Thermodynamics Statistical Mechanics Notes


       
 
  


   
 


                 

                          

              

  


   
  
                                                     

                         

  

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(" 2)

G25.2651: Statistical Mechanics Notes for Lecture 2

I. THE LIOUVILLE OPERATOR AND THE POISSON BRACKET From the last lecture, we saw that Liouville's equation could be cast in the form

@f + x x_ f = 0 @t r

The Liouville equation is the foundation on which statistical mechanics rests. It will now be cast in a form that will be suggestive of a more general structure that has a denite quantum analog (to be revisited when we treat the quantum Liouville equation). Dene an operator

iL = x_

x

r

known as the Liouville operator (i = 1 { the i is there as a matter of convention and has the e ect of making L a Hermitian operator). Then Liouville's equation can be written p

;

@f + iLf = 0 @t

The Liouville operator also be expressed as

@H @ iL = @ =1 p @ r N X

i

i

;

i

@H @
@r @p i

::: H

 f

i

g

where A B is known as the Poisson bracket between A(x) and B (x): f

g

A B =

f

@A @B @ =1 r @ p

N X

g

i

i

i

;

@A @B
@p @r i

i

Thus, the Liouville equation can be written as

@f + f H = 0 @t f

g

The Liouville equation is a partial di erential equation for the phase space probability distribution function. Thus, it species a general class of functions f (x t) that satisfy it. In order to obtain a specic solution requires more input information, such as an initial condition on f , a boundary condition on f , and other control variables that characterize the ensemble.

II. PRESERVATION OF PHASE SPACE VOLUME AND LIOUVILLE'S THEOREM Consider a phase space volume element dx0 at t = 0, containing a small collection of initial conditions on a set of trajectories. The trajectories evolve in time according to Hamilton's equations of motion, and at a time t later will be located in a new volume element dx as shown in the gure below: t

1

q

dx t

dx0 p FIG. 1.

How is dx0 related to dx ? To answer this, consider a trajectory starting from a phase space vector x0 in dx0 and having a phase space vector x at time t in dx . Since the solution of Hamilton's equations depends on the choice of initial conditions, x depends on x0 : t

t

t

t

x0 = (p1 (0) ::: p (0) r1 (0) ::: r (0)) x = (p1 (t) ::: p (t) r1 (t) ::: r (t)) x = x (x10  ::: x60 ) N

t i t

N

N

i t

N

N

Thus, the phase space vector components can be viewed as a coordinate transformation on the phase space from t = 0 to time t. The phase space volume element then transforms according to

dx = J (x  x0 )dx0 t

t

where J (x  x0 ) is the Jacobian of the transformation: t

1 J (x  x0 ) = @@ (x (x1

x) 0 x0 ) where n = 6N . The precise form of the Jacobian can be determined as will be demonstrated below. The Jacobian is the determinant of a matrix M, t

t

n t

n

J (x  x0 ) = det(M) = eTrlnM t

whose matrix elements are

M = @x @ x0 i t

ij

j

Taking the time derivative of the Jacobian, we therefore have 2

dJ = Tr M;1 dM  eTrlnM dt dt XX =J M;1 dM dt =1 =1 n

n

ji

ij

i

j

The matrices M;1 and dM=dt can be seen to be given by M;1 = @ x0 @x dM = @ x_ dt @ x0 i

ij

j t

j

ji

t

i

Substituting into the expression for dJ=dt gives

dJ = J X @ x0 @ x_ dt =1 @ x @ x0 X @ x0 @ x_ @ x =J =1 @ x @ x @ x0 n

i

j

j t

i

i j

t

n

i

j

i j k

j t

k t

k t

t

i

where the chain rule has been introduced for the derivative @ x_ =@ x0. The sum over i can now be performed: j

i

t

n X @ xi0 @ xkt

=1 @ x @ x0 j

i

t

i

=

n X

i

=1

M;1 M =

n X

ki

ij

i

=1

M M;1 =  ki

ij

kj

Thus,

dJ = J X  @ x_ dt =1 @ x X = J @ x_ = J x x_ x n

j t

jk

k t

j k n

j t

j

r

j t

=1

or

dJ = J x_ x dt The initial condition on this di erential equation is J (0) J (x0  x0 ) = 1. Moreover, for a Hamiltonian system x x_ = 0. This says that dJ=dt = 0 and J (0) = 1. Thus, J (x  x0 ) = 1. If this is true, then the phase space volume r



