternatively 0 and 2 electrons per site. M(0,1) has 2·2 2 || ground states of the chessboard type, one sublattice being empty, while the other has exactly one particle of spin ↑ or ↓; M(1,2) is similar, with 2 particles per site on one sublattice and one on the other. We are interested in the case where the temperature is small, but bigger than 0, and with small hopping. The phase diagrams for large β and small βt are presented in Fig. 3. µ νW

µ ν|W | β,t

β,t

β,t

M2

M1

M2 2

4

U ν|W |

β,t

4

M1

2

-2

β,t

Mcb β,t

M0

2

β,t

M0

U νW

(b) (a) Fig. 3. Phase diagrams of the extended Hubbard model at intermediate temperature and with small hopping, (a) when W < 0 and (b) when W > 0. Bold lines denote first-order phase transitions. White is the region PK that resists rigorous investigations, where second-order transitions are expected

In the case W < 0, all three domains survive at low temperature and with t  = 0; a first-order phase transition occurs when crossing the border between any two domains.

Quantum Lattice Models at Intermediate Temperature

55 β,t

µ U The point ( νW = 2, νW = 1) belongs to M1 : this phase has residual entropy (it also has more quantum fluctuations, although this has much less effect). The Gibbs β,t state corresponding to the domain M1 is thermodynamically stable and exponentially clustering. The restriction to intermediate temperatures (βt
ε) is important, because, for ν  3, a phase transition is expected when the temperature decreases, leading to an antiferromagnetic phase that breaks both symmetries of translations and of rotations of the spins. The phase diagram at finite β and nonzero t is especially interesting for W > 0. β,t β,t β,t β,t There are not six, but only four domains M0 , M1 , M2 and Mcb ; see Fig. 3. Indeed, the three domains corresponding to chessboard phases have merged into a single domain (this was first understood and proven in [BJK] in the absence of hopping). The β,t β,t free energy is real analytic in the whole domain Mcb . The transition between M2 β,t and Mcb is presumably second-order, but our results do not cover the intermediate β,t β,t region between these domains. The boundary between Mcb and M1 contains a part where a first-order phase transition occurs that can be rigorously described. Crossing the boundary elsewhere presumably results in a second-order transition. Due to the thermal β,t fluctuations, the segment from (2,2) to (2,4) belongs to M1 . Our results for this model are summarized in the next two theorems.

Theorem 5.1 (Hubbard model with attractive n.n. interactions). Let ν  2. There exist constants β0 < ∞ and ε0 > 0 (depending on ν) such that the phase diagram β,t (U, µ) for β|W |  β0 and βt
ε0 is regular; domains Ma , a ∈ {0, 1, 2} satisfy β,t β,t limβ→∞ limt→0 Ma = Ma . If (U, µ) belongs to a unique Ma , there is a unique Gibbs state. Furthermore, the density of the system is close to a,   %nx & − a 
ε(β, t), for all x. ε(β, t) can be made arbitrarily small by taking β large and t small. In order to describe the situation W > 0 we first introduce the region of the phase diagram PK where we have no results. Let      M0 ∪ M1 ∪ M2 \ M(0,2) ∩ M1 , (5.7) L = M(0,2) ∪ M(1,2) ∪ M(0,1) and for K > 0, PK =



BK (U, µ),

(5.8)

(U,µ)∈L

where BK (U, µ) is the open ball of radius K centered on (U, µ). We restrict our considerations to the complement of PK . Theorem 5.2 (Hubbard model with n.n. repulsions). Let ν  2 and K > 0. There exist constants β0 < ∞ and ε0 > 0 (depending on ν and K) such that if β0
βW < ∞ and βt
ε0 , we have the decomposition β,t

β,t

β,t

β,t

PKc = M0 ∪ M1 ∪ M2 ∪ Mcb , and

56

J. Fröhlich, L. Rey-Bellet, D. Ueltschi β,t

β,t

β,t

(i) M0 ⊂ M0 , M2 ⊂ M2 , M1 ( ⊂ M1 ) are domains with a unique Gibbs state. Densities are close to 0, 2, 1 respectively in the sense  β,t  in M0 %nx &
ε(β, t) β,t %nx &  2 − ε(β, t) in M2   β,t |%nx & − 1|
ε(β, t) in M1 with ε(β, t) arbitrarily close to 0 if β is large and t small. β,t (ii) Mcb ⊂ M(0,2) ∪ M(1,2) ∪ M(0,1) is a domain with two extremal Gibbs states of the chessboard type. The free energy is a real analytic function of β and µ in the domain  β,t (β, µ) : β0 /W
β
ε0 /t and (U, µ) ∈ Mcb . β,t

β,t

(iii) Mcb ∩ M1 states.

is a line of first-order phase transition, with exactly three extremal

Remarks. The proofs of Theorems 5.1 and 5.2 use Theorem 4.3. But using Theorem 4.1, one could establish stability of domains M0 , M2 , M(0,2) for all β|W |  β0 , without the restriction that the temperature be not too small. Another possible improvement, for U, W > 0, would use Theorem 4.4 to replace the condition βt
ε0 by βt 2 /U
ε0 . The latter clearly allows lower temperatures.5

6. Combined High-Low Temperature Expansions In this section we simultaneously perform a low and a high temperature expansion. The low ) are temperature is low, in such a way that excitations above the low energy states (H rare. At the same time, the temperature is high relatively to the quantum perturbations K and K  . These expansions allow to write the partition functions as one of a contour model, that can be treated by the Pirogov–Sinai theory, see Sect. 3.2. We rewrite the quantum model as a contour model, by making a mixed low and high temperature expansion (Sect. 6.1); we define suitable weights, so that the partition function takes the form required in Sect. 3.2. Section 6.2 is devoted to proving that the weights are small compared to their size. Finally, we explain in Section 6.3 how other requirements of Sect. 3.2 are fulfilled.

6.1. Expansion of the partition function. Our intention is to expand in K + K  + K  ; in order to simplify the notation, we introduce B = (B, i), B ⊂ Zν , i = 1, 2, 3, and we write KB = TB with B = (B, 1), KB = TB with B = (B, 2), and KB = TB with B = (B, 3). We refer to B as a transition. 5 Furthermore, the restriction to intermediate temperatures arises because of possible antiferromagnetism due to “quantum fluctuations” of strength t 2 /U ; it should be stable for βt 2 /U > const; therefore this new condition is qualitatively correct.

Quantum Lattice Models at Intermediate Temperature

57

Using Duhamel’s formula, we obtain Tr e−βH = Tr e−β



B⊂ VB

+

m

e−τ1



1 x∈ 0x (ωU (x) )












0<τ1 <···<τm <β

1 B 1 ,...,B m ω1 ,...,ωm  

1 2 %ω | TB 1 |ω & e−(τ2 −τ1 ) m . . . %ω | TB m



2 x∈ 0x (ωU (x) )

1 |ω & e−(β−τm )



dτ1 . . . dτm

...

1 x∈ 0x (ωU (x) )

.

(6.1)

At this point, it is natural to define the supports of contours as all sites that belong j / .D,U (x) . But two technical to ∪j Bj , or for which there exists ωj such that ωU (x) ∈ (1) (p) are periodic rather than translation invariant; and the difficulties arise: d , . . . , d weight of a contour should not depend on the configuration outside of its support (but it may depend on the labeling α). The later difficulty is specific to systems with phases given by a restricted ensemble instead of a single configuration. To account for these difficulties, we introduce a partition of the lattice into cubes of size =, where = is the lcm of the periods of {d (i) } (considering all spatial directions). Let B¯ = ∪x∈B U (x); we define excited cubes. • A cube C is quantum excited if there is B i such that C ∩ B¯ i  = ∅. j • Otherwise, it is classically excited if there is ωj and x ∈ C such that ωU (x) ∈ / .D,U (x) . Consider the set Q of quantum excited cubes, the set E of classically excited cubes, and the set N of cubes that are neighbors of Q ∪ E (two cubes C  = C  are neighbors iff there exist x ∈ C and y ∈ C  with |x − y|∞ = 1). Connected components of Q ∪ E ∪ N form the supports of the contours. Connected components of the complement of Q ∪ E ∪ N are characterized by a configuration d ∈ D, and this information may be stored in the labeling α. The union of all components corresponding to d is denoted Wd . Then  = Q ∪ E ∪ N ∪ (∪d∈D Wd ),

(6.2)

see Fig. 4 for illustration. Wd is a union of cubes, each cube C contributing in (6.1) by a factor [we use (D3)] 
β,µ ν d e−β x∈C 0x (ωx ) = e−βhd = . (6.3) ωC ∈.d,C

Summing first over admissible sets of contours {A1 , . . . , Ak }, we can rewrite (6.1) in the following way: Tr e−βH =









e

−β

{A1 ,...,Ak } d∈D ωWd ∈.d,Wd k



e

−τ1



x∈Aj







j =1 Q⊂A m  0 B 1 ,...,B m ω1 ,...,ωm Aj Aj B¯ i ⊂Q 1 0x (ωU (x) )



x∈Wd

0dx (ωx )



 0<τ1 <...<τm <β

1 2 m 1 %ωA | TB 1 |ωA & . . . %ωA | TB m |ωA &e j j j j

dτ1 . . . dτm

−(β−τm )



x∈Aj

1 0x (ωU (x) )

. (6.4)

58

J. Fröhlich, L. Rey-Bellet, D. Ueltschi

A2

A3

A1

A4  Fig. 4. The space  is divided into cubes; contours are formed by excited cubes (in black) and by their neighbors. There are four contours on this picture

We used here the fact that the contribution of different contours factorizes. There are i }: each cube several restrictions to the sums over transitions {B i } and configurations {ωA j of Q is intersected by at least one B¯ i ; {ωi } are compatible with the labeling αj ; excited Aj

cubes of Aj \ Q do not touch the boundary of Aj ; and non excited cubes in Aj have j at least one neighbor that is excited. In the last line appears configuration ωU (x) with U (x) ∩ Wd  = ∅, hence depending on ωWd . However, in such a case x belongs to a cube j j that is not excited, so that ωU (x) ∈ .d,U (x) . From (D3) we can substitute 0x (ωU (x) ) j

with 0dx (ωx ), which does not depend any more on the configuration outside the support of the contour.6 Then we obtain Tr e−βH =






β,µ

e−βhd

{A1 ,...,Ak } d∈D

|Wd |

k

z(Aj ),

(6.5)

j =1

where the sum is over admissible sets of contours, and z(A) is the weight of the contour A. The explicit expression for z(A) looks rather tedious, but the main point is to establish the properties of Sect. 3.2. The expression of z(A) is z(A) = e−τ1















m  0 Q⊂A B 1 ,...,B m ω1 ,...,ωm 0<τ1 <...<τm <β A A B¯ i ⊂Q

1 x∈A 0x (ωU (x) )

dτ1 . . . dτm

1 2 m 1 %ωA | TB 1 |ωA & . . . %ωA | TB m |ωA & e−(β−τm )



1 x∈A 0x (ωU (x) )

,

(6.6)

i }, see above. with some restrictions on the sums over {B i } and {ωA

Remark. We constructed contours out of cubes, while the supports of contours in Sect. 3.2 are any connected sets. There is no contradiction, if we define the weight z(A) to be 0 if the support of A is not a union of cubes. 6 This is why cubes that are neighbors of excited cubes need to be considered as part of contours.

Quantum Lattice Models at Intermediate Temperature

59

6.2. Bounds for the weights of the contours. We turn to the proof of the exponential decay of the weight of contours, as required in Sect. 3.2. We give the following “spacetime” interpretation to the collection of sums and integrals in (6.6): we view (B¯ j , τj ) as a subset of A × [0, β] per , with periodic boundary conditions along the “vertical” interval [0, β]. Furthermore, to each “time” τ ∈ [0, β] per corresponds the configuration ωj for which (τj −1 , τj ]  τ . We define B=

m 

B¯ j × {τj }



Q × {0};

j =1

E=

m+1  j =1

j

E(ωQ ) × [τj −1 , τj ],

|E| =

m+1
j =1

j

|E(ωQ )|(τj − τj −1 ),

(with τ0 ≡ 0, τm+1 ≡ β, and ωm+1 ≡ ω1 ). Here, we set B¯ = ∪x∈B U (x), and E(ωQ ) = {x ∈ Q : ωU (x) ∈ / .G,U (x) }. From assumptions (D1) and D2) we can bound µ

|z(A)|
e−βe0 |A|













m  0 Q⊂A B 1 ,...,B m ω1 ,...,ωm 0<τ1 <···<τm <β A A B¯ i ⊂Q ν

dτ1 . . . dτm

e−βD|E |/= e−D0 |E|

m

TB j ,

(6.7)

j =1 j

where the sums over {B j } and {ωA } satisfy the restrictions explained above. We view each B¯ j as a connected subset of Rν+1 (one can e.g. add links between nearest neighbors). Then B ∪ E is a subset of Rν+1 made out of vertical segments and horizontal sets. We consider connected components of B ∪ E. For a connected component with m horizontal sets and m  m − 1 vertical segments, we deleted m − m + 1 of the latter, in such a way that the component remains connected. One of these components contains Q× {0}, possibly with extra vertical segments and horizontal sets. Other components have m horizontal sets and m − 1 vertical segments. Because of the structure (D4), or (B3), components not linked with Q × {0}, either consists in a single transition of type K, or include at least two transitions of type K or K  . A connected object with m horizontal sets, and (m − 1) vertical segments that end on the horizontal sets, is called a gather and is denoted by the letter G. It is illustrated in Fig. 5. We introduce the following sets of gathers: • Gm : gathers with m horizontal sets, one containing the origin {x = 0} × {τ = 0}; G = G∞ . • G : gathers of G1 that consist in a unique transition of type K. • Gm : gathers of Gm , with at least two transitions of type K or K  ; G = G∞ . The connected component of B ∪ E that contains Q × {0} can be viewed as a set of gathers, each gather being connected to Q × {0} by a vertical segment. i } leads to a set of gathers, we obtain a bound by first Since a choice of {B i } and {ωQ integrating over sets of gathers, then summing over compatible space-time configurations

60

J. Fröhlich, L. Rey-Bellet, D. Ueltschi β

B¯ j

0

A Fig. 5. A “gather” with 6 transitions and 5 vertical segments

ωA , and choosing which gathers are linked to Q × {0}. Therefore
1  µ |z(A)|
e−βe0 |A| dG1 . . . dGk k! k0



ωA links

ν

e−βD|E |/= e−D0 |E|

k



 TB  .

(6.8)

j =1 B∈Gj

 The shortcut dG means a sum over the number m of transitions, a sum over transitions over ordered times τ1 , . . . , τm , and a sum over (m − 1) vertical B 1 , . . . , B m , an integral  segments that link B¯ i × {τi } together. We define

ν ¯ z˜ (G) = e−D0 |G | TB  e2ν= (log M+τ )|B| , (6.9) B∈G

where |G| is the total length of the vertical segments of G. If βD/=ν  2ν(log M + τ ), we can write  k
(2|Q|)k  β µ z(A)| eτ |A|
e−βe0 |A| dτ e−D0 τ dG z˜ (G) k! G 0 k0 (6.10)   k
|Q|k β dτ dG z˜ (G) , k! G ∪G 0 k0

where the first sum corresponds to the number of gathers linkedto Q × {0}, and the second sum is the number of independent gathers. The shortcut G dG is identical to dG, except for the absence of an integral over τ1 , which is set to 0; integrals over G and G are similar. One easily obtains an upper bound for the gathers with a unique transition: 
ν ¯ dG z˜ (G) = KB  e2ν= (log M+τ )|B|
(2R + 1)ν Kc (6.11) G

¯ B,B0

Quantum Lattice Models at Intermediate Temperature

61

with c = 2ν=ν (2R + 1)ν (log M + τ ); this is smaller than D0 if c is large enough in the assumptions of Theorem 4.4. For general gathers, we proceed by induction. First,   (6.12) dG z˜ (G)
(2R + 1)ν Kc + K  c + K  c . G1

Next, we use the recursion inequality: 
ν ¯ dG z˜ (G)
TB  e2ν= (log M+τ )|B| Gm

¯ B,B0

 k
1
 β dτ e−D0 τ dG z˜ (G) . 2 k! Gm−1,0 0

k0

(6.13)

y∈B¯

Integrating over τ , and since K · c /D0
1 for a large enough c, we get 
ν ¯ ¯ dG z˜ (G)
TB  e2ν= (log M+τ )|B| e2|B| Gm

¯ B,B0 ν









(6.14)


(2R + 1) Kc + K c + K c . This holds independently of m. This allows to estimate the integral over gathers that contain at least two transitions of type K or K  . Let Gm ⊂ Gm be gathers with at least one transition of type K or K  . One easily obtains   dG z˜ (G)
(2R + 1)ν Kc + K  c . (6.15) Gm

Then the integral over gathers with two transitions of type K or K  can be done by integrating first on the time for such a transition, then over vertical segments and gathers at their ends, at least one of which must belong to Gm−1 . We obtain 

β

dτ 0




ν (log M+τ )|B| ¯

(KB  + KB ) e2ν=

¯ Q B,B⊂

β

dτ  e−D0 τ

0



 Gm−1

dG z˜ (G)

 k
1
 β 2 dτ e−D0 τ dG z˜ (G) k! Gm−1,0 0

k0


β|Q|



y∈B¯

(Kc + K  c )2 . D0

Plugging these estimates in (6.10), one easily gets µ

z(A) eτ |A|
e−βe0 |A| e3|A| .

(6.16)

Exponential decay of the weights of the contours is now clear. The bound on the derivative can be proven in the same way. Looking at (6.6), we see that the integrand gets a factor bounded by β|A| supx,µ,ω,j | ∂µ∂ 0x (ωU (x) )|. j

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J. Fröhlich, L. Rey-Bellet, D. Ueltschi

6.3. Other properties of the weights. The weight of the contours can be viewed as a series in powers of {KB }, {KB }, {KB }. Since it is absolutely convergent uniformly in K, K  , K  (provided they be small enough), we have by the dominated convergence theorem lim

K,K  ,K  →0

z(A) = 0.

(6.17)

Analyticity of z(A) as a function of µ and β is clear, as well as a function of η if we add a new perturbation ηL, in a neighborhood of 0 that depends on βL. Periodicity is also obvious. Acknowledgements. It is a pleasure to thank Roberto Fernández, Roman Kotecký, and Charles-Édouard Pfister for several discussions, and the referee for useful comments.

References [Ara] [BI] [BJK] [BK] [BK2] [BK3] [BKU] [BW] [BR] [BKL] [BS] [DFF] [DFF2] [DFFR] [DLS] [EFS] [FL] [FR] [FSS]

Araki, H.: On the equivalence of the KMS condition and the variational principle for quantum lattice systems. Commun. Math. Phys. 38, 1–10 (1974) Borgs, C. and Imbrie, J.: A unified approach to phase diagrams in field theory and statistical mechanics. Commun. Math. Phys. 123, 305–328 (1989) Borgs, C., J¸edrzejewski, J. and Kotecký, R.: The staggered charge-order phase of the extended Hubbard model in the atomic limit. J. Phys. A 29, 733–747 (1996) Borgs, C. and Kotecký, R.: A rigorous theory of finite-size scaling at first-order phase transitions. J. Stat. Phys. 61, 79–119 (1990) Borgs, C. and Kotecký, R.: Surfaced induced finite size effects for first order phase transitions. J. Stat. Phys. 79, 43–115 (1994) Borgs, C. and Kotecký, R.: Low temperature phase diagrams of fermionic lattice systems. Commun. Math. Phys. 208, 575–604 (2000) Borgs, C., Kotecký, R. and Ueltschi, D.: Low temperature phase diagrams for quantum perturbations of classical spin systems. Commun. Math. Phys. 181, 409–446 (1996) Borgs, C. and Waxler, R.: First order phase transitions in unbounded spin systems. II. Completeness of the phase diagram. Commun. Math. Phys. 126, 483–506 (1990) Bratteli, O. and Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics I and II. Berlin–Heidelberg–New York: Springer-Verlag, 1981 Bricmont, J., Kuroda, J. and Lebowitz, J.L.: First order phase transitions in lattice and continuous systems: Extension of Pirogov–Sinai theory. Commun. Math. Phys. 101, 501–538 (1985) Bricmont, J. and Slawny, J.: Phase transitions in systems with a finite number of dominant ground states. J. Stat. Phys. 54, 89–161 (1989) Datta, N., Fernández, R. and Fröhlich, J.: Low-temperature phase diagrams of quantum lattice systems. I. Stability for quantum perturbations of classical systems with finitely-many ground states. J. Stat. Phys. 84, 455–534 (1996) Datta, N., Fernández, R. and Fröhlich, J.: Effective Hamiltonians and phase diagrams for tight-binding models. J. Stat. Phys. 96, 545–611 (1999) Datta, N., Fernández, R., Fröhlich, J. and Rey-Bellet, L.: Low-temperature phase diagrams of quantum lattice systems. II. Convergent perturbation expansions and stability in systems with infinite degeneracy. Helv. Phys. Acta 69, 752–820 (1996) Dyson, F.J., Lieb, E.H. and Simon, B.: Phase transitions in quantum spin systems with isotropic and nonisotropic interactions. J. Stat. Phys. 18, 335–383 (1978) van Enter, A., Fernández, R. and Sokal, A.D.: Regularity properties and pathologies of position-space renormalization-group transformations: Scope and limitations of gibbsian theory. J. Stat. Phys. 72, 879–1167 (1993) Fröhlich, J. and Lieb, E.H.: Phase transitions in anisotropic lattice spin systems. Commun. Math. Phys. 60, 233-267 (1978) Fröhlich, J. and Rey-Bellet, L.: Low-temperature phase diagrams of quantum lattice systems. III. Examples. Helv. Phys. Acta 69, 821-849 (1996) Fröhlich, J., Simon, B. and Spencer, T.: Infrared bounds, phase transitions and continuous symmetry breaking. Commun. Math. Phys. 50, 79–95 (1976)

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[GKU] Gruber, Ch., Kotecký, Ch. and Ueltschi, D.: Planar and lamellar antiferromagnetisms in Hubbard models. J. Phys. A 33, 7857–7871 (2000) [Isr] Israel, R.B.: Convexity in the Theory of Lattices Gases. Princeton, NJ: Princeton University Press, 1979 [J¸ed] J¸edrzejewski, J.: Phase diagrams of extended Hubbard models in the atomic limit. Physica A 205, 702–717 (1994) [KP] Kotecký, R. and Preiss, D.: An inductive approach to Pirogov–Sinai theory. Suppl. ai rendiconti del circilo matem. di Palermo, ser. II 3, 161–164 (1984) [KU] Kotecký, R. and Ueltschi, D.: Effective interactions due to quantum fluctuations. Commun. Math. Phys. 206, 289–335 (1999) [Lieb] Lieb, E.H.: The Hubbard model: Some rigorous results and open problems. In: XIth Internat. Congress of Math. Physics (Paris, 1994), Cmabridge, MA: Internat. Press, 1995, pp. 392–412 [Pei] Peierls, R.: On the Ising model of ferromagnetism. Proc. Cambridge Philos. Soc. 32, 477-481 (1936) [Pir] Pirogov, S.A.: Phase diagrams of quantum lattice systems. Soviet Math. Dokl. 19, 1096–1099 (1978) [PS] Pirogov, S.A. and Sinai, Ya.G.: Phase diagrams of classical lattice systems. Theoretical and Mathematical Physics 25, 1185–1192 (1975); 26, 39–49 (1976) [Rue] Ruelle, Ya.G.: Statistical Mechanics: Rigorous Results. Reading, MA: W. A. Benjamin, 1969 [Sim] Simon, B.: The Statistical Mechanics of Lattice Gases. Princeton: Princeton Univ. Press, 1993 [Sin] Sinai, Ya.G.: Theory of Phase Transitions: Rigorous Results. London: Pergamon Press, 1982 [Uel] Ueltschi, D.: Analyticity in Hubbard models. J. Stat. Phys. 95, 693–717 (1999) [Zah] Zahradník, M.: An alternate version of Pirogov–Sinai theory. Commun. Math. Phys. 93, 559–581 (1984) Communicated by H. Spohn

Commun. Math. Phys. 224, 65 – 81 (2001)

Communications in

Mathematical Physics

© Springer-Verlag 2001

Ergodicity of the 2D Navier–Stokes Equations with Random Forcing
J. Bricmont1 , A. Kupiainen2 , R. Lefevere2 1 UCL, Physique Théorique, 1348 Louvain-la-Neuve, Belgium 2 Helsinki University, Department of Mathematics, P.O.Box 4, Helsinki 00014, Finland

Received: 18 May 2000 / Accepted: 8 December 2000

Dedicated to Joel L. Lebowitz for his 70th birthday Abstract: We consider the Navier–Stokes equation on a two dimensional torus with a random force, acting at discrete times and analytic in space, for arbitrarily small viscosity coefficient. We prove the existence and uniqueness of the invariant measure for this system as well as exponential mixing in time. 1. Introduction A convenient mathematical model for the study of homogeneous isotropic turbulence is to consider the Navier–Stokes equation subject to a random stationary (in space and time) forcing. The turbulent situation is modelled by a smooth force, i.e. one whose Fourier transform decays fast for large wave numbers. One is then interested in various properties of the correlation functions of the velocity field in a stationary state of the ensuing stochastic process.An obvious first question concerns the large time convergence to such a stationary state starting from an arbitrary initial condition of the velocity field, i.e. the uniqueness of the stationary state. In this paper we prove the existence, uniqueness and exponential mixing of the stationary state in the case of two dimensional turbulence. We consider the Navier–Stokes equation for an incompressible velocity field u(t, x) defined on the torus T = (R/2π Z)2 : ∂t u + (u · ∇)u − ν u = f − ∇p

(1)

supplemented with the incompressibility condition ∇ · u = 0. The external force f(t, x) consists of random kicks at discrete times
e−2πix·k fk (t) f(t, x) = k∈Z2  Research partially supported by EC grant FMRX-CT98-0175 and by ESF/PRODYN.

(2)

(3)

66

J. Bricmont, A. Kupiainen, R.Lefevere

with fk (t) =




δ(t − n)fk,n .

(4)

n∈Z

The random variables fk,n will be taken Gaussian, with mean zero, f¯k = f−k and covariance β Efkα (m)fl (n) = δk,−l δm,n δ αβ φk . Furthermore, we  will assume φ0 = 0, which implies the vanishing of theaverage force over the torus: T f(t,  x) = 0. Assuming also zero average initial velocity T u(0, x) = 0 we conclude that T u(t, x) = 0 for all times t. It is convenient to solve the incompressibility condition (2) by expressing the Navier– Stokes equation (1) in terms of the vorticity ω = ∂1 u2 −∂2 u1 which satisfies the transport equation ∂t ω + (u · ∇)ω − ν ω = g,

(5)

where g = ∂1 f2 − ∂2 f1 .  Going to the Fourier transform ωk (t) = (2π )−2 T eik·x ω(t, x)dx with k ∈ Z2 , we may solve the velocity in terms of the vorticity as uk = i and write the vorticity equation as
∂t ωk = −νk2 ωk +

(−k2 ,k1 ) ωk k2

(k × l)|l|−2 ωk−l ωl +




δ(t − n)gk (n),

(6)

n∈Z

l∈Z2 \{0,k}

where k × l = k1 l2 − l1 k2 and gk (n) are Gaussian with mean zero, g¯k = g−k and covariance Egk (m)gl (n) = δk,−l δm,n γk with

γ k = k 2 φk .

We assume −1 |k|

b−1 e−κγ

−1 |k|

≤ γk ≤ be−κγ

,

(7)

where κγ > 0, and we think of b as being large. We will be interested in the turbulent region ν → 0; therefore, when it is convenient, we will always assume below that ν is small enough, although our results hold for all ν. Before stating our result, we need some definitions. First, we define the enstrophy as (a multiple of) the square of the L2 norm
1 1 "= 2 |ωk |2 = 2 ||ω||2L2 . (8) k

Next, we fix a number r > 1 and consider the Banach space $ = {ω | ω ≡ sup |ωk ||k|r < ∞} k

Ergodicity of 2D Navier–Stokes Equations with Random Forcing

67

as our probability space, with B the product σ -algebra. Note that $ is a subspace of L2 . Finally, due to the analyticity (with probability one) of the noise, ω(t) also will turn out to be analytic with probability one and it will be useful to introduce norms capturing this property. For any positive number κ, we define a norm (that we shall call the κ-norm), ||ω||κ = sup |ωk ||k|r eκ

−1 |k|

.

(9)

k

Functions with ||ω||κ < ∞ are analytic in a κ −1 neighbourhood of the torus. The factor |k|r is useful technically (and was already used in [5]). The stochastic equation (6) gives rise to a Markov chain ω(n), n ∈ N defined by ω(n + 1) = F (ω(n)) + g(n + 1),

(10)

where F is the map at time 1 of the Navier–Stokes flow (6) without the forcing. We denote by P (ω, E) the transition probability of this chain. Our main result is the Theorem. The Markov chain (10) is defined on ($, B) and has a unique invariant measure µ there. It satisfies  2 1(||ω||κ ≥νκ) µ(dω) ≤ C exp(−cν 4 κ α ) (11) for any α > 1 + r, and C, c < ∞, depending on α. Moreover, ∀ω ∈ $ and ∀E ∈ B, we have, |P t (ω, E) − µ(E)| ≤ C(ω)e−mt , C  . where m = m(ν) > 0 for all ν, and C(ω) ≤ C ω+1 ν

(12)

Remark 1. Since || · ||κ  < || · ||κ for κ  > κ, (11) holds for all κ  -norms with κ  > κ too, including the norm || · || defining $, which corresponds to κ  = ∞. Estimate (11) means that with high probability ω is analytic in a ν 2α -neighbourhood of the torus and bounded there by ν 1−2α . By taking r close to 1, α can be taken close to 2. Remark 2. Here and below, we denote by C or c a “generic” constant that can vary from place to place, even in the same equation. Remark 3. We obtain a lower bound on m in (12) of the form  c   m ≥ exp −Cν −3 log ν −1 (see Proposition 2 and Lemma 4 below), which means, however, that our estimate on the rate of convergence is unphysically small for ν small. Let us finish this section by a brief comment on previous work on the uniqueness question. There is a long history of proofs in cases that do not correspond to the turbulence problem. Either the forcing is taken to decay very slowly for large |k|, i.e. with a lower bound of the form |k|−p (see [3] and references therein), or the viscosity is taken large [6]. The only proof of uniqueness we know of in the turbulent situation is the recent one [4] where one considers a model like ours but with bounded noise (each gk has compact support). The proof of the Theorem, given in Sect. 4, will be based on probabilistic estimates (Sect. 3) and on properties of the deterministic Navier–Stokes equation, which we discuss now.

68

J. Bricmont, A. Kupiainen, R.Lefevere

2. The Deterministic Navier–Stokes Flow In this section we derive some properties of the flow of the deterministic Navier–Stokes equation, i.e. (6) without the forcing term gk (t). Let us define a family of subsets of $ that impose constraints on the size of the L2 -norm and of the κ-norm: U (κ, φ, A) = {ω|" ≤ φ, ||ω||κ ≤ A}.

(13)

Then, we introduce a one-parameter subfamily of U (κ, φ, A): Uκ ≡ U (κ, φ(κ), A(κ)),

(14)

2

where, φ(κ) = ν 2 ϕκ α and A(κ) = νaκ. This family is useful because, as we shall see, the flow maps one Uκ in that family into another one with a smaller κ. The parameter α will be taken to satisfy α > 1 + r and ϕ and a will be chosen small depending on some “geometric” constants that will appear in the course of the proof. Thus, if ω ∈ Uκ , then for all k we have |ωk | ≤ νaκ|k|−r e−κ

−1 |k|

(15)

and 2

" ≤ ν 2 ϕκ α .

(16)

κ , 1 + ηνt min(1, κ)

(17)

Let now κ(t) =

where η will be chosen suitably small below, and denote also by ω(t) the solution of (6) without the forcing term gk (t). Proposition 1. (a) Let ω(0) ∈ Uκ , then for all 0 ≤ t ≤ 1, ω(t) ∈ Uκ(t) . (b) Suppose ω(0) ∈ $ with ω(0) ≤ Dν. Then ω(1) ∈ Uκ for κ = C(D α + ν1 ). The point of part (a) of this proposition is that the domain of analyticity of the solution of the unforced Navier–Stokes equation increases with time and its L2 and κ-norms decrease with time. Part (b) says that, even if ω(0) is not analytic, but belongs to $, the solution after time 1 is analytic and its L2 and κ-norms are bounded in terms of the norm of the initial data in $. Our proof of Proposition 1 is inspired by [5] (see also [1]). For the proof we rewrite (6) (without forcing) in integral form  t
2 2 ds e(s−t)νk (k × l)|l|−2 ωk−l (s)ωl (s) (18) ωk (t) = e−tνk ωk (0) + 0

l∈Z2 \{0,k}

and solve this in a suitable Banach space. Let Yκ be the Banach space equipped with the norm || · ||κ and Xκ,τ = {ω ∈ C 0 ([0, τ ], Yκ ) | |||ω||| ≡ sup |||ω(t)||κ(t) < ∞}. t∈[0,τ ]

We have the following existence lemma.

(19)

Ergodicity of 2D Navier–Stokes Equations with Random Forcing

69

Lemma 1. Let ω(0) ∈ Uκ , then the solution ω of Eq. (18) exists in the set Xκ,τ , for 1

τ ≤ (Cν 2 κ)−2 . Moreover, |||ω||| ≤ 2νaκ. Proof. Let ωk0 (t) = e−tνk ωk (0), 2

(20)

and write (18) as a fixed point equation ω(t) = ω0 (t) + N (ω)(t) ≡ F(ω)(t)

(21)

with 

t

Nk (ω)(t) ≡

ds e(s−t)νk

0




2

(k × l)|l|−2 ωk−l (s)ωl (s).

(22)

l∈Z2 \{0,k}

We show that the map F is a contraction in the ball B = {ω ∈ Xκ,τ | |||ω − ω0 ||| ≤ νaκ}.

(23)

Let us first show that F maps B into itself. Obviously, |||ω0 ||| ≤ νaκ and if ω ∈ B, then |||ω||| ≤ 2νaκ, which means, −1 |k|

|ωk (t)| ≤ 2νaκ|k|−r e−κ(t)

.

(24)

We must prove that |||N (ω)||| ≤ νaκ, i.e. −1 |k|

|Nk (ω)(t)| ≤ νaκ|k|−r e−κ(t)

,

(25)

for all k ∈ Z2 \0 (recall that Nk = 0 for k = 0) and for all t ∈ [0, τ ]. Inserting (24) and |k × l| |l|−2 ≤ |k||l|−1 in (22), we get:  t 2 |Nk (ω)(t)| ≤ (2νaκ)2 |k| ds e(s−t)νk 0
−1 −1 e−κ(s) |k−l| e−κ(s) |l| |k − l|−r |l|−r−1 . ×

(26)

l∈Z2 \{0,k}

Writing (17) as κ(t)−1 = κ −1 + ηνt min(1, κ)κ −1 , we obtain, since k  = 0 means |k| ≥ 1, that 1 (s 2

− t)νk2 ≤ η(s − t)ν|k| ≤ (κ(s)−1 − κ(t)−1 )|k|

(27)

holds for 0 ≤ s ≤ t ≤ 1 and η ≤ 21 . Since −|k − l| − |l| ≤ −|k| and
l∈Z2 \{0,k}

|k − l|−r |l|−r−1 ≤ C|k|−r ,

(28)

70

J. Bricmont, A. Kupiainen, R.Lefevere

(since r > 1), we get   t 1 (s−t)νk2 −1 |k|−r e−κ(t) |k| ds e 2 |Nk (ω)(t)| ≤ C(2νaκ)2 |k| 0   −1 − 1 tνk2 = Cν(2aκ)2 2|k|−1 (1 − e 2 ) |k|−r e−κ(t) |k| . − 1 tνk2

1

(29)

1

) ≤ (νt) 2 , (25) follows for τ ≤ (Cν 2 κ)−2 . The contractive Since |k|−1 (1 − e 2 property is proven similarly. Thus we obtain a unique solution of (21) in B, which 1

satisfies (24), hence ωκ(t) ≤ 2νaκ, ∀t ≤ (Cν 2 κ)−2 .

 

Proof of Proposition 1. (a) It suffices to show that the solution constructed in Lemma 1 on the interval [0, τ ] satisfies the two bounds of the proposition there: the one on the κ(t) norm of the solution ω and the one on its enstrophy. This implies trivially that the solution can be extended to the whole interval [0, 1] and satisfies also there the bounds of the proposition. The bound on the enstrophy is easy to prove; as is well known, the enstrophy satisfies
d "(t) = − νk2 |ωk |2 ≤ −ν"(t), (30) dt k=0

leading to "(t) ≤ "(0)e−νt . Since e−νt ≤ (1 + ηνt min(1, κ))− α for η small, we get 2

2

"(t) ≤ ν 2 ϕκ(t) α ,

(31)

i.e. the claim of the proposition concerning "(t). To prove the bound ωκ(t) ≤ νaκ(t), we consider separately the cases κ < 1 and κ ≥ 1. If κ < 1, it is enough to use the bound (29) which, since |k| ≥ 1, gives   −1 − 1 tνk2 |Nk (ω)(t)| ≤ νaκλ 1 − e 2 (32) |k|−r e−κ(t) |k| , where λ can be chosen arbitrarily small by decreasing a. Now inserting this bound and e

− 21 tνk2

−1 |k|

|k|−r e−(κ)

−1 |k|

≤ |k|−r e−κ(t)

,

which follows from (27) with s = 0, in (21), we conclude that   1 2  −1 − tνk − 1 tνk2 |k|−r e−κ(t) |k| +λ 1−e 2 |ωk (t)| ≤ νaκ e 2 −1 |k|

≤ νaκ(t)|k|−r e−κ(t)

(33)

(34)

,

since e

− 21 tνk2

+ λ(1 − e

− 21 tνk2

) ≤ (1 + ηνt min(1, κ))−1 ,

(35)

which holds, since |k| ≥ 1, for λ and η small enough and 0 ≤ t ≤ 1. Inequality (34) is the claim of part (a) of the proposition concerning the κ(t) norm of the solution, namely ω(t)κ(t) ≤ νaκ(t).

Ergodicity of 2D Navier–Stokes Equations with Random Forcing

Turning to κ ≥ 1, fix a number 1 < β < β α

α−1 r

71

(recall that α > 1 + r) and consider

first 1 ≤ |k| ≤ κ . Using 1

2

|ωk (t)| ≤ (2"(t)) 2 , "(t) ≤ ν 2 ϕκ(t) α , we get immediately, 1

−1 |k|

|ωk (t)| ≤ (2ν 2 ϕ) 2 κ(t) α ≤ νaκ(t)|k|−r e−κ(t) 1

rβ

(36)

for ϕ small enough and t ≤ 1 since |k|r ≤ κ α ≤ κ 1− α and κ(t)−1 |k| ≤ 1. To conclude the proof, it suffices to show |Nk (ω)(t)| ≤ νaκλ(1 − e

1

− 21 tνk2

−1 |k|

)|k|−r e−κ(t)

,

(37)

β

for |k| ≥ κ α since we may then proceed as in (33–35). β Consider first the case κ α < |k| ≤ κ. We bound |k × l||l|−2 ≤ |k||l|−1 and split the sum in (22) into 

 |ωk−l (s)||ωl (s)||k||l|−1 ≡ 91 + 92 . + (38) 0=|l|≤ |k| 2

l=k,|l|> |k| 2

In the first sum, we bound, using Lemma 1, |ωk−l (s)| ≤ 2νaκ|k − l|−r ≤ Cνaκ|k|−r , since |k − l| ≥ 21 |k|. Then Schwartz’ inequality and (31) yield


1

|ωl (s)||l|−1 ≤ (2ν 2 ϕ) 2 κ α 1



0=|l|≤ |k| 2




|l|−2

 21

1

1

1

≤ Cνϕ 2 κ α (log |k|) 2 .