r

t

element transforms according to

dx0 = dx

t

which is another conservation law. This conservation law states that the phase space volume occupied by a collection of systems evolving according to Hamilton's equations of motion will be preserved in time. This is one statement of Liouville's theorem. Combining this with the fact that df=dt = 0, we have a conservation law for the phase space probability:

f (x0  0)dx0 = f (x  t)dx t

which is an equivalent statement of Liouville's theorem.

3

t

III. LIOUVILLE'S THEOREM FOR NON-HAMILTONIAN SYSTEMS The equations of motion of a system can be cast in the generic form x_ =  (x) where, for a Hamiltonian system, the vector function  would be 

 @H @H @H @H  (x) = @ r  ::: @ r  @ p  ::: @ p 1 1 and the incompressibility condition would be a condition on  : x x_ = x  = 0 A non-Hamiltonian system, described by a general vector funciton  , will not, in general, satisfy the incompressibility ;

;

N

r

N

r

condition. That is:

r

=0

x x_ = rx

6

Non-Hamiltonian dynamical systems are often used to describe open systems, i.e., systems in contact with heat reservoirs or mechanical pistons or particle reservoirs. They are also often used to describe driven systems or systems in contact with external elds. The fact that the compressibility does not vanish has interesting consequences for the structure of the phase space. The Jacobian, which satises

dJ = J x_ x dt r

will no longer be 1 for all time. Dening  = x x_ , the general solution for the Jacobian can be written as r

J (x  x0 ) = J (x0  x0 ) exp

Z

t

0

t



ds(x ) s

Note that J (x0  x0 ) = 1 as before. Also, note that  = d ln J=dt. Thus,  can be expressed as the total time derivative of some function, which we will denote W , i.e.,  = W_ . Then, the Jacobian becomes

J (x  x0 ) = exp

Z

t

dsW_ (x )

t



s

0

= exp (W (x ) W (x0 )) t

;

Thus, the volume element in phase space now transforms according to

dx = exp (W (x ) W (x0 )) dx0 t

t

;

which can be arranged to read as a conservation law:

e; (x ) dx = e; (x0 ) dx0 W

t

W

t

Thus, we have a conservation law for a modied volume element, involvingpa \metric factor" exp( W (x)). Introducing p the suggestive notation g = exp( W (x)), the conservation law reads g(x dx = g(x0 dx0 . This is a generalized version of Liouville's theorem. Furthermore, a generalized Liouville equation for non-Hamiltonian systems can be derived which incorporates this metric factor. The derivation is beyond the scope of this course, however, the result is p

;

;

t

t

@ (f g) + x (_xf g) = 0 p

p

r

We have called this equation, the generalized Liouville equation Finally, noting that g satises the same equation as J , i.e., p

4

d g = g dt p

p

the presence of g in the generalized Liouville equation can be eliminated, resulting in p

@f + x_ @t

xf =

r

df = 0 dt

which is the ordinary Liouville equation from before. Thus, we have derived a modied version of Liouville's theorem and have shown that it leads to a conservation law for f equivalent to the Hamiltonian case. This, then, supports the generality of the Liouville equation for both Hamiltonian and non-Hamiltonian based ensembles, an important fact considering that this equation is the foundation of statistical mechanics.

IV. EQUILIBRIUM ENSEMBLES An equilibrium ensemble is one for which there is no explicit time-dependence in the phase space distribution function, @f=@t = 0. In this case, Liouville's equation reduces to

f H = 0

f

g

which implies that f (x) must be a pure function of the Hamiltonian

f (x) = F (H (x)) The specic form that F (H (x)) has depends on the specic details of the ensemble. The integral over the phase space distribution function plays a special role in statistical mechanics: F

=

Z

dxF (H (x))

(1)

It is known as the partition function and is equal to the number of members in the ensemble. That is, it is equal to the number of microstates that all give rise to a given set of macroscopic observables. Thus, it is the quantity from which all thermodynamic properties are derived. If a measurement of a macroscopic observable A(x) is made, then the value obtained will be the ensemble average: Z 1 dxA(x)F (H (x)) (2) A = h

i

F

Eqs. (1) and (2) are the central results of ensemble theory, since they determine all thermodynamic and other observable quantities.

A. Introduction to the Microcanonical Ensemble The microcanonical ensemble is built upon the so called postulate of equal a priori probabilities: Postulate of equal a priori probabilities: For an isolated macroscopic system in equilibrium, all microscopic states corresponding to the same set of macroscopic observables are equally probable. 1. Basic denitions and thermodynamics

Consider a thought experiment in which N particles are placed in a container of volume V and allowed to evolve according to Hamilton's equations of motion. The total energy E = H (x) is conserved. Moreover, the number of particles N and volume V are considered to be xed. This constitutes a set of three thermodynamic variables N V E that characterize the ensemble and can be varied to alter the conditions of the experiment. 5

The evolution of this system in time generates a trajectory that samples the constant energy hypersurface H (x) = E . All points on this surface correspond to the same set of macroscopic observables. Thus, by the postulate of equal a priori probabilities, the corresponding ensemble, called the microcanonical ensemble, should have a distribution function F (H (x)) that reects the fact that all points on the constant energy hypersurface are equally probable. Such a distribution function need only reect the fact that energy is conserved and can be written as F (H (x)) = (H (x) E ) where (x) is the Dirac delta function. The delta function has the property that ;

Z 1

(x a)f (x)dx = f (a) ;

;1

for any function f (x). Averaging over the microcanonical distribution function is equivalent to computing the time average in our thought experiment. The microcanonical partition function (N V E ) is given by Z

(N V E ) = C

dx (H (x) E ) ;

N

In Cartesian coordinates, this is equivalent to (N V E ) = C

Z

dp

Z

n

N

( )

D V

where C is a constant of proportionality. It is given by

d r(H (p r) E ) N

;

N

C = N E!h03 Here h is a constant with units Energy Time, and E0 is a constant having units of energy. The extra factor of E0 is needed because the  function has units of inverse energy. Such a constant has no e ect at all on any properties). Thus, (N V E ) is dimensionless. The origin of C is quantum mechanical in nature (h turns out to be Planck's N

N

N

constant) and must be put into the classical expression by hand. Later, we will explore the e ects of this constant on thermodynamic properties of the ideal gas. The microcanonical partition function function measures the number of microstates available to a system which evolves on the constant energy hypersurface. Boltzmann identied this quantity as the entropy, S of the system, which, for the microcanonical ensemble is a natural function of N , V and E : S = S (N V E ) Thus, Boltzmann's relation between (N V E ), the number of microstates and S (N V E ) is S (N V E ) = k ln (N V E ) where k is Boltzmann's constant 1=k = 315773.218 Kelvin/Hartree. The importance of Boltzmann's relation is that it establishes a connection between the thermodynamic properties of a system and its microscopic details. Recall the standard thermodynamic denition of entropy:

S=

Z

dQ T

where an amount of heat dQ is assumed to be absorbed reversibly, i.e., along a thermodynamic path, by the system. The rst law of thermodynamics states that the energy, E of the system is given by the sum of the heat absorbed by the system and the work done on the system in a thermodynamic process: E = Q+W If the thermodynamic transformation of the system is carried reversibly, i.e., along a thermodynamic path, then the rst law will be valid for the di erential change in energy, dE due to absorption of a di erential amount of heat, dQrev and a di erential amount of work, dW done on the system: 6

dE = dQ + dW The work done on the system can be in the form of compression/expansion work at constant pressure, P , leading to a change, dV in the volume and/or the insertion/deletion of particles from the system at constant chemical potential, , leading to a change dN in the particle number. Thus, in general

dW = PdV + dN ;

(The above relation for the work is true only for a one-component system. If there are M types of particles present, P then the second term must be generalized according to =1  dN ). Then, using the fact that dQ = TdS , we have M

k

k

k

dE = TdS PdV + dN ;

or

P dV dS = dE + T T

 dN T

;