(39)

0 =|l|≤ |k| 2

Combining these two bounds, we get 1

1

91 ≤ Cν 2 |k|ϕ 2 κ α (log |k|) 2 aκ|k|−r . 1

(40)

For the second sum, we use |ωl (s)| ≤ 2νaκ|l|−r (coming from Lemma 1 again), together with (31) and Schwartz’ inequality to bound it by 1 2

1 α

92 ≤ Cν |k|ϕ κ aκ 2






|l|

−2(r+1)

 21

1

≤ Cν 2 |k|ϕ 2 κ α aκ|k|−r . 1

(41)

l=k,|l|> |k| 2

Inserting (38), (40) and (41) into Nk (ω)(t) and performing the integral over time, we get the bound   1 1 1 2 |Nk (ω)(t)| ≤ Cνϕ 2 κ α (log |k|) 2 |k|−1 1 − e−tνk aκ|k|−r (42)   1 1 β 1 −1 − 1 tνk2 aκ|k|−r e−κ(t) |k| ≤ Cνe1+ην ϕ 2 κ α κ − α (log κ) 2 1 − e 2

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J. Bricmont, A. Kupiainen, R.Lefevere β

where we used κ α < |k| ≤ κ and −1 |k|

1 ≤ e−κ(t)

e1+ην

which holds since |k| ≤ κ and, see (17), κ(t)−1 ≤ κ −1 (1 + ην) if 0 ≤ t ≤ 1. Thus we obtain (37) for ϕ small enough, because β > 1, log x ≤ 1; x ; , for x > 1 and ; > 0. Finally, in the case |k| > κ the bound (29) yields immediately (37) for a small. This finishes the proof of part (a) of Proposition 1. For part (b), we can proceed as in Lemma 1, but replace aκ by D and in the definition (19) κ(t) by νt2 . The inequality (27) is then replaced by 21 (s − t)νk2 ≤ 21 (s − t)ν|k| and the proof goes as before to the conclusion ω(t) νt2 ≤ 2νD

(43)

1

for t ≤ (Cν 2 D)−2 ≡ τ . We want to rewrite this bound in the form ω(τ )κ ≤ νaκ, for −1 a suitable κ. If τ ≤ 1, i.e. D ≥ C −1 ν 2 we have ντ2 = CD 2 and we can write (43) as ω(τ )ρ ≤ 2νD where ρ = CD 2 (remember that C is allowed to vary). Choosing now κ = Cρ = C  D 2 , we obtain (since D is bounded away from zero, hence D ≤ CD 2 ) that ω(τ )κ ≤ νaκ. Applying now part (a) yields the same claim for ω(1). For τ > 1, i.e. D < C −1 ν

− 21

we get ω(1) 2 ≤ 2νD ≤ C, given the bound on D; so, ω(1)κ ≤ νaκ, ν

C ν.

with κ = Finally, for the enstrophy, we have, by (31), 2

"(t) ≤ "(0) ≤ ν 2 CD 2 ≤ ν 2 ϕκ α  if we take κ > CD α . Since α > 2, taking κ = C D α + ν1 gives an upper bound covering all cases, i.e. ω(1) ∈ Uκ .  

3. Probabilistic Estimates We define a region U ≡ Uν −p , where p > 27 α, and in which the solution of (6) is confined with high probability. Let us divide the transition probability into a likely and unlikely part: P (ω, E) = Q(ω, E) + R(ω, E),

(44)

Q(ω, E) = χU (ω)P (ω, U ∩ E).

(45)

where The following proposition about the dynamics in U and the unprobability of excursions outside U will play a central role in the proof of our uniqueness result1 . Proposition 2. (a) There exist constants c, C < ∞, c > 0, such that for all ω ∈ U , E ∈ B, |Qt (ω, E) − Qt (0, E)| ≤ 4e−mt ,   c  and t ≤ c m−1 ν −q , with q ≡ where m ≥ exp −Cν −3 log ν −1 1 Here and below, the kernel AB(ω, E) is defined in the obvious way by



(46) 2p α

− 4 > 3.

A(ω, dω )B(ω , E).

Ergodicity of 2D Navier–Stokes Equations with Random Forcing

73

(b) There exists ζ < 1, c > 0, C < ∞, such that ∀κ ≥ 0, for all ω ∈ Uκ and for κ  ≥ ζ κ,   2 P (ω, Uκc ) ≤ C exp −cν 4 κ α . (47) The proof of (46) is based on a standard argument for exponential convergence of Markov chains (given in Doob [2]), and the idea is fairly simple. If Q was a genuine transition probability, it would be enough, in order to prove the proposition, to show that Q has good mixing properties. The precise properties are stated in the lemmas below. First, Lemma 2 says that, for any point in U there is a nonzero probability to go in a finite time to a smaller region U¯ ⊂ U determined by the covariance of the noise and thus by κγ : U¯ ≡ U2κγ +ρν ,

(48)

where ρ > 0 will be chosen below (sufficiently small)2 . This is an easy consequence of Proposition 1. On each time interval, the solution increases its domain of analyticity (which is determined by κ, i.e. κ decreases); then, if the “kicks” of the noise are sufficiently small (but not too small, so that this event is not too unprobable), the solution reaches U¯ in a finite time (of order ν −1 log ν −1 ). Secondly, we show in Lemma 3 that, in the region U¯ , the stochastic dynamics is sufficiently mixing; this is again due to the fact that the deterministic Navier–Stokes evolution increases the domain of analyticity of the solution. Third, the fact that Q is not a bona fide transition probability is what limits the proposition to finite times. For longer times, we will need to have some estimate on the probability of escaping the region U , which follows from part (b) of the Proposition. Indeed, the latter implies, using (44, 45) and taking κ = κ  = ν −p that, for all ω ∈ U , −q

P (ω, U c ) = R(ω, $) ≤ e−cν , with q =

2p α

(49)

− 4 > 3 (remember that p > 27 α and that ν is small).

Lemma 2. There exist constants c, C < ∞, such that ∀ω ∈ U ,   P T1 (ω, U¯ ) ≥ exp −Cν −3 (log ν −1 )c ,

(50)

with T1 = Cν −1 log ν −1 . Lemma 3. There exist constants c, C < ∞, such that, ∀ω, ω ∈ U¯ , ∀B ⊂ U¯ ,   P (ω, B) + P (ω , U¯ \B) ≥ exp −Cν −2 (log ν −1 )c . Lemmas 2, 3 imply that there exist

  δ(ν) ≡ exp −Cν −3 (log ν −1 )c

and

T ≡ T (ν) = Cν −1 log ν −1

2 Similar ideas were used by Kuksin and Shirikyan in [4].

(51)

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J. Bricmont, A. Kupiainen, R.Lefevere

with C, c < ∞, such that ∀ω, ω ∈ U and ∀B ⊂ U¯ , P T (ω, B) + P T (ω , U¯ \B) ≥ δ(ν),

(52)

which implies in turn, since U¯ ⊂ U , that ∀ω, ω ∈ U and ∀B ⊂ U , P T (ω, B) + P T (ω , U \B) ≥ δ(ν).

(53)

This is the main inequality that we shall use now.

3.1. Proof of Proposition 2. We start with the proof of part (a), where we shall use (49), which is a consequence of part (b), to be proven independently below. To get (46) we follow, with slight modifications, an argument in [2, pp. 197–198]. Let Q(t, E) = inf Qt (ω, E), Q(t, E) = sup Qt (ω, E). ω∈U

ω∈U

Fix ω, ω ∈ U and consider the function defined on subsets E ⊂ $: ψω,ω (E) = QT (ω, E) − QT (ω , E). Let S + be the set such that ψω,ω (E) ≥ 0 for E ⊂ S + and ψω,ω (E) ≤ 0 for E ⊂ U \S + ≡ S − (S ± depend on ω, ω , but we suppress this dependence). Observe that writing, see (44), P = Q + R, and using (49), we have, for any ω ∈ U , E ⊂ $, that |P T (ω, E) − QT (ω, E)| = |

T
−1

Qt RP T −t−1 (ω, E)| ≤ T e−cν

t=0

−q

1

≡ 2 ;(ν).

(54)

Then, |ψω,ω (S + ) + ψω,ω (S − )| = |QT (ω, $) − QT (ω , $)| ≤ ;(ν),

(55)

since P T (ω, $) = P T (ω , $) = 1. Moreover, using (54, 53), ψω,ω (S + ) = QT (ω, S + ) − QT (ω , S + ) ≤ 1 − (P T (ω, S − ) + P T (ω , S + )) + ;(ν) ≤ 1 − δ(ν) + ;(ν).

(56)

Thus,  Q(t + T , E) − Q(t + T , E) = sup

ω,ω

= sup

ω,ω



(QT (ω, dω ) − QT (ω , dω ))Qt (ω , E) ψω,ω (dω )Qt (ω , E)

≤ sup (ψω,ω (S + )Q(t, E) + ψω,ω (S − )Q(t, E)) ω,ω

≤ (1 − δ(ν) + ;(ν))(Q(t, E) − Q(t, E)) + ;(ν),

Ergodicity of 2D Navier–Stokes Equations with Random Forcing

75

where, to get the last inequality, we write ψω,ω (S − ) = −ψω,ω (S + ) + ψω,ω (S + ) + ψω,ω (S − ), bound ψω,ω (S + ) by (56), |ψω,ω (S + ) + ψω,ω (S − )| by (55) and use Q(t, E) ≤ 1. We conclude that, for ;(ν) < δ(ν), |QnT (ω, E) − QnT (0, E)| ≤ Q(nT , E) − Q(nT , E) ≤ 2(1 − δ(ν) + ;(ν))n−1 +

;(ν) . δ(ν) − ;(ν)

Recall that δ(ν) = exp(−Cν −3 (log ν −1 )c ) and that T = T (ν) = Cν −1 log ν −1 , hence,  −q see (54), ;(ν) ≤ e−c ν ; so, since we assume ν to be small and q > 3, part (a) of the proposition follows. Let us now prove part (b). It suffices to assume κ ≥ Cν −2α since the LHS of (47) is bounded by one. Using (10) for n = 0, we have ω(1)κ  ≤ F (ω(0))κ  + g(1)κ 

(57)

and 1

"(1) = 2 ω(1)2L2 ≤ F (ω(0))2L2 + g(1)2L2 ,

(58)

and, by Proposition 1, we know that, if ω(0) ∈ Uκ , F (ω(0)) ∈ Uκ(1) , with κ(1) = (recall that κ ≥ Cν −2α ≥ 1 here). Then, letting ζ = (1 + ην) κ  ≥ ζ κ, that F (ω(0))κ  ≤ (1 + ην)

− 21

− 21

κ 1+ην

< 1, we get, for

νaκ 

and 2

F (ω(0))2L2 ≤ (1 + ην)− α ν 2 ϕκ  α . 2

Now, assume that g(1) satisfies, ∀k, 1

|gk (1)| ≤ ;1 ν 2 κ  α b

− 21

1

|k|(γk ) 2 ,

(59)

with ;1 small (depending on η but independent of ν and κ). Then, we get that ω(1) ∈ Uκ  , using the upper bound in (7) and the fact that, for ν small, κ  ≥ ζ κ ≥ Cζ ν −2α is much larger than 2κγ . Hence, the probability in (47) is bounded by the probability that at least one of the inequalities in (59) is violated. Since the gk ’s are Gaussian random variables with covariance γk , this event has a probability less than:  

 2 1 − C exp −C;12 ν 4 (κ  ) α b−1 |k|2 . (60) 1− k 2

For ν 4 (κ  ) α large, each is small. The factor |k|2 controls  exponential2 in the product  2 2 4  −1 2 the sum over k of exp −C;1 ν (κ ) α b |k| , a sum which is small for ν 4 (κ  ) α large    2 enough, and therefore (60) is bounded from above by 1 − 1 − exp −cν 4 κ  α , i.e. by the RHS of (47).  

76

J. Bricmont, A. Kupiainen, R.Lefevere

3.2. Proofs of Lemmas 2 and 3. Proof of Lemma 2. Let ω(0) ∈ Uκ ⊆ U and consider (10) for n = 0. Choose g(1) such that, ∀k, |gk (1)| ≤ ;1 ν 2 e;2 ν|k| b

− 21

1

(γk ) 2 .

(61)

From Proposition 1, (61) and (7), one obtains, −1 |k|

|ωk (1)| ≤ νaκ(1)|k|−r e−κ(1)

−1 |k| 2

+ ;1 ν 2 e;2 ν|k| e−κγ

.

(62)

Then, from (17), one gets, for any ρ > 0, by choosing ;1 , ;2 small enough, that ∃λ¯ < e−cν < 1 such that, ∀k,  −1 |k|

|ωk (1)| ≤ νaκ  |k|−r e−(κ )

(63)

¯ 2κγ + ρν). with κ  = max(λκ, From (31), (58), (59), one also easily obtains that "(1) ≤ ν 2 ϕ(κ  ) α , 2

and thus that ω(1) ∈ Uκ  . Thus, 

 1 −1 P |gk (1)| ≤ ;1 ν 2 e;2 ν|k| b 2 (γk ) 2 . P (ω, Uκ  ) ≥ k

Now, since the gk ’s are Gaussian random variables with covariance γk , we have that, P (|gk (1)| ≤ ;1 ν 2 e;2 ν|k| b

− 21

1

(γk ) 2 ) ≥ 1 − exp(−c;12 ν 4 e2;2 ν|k| b−1 )

(64)

for |k| ≥ Cν −1 log ν −1 , if C is chosen so that bν −4 ≤ e;2 ν|k| (note that the product over such k’s of the RHS of (64) is strictly positive uniformly in ν), while for |k| ≤ Cν −1 log ν −1 ,     1 1 −1 −1 P |gk (1)| ≤ ;1 ν 2 e;2 ν|k| b 2 (γk ) 2 ≥ P |gk (1)| ≤ ;1 ν 2 b 2 (γk ) 2 ≥ Cν 4 , (65) which follows from the fact that the gk ’s are (complex) Gaussian random variables with covariance γk and therefore that    ;k 1 2rdr −1 P |gk (1)| ≤ ;1 ν 2 b 2 (γk ) 2 ≥ ≥ Cν 4 γ 0

−1

k

1

with  ;k ≡ ;1 ν 2 b 2 (γk ) 2 . The bound (65) readily implies that there are constants C, c1 < ∞ such that ∀ω ∈ Uκ ,   (66) P (ω, Uκ  ) ≥ exp −Cν −2 (log ν −1 )c1 . Since U = Uν −p , and since κ decreases by a factor λ¯ < 1 at each step, as long as ¯ ≥ 2κγ + ρν, one may iterate the above argument and reach U¯ = U2κγ +ρν , see λκ (48), in a time less than T1 (ν) = Cν −1 log ν −1 , ∀ω(0) ∈ U . Therefore, the claim of the lemma follows (with a different C than in (66), and with c = c1 + 1).  

Ergodicity of 2D Navier–Stokes Equations with Random Forcing

77

Proof of Lemma 3. Let ω0 ∈ U¯ and B ⊂ U¯ . Since the gk ’s are Gaussian random variables with covariance γk , we have, 

  |ωk −Fk (ω0 )|2 d ω¯ k ∧dωk P (ω0 , B) = , (67) exp − 2πiγ γ k

B k

k

where we recall from (10) that F (ω0 ) denotes the value at time 1 of the solution of (18) with initial condition ω0 . In view of Proposition 1, and the definition of U¯ = U2κγ +ρν , we can bound, ∀ω0 ∈ U¯ , |Fk (ω0 )| ≤ Cνae

|k| − 2κ γ

e−ρν|k| ≡ ;k ,

(68)

provided we choose ρ sufficiently small so that 1 + ην min(1, 2κγ + ρν) 1 + ρν. ≥ 2κγ + ρν 2κγ Thus, we can bound |ωk − Fk (ω0 )|2 ≤ (|ωk | + ;k )2 ; this gives a lower bound on (67) independent of ω0 and we may use this bound on each term of the LHS of (51), with ω0 = ω, ω . We get that the LHS of (51) is bounded from below by 

  (|ω |+; )2 d ω¯ k ∧dωk . (69) exp − k γ k 2πiγ U¯ k

k

k

In order to estimate that latter integral, observe that, by (7), ω ∈ U¯ = U2κγ +ρν provided that, ∀k, |ωk | ≤ ;1 νe;2 ν|k| b

− 21

1

(γk ) 2 ≡ ;¯k ,

(70)

if we take ;1 , ;2 small enough. Thus, by restricting the domain of integration, we get a lower bound on (69):  

 ;¯k 2rdr (r+;k )2 exp − . (71) γ γ k

0

k

k

Each factor is bounded from below by 1−C

  ;k2 − exp −c;12 ν 2 e2;2 |k| (b)−1 γk

(72)

for |k| ≥ Cν −1 log ν −1 . To bound the product over those k’s of the factors given by (72) by a strictly positive constant, independent of ν, observe first that the last term is summable over k, for |k| ≥ Cν −1 log ν −1 and that the sum is small. Moreover, using (68) and the lower bound in (7), we get, ;k2 ≤ Cabν 2 exp (−2ρν|k|) . γk

(73)

Then, (73) is also summable over k, for |k| ≥ Cν −1 log ν −1 and the sum is also small. Finally, for |k| ≤ Cν −1 log ν −1 , each factor in (71) is bounded from below, using (7,  ;¯ 2   70), by C 0 k 2rdr γk ≥ Cν , which yields the claim of the lemma.

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J. Bricmont, A. Kupiainen, R.Lefevere

4. Proof of the Theorem We deduce the theorem from Proposition 2. Let us choose a number ;¯ small enough and a time τ large enough, i.e. τ = cm−1 ν −q so that (46) is less than 2;¯ . Then, for T an integer multiple of τ , write T

P T (ω, E) = (P τ ) τ (ω, E).

(74)

π(ω, E) ≡ π(E) = Qτ (0, E)

(75)

Next, let

and ¯ R(ω, E) = P τ (ω, E) − Qτ (ω, E), R  (ω, E) =τ (ω, E) − Qτ (0, E) = Qτ (ω, E) − π(E), ¯ r(ω, E) = R(ω, E) + R  (ω, E).

(76) (77)

One may then write T

P T (ω, E) = (π + r) τ (ω, E).

(78)

T

We can expand (π + r) τ in powers of r:
T T T (π + r) τ = π k1 r k2 . . . π kl + r τ ≡ 9 1 + r τ ,

(79)

ki

 where the sum 9 1 runs over ki ≥ 0, ki = Tτ and collects all the terms with at least one factor π. Now observe that, using (78) with T = τ , we have that r(ω, $) = P τ (ω, $) − π($) = 1 − π($) = 1 − Qτ (0, U ) is independent of ω; hence, by (75),  (rπ )(ω, dω2 ) = r(ω, dω1 )π(ω1 , dω2 )

(80)

= r(ω, $)π(dω2 ) = (1 − Qτ (0, U ))π(dω2 )

is also independent of ω. From this, we conclude that, since there is at least one factor π in each term of 9 1 , 9 1 (ω, E) = 9 1 (ω , E), ∀ω, ω , ∀E, and, using (78), that T

T

|P T (ω, E) − P T (ω , E)| ≤ |r τ (ω, E)| + |r τ (ω , E)|,

(81)

where the RHS is controlled by: Lemma 4. For T an integer multiple of τ , T

|r τ (ω, E)| ≤ C(ω)e−mT , where m = m(ν) ≥ exp(−Cν −3 (log ν −1 )c ) and where C(ω) ≤ C C, c < ∞.