But since S = S (N V E ) is a natural function of N , V , and E , the di erential, dS is also given by  @S  dS = @E

N V

 @S  dE + @V

Comparing these two expressions, we see that

N E

 @S  dV + @N

dN

E V



@S  = 1 @E T   P @S = @V T   @S = T @N Finally, using Boltzmann's relation between the entropy S and the partition function , we obtain a prescription for N V

N E

;

E V

obtaining the thermodynamic properties of the system starting from a microscopic, particle-based description of the system:   1 = k @ ln 

T @E  P = k @ ln   T @V   = k @ ln   T @N

N V

N E

;

V E

Of course, the ultimate test of Boltzmann's relation between entropy and the partition function is that the above relations correctly generate the known thermodynamic properties of a given system, e.g. the equation of state. We will soon see several examples in which this is, indeed, the case.

7

G25.2651: Statistical Mechanics Notes for Lecture 3

I. MICROCANONICAL ENSEMBLE: CONDITIONS FOR THERMAL EQUILIBRIUM Consider bringing two systems into thermal contact. By thermal contact, we mean that the systems can only exchange heat. Thus, they do not exchange particles, and there is no potential coupling between the systems. In this case, if system 1 has a phase space vector x1 and system 2 has a phase space vector x2 , then the total Hamiltonian can be written as H (x) = H1 (x1 ) + H2 (x2 ) Furthermore, let system 1 have N1 particles in a volume V1 and system 2 have N2 particles in a volume V2 . The total particle number N and volume V are N = N1 + N2 and V = V1 + V2 . The entropy of each system is given by S1 (N1 V1 E1 ) = k ln 1 (N1 V1 E1 ) S2 (N2 V2 E2 ) = k ln 2 (N2 V2 E2 ) The partition functions are given by 1 (N1 V1 E1 ) = CN1 2 (N2 V2 E2 ) = CN2 (N V E ) = CN

Z Z

Z

dx1
(H1 (x1 ) ; E1 ) dx2
(H2 (x2 ) ; E2 ) dx
(H1 (x1 ) + H2 (x2 ) ; E ) 6= 1 (N1 V1 E1 )2 (N2 V2 E2 )

However, it can be shown that the total partition function can be written as (E ) = C

0

ZE 0

dE1 1 (E1 )2 (E ; E1 )

where C is an overall constant independent of the energy. Note that the dependence of the partition functions on the volume and particle number has been suppressed for clarity. Now imagine expressing the integral over energies in the above expression as a Riemann sum: 0

(E ) = C

0

P X i=1

1 (E1(i) )2 (E ; E1(i) )

where is the small energy interval (which we will allow to go to 0) and P = E= . The reason for writing the integral this way is to make use of a powerful theorem on sums with large numbers of terms. Consider a sum of the form

=

P X i=1

ai

where ai > 0 for all ai . Let amax be the largest of all the ai 's. Clearly, then

amax  Pamax  1

P X i=1 P X i=1

ai ai

Thus, we have the inequality

amax    Pamax or ln amax  ln   ln amax + ln P This gives upper and lower bounds on the value of ln . Now suppose that ln amax >> ln P . Then the above inequality implies that ln   ln amax This would be the case, for example, if amax  eP . In this case, the value of the sum is given to a very good approximation by the value of its maximum term. Why should this theorem apply to the sum expression for (E )? Consider the case of a system of free particles P N H = i=1 p2i =2mi, i.e., no potential. Then the expression for the partition function is (E ) 

Z

D(V )

dN r

Z

dN p


! N p2 X i ;E  VN 2m i=1

i

since the particle integrations are restricted only the volume of the container. Thus, the terms in the sum vary exponentially with N . But the number of terms in the sum P also varies like N since P = E= and E  N , since E is extensive. Thus, the terms in the sum under consideration obey the conditions for the application of the theorem. Let the maximum term in the sum be characterized by energies E1 and E2 = E ; E1 . Then, according to the above analysis,   S (E ) = k ln (E ) = k ln + k ln 1 (E1 )2 (E ; E1 ) + k ln P + k ln C Since P = E= , ln + ln P = ln + ln E ; ln = ln E . But E  N , while ln 1  N . Since N >> ln N , the above expression becomes, to a good approximation   S (E )  k ln 1 (E1 )2 (E ; E1 ) + O(ln N ) + const Thus, apart from constants, the entropy is approximately additive: S (E ) = k ln 1 (E1 ) + k ln 2 (E2 ) = S1 (E1 ) + S2 (E2 ) + O(ln N ) + const Finally, in order to compute the temperature of each system, we make a small variation in the energy E1 dE1 . But since E1 + E2 = E , dE1 = ;dE2 . Also, this variation is made such that the total entropy S and energy E remain constant. Thus, we obtain 0

0 = @S1 + @S2 @ E1 @ E1 0 = @S1 ; @S2 @ E1 @ E2 0= 1 ; 1

T1

T2

from which it is clear that T1 = T2 , the expected condition for thermal equilibrium. It is important to point out that the entropy S (N V E ) dened via the microcanonical partition function is not the only entropy that satises the properties of additivity and equality of temperatures at thermal equilibrium. Consider an ensemble dened by the condition that the Hamiltonian, H (x) is less than a certain energy E . This is known as the uniform ensemble and its partition function, denoted (N V E ) is dened by (N V E ) = C

Z

H (x)<E

dx = C 2

Z

dx (E ; H (x))

where (x) is the Heaviside step function. Clearly, it is related to the microcanonical partition function by

@ (N V E ) (N V E ) = @E

Although we will not prove it, the entropy S~(N V E ) dened from the uniform ensemble partition function via S~(N V E ) = k ln (N V E ) is also approximately additive and will yield the condition T1 = T2 for two systems in thermal contact. In fact, it diers from S (N V E ) by a constant of order ln N so that one can also dene the thermodynamics in terms of S~(N V E ). In particular, the temperature is given by

!

 @ ln   1 = @ S~ = k T @E N
V @E N
V

II. REVERSIBLE LAWS OF MOTION AND THE ARROW OF TIME Hamilton's equations of motion

q_i = @H @pi

p_i = ; @H @q i

are invariant under a reversal of the direction of time t ! ;t. Under such a transformation, the positions and momenta transform according to

qi ! qi i ! m dqi = ;p pi = mi dq i d(;t) i dt Thus, or

d(;pi ) = ; @H d(;t) @qi

dqi @H d(;t) = ; @pi q_i = @H @pi

p_i = ; @H @q i

The form of the equations does not change! One of the implications of time reversal symmetry is as follows: Suppose a system is evolved forward in time starting from some initial condition up to a maximum time t at t, the evolution is stopped, the sign of the velocity of each particle in the system is reversed, i.e., a time reversal transformation is performed, and the system is allowed to evolve once again for another time interval of length t the system will return to its original starting point in phase space, i.e., the system will return to its initial condition. Now from the point of view of mechanics and the microcanonical ensemble, the initial conditions (for the rst segment of the evolution) and the conditions created by reversing the signs of the velocities for initiating the second segment are equally valid and equally probably, both being points selected from the constant energy hypersurface. Therefore, from the point of view of mechanics, without a priori knowledge of which segment is the forward evolving trajectory and which is the time reversed trajectory, it should not be possible to say which is which. That is, if a movie of each trajectory were to be made and shown to an ignorant observer, that observer should not be able to tell which is the forwardevolving trajectory and which the time-reversed trajectory. Therefore, from the point of view of mechanics, which obeys time-reversal symmetry, there is not preferred direction for the ow of time. Yet our everyday experience tells us that there are plenty of situations in which a system seems to evolve in a certain direction and not in the reverse direction, suggesting that there actually is a preferred direction in time. Some common examples are a glass falling to the ground and smashing into tiny pieces or the sudden expansion of a gas into a large box. These processes would always seem to occur in the same way and never in the reverse (the glass shards never reassemble themselves and jump back onto the table forming an unbroken glass, and the gas particles never 3