(82) 

||ω||+1 ν

c

, for

Ergodicity of 2D Navier–Stokes Equations with Random Forcing

79

To conclude the proof of (12), it is enough to show that limT →∞ P T (0, E) = µ(E) exists. And, to prove that, we write, for T > T  ,      |P T (0, E) − P T (0, E)| ≤ P T −T (0, dω)|P T (ω, E) − P T (0, E)|. (83) −p We may write the integral as an integral over U plus a sum over κ ∈ N,  κ ≥ νc , of integrals over Uκ+1 \Uκ and, combining (81), Lemma 4, C(ω) ≤ C ||ω||+1 ≤ ν  c  C ||ω||νκ +1 , ∀κ, and (47) (which implies a similar bound for P T −T (0, Uκc )), we bound (83) by  
2   Ce−mT (ν −(p+1)c + κ c exp −c ν 4 κ α ≤ C(ν)e−mT , (84) κ∈N,κ≥ν −p

which proves the existence of limT →∞ P T (0, E). Finally, the bound (11) follows from (47), for κ large, and we bound the LHS of (11) by 1 for κ small.   Proof of Lemma 4. Define, for n ≥ 0, U (n) ≡ Uζ −n ν −p , (so that U = U (0)) with ζ < 1 as in Proposition 2, and define V (n) by V (n) = U (n)\U (n − 1), for n ≥ 1, and V (0) = U (0). Next, let ρmn ≡ sup |r(ω, V (n))|, ω∈V (m)

where r is defined in (77). Observe that we have the following bounds on ρmn : ρ00 ≤ ;¯ ,

 ρmn ≤ exp −cξ n ν −q ρmn ≤ 4

n ≥ m,

(85)

n < m,

2 where ξ ≡ ζ − α > 1. To check this, use, for m, n = 0, (54) to bound R¯ and (46) to bound  R . For the second inequality, n  = 0, only P contributes to r and the bound follows immediately from (47), with κ  = ζ 1−n ν −p (remember that q = 2p α − 4). Finally, for n < m, we use the fact that r is the sum of four terms, each less than 1. Write now  N−1

N r (ω, E) = r(ωi , dωi+1 )χ (ωN ∈ E) (86)

i=0

with ω0 = ω, and insert a decomposition of the identity for each i = 1, . . . , N,
1= χ (ωi ∈ V (ni )) . ni ≥0

This leads to sup sup |r N (ω, E)| ≤

ω∈U (n0 ) E




N−1

i=0 (ni )N i=1 ,ni ≥0

ρni ni+1 ≡


n

ρnN0 n .

(87)

80

J. Bricmont, A. Kupiainen, R.Lefevere

Note that the RHS describes “random walks” on nonnegative integers, where only steps strictly down (ni+1 < ni ) are not suppressed. To estimate it, write ρ = d + u, where d is the “down” part of ρ, i.e. the matrix whose elements are given by ρmn with n < m and zero otherwise, and u is the rest (“up”). We shall first prove the simple estimates
k dmn ≤ Cm k ≤ m (88) n

and zero otherwise (where the restriction k ≤ m comes from the fact that the indices of dmn must be positive and, whenever dmn  = 0, must satisfy n < m), and
(uk d l )mn ≤ (C ;¯ )k+l . (89) n

Indeed, (88) is estimated by
n1 ...nk

dmn1 . . . dnk−1 nk ≤


k

4k = 4k

1 pi ≤m

 m 
l−1 ≤ 4k 2 m , k−1 l=k

with pi = ni−1 − ni ≥ 1, n0 = m, yielding the claim since k ≤ m. To prove (89), write
l (uk d l )mn = ukmm dm n, l≤m

where the constraint in the sum comes from the second inequality in (88), and note that ukmm is bounded by (¯; )k if m = m = 0 and by    exp −cξ m ν −q (C ;¯ )k−1 otherwise (both bounds following from (85) and the fact that, by definition, umn = 0, unless m ≥ n). The bound (89) follows by combining these with (88), since l ≤ m , we  can therefore use the factor exp(−cξ m ν −q ) and ν small to obtain the factor ;¯ l+1 in (89) (¯; is a fixed small number). Inserting (88), (89) into
ρ N = (u + d)N = d l0 uk1 d l1 . . . uks d ls , where li ≥ 0, li > 0 for i  = 0, s, we obtain the bound
ρnN0 n ≤ C N ;¯ N−n0 , n

where ;¯ −n0

comes from the fact that we have no ;¯ bound on d l0 , but we can use l0 ≤ n0 . This proves the lemma, if we choose in (82),   1 m = − log C ;¯ = −cm−1 ν −q log C ;¯ ≥ exp −Cν −3 (log ν −1 )c τ (given our choice of τ at the beginning of this section, our bound on m in Proposition 2, and changing the constants), and, for ω ∈ U (n0 ), choose C(ω) = (C ;¯ )−n0 ; indeed, let, for ω ∈ Uκ , n0 be the smallest integer such that κ ≤ ζ −n0 ν −p ; then, n0 ≤ C log κ and C(ω) ≤ Cκ c . Moreover, that,  from  part (b) of Proposition 1, we know   ∀ω ∈ $, F (ω) ∈ Uκ , with κ ≤ C

||ω|| ν

α

+

1 ν

; so, altogether, C(ω) ≤ C

||ω||+1 ν

c

.

 

Ergodicity of 2D Navier–Stokes Equations with Random Forcing

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References 1. Bricmont, J., Kupiainen, A., Lefevere, R.: Probabilistic estimates for the two dimensional stochastic Navier–Stokes equations. J. Stat. Phys. 100, 743–756 (2000) 2. Doob, J.L.: Stochastic Processes. New-York: John Wiley, 1953 3. Flandoli, F., Maslowski, B.: Ergodicity of the 2-D Navier–Stokes equation under random perturbations. Commun. Math. Phys. 171, 119–141 (1995) 4. Kuksin, S., Shirikyan, A.: Stochastic dissipative PDE’s and Gibbs measures. Commun. Math. Phys. 213, 291–330 (2000) 5. Mattingly, J.C., Sinai, Y.: An elementary proof of the existence and uniqueness theorem for the Navier– Stokes equations. Preprint 6. Mattingly, J.C.: Ergodicity of 2D Navier–Stokes equations with random forcing and large viscosity. Commun. Math. Phys. 206, 273–288 (1999) Communicated by G. Gallavotti

Commun. Math. Phys. 224, 83 – 106 (2001)

Communications in

Mathematical Physics

© Springer-Verlag 2001

Gibbsian Dynamics and Ergodicity for the Stochastically Forced Navier–Stokes Equation Weinan E1 , J. C. Mattingly2 , Ya. Sinai3 1 Department of Mathematics and Program in Applied and Computational Mathematics, Princeton University,

Princeton, NJ 08544, USA and School of Mathematics, Peking University, Beijing, P.R. China

2 Department of Mathematics, Stanford University, Stanford, CA 94305, USA 3 Department of Mathematics, Princeton University, Princeton, NJ 08544, USA and Landau Institute of

Theoretical Physics, Moscow, Russia Received: 21 November 2000 / Accepted: 9 December 2000

Dedicated to Joel L. Lebowitz, on the occasion of his 70th birthday Abstract: We study stationary measures for the two-dimensional Navier–Stokes equation with periodic boundary condition and random forcing. We prove uniqueness of the stationary measure under the condition that all “determining modes” are forced. The main idea behind the proof is to study the Gibbsian dynamics of the low modes obtained by representing the high modes as functionals of the time-history of the low modes. 1. Introduction and Main Results We are interested in determining conditions sufficient to insure that the stochasticallyforced Navier–Stokes equation (SNS) possesses a unique stationary measure, or equivalently, that the dynamics is ergodic in the phase space. Our main result is that this holds if all the “determining modes” are forced. To prove this, we show that the dynamics of the Navier–Stokes equation can be reduced to the dynamics of the low modes, the so-called determining modes, with memory. This is the stochastic analog of results proved for the deterministic case by Foias et al. [FMRT]. We will work with the periodic boundary condition. But in principle our techniques should also apply for the more physical no-slip boundary condition. Consider the two-dimensional Navier–Stokes equation with stochastic forcing:
∂u ∂W (x, t) + (u · ∇)u + ∇p − νu = . (1) ∂t ∂t ∇ ·u=0 For simplicity of presentation we will take W to be of the form  σk wk (t, ω)ek (x)m W (x, t) =

(2)

|k|≤N

where the wk ’s are standard i.i.d complex valued Wiener process satisfying w−k (t) = w k (t), and σk ∈ C, with |σk | > 0 and σ−k = σ k , are the amplitudes of

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 2  eik·x 2 the forcing, {ek (x) = −ik ik1 |k| , k ∈ Z} are the basis in the space of L divergence2 free, mean zero vector fields on T , the two dimensional torus. Our techniques apply to more general cases when the higher modes are also forced, as long as |σk | decays sufficiently fast as |k| → ∞ or to forcing which is not diagonal in Fourier space. But we will restrict ourselves to the form in (2) for clarity. Define B(u, v) = −Pdiv (u · ∇)v, 2 u = −Pdiv u, where Pdiv is the L2 projec2 tion operator onto vector fields. Let σmax = max{|σk |2 :  the space2 of divergence-free  2 2 |k| ≤ N }. E0 = |k|≤N |σk | and E1 = |k|≤N |k| |σk | . Writing u(x) = k uk ek (x),    we will define Hα = u = (uk )k∈Z2 , u0 = 0, k |k|2α |uk |2 < ∞ and L2 = H0 . We will work on a probability space (, F, Ft , P, θt ). We associate  with the canonical space generated by all dωk (t). F and Ft are respectively the associated global σ -algebra and filtration generated by W (t). Lastly, θt is the shift on  defined by θt dωk (s) = dωk (s + t). Notice that θt is an ergodic group of measure-preserving transformations with respect to P. Expectations with respect to P will be denoted by E. Projecting (1) onto L2 , we obtain the the following system of Itô stochastic equation du(x, t) + ν2 u(x, t)dt = B(u, u)dt + dW (x, t).

(3)

It can be shown that (3) generates a continuous Markovian stochastic semi-flow on L2 defined by ω ϕs,t u0 = u(t, ω; s, u0 ).

(4)

When s = 0, we simply write ϕtω (see [Fla94, DPZ96]). We will take the state space of (3) to be L2 equipped with the Borel σ -algebra. A measure µ(du) on L2 is stationary for the stochastic flow (3) if for all bounded continuous functions F on L2 and t > 0,     F (u)µ(du) = EF ϕtω u µ(du). (5) L2

L2

Our main result is: Theorem 1. There exists some absolute constant C such that if N 2 ≥ C Eν 30 then (3) has a unique stationary measure on L2 . The existence of at least one stationary measure was proved in [Fla94] and [VF88]. The proof proceeds by establishing compactness for a family of empirical measures. The limiting points of these empirical measures are the stationary measures. Uniqueness has been proved under restrictive assumptions when ALL modes are forced. Flandoli and Maslowski [FM95] proved that if the σk ’s decay algebraically, i.e. if the forcing is sufficiently rough spatially, then the system has a unique stationary measure. These results were extended and refined in [Fer97]. In [Mat99], it was proven that if the viscosity was large enough the contraction induced by the Laplacian dominates and the system possesses a trivial random attractor; and hence, a unique stationary measure. We do not address convergence to the stationary measure. This and the coupling construction used to prove convergence are discussed in [Mat00]. Recently Kuksin and Shirikyan [KS] proved uniqueness of stationary measure when the Navier–Stokes equation is perturbed by a bounded degenerate kicked noise. Results similar to ours have also been obtained independently by Bricmont et al. [BKL].

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Our main strategy is to reduce the dynamics of the Navier–Stokes equation to the dynamics of a finite dimensional set of low modes with memory. The reduced dynamics is no longer Markovian, but rather Gibbsian (see §2, §4). The finite dimensional Gibbsian dynamics has a non-degenerate noise, and have a unique stationary measure if the memory is short ranged. Before proceeding further, let us observe that any given stationary measure µ can be extended to a measure on the path space, denoted by µp , where p stands for path or   past. Consider the example of the path space C (−∞, 0], L2 . Let A be a cylinder set of the type: For some t0 , t1 , · · · tn , t0 < t1 < t2 · · · tn ≤ 0,

 A = u(s) ∈ C (−∞, 0], L2 , u(ti ) ∈ Ai , i = 0, · · · n , (6) where the Ai ’s are Borel sets of L2 . Corresponding to A, let B ⊂  × L2 , B = {(u, ω), u ∈ A0 , ϕtω0 ,ti u ∈ Ai , i = 1, · · · n}.

(7)

µp (A) = (P × µ)(B),

(8)

We will define

where (P × µ) is the product measure on  × L2 . Clearly µp is consistent on cylinder sets and can be extended to the natural σ -algebra using the Kolmogorov extension theorem. The natural σ -algebra is the one generated by the cylinder sets. The dynamics of the stochastic semi-flow {ϕtω } can be trivially extended to return   a function from C (−∞, t], L2 , given an initial function from C (−∞, 0], L2 . One simply flows forward with ϕ from the initial condition  avoid confusion, we will call  at time 0. To this map ψtω . Symbolically, if u(·) ∈ C (−∞, 0], L2 , then (ψtω u)(s) = ϕsω u(0) for s ∈ [0, t] and (ψtω u)(s) = u(s) for s ≤ 0. If we define  the shift on trajectories by (θt v)(s) = v(s + t), we can define a dynamics on C (−∞, 0], L2 by θt ψtω . In other words, θt ψtω u takes a trajectory u from   C (−∞, 0], L2 , extends it t units of time by flowing forward and then shifts the entire   resulting trajectory back t units of time so it again lives on C (−∞, 0], L2 . It is easy to check directly that if µ is invariant then µp is invariant in the sense that   (9) F (u)dµp (u) = E F (θt ψtω u)dµp (u) 2 2 C ((−∞,0],L ) C ((−∞,0],L )   for all bounded functions on C (−∞, 0], L2 , and t ≥ 0. Assume that µ and ν are two stationary measures for the stochastic  flow (3), and  µp 2 and νp are respectively their induced measure on the path space C (−∞, 0], L . It is obvious that µp = νp implies µ = ν. 2. Reduction to the Gibbsian Dynamics Define two subspaces of L2 : L2( = span{ek , |k| ≤ N },

L2h = span{ek , |k| > N }.

(10)

We will call L2( the set of low modes and L2h the set of high modes. Obviously L2 = L2( ⊕ L2h . Denote by P( and Ph the projections onto the low and high mode spaces.

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Since we are concerned with stationary measures of (3), we are interested in (statistically) stationary solutions of (3) that exist for time from −∞ to +∞. We will show in this section that for such solutions, the high modes are completely by the  determined  past history of the low modes. For this purpose, we write u(t) = ((t), h(t) and

 d((t) = −ν2 ( + P( B((, () dt

 + P( B((, h) + P( B(h, () + P( B(h, h) dt + dW (t), (11) 

dh(t) (12) = −ν2 h + Ph B(h, h) + Ph B((, h) + Ph B(h, () + Ph B((, (). dt   Define the set of “nice pasts” U ⊂ C (−∞, 0], L2 to consist of all v : (−∞, 0] → L2 such that: i) v(t) is in H2 for all t ≤ 0. ii) The energy averages correctly. More precisely, 1 t→−∞ |t| lim

 t

0

|v(s)|2L2 ds =

E0 . 2ν

iii) The energy fluctuations are typical. More precisely, there exists a T = T (v) such that 2

|v(t)|2L2 ≤ E0 + max(|t|, T ) 3 for t ≤ 0. The following lemma shows that U contains almost all of the trajectories defined on the whole time interval.   Lemma 2.1. Let µp be themeasure on C (−∞, 0], L2 induced by a stationary measure µ for (3). Then µp U = 1. Proof of Lemma 2.1. It is proved in [Mat98] or [Fer97] that with probability one, a solution to (3) is in H2 for all t. The fact that the last condition is satisfied by a set of full measure is proved in Lemma B.3. All that remains to show is ii).  From Lemma B.2 |v|2L2 is in L1 (µ) for any stationary measure µ and |v|2L2 dµ = E0 2ν . Since the measure is invariant under shifts back in time and each ergodic component has the same average enstrophy, the ergodic theorem implies that for µp –almost every  trajectory time average converges to the average of |u|2L2 against µ.  Given an arbitrary continuous function of time ((t) on L2( , we can view (12) as a closed equation with some exogenous forcing ((t). By ,s,t ((, h0 ), we mean the solution to (12) at time t given the initial condition h0 at time s and the “forcing” (.  Denote by P the set of all ( ∈ C (−∞, 0], L2( such that the following two conditions hold. First, ( = P( u for some u = ((, h) ∈ U . Second, h(t) = ,s,t ((, h(s)) for any s < t ≤ 0, where h was the matching high mode so ((, h) ∈ U . That is to say h(t) solves (12) with low modes ((t) and the total solution ((, h) is in our space of “nice pasts”. In light of Lemma 2.1 the set P is not empty. We now will show that this h is uniquely determined by (.

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Lemma 2.2. There exists an absolute positive constant C such that if N 2 > C Eν 30 then the following holds:     If there exists two solutions u1 (t) = ((t), h1 (t) , u2 (t) = ((t), h2 (t) corresponding to some (possibly different) realizations of the forcing and such that u1 , u2 ∈ U , then u1 = u2 , i.e. h1 = h2 .   Furthermore given a solution u(t) = ((t), h(t) ∈ U , any h0 ∈ L2h , and t ≤ 0, the following limit exists: lim ,t0 ,t ((, h0 ) = h∗

t0 →−∞

and h∗ = h(t). Proof of Lemma 2.2. We begin with the first clam. Denote by ρ(t) = h1 (t) − h2 (t). From (12) we have dρ = − ν2 ρ + Ph B(h1 , h1 ) − Ph B(h2 , h2 ) + Ph B((, ρ) + Ph B(ρ, () dt = − ν2 ρ + Ph B(( + h1 , ρ) + Ph B(ρ, ( + h2 )

(13)

= − ν2 ρ + Ph B(u1 , ρ) + Ph B(ρ, u2 ). Taking the inner product with ρ, using the fact that Ph B(u1 , ρ), ρL2 = 0, gives 1 d |ρ|2 2 = −ν|ρ|2L2 + Ph B(ρ, u2 ), ρL2 . 2 dt L Since |Ph B(ρ, u2 ), ρL2 | ≤Cˆ |ρ|L2 |ρ|L2 |u2 |L2 ν Cˆ 2 |ρ|2L2 |u2 |2L2 , ≤ |ρ|2L2 + 2 2ν we get ν Cˆ 2 1 d |ρ|2L2 ≤ − |ρ|2L2 + |u2 |2L2 |ρ|2L2 . 2 dt 2 2ν Since ρ only contains modes with |k| > N , the Poincaré inequality implies   Cˆ 2 d 2 2 2 |u2 |L2 |ρ|2L2 . |ρ|L2 ≤ −νN + ν dt Therefore we have, for t0 < t < 0,
|ρ(t)|2L2

≤

|ρ(t0 )|2L2

  Cˆ 2 t 2 |u2 (s)|L2 ds . exp −νN (t − t0 ) + ν t0 2

From the third assumption on functions in U , we know that lim E0 2ν . Hence for t0 < T1 , where T1 depends on t and u2 , we have −νN 2 (t − t0 ) +

 1 0 t −t

(14)

|u2 (s)|2L2 ds =

 Cˆ 2 t γ |u2 (s)|2L2 ds ≤ − (t − t0 ), ν t0 2

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ˆ2

where γ = νN 2 − C2νε20 . If we set C = C2 , then our assumption on N implies γ > 0. Now using the last property of paths in U we have for any t0 ≤ T2 ,  γ  |ρ(t)|2L2 ≤ |ρ(t0 )|2L2 exp − (t − t0 ) 2   γ  2 3 ≤2 E0 + |t0 | ] exp − (t − t0 ) → 0 2 as t0 → −∞, where T2 is some finite constant depending on u1 and u2 . This completes the proof of the first part of Lemma 2.2. t To see the second part, observe that (14) only required control of t0 |u(s)|2L2 ds for one of the two solutions. If we proceed as before letting the given solution u(t) play the role of u2 and the solution to (12) starting from h0 play the role of u1 , the we obtain the estimate
  Cˆ 2 t 2 2 2 2 |ρ(t)|L2 ≤ |h(t0 ) − h0 |L2 exp −νN (t − t0 ) + |u(s)|L2 ds . (15) ν t0 Since u(t) = (((t), h(t)) ∈ U , the same reasoning as before shows that ρ(t) goes to zero as t0 → −∞. Hence the limit exists and equals h(t).   In fact the splitting into high and low modes can be accomplished even when all of the modes are forced. One replaces (12) with an Itô stochastic differential equation. This causes little complication as (13) remains a standard PDE. See [Mat98].The ideas in this section are related to the ideas of Lyapunov-Schmidt reduction and those around center and inertial manifolds. See [EFNT94] for a discussion and other references. From now on we assume that N satisfies N2 > C

E0 , ν3

(16)

where C is the constant from Lemma 2.2. Because of Lemma (2.2), we can define a map ,0 which reconstructs the high modes at time zero from a given low mode trajectory stretching from zero back to −∞. Before making this more precise, let us fix some notation. In general, we will use ((t) to refer to the value of the low modes at time t and will use Lt to mean the entire trajectory from −∞ to t. Hence ((t) ∈ L2 and Lt ∈ C (−∞, t], L2 and ((s) = Lt (s) for s ≤ t. In this   notation h(0) = ,0 L0 , where L0 is some “low mode past” in P which is the projection of U to the low modes. By ,s (Lt , h(0)) with s ≤ t, we mean the solution to (12) at time s with initial condition h(0) and low mode forcing Lt . Of course ,s (Lt , h(0)) only depends on the information in Lt between 0 and s. We can extend the definition of , beyond time zero by defining ,t (Lt ) = ,t (Lt , h(0)), where h(0) = ,0 (L0 ). Given the initial low mode past of L0 ∈ P, we can solve for the future of ( using

    d((t) = −ν2 ((t) + P( B ((t), ((t) + G ((t), ,t (Lt ) dt + dW (t), (17) where G ((, h) = P( B((, h) + P( B(h, () + P( B(h, h).