suddenly all gather in one corner of the large box). This seeming inconsistency with the reversible laws of mechanics is known as Loschmidt's paradox. Indeed, the second law of thermodynamics, itself, would seem to be at odds with the reversibility of the laws of mechanics. That is, the observation that a system naturally evolves in such a way as to increase its entropy cannot obviously be rationalized starting from the microscopic reversible laws of motion. Note that a system being driven by an external agent or eld will not be in equilibrium with its surroundings and can exhibit irreversible behavior as a result of the work being done on it. The falling glass is an example of such a system. It is acted upon by gravity, which drives it uniformly toward a state of ever lower potential energy until the glass smashes to the ground. Even though it is possible to write down a Hamiltonian for this system, the treatment of the external eld is only approximate in that the true microscopic origin of the external eld is not taken into account. This also brings up the important question of how one exactly denes non-equilibrium states in general, and how do they evolve, a question which, to date, has no fully agreed upon answer and is an active area of research. However, the expanding gas example does not involve an external driving force and still seems to exhibit irreversible behavior. How can this observation be explained for such an isolated system? One possible explanation was put forth by Boltzmann, who introduced the notion of molecular chaos. Under this assumption, the momenta of two particles do not become correlated as the result of a collision. This is tantamount to the assumption that microscopic information leading to a correlation between the particles is lost. This is not inconsistent with microscopic reversibility from a probabilistic point of view, as the momenta of two particles before a collision are certainly uncorrelated with each other. The assumption of molecular chaos allows one to prove the so called Boltzmann H-theorem, a theorem that predicts an increase in entropy until equilibrium is reached. For more details see Chapters 3 and 4 of Huang. Boltzmann's assumption of molecular chaos remains unproven and may or may not be true. Another explanation due to Poincare is based on his recurrence theorem. The Poincare recurrence theorem states that a system having a nite amount of energy and conned to a nite spatial volume will, after a suciently long time, return to an arbitrarily small neighborhood of its initial state.

Proof of the recurrence theorem (taken almost directly from Huang, pg. 91):

Let a state of a system be represented by a point x in phase space. As the system evolves in time, a point in phase space traces out a trajectory that is uniquely determined by any given point on the trajectory (because of the deterministic nature of classical mechanics). Let g0 be an arbitrary volume element in the phase space in the volume !0 . After a time t all points in g0 will be in another volume element gt in a volume !t , which is uniquely determined by the choice of g0. Assuming the system is Hamiltonian, then by Liouville's theorem: !0 = !t Let ;0 denote the subspace of phase space that is the union of all gt for 0  t < 1. Let its volume be 0 . Similarly, let ; denote the subspace that is the union of all gt for  t < 1. Let its volume be  . The numbers 0 and  are nite because, since the energy of the system is nite and the spatial volume occupied is nite, a representative point is conned to a nite region in phase space. The denitions immediately imply that ;0 contains ; . We may think of ;0 and ; in a dierent way. Imagine the region ;0 to be lled uniformly with representative points. As time progresses, ;0 will evolve into some other regions that are uniquely determined. It is clear, from the denitions, that after a time , ;0 will become ; . Also, by Liouville's theorem: 0 =  Recall that ;0 contains all the future destinations of the points in g , which in turn is evolved from g0 after a time . It has been shown that ;0 has the same volume as ; since 0 =  by Liouville's theorem. Therefore, ;0 and ; must contain the same set of points (except possibly for a set of zero measure). In particular, ; contains all of g0 (except possibly for a set of zero measure). But, by denition, all points in ; are future destinations of the points in g0 . Therefore all points in g0 (except possibly for a set of zero measure) must return to g0 after a suciently long time. Since g0 can be made arbitrarily small, Poincare's theorem follows. Now consider an initial condition in which all the gas particles are initially in a corner of a large box. By Poincare's theorem, the gas particles must eventually return to their initial state in the corner of the box. How long is this recurrence time? In order to answer this, consider dividing the box up into M small cells of volume v. The total number of microstates available to the gas varies with N like V N . The number of microstates corresponding to all the particles occupying a single cell of volume v is vN . Thus, the probability of observing the system in this microstate is approximately (v=V )N . Even if v = V=2, for N  1023 , the probability is vanishingly small. The Poincare recurrence time, on the other hand, is proportional to the inverse of this probability or (V=v)N . Again, since N  1023, if v = V=2, the required time is 4

t  21023 which is many orders of magnitude longer than the current age of the universe. Thus, although the system will return arbitrarily close to its initial state, the time required for this is unphysically long and will never be observed. Over times relevant for observation, given similar such initial conditions, the system will essentially always evolve in the same way, which is to expand and ll the box and essentially never to the opposite.

5

G25.2651: Statistical Mechanics Notes for Lecture 4

I. THE CLASSICAL VIRIAL THEOREM (MICROCANONICAL DERIVATION) Consider a system with Hamiltonian H (x). Let xi and xj be specic components of the phase space vector. The classical virial theorem states that

@H i = kT
hxi @x ij j

where the average is taken with respect to a microcanonical ensemble. To prove the theorem, start with the denition of the average:

Z @H C @H
(E ; H (x)) hxi @x i = (E ) dxxi @x j j where the fact that
(x) =
(;x) has been used. Also, the N and V dependence of the partition function have been

suppressed. Note that the above average can be written as

@H i = C @ hxi @x (E ) @E j

Z

@H (E ; H (x)) dx xi @x j

@ Z @H = (CE ) @E dx xi @x j H (x)<E Z @ = (CE ) @E dx xi @ (H@x; E ) j H (x)<E

However, writing

xi @ (H@x; E ) = @x@ xi (H ; E )] ;
ij (H ; E ) j j allows the average to be expressed as


dx @x@ xi (H ; E )] +
ij (E ; H (x)) j j H (x)<E I  Z @ xi (H ; E )dSj +
ij dx (E ; H (x)) = (CE ) @E

@H i = C @ hxi @x (E ) @E

Z

H =E

H<E

The rst integral in the brackets is obtained by integrating the total derivative with respect to xj over the phase space variable xj . This leaves an integral that must be performed over all other variables at the boundary of phase space where H = E , as indicated by the surface element dSj . But the integrand involves the factor H ; E , so this integral will vanish. This leaves:

@H i = C
ij @ hxi @x (E ) @E j C
Z = (Eij)

Z

H (x)<E

H (x)<E

= (
ijE ) (E )

1

dx

dx(E ; H (x))

where (E ) is the partition function of the uniform ensemble. Recalling that we have

@ (E ) (E ) = @E @H i =
(E ) hxi @x ij @ E j

( )

@E

=
ij @ ln 1(E) @E

= k
ij @1S~

@E

= kT
ij which proves the theorem. Example: xi = pi and i = j . The virial theorem says that

hpi @H @p i = kT i

2 h mpi i = kT i 2 p h 2mi i = 12 kT

i

Thus, at equilibrium, the kinetic energy of each particle must be kT=2. By summing both sides over all the particles, we obtain a well know result 3N 3N 2 X X h 2pmi i = h 21 mi vi2 i = 23 NkT i i=1 i=1

II. LEGENDRE TRANSFORMS The microcanonical ensemble involved the thermodynamic variables N , V and E as its variables. However, it is often convenient and desirable to work with other thermodynamic variables as the control variables. Legendre transforms provide a means by which one can determine how the energy functions for dierent sets of thermodynamic variables are related. The general theory is given below for functions of a single variable. Consider a function f (x) and its derivative

df
g(x) y = f (x) = dx The equation y = g(x) denes a variable transformation from x to y. Is there a unique description of the function f (x) in terms of the variable y? That is, does there exist a function (y) that is equivalent to f (x)? Given a point x0 , can one determine the value of the function f (x0 ) given only f (x0 )? No, for the reason that the function f (x0 ) + c for any constant c will have the same value of f (x0 ) as shown in the gure below. 0

0

0

2

f(x)

x0

x

b(x 0 )

FIG. 1.