(18)

Thus we have a closed formulation of the dynamics on the low modes given an initial past in L0 ∈ P. We write Lt = Sωt L0 . We reiterate that Lt is the entire trajectory from time t back to −∞, whereas ((t) is simply the value of the low modes at time t.

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89

Except for the fact the G-term in (17) is history-dependent, (17) has the form of a standard finite dimensional stochastic ODE with non-degenerate forcing, which of course has a unique stationary measure. Our task is reduced to showing that the memory effort in (17) is not strong enough to spoil ergodicity. Existence of the solution for memory-dependent stochastic ODEs of the type (17) was considered in the work of Ito et al. [IN]. 3. Uniqueness of the Invariant Measure 3.1. Proof of the Main Theorem. Given any “nice low mode past” L ∈ P, we can reconstruct the “high modes” and hence define a closed dynamics on the paths of the low modes. However, this dynamics is no longer Markovian which will produce difficulties. 2 Let µ be an ergodic  stationary measure on L and µp be its extension   to the path 2 space C (−∞, 0], L . We will also consider the restriction of µp to C (−∞, 0], L2( , still denoted by µp . Lemma 2.1 says that µp (P) = 1.   Given any L0 ∈ P, let Qt (L0 , · ) be the measure induced on C [0, t], L2( by the dynamics starting from L0 . In other words,Qt (L0 , · ) is the distribution of Sωt L0 viewed as a random variable taking values in C [0, t], L2( . Similarly let Q∞ (L0 , · ) be the   distribution induced on C [0, ∞), L2( starting from L0 . Consider the stochastic process defined by θt Sωt L0 , where L0 is a random variable on P distributed according to the invariant measure µp . For t ≥ 0 it is a random process with values in P. This is clear as all of the defining properties of U are asymptotic in t; and hence the addition of a segment of finite length does not destroy them. Since µp is invariant with respect to the dynamics, θt Sωt L0 is a stationary random process. Hence 0 with probability one there exist time averages along trajectories θt Sωt L  .  2 Take any bounded measurable functional F from C (−∞, 0], L( → R such that   F (L0 ), L0 ∈ C (−∞, 0], L2( depends only on a finite range of L0 . Let  F¯ = F (L)dµp (L). (19)

Theorem 2. The SNS equation (1) has a unique stationary measure. The proof of Theorem 2 is based on the following two lemmas whose proofs will be given later. Lemma 3.1. Let L01 and L02 be two initial pasts in P, such that (1 (0) = (2 (0). Then Q∞ (L01 , ·) and Q∞ (L02 , ·) are equivalent. Recall that ((τ ) is the solution of (16) with initial condition L. Lemma 3.2. For any past L ∈ P and any t > 0, the distribution of ((t) ∈ L2( conditioned at starting from L at time zero, denoted by Rt (L, ·), satisfies the following: there exists a strictly positive function fL,t ∈ L1 (L2( ), such that dRt (L, ·) ≥ fL,t (·)dm(·). where m(·) is the Lebesgue measure on L2( .

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For any measure µ on L2 let P( µ denote its projection to a measure on the low modes L2( . Namely, (P( µ)(B) = µ(P(−1 (B)). Then we have the following direct consequence of Lemma 3.2. Corollary 3.3. If µ is a stationary measure then P( µ has a component which is equivalent to the Lebesgue measure. Proof of Theorem 2. Assume that there are two different ergodic stationary measures on L2 called µ1 and µ2 . They must be mutually singular. Let µ1,p and µp,2 be the extensions of these two measures onto the path space P. Let L0i be a random variable on P distributed as µi,p . Since θt Sωt L0i is stationary with respect to µp,i we can pick a set Pi , of full µp,i -measure, such that for all L ∈ Pi One can find a functional F such as  above so that F¯1 = F (L)dµp,1 (L)  = F¯2 = F (L)dµp,2 (L). This assumption will lead to a contradiction. The limit  1 T F (θt Sωt Loi )dt = F¯i (20) lim T →∞ T 0 is well defined for P-almost every ω. For ( ∈ L2( define Pi (() = {L ∈ Pi : L(0) = (} and let µp,i ( · |() be the conditional measure that L(0) = (. By Fubini’s theorem, we know that for P( µi -almost every ( ∈ L2( we have µp,i (Pi (() | () = 1. Hence we can find a set Ai ⊂ L2( such that µp,i (Pi (() | () = 1 for all ( ∈ Ai and P( µi (Ai ) = 1. Define A = A1 ∩ A2 . Corollary 3.3 implies that P( µi (A) > 0 for i = 1, 2. Hence there exists some (∗ ∈ A. Since (∗ ∈ A1 ∩A2 , we know that µp,i (Pi ((∗ ) | (∗ ) = 1 for i = 1, 2. Thus there exist some L∗,1 ∈ P1 ((∗ ) and L∗,2 ∈ P2 ((∗ ). Notice that by construction L∗,1 (0) = (∗ = L∗,2 (0), and hence it follows from Lemma 3.1 that Q∞ (L∗,1 , ·) and Q∞ (L∗,2 ,·) are equivalent. Since L∗,i ∈ Pi ((∗ ), we know that we can pick Bi ⊂ C [0, ∞), L2 such that the time average of F converges to F¯i for all futures in Bi and Q∞ (L∗,i , Bi ) = 1 for i = 1, 2. Since the Q’s are equivalent, Q∞ (L∗,1 , B1 ∩ B2 ) > 0 and hence B1 ∩ B2 is non-empty. This in turn implies that F¯1 = F¯2 which contradicts the assumption that they were not equal.  

3.2. Proofs of the lemmas. We first prove Lemma 3.1. Fix L01 and L02 . Most of our construction will depend explicitly on them. With probability one, we can extend each of the initial pasts into the infinite future by Lsi = Sωs L0i and setting (i (s) = Lti (s) for s ≤ t. We can also reconstruct the entire  solution by  using ,t to obtain the high modes. Set hi (s) = ,s (Lsi ) and ui (s) = (i (s), hi (s) . Fix a constant C0 such that |ui (0)|2L2 ≤ C0 . We begin by constructing a set of nice future paths which will contain most trajectories. For any positive K we define 

  Ai (K) = f ∈ C [0, ∞), L2( : |v(t)|2L2 + 2ν

and A(K) = A1 (K) ∩ A2 (K).



t

4

|v(s)|2L2 ds < C0 + E0 t + Kt 5 0  where v(s) = f (s) + ,s (f, hi )

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By Lemma A.5, we know that for any a ∈ (0, 1) there exists a K such that a for i = 1, 2, P ω : Sωt L0i ∈ Ai (K) > 1 − 2 and hence

P ω : Sωt L0i ∈ A(K)

for i = 1, 2 > 1 − a > 0.

This is just another way of saying Q∞ (L0i , A(K)) > 1 − a. 0 0 0 Lemma 3.4. Let L01 and Let  L2 be two initial pasts in P such that L1 (0) = L2 (0). 2 0 A(K) ⊂ C [0, ∞), L( be as defined above. For any choice of K > 0, Q∞ (L1 , · ∩ A(K)) is equivalent to Q∞ (L02 , · ∩ A(K)).

Proof of Lemma 3.1. Since we can choose K so that A(K) has measure arbitrarily close to 1, we have that Q∞ (L01 , ·) is equivalent to Q∞ (L02 , ·).   Proof of Lemma 3.4. We intend to use Girsanov’s theorem to compare the two induced measures, Q∞ (L01 , · ) and Q∞ (L02 , · ). However we do not do so directly. To aid in our analysis, we consider the following surrogate processes y which will agree with ( on the set A = A(K). As before, we will use y(t) to denote the value of the process at time t and Y t to be the entire trajectory up to time t.

  dyi (t) = −ν2 yi (t) + P( B yi (t), yi (t)   + :t (Yit )G yi (t), ,t (Yit , hi (0)) dt + dW (t) (21) yi (0) = (i (0), where hi (0) = ,t (L0i ),
1 if f ∈ A|[0,t] , :t (f ) = 0 if f  ∈ A|[0,t] and A|[0,T ] isthe low mode paths which agree with a path in A up to time T . Recall that ,t Yit , hi (0) is the solution to (12) with ( = Y and h(0) = hi (0). Equation (21) is the same as (17) except for the insertion of :t (Yit ). As long as :s (Yit ) = 1 for s ∈ [0, t], then yi (s) = (i (s) for s ∈ [0, t]. y y Let Q∞ (L01 , · ) and Q∞ (L02 , · ) be the measures induced by Y1 and Y2 respectively. If applicable, Girsanov’s theorem would imply that these measure are equivalent, that y y is Q∞ (L01 , · ) ∼ Q∞ (L02 , · ). For Girsanov’s theorem to apply, it is sufficient that the Novikov condition holds. Namely,
       2  t  t 1 ∞  −1 t ; :t (Y1 )D y1 (t), ,t Y1 , h1 (0) , ,t Y1 , h2 (0)  dt < ∞, E exp 2 0  (22) where D(g, f1 , f2 )=G(g, f1 ) − G(g, f2 ) and ; is a diagonal matrix with the σk ’s on its diagonal. Here we have written the condition in terms of the y1 process. One can also def

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write the condition in terms of the y2 process; the finiteness of one implies the finiteness of the other. We will in fact show something much stronger than (22). Since |; −1 | < ∞, it would be enough to show that 

∞

sup ω

     :t (Y t )D y1 (t), ,t Y t , h1 (0) , ,t Y t , h2 (0) 2 dt < ∞. 1 1 1

0

(23)

Putting hi (s) = ,s (Y1s , hi (0)), ui (s) = (i (s) + hi (s), ρ(s) = h1 (s) − h2 (s) and using Lemma A.4, we have

    D (1 (s), h1 (s), h2 (s) 2 2 ≤ C  |ρ(s)|2 2 |u1 (s)|2 2 + |u2 (s)|2 2 . (24) L L L L Notice that if (i ∈ A|[0,T ] then for all t ∈ [0, T ], 4

|ui (t)|2L2 < C0 + E0 t + Kt 5 ,  t 4 1 |ui (s)|2L2 ds < C0 + E0 t + Kt 5 , 2ν 0   |ρ(0)|2L2 = |u1 (0) − u2 (0)|2L2 ≤ 2 |u1 (0)|2L2 + |u2 (0)|2L2 ≤ 4C0 . In addition, we can apply the same analysis as in Sect. 2. Starting from (14) and using the above estimates produces


|ρ(t)|2L2

  Cˆ 2 t 2 |u2 (s)|L2 ds ≤ exp −νN t + ν 0
 4 Cˆ 2  2 ≤ 4C0 exp −νN t + 2 C0 + E0 t + Kt 5 . 2ν |ρ(0)|2L2

2

ˆ2

Since by assumption νN 2 > C Eν 20 = C2νE20 , the second term goes to zero sufficiently fast and hence the estimate on the right-hand side of (24) decays exponentially fast. Thus,  ω

   2 :t (Y1 )D y1 (t), ,t (Y t , h1 (0)), ,t (Y t , h2 (0))  dt 1 1    ∞ |D (f (r), ,t (f, h1 (0)), ,t (f, h2 (0)))|2 dt ≤ sup

∞

sup 0

f ∈A 0

< const(C0 ) < ∞, y

y

which implies, Q∞ (L01 , · ) ∼ Q∞ (L02 , · ). As long as Yi stays in A, yi = (i . Hence y Q∞ (L0i , · ∩ A) = Q∞ (L0i , · ∩ A) and finally Q∞ (L01 , · ∩ A) ∼ Q∞ (L02 , · ∩ A).   In fact our proof provided more information than stated in Lemma 3.4. It contains some estimates uniform over a class of initial pasts which will be useful in later investigations of the convergence rate. (See [Mat00]. ) We state the extra information in the following corollary.

Gibbsian Dynamics and Ergodicity for the Stochastically Forced Navier–Stokes Equation

93

Corollary 3.5. In the setting of the proof of Lemma 3.4, define P  = {L ∈ P : |L(0) + ,0 (L)|L2 < C0 }. Then there exists a constant, depending on C0 and K, so that 2   y  dQ∞ (L1 , g)  y  sup 1 − dQy (L , g)  dQ∞ (L2 , g) < const(C0 , K1 ) < ∞. L1 ,L2 ∈P  ∞ 2 We now move to the proof of Lemma 3.2. Fix L ∈ P. The proof proceeds by comparing the process ((t) to the associated Galerkin approximation living on L2( which we will denote by x(t). The advantage is that x(t) is a standard non-degenerate diffusion and hence it is Markovian and well understood. Take x(t) as the solution defined by the following stochastic ODEs:   dx(t) = −ν2 x + P( B(x, x) dt + dW (t), x(0) = ((0). As in the previous section, we do not compare x(t) directly to ((t) but instead to a modified version of ((t) which we will denote by z(t). In analogy to before, we will denote the path of this process up to time t by Z t . Before continuing let us assume without loss of generality that |((0)|L2 ≤ C0 and t ≤ T for some positive C0 and T . This will give our estimates some uniformity over all initial conditions inside this ball and for times t ≤ T . The evolution of z(t) is given by

   dz(t) = −ν2 z + P( B(z, z) + :t (Z t )G z, ,t Z t , h0 dt + dW,   z(0) = ((0) = L(0) , where h0 = ,0 (L) and G is defined in (18). As in the last section, :t (Z t ) is a cut-off function. For any fixed b0 > 1, we define
s 1 if 0 |Z s (r)|4L2 dr < (b0 C0 )4 T s :s (Z ) = . 0 otherwise Here b0 is a fixed constant to be chosen below. For any B ⊂ L2( , define     [B] = v ∈ C [0, t], L2( : v(t) ∈ B . Then Rt (L(0), B) = Qt (L, [B]).   Letting Qxt (L, · ) and Qzt (L, · ) be the two measures induced on C [0, t], L2( by the dynamics of x and z respectively. Lemma 3.2 will be a consequence of the following two lemmas. Lemma 3.6. Fix any b0 > 1. (The constant used in defining the z process.) Then the following holds: For any L ∈ P and t ≥ 0, Qxt (L(0), · ) is equivalent to Qzt (L, · ). Lemma 3.7. For any b0 the following holds: For any L ∈ P and t ≥ 0, there exists a positive function g( · ) so that Qxt (L(0), [B] ∩ A) ≥ B g(y)dm(y), where m( · ) is the Lebesgue measure.

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We now use these two lemmas to prove Lemma 3.2. Proof of Lemma 3.2. Observe that by construction as long as the trajectories stay in A, x(t) = ((t). Hence using Lemma 3.7, we have Rt (L, B) = Qt (L, [B]) ≥ Qt (L, [B] ∩ A) = Qzt (L, [B] ∩ A),  g(L(0), y)dm(y), Qxt (L(0), [B] ∩ A) ≥ B

where g(L(0), y) is a positive function in y. Since Lemma 3.6 says that Qzt ((, · ∩ A) is equivalent to Qxt (L(0), · ∩ A), we know that Rt (L(0), B) is also bounded from below by a positive measure equivalent to the Lebesgue measure.   We now turn to Lemma 3.6. Our construction gives some measure of uniform control which is useful for estimating the rate the system converges to the stationary measure. (See [Mat00]. ) We state these more precise estimates in the following corollary. Corollary 3.8. Fix a C0 > 0 and define P  = {L ∈ P : |L(0) + ,0 (L)|L2 < C0 }. Then for any α ∈ (0, 1) there exists a b0 > 0 (the constant used to define A) so that:   inf inf P Sωt L ∈ A > 1 − a, t∈[0,T ] L∈P 

2   z   1 − dQt (L, g)  dQx (L, g) < K(C0 , t) sup t x  dQt (L, g)  L∈P  for t ∈ [0, T ], where K is a constant depending on C0 and t such that for each C0 , K → 0 as t → 0. Proof of Lemma 3.6 and Corollary 3.8. Girsanov’s theorem would imply the result if the Novikov condition    t   2 1 s 2 s  |:s (Z )| G z(s), ,s (Z , h0 ) L2 ds < ∞ E exp 2 0 holds. As in the proof of Lemma 3.4, we will prove the stronger condition  t    G z(s), ,s (Z s , h0 ) 2 2 ds < ∞. sup L z(·)∈A 0

Using Lemma A.4, we obtain the following estimate on G:      G z(s), ,s (Z s , h0 ) 2 2 ≤ C  |z(s)|2 2 |h(s)|2 2 + |h(s)|4 2 , L L L L  where h(s) = ,s (Z s , h0 ) . By Lemma C.1 we know that if z is in A then sups∈[0,t] |h(t)|L2 is less than some C1 , where C1 depends on |h0 |L2 and the b0 , C0 and T used to define A. Hence for any z ∈ A, we have  t  t      G z(s), ,s (Z s , h0 ) 2 2 ds ≤ C  |z(s)|2L2 |h(s)|2L2 + |h(s)|4L2 ds L 0

0

≤ C



t 0



|z(s)|4L2 ds

≤ C (b0 C0 ) T 2

1 2

1 C12 t 2

 21 

t 0



|h(s)|4L2 ds

+ C C14 t.

 21

+ C  C14 t

Gibbsian Dynamics and Ergodicity for the Stochastically Forced Navier–Stokes Equation

Hence Novikov’s condition holds and the lemma is proven.

95

 

Proof of Lemma 3.7. The basic idea is as follows. Some of the paths which satisfy the condition defining A can be described by requiring that some norm of the paths be less than some fixed fk∗ (t) at time t. Such a condition has the advantage that it corresponds to fixing a zero boundary condition along the boundary of some region for the associated Fokker-Planck equation. Since the diffusion is nondegenerate this process has a positive density on the interior of this region. By carefully picking fk∗ we can have the region contain sets arbitrarily far away from the origin. We now make this precise. Fix a L ∈ P, and a t > 0. For k = 0, 1, 2, . . . define the disk Dk by   Dk = f ∈ L2( : |f |4L2 ∈ [2k , 2k+1 )  and let D¯ k be the closure of Dk . We will construct g( · ) = gk ( · )1Dk , where gk is strictly positive on D¯ k and zero outside of D¯ k . Let fk∗ be a non-decreasing, positive, real-vaued C ∞ function fk∗ such that fk∗ (s) = 1

1

(C04 + αk ) 4 for s ∈ [0, (1 − αk )t − ε] and fk∗ (s) = (100 · 2k+1 ) 4 for s ∈ [(1 − αk )t, t] and linearly  t interpolates in [(1−αk )t −ε, (1−αk )t]. αk is some number in (0, 1) chosen so that 0 (fk∗ (r))4 dr < (b0 C0 )4 T . This is possible as long as b0 > 1 and t ≤ T .   Now define the subset Hk of C [0, t], L2( by
Hk = f ∈ C



[0, t], L2(



 : sup |f (s)|L2 ≤ s∈[0,t]

fk∗ (s)

.

By the choice of fk∗ it is clear that Hk ⊂ A, where A is the same set used in the definition of z. Now consider the process xk (t) which follows the same equation as x(t) except that it is killed whenever the trajectory leaves Hk . Another way of saying this is xk (t) is the process x(t) conditioned on staying in Hk . The transition density of this process gk (s, ((0), y) is the solution to the Kolmogorov equation with the same generator as x but with zero boundary conditions along the boundary of Hk . Since the generator is elliptic, we know that gk (t, ((0), y) is strictly positive everywhere in the interior of Hk . Since the trace of Hk at time t strictly contains Dk , we know that gk (t, ((0), y) is strictly positive for y ∈ D¯ k . Also by construction it is clear that Qxt (((0), Hk ) > 0 for all k. Let ak = Qxt (((0), Hk ) and set gk ( · ) = ak gk (t, ((0), · )1Dk ( · ). All that remains is to verify that this choice of gk constructs a g with the desired minorization property since it is clearly everywhere positive. Without loss of generality it is enough to show it for a B contained in some arbitrary Dk . Then Qxt (((0), [B] ∩ A) ≥ Qxt (((0), [B] ∩ Hk ) ≥ P((0) {x ∈ [B] & x ∈ Hk }  ≥ P((0) {x ∈ [B]  x ∈ Hk }P((0) {x ∈ Hk }   ≥ ak gk (t, ((0), y)dm(y) = gk (y)dm(y).   B

B

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4. Stationary Measures and Thermodynamical Formalism In this section we make a few general heuristic remarks about the methodology behind our approach. The starting point of our construction is rewriting the original Navier–Stokes equation with random forcing as a finite-dimensional system of ordinary stochastic differential equations whose drift coefficients depends on the whole past: d( = [−ν2 ( + P( B((, () + G((, ,t (Lt ))]dt + dW.

(25)

dW = d( − [−ν2 ( + P( B((, () + G((, ,t (Lt ))]dt.

(26)

From (25)

The measure corresponding to all dwk (t), k ∈ Zν , −∞ < t < ∞ can be symbolically written as     1  1  ∞  dw (t) 2    k  dt exp − dwk (t).  2  |σk |2 −∞  dt  k∈Zν

k

Here Zν is the set of modes that are forced. The substitution of the expression for dwk from (26) gives  exp

∞

−∞

 L1 (((t))dt +

∞ −∞

L2 (((t))dt −

   ∞  d(k (t) 2 1  1   dt 2 |σk |2 −∞  dt  k∈Zν



d(k (t),

k

where 2 1 L1 (((t)) = − −ν2 ( + P( B((, () + G((, ,t (Lt )) ,  2∞   ∞  1  L2 (((t))dt = −ν2 ( + P( B((, () + G((, ,t (Lt )) k d(k (t). 2 |σk | −∞ −∞ k∈Zν

2    ∞     The factor exp − 21 k∈Zν |σ1|2 −∞  d(dtk (t)  dt k d(k (t) can be considered as the k differential of a “free measure” which in our case is a finite-dimensional white noise. The “Lagrangians” L1 , L2 describe the non-local interaction of ((t) with the past. The whole expression shows that the stationary measure for the SNS system is actually a Gibbs state constructed with the help of Lagrangians L1 , L2 . The estimations of the growth of L1 , L2 as a function of the growth of |(k (s)|L2 , s → −∞ show the class of realizations for which the conditional distributions can be defined. Therefore we have a weaker form of the Gibbs state. R. L. Dobrushin in his last papers and talks stressed the importance of this class of probability distributions. Since we are dealing all the time with probability distributions, the free energy of our Gibbs state is zero. It would be interesting to develop a general theory of existence and uniqueness of Gibbs states for general Lagrangians L1 , L2 so that our result becomes a particular case of a more general statement.

Gibbsian Dynamics and Ergodicity for the Stochastically Forced Navier–Stokes Equation

97

5. Conclusion When analyzing the ergodic properties of an infinite dimensional stochastic process, one of the most delicate aspects is often finding the correct topology in which to work. One of the principle advantages of the approach presented in this paper is that it evades this difficulty. We trade an infinite dimensional diffusion process for a finite dimensional Itô process with memory. We have tried to present the simplest case of our theory, so that the exposition would be unencumbered. In fact the proofs contained in this work have proved a more general theorem than originally stated. Consider forcing defined by  W (x, t) = σk wk (t, ω)ek (x), k∈Z

where Z is some finite subset of Z2 such that (0, 0)  ∈ Z and k ∈ Z if and only if σk > 0. If we define L2( = span{ek , k ∈ Z}, and

L2h = span{ek , k  ∈ Z}

  N− = sup N : k ∈ Z for all k with 0 < |k| ≤ N .

With these definitions all of the previous lemmas and theorems hold with the role of N replaced by N− . In particular, if N−2 > C Eν 30 the system has a unique invariant measure. This formulation emphasizes the nature of our principle assumption. By requiring that all of the low modes are forced, we are essentially requiring that the reduced Gibbsian dynamics are elliptic in nature. Some steps towards dealing with a hypo-elliptic setting have been made. In [EMatt], finite dimensional truncations of the two dimensional SNS equation were studied and shown to be ergodic under minimal assumptions. In [EM], a reaction diffusion equation was studied under degenerate forcing. Our arguments can be easily extended to the case where the forcing of the k th mode has the form fk + σk dwk (t), fk is a constant, fk = 0 and σk = 0 for k ∈ / Z or the case when the forcing is not diagonal in Fourier space. Our approach can also be extended in several other different directions. We can consider the case when the high modes are also forced. As long as the forcing of the high modes decays sufficiently fast, our argument still applies with almost no change. The Wiener process in the forcing can be replaced by other diffusion processes such as the Ornstein-Uhlenbeck process. Dissipative PDEs such as the Cahn-Hilliard equation and the Ginzburg-Landau equations can also be studied using the same method. Finally, exponential convergence of empirical distributions to the stationary distribution can be proved. A. Energy Estimates In this Appendix, we prove a number of estimates controlling the evolution of the energy and enstrophy. Estimates for higher Sobolev norms are also possible, see [Mat98] for examples. In all cases, they are analogous to the standard results in the deterministic setting. Here we do not limit ourselves to forcing with only finitely many active modes. def  We will characterize the forcing in terms of the El defined by El = |k|2l |σk |2 . We begin with the basic energy and enstrophy estimates in the stochastic setting.

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W. E, J.C. Mattingly, Ya. Sinai

Lemma A.1. For any p > 1, we have 

2p

t

2(p−1)

E |u(s)|2L2 |u(s)|L2 ds  t 2p 2(p−1) ≤ E |u(0)|L2 + C0 E |u(s)|L2 ds, 0  t  2   2p 2(p−1) E |u(t)|L2 + 2pν E 2 u(s) 2 |u(s)|L2 ds L 0  t 2p 2(p−1) ≤ E |u(0)|L2 + C1 E |u(s)|L2 ds. E |u(t)|L2 + 2pν

0

0

2 and σ 2 = sup |σ |2 . In the case p = 1, we have the Here Ci = pEi + 2p(p − 1)σmax k max equalities



E |u(t)|2L2

t

+ 2ν E |u(s)|2L2 =E |u(0)|2L2 + E0 t, 0  t  2   + 2ν E 2 u(s) 2 =E |u(0)|2L2 + E1 t.

E |u(t)|2L2

L

0

(27) (28)

Proof. We begin by fixing a positive integer M and considering the Galerkin approxima (M) tion defined by u(M) (t) = |k|≤M uk (t)ek . u(M) (t) satisfies an equation of exactly the same form as the full solution except the nonlinearity has been projected to those terms def  |k|2l |σk |2 . Our estimates of order less than or equal to M. We will also need ElM = |k|≤M

will be independent of the order of approximation M. For simplicity, we will sometimes neglect the superscript M. p  Applying Itô’s formula to the map {uk } → |uk |2 produces, 2p

 −ν |u(t)|2L2 dt + u(t), dW L2 (29)

  2(p−2) 2(p−1) M + 2p(p − 1) |u(t)|L2 |uk (t)|2 |σk |2 dt + p |u(t)|L2 E0 dt 2(p−1)

d |u(t)|L2 = 2p |u(t)|L2

k

for the energy moments and d

2p |u(t)|L2

!  2  2  2 −ν  u(t) 2 dt +  u(t), dW L2 = L

  2(p−2) + 2p(p − 1) |u(t)|L2 |k|2 |σk |2 |uk (t)|2 dt 2(p−1) 2p |u(t)|L2

(30)

k

2(p−1) M + p |u(t)|L2 E1 dt

for the enstrophy moments.  Here α u(t), dW (t)L2 is shorthand for |k|α uk (t)σk dwk (t). In the first, we have used the fact that B(u, u), uL2 = 0 and in the second the fact that B(u, u), 2 uL2 = 0. Since, on the torus, the structure of the energy and the enstrophy equations are the same we will continue giving all of the details for analysis of the enstrophy equation.

Gibbsian Dynamics and Ergodicity for the Stochastically Forced Navier–Stokes Equation

99

The analysis for the energy equation proceeds analogously, see [Mat99, Mat98]. For a fixed H > 0, we introduce the stopping time  2     T = inf t ≥ 0 : 2 u(t) 2 ≥ H 2 . L

Denoting by Mt the local martingale term in (30) , we define the stopped martingale MtT by  t 2(p−1) MtT = 2p |u(s ∧ T )|L2 2 u(s ∧ T ), dW (s)L2 . 0

MtT has the advantage that its quadratic variation, denoted by [M T , M T ]t , is clearly finite.  t 2p  2  T T 2 [M , M ]t ≤ 2pσmax  u(s ∧ T ) 2 ds L 0  t 2p  2  2 2 ≤ 2pσmax H 2p t < ∞.  u(s ∧ T ) 2 ds ≤ 2pσmax L

0

Because E[M T , M T ]t < ∞ we know that EMtT = 0. And because t ∧ T is a bounded T stopping time the Optional Stopping Time Lemma says that EMt∧T = 0. Since Mt∧T = T Mt∧T , we have  E |u(t

∧ T )|2L2

t∧T

+ 2νE

 2  2   u(s) 2 ds = E |u(0)|2L2 + E1M E(t ∧ T ), L

0

and when p > 1, 

2p

E |u(t ∧ T )|L2 + 2pνE 2p

= E |u(0)|L2 + E



t∧T

0 t∧T

0

 2  2 u(s) 2 ds

2(p−1) 

|u(t)|L2

L

2(p−2)

2p(p − 1) |u(s)|L2



|k|2 |σk |2 |uk (s)|2



k 2(p−1)

+ p |u(s)|L2

E1M ds.

Hence 

 2  2 u(s) 2 ds L 0  t∧T   2p 2(p−1) 2 |u(s)|L2 ≤ E |u(0)|L2 + 2p(p − 1)σmax + pE1M E ds. 2p

E |u(t ∧ T )|L2 + 2pνE

t∧T

2(p−1) 

|u(t)|L2

0

Since u(t) is continuous in time, T → ∞ as H → ∞ and hence T ∧ t → t. Thus we obtain  t 2  2  E |u(t)|2L2 + 2νE  u(s) 2 ds = E |u(0)|2L2 + E1M t, 0

L

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W. E, J.C. Mattingly, Ya. Sinai

2p E |u(t)|L2



t

+ 2pνE 0

 2  2 u(s) 2 ds

2(p−1) 

|u(t)|L2

L

 2p  2 ≤ E |u(0)|L2 2p(p − 1)σmax + pE1M E



t

2(p−1)

|u(s)|L2

0

ds.

Recall that we have been calculating with an M th order Galerkin approximation. For the p = 1 equation, the right hand side converges to the desired right hand side. With this bound on E |u(t)|2L2 in hand we can take the M → ∞ limit of the p = 2 equation. Analogously, once we have taken the limit in the pth equation we have the dominating bound needed to take the limit in the p + 1 equation.    2 In our setting, the Poincaré inequality reads |f |2L2 > |f |2L2 and 2 f L2 > |f |2L2 . This allows us to close the above inequalities. After applying Gronwall’s inequality, we obtain the following estimates which are uniform in time. Corollary A.2. E |u(t)|2L2 ≤ e−2νt E |u(0)|2L2 + E |u(t)|2L2 ≤ e−2νt E |u(0)|2L2

 E0

1 − e−2νt , 2ν  E1

+ 1 − e−2νt . 2ν

For any p > 1, E |u(t)|L2 ≤ e−2νt E |u(0)|L2 + C0 2p

2p



t 0

E |u(t)|L2 ≤ e−2νt E |u(0)|L2 + C1 2p

2p



e−2ν(t−s) E |u(s)|L2

2(p−1)

t 0

e−2ν(t−s) E |u(s)|L2

ds,

2(p−1)

ds.

We use standard estimates in the tri-linear term B(u, v), wL2 specialized to our two dimensional setting. Its proof can be found in [CF88] for example. Lemma A.3. Let α, β, γ be positive real numbers such that α + β + γ ≥ 1 and (α, β, γ )  = (0, 0, 1), or (0, 1, 0), or (1, 0, 0),      |B(u, v), wL2 | ≤ C α uL2 β+1 v 

L2

 γ   w 

L2

.

Using this lemma we prove the following estimate specialized to the two dimensional setting with periodic boundary conditions. Lemma A.4. Let {ek , k ∈ Z2 } be a basis for L2 . Consider a splitting of L2 = L2( + L2h . Let N + be in sup{|k| : ∃ ek with ek ∈ L2( } and P( be the projector onto L2( . If u, v ∈ L2 then |P( B(u, v)| ≤ C(N + )3 |u|L2 |v|L2 .

Gibbsian Dynamics and Ergodicity for the Stochastically Forced Navier–Stokes Equation

101

Proof of Lemma A.4. In the periodic setting, P( , Pdiv , and (−)s all are simply Fourier multipliers and hence commute with one other. Recall that B(u, v) = Pdiv (u · ∇)v and hence, |P( B(u, v)| = sup |P( B(u, v), wL2 | = sup |B(u, v), P( wL2 | w∈L2 |w|=1

w∈L2 |w|=1

    = sup |B(u, P( w), vL2 | ≤ C |u|L2 |v|L2 sup 3 P( w  w∈L2 |w|=1

L2

w∈L2 |w|=1

≤ C(N + )3 |u|L2 |v|L2 sup |w|L2 ≤ C(N + )3 |u|L2 |v|L2 . w∈L2 |w|=1

 

Lemma A.5. Fix any δ > 21 , a ∈ (0, 1) and C1 > 0. Let u(t) = ϕtω u0 . There exists a K1 > 0 such that whenever |u0 |2L2 < C0 ,    t |u(s)|2L2 ds ≤ C0 + E0 t + K1 (t + 1)δ for all t ≥ 0 ≥ 1 − a. P |u(t)|2L2 + 2ν 0

Proof of Lemma A.5. The energy equation reads  |u(t)|2L2

t

+ 2ν 0

 |u(s)|2L2

ds =

|u0 |2L2

t

+ E0 t + 0

u(s), dW (s)L2 .

Since |u0 |2L2 < C0 , all we need to show is that   P Mt ≤ K1 (t + 1)δ for t ≥ 0 ≥ 1 − a t for K1 large enough, where Mt = 0 u(s), dW (s)L2 . The quadratic variation [M, M]t can be calculated and one sees that  t 2 |u(s)|2L2 , [M, M]t ≤ σmax 0

and hence p

([M, M]t ) ≤

 2p σmax

t 0

|u(s)|2L2

p

≤

2p p−1 σmax t



t 0

2p

|u(s)|L2 ds.

From Corollary A.2, we know that if |u(0)|2L2 < C0 , then there exists a constant Cp (C0 ) 2p

so that E |u(t)|L2 ≤ Cp for all t ≥ 0 and p ≥ 1. Now define the events
Ak =

sup |Ms | > K1 k

s∈[0,k]

 δ

.

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W. E, J.C. Mattingly, Ya. Sinai

By the Doob–Kolmogorov martingale inequality we have 2p   E([M, M]t )p σmax Cp k p P Ak ≤ ≤ . 2p 2p k 2pδ K1 k 2pδ K1

Lastly observe that 

P Mt ≤ K1 (t + 1) 

δ



≥1−P



"

 Ak ≥ 1 −

k

   P Ak . k

By the previous estimate on P Ak , for any δ > 21 we see that the sum is finite for p sufficiently large. Specifically, we need δ > 21 (1 + p1 ). Lastly, the sum can be made as small as we want by increasing K1 .  

B. Properties of Stationary Measures We now establish a number of properties, derived from the dynamics, which any stationary measure must possess. Lemma B.1. For any stationary measure all energy moments are finite. In fact for any p ≥ 1 there exist a constant Cp < ∞ such that  2p |u|L2 dµ(u) < Cp L2

for all stationary measures µ. In particular C1 = E2ν0 . Proof. We will consider the case when p = 1. The other cases follow by the same method. For any H > 0 there exists a bH such that µ{u ∈ L2 : |u|2L2 ≤ bH } > 1 − H. Let BH denote {u ∈ L2 : |u|2L2 ≤ bH }. For any H > 0 and t > 0, we have  



   ω 2 |u|2L2 ∧ H dµ(u) = E ϕ0,t u L2 ∧ H dµ(u) L2 L2 

   ω 2 ≤ HH + E ϕ0,t u L2 ∧ H dµ(u) B  H

 ω 2  ≤ HH + E ϕ0,t uL2 dµ(u). BH

Applying the first bound in Corollary A.2 gives   

 E0 E0 2 −2νt |u|L2 ∧ H dµ(u) ≤ H H + bH − +e . 2ν 2ν L2 Taking the limit as t → ∞ and then observing that H was arbitrary, we obtain 



  E0 |u|2L2 ∧ H dµ(u) = |u|2L2 ∧ H dµ(u) ≤ . 2ν L2 U Taking H → ∞ gives that the energy of any stationary measure is bounded by E2ν0 . The argument for higher moments of the energy is the same  

Gibbsian Dynamics and Ergodicity for the Stochastically Forced Navier–Stokes Equation

103

Lemma B.2. For any stationary measure µ,  L2

|u|2L2 dµ(u) =

E0 . 2ν

In addition if the forcing is such that E1 < ∞ then 



  E1  2 2  u 2 dµ(u) = 2 L 2ν L

and

2p

L2

|u|L2 dµ(u) < C1 (p) < ∞

for all p ≥ 1. Proof. Using Eq. (27), we have that for any initial condition u0 ∈ L2 ,  2 E ϕ0,t u0 L2 + 2ν



t 0

2  E ϕ0,s u0 L2 ds = |u0 |2L2 + E0 t.

Here we have switched the time integral and the expectation by the Fubini–Tonelli theorem because the integrand is non-negative. We know from Lemma B.1 that any stationary measure has finite energy moments. Hence averaging with respect to the stationary measure gives 

  t   2 2 E ϕ0,t u0 L2 dµ(u0 ) + 2ν E ϕ0,s u0 L2 ds dµ(u0 ) 2 L2  L 0 |u0 |2L2 dµ(u0 ) + E0 t. = L2

Because µ was a stationary measure, we have that  L2

and

 L2

 2 E ϕ0,t u0  2 dµ(u0 ) =



L



t 0

2  E ϕ0,s u0 L2 ds = t

L2

|u0 |2L2 dµ(u0 )

 L2

|u0 |2L2 dµ(u0 ).

 Hence 2ν L2 |u0 |2L2 dµ(u0 ) = E0 , concluding the proof of the first claim. We now turn to the enstrophy moments. By the first part of this lemma, we know that there exist a U ⊂ H1 such that µ(U ) = 1. We now can proceed just as in Lemma B.1 to prove that all of the enstrophy moments are finite. To find the expected value of the H2 norm we use Eq. (28). Then we proceed exactly as we did to obtain the expected value of the enstrophy (the H1 norm).     Lemma B.3. Let µp be the measure induced on C (−∞, 0], L2( by any given stationary measure µ. Fix any K0 > 0 and δ > 21 . Then for µp -almost every trajectory in   C (−∞, 0], L2( , v(s), there exists a constant T such that for s ≤ 0, |v(s)|2L2 ≤ E0 + K0 min(T , |s|)δ .

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Proof. The basic energy estimate, derived from (29), reads:  t  t |v(s)|2L2 ds + |v(t)|2L2 = |v(t0 )|2L2 + E0 (t − t0 ) − 2ν v(s), dW (s)L2 , t0

t0

for any t0 < t ≤ 0. There is no problem writing the integration against the Wiener path in the above integral. Our stochastic PDE had pathwise defined solutions. Therefore if we know the initial condition v(t0 ) and the trajectory of v(s) for s ∈ [t0 , t] the increments of the Wiener process on the interval [t0 , t] are uniquely defined. For any k ≥ 1, the above estimate implies sup

s∈[−k,−k+1]

where Fk (s) = −2ν Now define

s

−k

|v(s)|2L2 ≤ |v(−k)|2L2 + E0 +

sup

s∈[−k,−k+1]

|v(r)|2L2 dr + Mk (s) and Mk (s) =

s

−k

Fk (s),

v(r), dW (r)L2 . 




Ak = v(s) :

sup

s∈[−k,−k+1]

|v(s)|2L2

≤ E0 + K0 |k − 1|

δ

and UT = ∩k>T Ak . Since the UT are an increasing collection of sets it will be sufficient to prove that the limT →∞ µp (UT ) =  1. This is the same as showing that c c c limT →∞ µp (UT ) = 0. Now since µp (UT ) ≤ k>T µp (Ak ), we need only to show c that k>0 µp (Ak ) < ∞:   K0 c 2 δ |k − 1| µp (Ak ) ≤ µp v(s) : |v(−k)|L2 ≥ 2
 K0 + µp v(s) : sup Fk (s) ≥ |k − 1|δ , 2 s∈[−k,−k+1] The first term is the most straightforward. Lemma B.2 implies that the second moment of the energy is uniformly bounded by some constant C2 . Hence Chebyshev’s inequality produces   4 K0 4C E |v(−k)|4L2 ≤ 2 |k − 1|δ ≤ 2 µp v(s) : |v(−k)|2L2 ≥ 2δ 2 K0 |k − 1| K0 |k − 1|2δ which is summable as long as δ > 21 . The second term proceeds in the same way but with Chebyshev’s inequality replaced by the exponential martingale estimate. The exponential martingale inequality controls the size of a martingale minus something proportional to its quadratic variation (see [RY94, Mao97] for example). The details are given in the following. The key observation is that we can control Fk (s) by controlling Mk (s)− α[Mk , Mk ](s), where [Mk , Mk ](s) is the quadratic variation of the martingale Mk (s) and α is a constant we will choose presently. First notice that with probability one,  s  s  2 |v(r)|2L2 dr [Mk , Mk ](s) = |σl |2 |vl (r)|2 dr ≤ σmax −k

2 ≤ σmax

l



−k

s −k

|v(r)|2L2 dr

Gibbsian Dynamics and Ergodicity for the Stochastically Forced Navier–Stokes Equation

105

and hence Fk (s) ≤ Mk (s) −

2ν [Mk , Mk ](s) 2 σmax

almost surely. In this setting, the exponential martingale inequality states that for positive α and β,   α P sup Mk (s) − [Mk , Mk ](s) > β ≤ e−αβ . 2 s∈[−k,0] Taking α =
µp

4ν 2 σmax

we find

K0 v(s) : sup Fk (s) ≥ |k − 1|δ 2 s∈[−k,−k+1]



  2νK0 δ ≤ exp − 2 |k − 1| . σmax

Since this is summable for any δ > 0, the proof is complete.

 

C. Control of High Modes by Low Modes

  Lemma C.1. If h(t) is the solution to (12) with some low mode forcing ( ∈ C [0, t], L2( , t 4 then sups∈[0,t] |h(s)|L2 is bounded by a constant depending on |h(0)|L2 and 0 |(|L2 ds. Proof. Taking the inner product of (12) with h produces 1d |h(t)|2L2 = −ν |h|2L2 + Ph B(h, (), hL2 + Ph B((, (), hL2 2 dt because Ph B((, h), hL2 = Ph B(h, h), hL2 = 0. Next using Lemma A.3 produces, 1d |h(t)|2L2 ≤ −ν |h|2L2 + C |h|L2 |h|L2 |(|L2 + C |h|L2 |(|2L2 2 dt C C |h|2 2 |(|2L2 + |(|4L2 ≤ 2ν L 2ν Since ( ∈ L2( we have |(|L2 ≤ (N + ) |(|L2 , where N + = sup{|k| : ∃ ek with ek ∈ L2( }, and hence after applying Gronwall’s Lemma we have   t  |h(t)|2L2 ≤ C1 |h(0)|2L2 exp a1 |(|2L2 ds 0  t    t  |(|4L2 ds exp a1 |(|2L2 ds . + C2 0

Since by Hölder inequality,



t 0

0

 |(|2L2 ds ≤ t

0

t

|(|4L2 ds,

the proof is complete.   Acknowledgements. The authors would like to thank Gérard Ben Arous, Amir Dembo, Perci Diaconis,Yitzhak Katznelson, Di Liu, George Papanicolaou and Andrew Stuart for useful discussions. The work of the first author is partially supported by a Presidential Faculty Fellowship from the NSF. The work of the second author is partially supported by NSF grant DMS-9971087. The work of the third author is partially supported by NSF grant DMS-9706794 and RFFI grant 99-01-00314.

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References [BKL] [CDF97]

Bricmont, J., Kupiainen, A., and Lefevere, R.: Preprint Crauel, H., Debussche, A., and Flandoli, F.: Random attractors. J. Dynam. Diff. Eqs. 9 no. 2, 307–341 (1997) [CF88] Constantin, P. and Foia¸s, C.: Navier–Stokes equations. Chicago: University of Chicago Press, 1988 [EMatt] E, W. and Mattingly, J.C.: Ergodicity for the Navier–Stokes Equation with Degenerate Random Forcing: Finite Dimensional Approximation. Submitted [EM] Eckmann, J.P., and Hairer, M.: Uniqueness of the invariant measure for a stochastic PDE driven by degenerate noise. Preprint [EFNT94] Eden, A., Foias, C., Nicolaenko, B., and Temam, R.: Exponential attractors for dissipative evolution equations. Research in Applied Mathematics, New York: John Wiley and Sons and Masson, 1994 [Fer97] Ferrario, B.: Ergodic results for stochastic Navier–Stokes equation. Stochastics and Stochastics Rep. 60, no. 3–4, 271–288 (1997) [Fla94] Flandoli, F.: Dissipativity and invariant measures for stochastic Navier–Stokes equations. NoDEA 1, 403–426 (1994) [FM95] Flandoli, F. and Maslowski, B.: Ergodicity of the 2-D Navier–Stokes equation under random perturbations. Commun. Math. Phys. 171, 119–141 (1995) [FMRT] Foias, C., Manley, O., Rosa, R., Temam, R.: Navier–Stokes Equations and Turbulence. To be published [IN] Ito, K., Nisio, M.: On stationary solutions of a stochastic differential equation. J. Math. Kyoto Univ. 4, 1–75 (1964) [KS] Kuksin, S. and Shirikyan, A.: Stochastic Dissipative PDE’s and Gibbs Measures. Commun. Math. Phys. 213, 291–330 (2000) [Mao97] Mao, X.: Stochastic differential equations and their applications. Horwood Series in Mathematics & Applications, Chichester: Horwood Publishing Limited, 1997 [Mat98] Mattingly, J.C.: The stochastically forced Navier–Stokes equations: Energy estimates and phase space contraction. Ph.D. thesis, Princeton University, 1998 [Mat99] Mattingly, J.C.: Ergodicity of 2D Navier–Stokes equations with random forcing and large viscosity. Commun. Math. Phys. 206 no. 2, 273–288 (1999) [Mat00] Mattingly, J.C.: Exponential convergence for the stochastically forced Navier–Stokes equations and other partially dissipative dynamics. Submitted [RY94] Revuz, D. and Yor, M.: Continuous martingales and Brownian motion. Second ed., Grundlehren der Mathematischen Wissenschaften, Vol. 293, Berlin: Springer-Verlag, 1994 [Str82] Stroock, D.W.: Lectures on topics in stochastic differential equations. Bombay: Tata Institute of Fundamental Research, 1982, with notes by Satyajit Karmakar [SV79] Stroock, D.W. and Varadhan, S.R.S.: Multidimensional diffusion processes. Berlin: SpringerVerlag, 1979 [VF88] Vishik, M. and Fursikov, A.: Mathematical problems of statistical hydrodynamics. Dordrect: Kluwer Academic Publishers, 1988 Communicated by G. Gallavotti

Commun. Math. Phys. 224, 107 – 112 (2001)

Communications in

Mathematical Physics

© Springer-Verlag 2001

Counting Phase Space Cells in Statistical Mechanics
Giovanni Gallavotti Fisica, I.N.F.N., Università di Roma “La Sapienza”, P. le Moro 2, 00185 Roma, Italy. E-mail: [email protected] Received: 16 November 2000 / Accepted: 22 April 2001

To Joel L. Lebowitz on his 70th birthday Abstract: The problem of counting the number of phase space cells is analyzed with the purpose of interpreting the variational principle for the SRB statistics as an equidistribution property, in equilibrium as well as in nonequilibrium statistical mechanics. 1. Phase Space Cells When Volume is Not Conserved. Variational Properties of Stationary States Consider a transitive Anosov map S on a bounded surface M (“phase space”) modeling, for instance, a simple gas of identical particles subject to nonconservative external forces and to thermostating forces balancing them in the average: a simple but important example that seems well modeled in this way can be found in [CELS93]. Such models would be “typical” for non equilibrium systems if one accepted the chaotic hypothesis, see [GC95a], and Ch. 9 in [Ga99]; for a general discussion see [Ru99]. The general theory of Anosov systems, see [Si68], implies the existence of a “statistics” µSRB describing the asymptotic behavior of almost all initial data in phase space (in the sense of the Liouville measure). This means thatexcept for a volume zero set of ini
−1 tial data x it will be limT →∞ T −1 Tj =0 F (S j x) = µSRB (dy)F (y) for all continuous functions (“observables”) F on M. The SRB distribution admits a rather simple representation which can be interpreted in terms of “coarse graining” of the phase space, and it is convenient to introduce it at this point for later use. Let P be a “Markov partition” of phase space P = (P1 , . . . , Pm ) with sets Pσ , see (for instance) Ch. 9 in [Ga99]. Let T be a time such that the size of the Eσ−T /2 ,... ,σT /2 ∈ def T /2 P T = −T /2 S j P, σj = 1, 2, . . . , m, is so small that the physically interesting observables can be viewed as a constant inside Eσ−T /2 ,... ,σT /2 = E( σ ). Then the SRB  Work partially supported by IHES and Rutgers University

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probability µ( σ ) of E( σ ) and the Liouville distribution are described in terms of the functions λ1u (x) = log | det(∂S)u (x)|,

λ1s (x) = log | det(∂S)s (x)|,

(1)

where (∂S)u (x) (resp. (∂S)s (x)) is the Jacobian of the evolution map S restricted to the unstable (stable) manifold through x and mapping it to the unstable (stable) manifold
±T /2 1 j
±T /2 1 j T /2 T /2 through Sx. Defining Uu,± (x) = j =0 λu (S x) and Us,± (x) = j =0 λs (S x) and selecting a point x( σ ) ∈ E( σ ) for each σ , the SRB distribution and the volume distribution µL , on the phase space M, which we suppose to have volume W = V (M), attribute to the nonempty sets E( σ ) the probabilities   def T /2 T /2 µ( σ ) = µSRB (E( σ )) = hTu,u ( σ ) · exp − Uu,− (x( σ )) − Uu.+ (x( σ ))   def T /2 T /2 µL ( σ ) = V (E( σ ))/W = hTs,u ( σ ) · exp − Us,− (x( σ )) − Uu,+ (x( σ )) ,

(2)

where V (E) is the Liouville volume of E, and hTu,u ( σ ), hTs,u ( σ ) are suitable functions of σ uniformly bounded as σ , T vary, c.f.r. Ch. 9 in [Ga99]. One can read (2) by saying that the “difference” between the Liouville volume and the SRB volume is that the first weighs asymmetrically the past and the future while the second weighs them symmetrically. As mentioned above we have in mind that the sets E( σ ) represent macroscopic states, being small enough so that the physically interesting observables have a constant value inside them; and we would like to think that they provide us with a model for a “coarse grained” description of the microscopic states. The dynamics will, in general, be nonconservative: hence the phase space volume will generally contract under time evolution. We want to describe the time evolution in terms of evolution of microscopic states, with the aim of counting the microscopic states relevant for a given stationary state of the system, i.e. for the SRB distribution. Therefore we divide phase space, supposed of dimenson d, into parallelepipedal cells  of size εd V (E( σ )) and try to discuss time evolution in terms of them. This is a situation that arises in computer simulations: where the cells  are the computer points with coordinates given by a set of integers and the evolution S is a program or code (simulating the solution of equations of motion suitable for the model under study) which operates exactly on the coordinates (i.e. imagining that the deterministic round offs are part of the program). It is clear, or at least it is a widely held belief, that the simulation will produce a chaotic evolution “for all practical purposes”, i.e. if we only look at “macroscopic observables” on the coarse graining scale e−λT 0 of the partition P T , if 0 is the phase space size1 , 1/d and −λ is the most contractive line element exponent, or even at finer 0 = W observables corresponding to a finer coarse graining, which are constant on elements of  the pavement P T for T  > T ,: provided the latter size is greater than the size ε of the cells : T  < T with e−λ T 0 ≥ ε. The question we ask on general grounds is, see also [Ga95] 1 Here the phase space size 0 should be thought of as measured in dimensionless units, i.e. in terms of the sizes δp, δq in momentum and position of the cells . Assuming that we consider N mass–m particles in a gas at temperature T and density ρ, so that d = 6N, then W = 6N 0 with 0 proportional to √ (ρ −1/3 2mkB T /δpδq)1/2 .

Counting Phase Space Cells in Statistical Mechanics

109

Question: Can we count the number of ways in which the asymptotic state of the system can be realized microscopically? In equilibrium the (often) accepted answer is simple: the number is N0 = W/ε d , i.e. just the number of cells (“ergodic hypothesis”). This means that we think that our program will generate a one cycle permutation of the N0 cells , each of which is therefore representative of the equilibrium state. Average values of macroscopic observables will be obtained simply as: lim N −1

N→∞

N −1 

F (S j x) = N0−1



 F () =



j =0

M

F (y)µL (dy)

(3)

According to Boltzmann the quantity: SB = log (W/εd ) def

(4)

is then, see [Bo77] (where however w’s denote integers rather than phase space volumes), proportional to the physical entropy of our equilibrium system. Can one extend the above view to systems out of equilibrium? In such systems the volume will no longer be preserved by time evolution and, in fact, its contraction rate η(x) = − log | det ∂S(x)|

(5)

not only does not vanish but, in general, will have a positive time average η+ , η+ =
j limN→∞ N −1 N−1 j =0 η(S x) = M η(y)µSRB (dy), see [Ru96]. If η+ > 0 the volume will contract indefinitely (hence the system is called dissipative). Out of equilibrium we may imagine that a similar kind of “ergodicity” holds: namely that the cells that represent the stationary state form a subset of all the cells, on which evolution acts as a one cycle permutation. If so the statistical properties of motions will be determined by the equidistribution among
such cells, which thus attributes probabilities ρ() which maximize the quantity −  ρ() log ρ(). Hence the above counting question can be related to a problem ... which necessarily follows from Boltmann’s train of thought, [and] has remained untouched. Consider an irreversible process which, with fixed outside constraints, is passing by itself from the nonstationary to the stationary state. Can we characterize in any sense the resulting distribution of state as the “relatively most probable distribution”, and can this be given in terms of the minimum of a function which can be regarded as the generalization..., [EE11], footnote 239, p. 103. Before proceeding it is convenient to note a nontrivial relation between η and λ1u , λ1s valid for all T , T  > 0, T /2  j =−T /2

η(S j x) =



T /2

T /2

(Uu,α (x) + Us,α (x)) + O(1)

(6)

α=±

see Eq. (1), with the error O(1) being uniformly bounded in T and x: this is a property which is obtained in proving (2), see [Si68] and [Ga99], Chap. 9. Considering simulations of a dissipative system we must recognize that no code can be an invertible code: it must happen (many times) that S = S with   =  . Clearly

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˜ we can think that both  and  are not really different and only one if S = S =  of the two can be taken as a representative of the microscopic state. We can imagine “pruning” one after the other the “unnecessary” cells until the map S becomes invertible. More formally each cell  will have a motion that is eventually periodic and we discard as “transients” all cells whose evolution is not strictly periodic. The remaining cells will form a discrete model of the attractor. The above question becomes now a precise one: which is the number N of leftover cells? It will be only a fraction of the initial number N0 of cells: and we can attempt to estimate it assuming that the evolution is a one cycle permutation of them. The number N ( σ ) of cells leftover in E( σ ) will have to be proportional to the SRB probability µ( σ ) of E( σ ) otherwise the time average of the observables (i.e. the SRB average introduced above) will not be correctly given by the sum over the cells. The just described pruning process will have to leave N ≤ N0 cells; and furthermore inside each “coarse grain” set E( σ ) a number of cells equal to N ( σ ) = N µSRB (E( σ )).
If V ( σ ) is the volume of E( σ ), so that σ V ( σ ) = W , it must be: V ( σ )/εd ≥ N ( σ ) = N µSRB (E( σ ))

(7)

(8)

for all σ ’s. This gives, using that W = ε d N0 : N ≤ N0 min σ

V ( σ )/ε d . N0 µSRB (E( σ ))

(9)

T (x( σ )) − U T (x( σ ))) differs by O(T −1 ) from the The quantity η = max σ T2 (Us,− u,− maximal average phase space contraction maxx∈ attractor − T2 log | det ∂S T /2 (x)|, and Eqs. (2), (6), give

N ≤ N0 e− 2 T η+O(1) , 1

(10)

where the O(1) is uniform in T , and η can be identified with the infinite time average η+ of the phase space contraction rate − log | det ∂S(x)|. The picture must hold for all Markovian pavements P and for all T ’s such that e−λT δ > ε if δ  0 is the typical size of an element of the partition P: this restricts T to be of −1 the order of T = λ log 0 /ε. And, as in equilibrium, once this requirement is fulfilled we shall think that N has the maximal allowed value, i.e. that in (10) the inequality is saturated for T = T . This is a kind of “ergodicity” assumption which is similar to the corresponding assumption that in equilibrium all cells are actually visited (while assuming that only a fraction of them is visited would give the same statistics as long as the fraction is taken to be the same in each coarse grain volume, but a different cell count hence a different entropy assignment). + α We call −λ− (equal in i , λi the Lyapunov exponents, λi > 0, i = 1, 2, . . . , d/2
− , d λ ≥ η ≥ η = number by the transitivity assumption), so that λ ≤ mini λ− + i (λi − i + λi ), and define:  1η 1 η 0 Scells = log N = log N0 − log . (11) = (log N0 ) 1 − 2λ ε 2d λ

Counting Phase Space Cells in Statistical Mechanics

111

This will depend on ε and, unlike the equilibrium case when η = 0, nontrivially so because η/λ is a dynamical quantity and changing (i.e. our representation of the microscopic motion) ε will change Scells as Scells /Scells = | log ε  /ε|. Given a precision ε of the observations, the quantity Scells measure, how many “non transient” phase space cells must be used to obtain a faithful representation of the attractor and of its statistical properties on scale ε. Here by “faithful” on scale ε we mean that all observables which are constant on such a scale will show the correct statistical properties, i.e. that cells of size larger than ε will be visited with the correct SRB frequency. Since the quantity η/λ is bounded by 1 we see that dissipation does not “simplify” much the motion. Note, however, that we also assume that the system isAnosov transitive: which implies that the attactor is dense; so that the small reduction due to the dissipation, estimated above, holds only as long as this is a correct assumption: at high forcing the attractor is likely (i.e. examples abound) to be no longer dense on phase space and the number N0 will have to be replaced by the smaller power of 0 , affecting correspondingly the analysis leading to (11). One can ask how many phase space cells are required for a faithful representation of the dynamics by a permutation of cells if one just asks faithfulness to hold only for “most” observations on scale ε or higher: depending on the meaning attributed to “most” we can expect that η/λ is replaced by other similar quantities (i.e.g. by some averages of η+ and of Lyapunov exponents, respectively). Since we are not interested in all observables but only in very few ones, it might be interesting to attempt concrete estimates in this more general sense. 2. Remarks (1) Although Eq. (11) gives the cell count it does not seem to deserve to be taken as a definition of entropy also for systems out of equilibrium, not even for systems simple enough to admit a transitive Anosov map as a model for their evolution. It seems a notion distinct from what has become known as the “Boltzmann entropy”, [Le93], see also [EE22]. The notion is also different from the Gibbs’entropy, to which it is equivalent only in equilibrium systems: in nonequilibrium (dissipative) systems the latter can only be defined as −∞ and perpetually decreasing; because in such systems one can define the rate at which (Gibbs’) entropy is “created” or “ceded to the thermostats” by the system to be η+ , i.e. to be the average phase space contraction η+ , see [An82, Ru99]. (2) We also see, from the above analysis, that the variational principle that determines the SRB distribution can be identified with the one that leads to equal probability of the phase space cells. The SRB distribution appears to be the equal probability distribution among the N cells which are not transient. In equilibrium all cells are non transient and the SRB distribution coincides with the Liouville distribution. (3) If we could take T → ∞ (hence, correspondingly, ε → 0) then the distribution µ which is uniform inside each E( σ ) but which attributes a total weight to E( σ ) equal to N ( σ ) = µSRB (E( σ ))N would become the exact SRB distribution. However it seems conceptually more satisfactory, imitating Boltzmann, to suppose that ε is very small but > 0 so that T will be large but not infinite. (4) A deeper understanding of the above analysis appears to be linked to an important question raised by Ruelle asking whether (and how) one could possibly relate an entropy notion to the logarithm of the Hausdorff measure of the attractor: and a pertinent possibility is that the Hausdorff measure on the attractor is absolutely continuous with

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respect to the SRB measure. The above analysis in terms of cells is reminiscent, in fact, of the methods to study Hausdorff dimension, Hausdorff measure and Pesin’s formula in general hyperbolic systems, [Yo94]. Acknowledgements. I am grateful to E. Speer for pointing out an inconsistency in a preliminary version of this work and to F. Bonetto for very stimulating and clarifying discussions. The references to [EE11, EE33] and their relevance were pointed out to me by E.G.D. Cohen.

References [An82]

Andrej, L.: The rate of entropy change in non-Hamiltonian systems. Phys. Lett: A 111, 45–46 (1982) [Bo77] Boltzmann, L.: Über die Beziehung zwischen dem zweiten Hauptsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung, respektive den Sätzen über das Wärmegleichgewicht. In: Wissenschaftliche Abhandlungen, Vl. II, Chelsea, New York: F. Hasenöhrl, 1968, reprint, pp. 164–223 [CELS93] Chernov, N. I., Eyink, G. L., Lebowitz, J.L., Sinai, Y.: Steady state electric conductivity in the periodic Lorentz gas. Commun. Math. Phys. 154, 569–601 1993 [EE11] Ehrenfest, P., Ehrenfest, T.: The conceptual foundations of the statistical approach in Mechanics. New York: Dover, 1990, reprint [EE22] Einstein, E.: Zur Theorie des Radiometers. Annalen der Physik, 69, 241–254, 1922. And: Epstein, P.S.: On the resistance experienced by spheres in their motion through gases. Physical Review 23, 710–733 (1924). See also Epstein, P.S.: Theory of the radiometer. Zeitschrift für Physik 54, 537–563 (1929) [Ga95] Gallavotti, G.: Ergodicity, ensembles, irreversibility in Boltzmann and beyond. J. Stat. Phys. 78, 1571–1589 1995 [GC95a] Gallavotti, G., Cohen, E.G.D.: Dynamical ensembles in nonequilibrium statistical mechanics. Phys. Rev. Lett. 74, 2694–2697 (1995) Dynamical ensembles in stationary states. J. Stat. Phys. 80, 931–970, (1995) [Ga99] Gallavotti, G.: Statistical mechanics. A short treatise. Berlin–Heidelberg–New York: Springer Verlag, 1999, pp. 1–345 [Le93] Lebowitz, J.L.: Boltzmann’s entropy and time’s arrow. Phys. Today, 32–38, 1993 [Ru68] Ruelle, D.: Statistical mechanics of one-dimensional lattice gas. Commun. Math. Phys. 9, 267–278 1968 [Ru96] Ruelle, D.: Positivity of entropy production in non equilibrium statistical mechanics. J. Stat. Phys. 85, 1–25 (1996); Entropy production in nonequilibrium statistical mechanics. Commun. Math. Phys. 189, 365–371 (1997) [Ru99] Ruelle, D.: Smooth dynamics and new theoretical ideas in non-equilibrium statistical mechanics. J. Stat. Phys. 95, 393–468 (1999) [Si68] Sinai, Y.G.: Markov partitions and C-diffeomorphisms. Funct. Anal. and Appl. 2, no. 1, 64–89 (1968); Construction of Markov partitions. Funct. Anal. and Appl. 2, no. 2, 70–80 (1968) See also Gibbs measures inergodic theory, Russ. Math. Surv. 27, 21–69 (1972) and Lectures in ergodic theory, Lecture notes in Mathematics, Princeton, NJ: Princeton University Press, 1977 [Yo94] Young, L.S.: Ergodic theory of differentiable dynamical systems. In: Real and complex dynamical systems, ed. B. Branner, P. Hjorth, Nato ASI series, Dordrecht: Kluwer, 1995; Ergodic theory of chaotic dynamical systems. In: Mathematical Physics XII (M ∩ 5 Conference Proceedings), editors D. de Witt, A.J.B. Bracken, M.D. Gould, P.A. Pearce, Cambridge, MA: International Press, 1999 Communicated by Ya. G. Sinai

Commun. Math. Phys. 224, 113 – 132 (2001)

Communications in

Mathematical Physics

© Springer-Verlag 2001

Adiabatic Decoupling and Time-Dependent Born–Oppenheimer Theory Herbert Spohn, Stefan Teufel Zentrum Mathematik and Physik Department, Technische Universität München, 80290 München, Germany. E-mail: [email protected]; [email protected] Received: 10 July 2000 / Accepted: 30 July 2001

Dedicated to Joel Lebowitz on the occasion of his 70th birthday Abstract: We reconsider the time-dependent Born–Oppenheimer theory with the goal to carefully separate between the adiabatic decoupling of a given group of energy bands from their orthogonal subspace and the semiclassics within the energy bands. Band crossings are allowed and our results are local in the sense that they hold up to the first time when a band crossing is encountered. The adiabatic decoupling leads to an effective Schrödinger equation for the nuclei, including contributions from the Berry connection. 1. Introduction Molecules consist of light electrons, mass me , and heavy nuclei, mass M which depends on the type of nucleus. Born and Oppenheimer [3] wanted to explain some general features of molecular spectra and realized that, since the ratio me /M is small, it could be used as an expansion parameter for the energy levels of the molecular Hamiltonian. The time-independent Born–Oppenheimer theory has been put on firm mathematical grounds by Combes, Duclos, and Seiler [5], Hagedorn [8], and more recently in [16]. With the development of tailored state preparation and ultra precise time resolution there is a growing interest in understanding and controlling the dynamics of molecules, which requires an analysis of the solutions to the time-dependent Schrödinger equation, again exploiting that me /M is small. The molecular Hamiltonian is of the form H =

2 2 h¯ 2
h¯ 2
− i∇x − Aext (x) + − i∇X + Aext (X) 2me 2M + Ve (x) + Ven (X, x) + Vn (X).

(1)

For notational simplicity we ignore spin degrees of freedom and assume that all nuclei have the same mass. We have k electrons with positions {x1 , . . . , xk } = x and l nuclei with positions {X1 , . . . , Xl } = X. The first and second term of H are the kinetic energies of the electrons and of the nuclei, respectively. An external magnetic field is

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σ (He (R))



(R)

E3 (R) E2 (R) E1 (R)

R0

R

Fig. 1. The schematic spectrum of He (R) for a diatomic molecule as a function of the separation R of the two nuclei

included through the vector potential Aext . Electrons and nuclei interact via the static Coulomb potential. Therefore Ve is the electronic, Vn the nucleonic repulsion, and Ven the attraction between electrons and nuclei. Ve and Vn may also contain an external electrostatic potential. In atomic units (me = h¯ = 1) the Hamiltonian (1) can be written more concisely as H =

2 me 1
− i∇X + Aext (X) + He (X), M 2

(2)

emphasizing that the nuclear kinetic energy will be treated as a “small perturbation”. He (X) is the electronic Hamiltonian for a given position X of the nuclei, He (X) =

2 1
− i∇x − Aext (x) + Ve (x) + Ven (X, x) + Vn (X). 2

(3)

He (X) is a self-adjoint operator on the electronic Hilbert space L2 (R3k ) restricted to its antisymmetric subspace. Later on we will need some smoothness of He (X), which can be established easily if the electrons are treated as point-like and the nuclei have an extended, rigid charge distribution. Generically He (X) has, possibly degenerate, eigenvalues E1 (X) < E2 (X) < . . . which terminate at the continuum edge (X). Thereby one obtains the band structure as plotted schematically in Fig. 1. The discrete bands Ej (X) may cross and possibly merge into the continuous spectrum as indicated in Fig. 2. Comparing kinetic energies, we find for the speeds |vn | ≈ (me /M)1/2 |ve |, which means that on the atomic scale the nuclei move very slowly. If we regard X(t) as a given nucleonic trajectory, then He (X(t)) is a Hamiltonian with slow time variation and the time-adiabatic theorem [15, 14, 1] can be applied [2]. For us X are quantum mechanical degrees of freedom. The Hamiltonian H of (2) is time-independent and we can only exploit that the nucleonic Laplacian carries a small prefactor. To distinguish, we refer to our situation as space-adiabatic. Since the nuclei move very slowly, their dynamics must be followed over sufficiently long times. From the speed ratio we conclude that

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these times are of order (me /M)1/2 in atomic units. To simplify notation we define  me (4) ε= M as the small dimensionless parameter. Then 2 1
− i∇X + Aext (X) + He (X), H ε = ε2 2

(5)

and we want to study the solutions of the time-dependent Schrödinger equation iε

∂ψ = H εψ ∂t

(6)

in the limit of small ε. The crude physical picture underlying the analysis of (6) is that the nuclei behave semiclassically because of their large mass and that the electrons rapidly adjust to the slow nucleonic motion. Thus, in fact, the time-dependent Born–Oppenheimer approximation involves two limits. If the electrons are initially in the eigenstate χj (X0 ) of the j th band with energy Ej (X0 ), where X0 is the approximate initial configuration of the nuclei, then the j th band is adiabatically protected provided there is an energy gap separating it from the rest of the spectrum. Thus at later times, up to small error, the electronic wave function is still in the subspace corresponding to the j th band. But this implies that the nuclei are governed by the Born–Oppenheimer Hamiltonian 2 1
ε = ε2 (7) − i∇X + Aext (X) + Ej (X). HBO 2 ε can be analyzed through semiclassical methods where to leading Since ε 1, HBO order the contributions come from the classical flow t corresponding to the classical cl = 1 p 2 + E (q) on nucleonic phase space. Hamiltonian HBO j 2 In general, Ej (X) may touch another band as X varies. To allow for such band crossings we introduce the region ! ⊂ Rn , n = 3l, in nucleonic configuration space, such that Ej restricted to ! does not cross or touch any other energy band. The classical flow t then has ! × Rn as phase space and is defined only up to the time when it first hits the boundary ∂! × Rn . Up to that time (7) still correctly describes the quantum evolution. To follow the tunneling through a band crossing other methods have to be used [11, 7], in particular, the codimension of the crossing is of relevance. The mathematical investigation of the time-dependent Born–Oppenheimer theory was initiated and carried out in great detail by Hagedorn. In his pioneering work [9] he constructs approximate solutions to (6) of the form φq(t),p(t) ⊗ χj (q(t)), where φq(t),p(t) is a coherent state carried along the classical flow, (q(t), p(t)) = t (q0√ , p0 ). The difference to the true solution with the same initial condition is of order ε in the L2 -norm over times of order ε −1 in atomic units and the approximation holds until the first hitting time of ∂! × Rn . In a recent work Hagedorn and Joye [10] construct solutions to (6) satisfying exponentially small error estimates. In Hagedorn’s approach the “adiabatic and semiclassical limits are being taken simultaneously, and they are coupled [10]”. In our paper we carefully separate the space-adiabatic and the semiclassical limit. One immediate benefit is the generalization of the first order analysis of Hagedorn from coherent states to arbitrary wave functions.

116

H. Spohn, S. Teufel σ (He (X))

(a) (b)

(b)

}

σ∗ (X)

(a)

Ran P∗

!

X

Fig. 2. The wave function can leave RanP∗ in two different ways: either by transitions to other bands (a) or through the boundary of ! (b)

Let us explain our result for the space-adiabatic part in more detail. We assume that there is some region ! ⊂ Rn in the nucleonic configuration space, such that some subset σ∗ (X) of σ (He (X)) is separated from the remainder of the spectrum by a gap for all X ∈ !, i.e.   dist σ∗ (X), σ (He (X)) \ σ∗ (X) ≥ d > 0 for all X ∈ !. ! could be punctured by small balls (for n = 2) because of band crossings. ! could also terminate because the point spectrum merges in the continuum, which physically means that the molecule loses an electron through ionization. Let P∗ (X) be the spectral ⊕ projection of He (X) associated with σ∗ (X) and P∗ = ! dX P∗ (X). We will establish ε that the unitary time evolution e−iH t/ε agrees on RanP∗ with the diagonal evolution −iH ε t/ε ε e diag generated by Hdiag := P∗ H ε P∗ up to errors of order ε as long as the leaking through the boundary of ! is sufficiently small. To complete the analysis one has to control the flow of the wave function through ∂!. One possibility is to simply avoid the problem by assuming that ! = Rn , hence ∂! = ∅. We will refer to this case as a globally isolated band. Of course, the set {(X, y) ∈ Rn × R : y ∈ σ∗ (X)} may contain arbitrary band crossings. As one of our main results, we prove that the subspace RanP∗ is adiabatically protected. In particular for the purpose of studying band crossings the full molecular Hamiltonian may be replaced by a simplified model with two bands only. In general one has ∂!  = ∅, to which we refer as a locally isolated band. To estimate the flow out of ! the only technique available seems to be semiclassical analysis. But this requires a control over the semiclassical evolution, for which one needs, at present, that {(X, y) ∈ ! × R : y ∈ σ∗ (X)} contains no band crossings. Then {(X, y) ∈ ! × R : y ∈ σ∗ (X)} = ∪j {(X, y) ∈ ! × R : y = Ej (X)} is the disjoint union of possibly degenerate energy bands Ej (X). We will prove that each band separately is adiabatically protected. In the special case where σ∗ (X) = Ej (X) is a nondegenerate eigenvalue for X ∈ !, ε −iH ε t/ε ε is a standard e diag is well approximated through e−iHBO t/ε on L2 (Rn ). Since HBO

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semiclassical operator, one can easily control the X-support of the wave function and therefore prove a result for rather general ! ⊂ Rn , for the details see Theorem 2. Roughly speaking, it says that if φt is a solution of the effective Schrödinger equation for the nuclei iε

∂φt ε = HBO φt , ∂t

(8)

with suppφ0 ⊂ !, then, modulo an error of order ε, ψt := φt (X)χj (X, x) is a solution of the full Schrödinger equation (6) with initial condition ψ0 (X, x) = φ0 (X)χj (X, x) as long as φt is supported in ! up to L2 -mass of order ε. This maximal time span can be computed using the classical flow t . ε acquires as a first order As first observed by Mead and Truhlar [19], in general HBO correction an additional vector potential Ageo (X) = −iχj (X), ∇X χj (X) and (7) has to be replaced by 2 1
ε HBO = ε2 (9) − i∇X + Aext (X) + Ageo (X) + Ej (X). 2 Multiplying χj (X) with a smooth X-dependent phase factor induces a gauge transformation for Ageo , which implies that the physical predictions based on (9) do not change, as it should be. As noticed in [19], in general Ageo cannot be removed through a gauge transformation and (9) and (7) describe different physics. Berry realized that geometric phases appear whenever the Hamiltonian has slowly changing parameters. Therefore Ageo (X) is referred to as Berry connection, cf. [22] for an instructive collection of reprints. In fact, the motion of nuclei as governed by the Born–Oppenheimer Hamiltonian (9) is one of the paradigmatic examples for geometric phases. ε If σ∗ (X) = E(X) is k-fold degenerate, not much of the above analysis changes. HBO 2 n ⊕k becomes matrix-valued and acts on L (R ) , i.e.  2
2 ε ε HBO = − i∇X + Aext (X) + Ej (X) 1k×k 2 
ε + (−iε∇X ) · Ageo (X) + Ageo (X) · (−iε∇X ) . 2 The connection Ageo (X) contains in general also off-diagonal terms and matrix-valued semiclassics must be applied. However, since the only nondiagonal term is in the subprincipal symbol, the leading order semiclassical analysis reduces to the scalar case and, in particular, agrees with the nondegenerate band case. We do not carry out the straightforward extension of Theorem 2 below to the degenerate band case, because the technicalities of matrix-valued semiclassics would obscure the simple ideas behind our analysis. In their recent work [18] Martinez and Sordoni independently study the time-dependent Born–Oppenheimer approximation as based on techniques developed by Nenciu and Sordoni [20]. They consider the case of a globally isolated band for a Hamiltonian of the form (1) with smooth V and Aext = 0. They succeed in proving the adiabatic decoupling to any order in ε for subspaces P∗ε which are ε-close to the unperturbed subspaces P∗ considered by us. With this result, in principle, higher order corrections to the effective Hamiltonian (7) could be computed.

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The paper is organized as follows. Section 2 contains the precise formulation of the ε and on results. Section 3 gives a short discussion of the semiclassical limit of HBO how such results extend to the full molecular system. Proofs are provided in Sect. 4. In spirit they rely on techniques developed in [23] in the context of the semiclassical limit for dressed electron states. In practice the Born–Oppenheimer approximation requires 2 several novel constructions, since the “perturbation”