However, the value f (x0 ) can be determined uniquely if we specify the slope of the line tangent to f at x0 , i.e., f (x0 ) and the y-intercept, b(x0 ) of this line. Then, using the equation for the line, we have f (x0 ) = x0 f (x0 ) + b(x0 ) This relation must hold for any general x: f (x) = xf (x) + b(x) 1 Note that f (x) is the variable y, and x = g (y), where g 1 is the functional inverse of g, i.e., g(g 1 (x)) = x. Solving for b(x) = b(g 1 (y)) gives b(g 1(y)) = f (g 1 (y)) ; yg 1(y)
(y) where (y) is known as the Legendre transform of f (x). In shorthand notation, one writes (y) = f (x) ; xy however, it must be kept in mind that x is a function of y. 0

0

0

0

;

;

;

;

;

;

;

III. THE CANONICAL ENSEMBLE A. Basic Thermodynamics In the microcanonical ensemble, the entropy S is a natural function of N ,V and E , i.e., S = S (N V E ). This can be inverted to give the energy as a function of N ,V , and S , i.e., E = E (N V S ). Consider using Legendre transformation to change from S to T using the fact that 3

  T = @E @S NV The Legendre transform E~ of E (N V S ) is

E~ (N V T ) = E (N V S (T )) ; S @E @S = E (N V S (T )) ; TS The quantity E~ (N V T ) is called the Hemlholtz free energy and is given the symbol A(N V T ). It is the fundamental energy in the canonical ensemble. The dierential of A is

 @A   @A    dT + dV + dA = @A @T NV @V NT @N TV dN

However, from A = E ; TS , we have

dA = dE ; TdS ; SdT From the rst law, dE is given by

dE = TdS ; PdV + dN Thus,

dA = ;PdV ; SdT + dN Comparing the two expressions, we see that the thermodynamic relations are

 @A  S = ; @T  @A NV P = ; @V  @A  NT  = @N VT

B. The partition function Consider two systems (1 and 2) in thermal contact such that

N2  N1 E2  E1 N = N1 + N2 

E = E1 + E 2

dim(x1 )  dim(x2 ) and the total Hamiltonian is just H (x) = H1 (x1 ) + H2 (x2 ) Since system 2 is innitely large compared to system 1, it acts as an innite heat reservoir that keeps system 1 at a constant temperature T without gaining or losing an appreciable amount of heat, itself. Thus, system 1 is maintained at canonical conditions, N V T . The full partition function (N V E ) for the combined system is the microcanonical partition function (N V E ) =

Z

dx
(H (x) ; E ) =

Z

dx1 dx2
(H1 (x1 ) + H2 (x2 ) ; E )

Now, we dene the distribution function, f (x1 ) of the phase space variables of system 1 as 4

f (x1 ) = Taking the natural log of both sides, we have

Z

dx2
(H1 (x1 ) + H2 (x2 ) ; E )

Z

ln f (x1 ) = ln dx2
(H1 (x1 ) + H2 (x2 ) ; E ) Since E2  E1 , it follows that H2 (x2 )  H1 (x1 ), and we may expand the above expression about H1 = 0. To linear order, the expression becomes



Z

Z

ln f (x1 ) = ln dx2
(H2 (x2 ) ; E ) + H1 (x1 ) @H @(x ) ln dx2
(H1 (x1 ) + H2 (x2 ) ; E ) 1 1 H1 (x1 )=0

Z

Z @ = ln dx2
(H2 (x2 ) ; E ) ; H1 (x1 ) @E ln dx2
(H2 (x2 ) ; E ) where, in the last line, the dierentiation with respect to H1 is replaced by dierentiation with respect to E . Note

that

Z

ln dx2
(H2 (2 ) ; E) = S2k(E )

@ ln Z dx
(H (x ) ; E ) = @ S2 (E ) = 1 2 2 2 @E @E k kT where T is the common temperature of the two systems. Using these two facts, we obtain ln f (x ) = S2 (E ) ; H1 (x1 )

k

1

kT

f (x1 ) = eS2(E)=k e H1 (x1 )=kT ;

Thus, the distribution function of the canonical ensemble is

f (x) / e H (x)=kT ;

The prefactor exp(S2 (E )=k) is an irrelevant constant that can be disregarded as it will not aect any physical properties. The normalization of the distribution function is the integral:

Z

dxe H (x)=kT
Q(N V T ) ;

where Q(N V T ) is the canonical partition function. It is convenient to dene an inverse temperature  = 1=kT . Q(N V T ) is the canonical partition function. As in the microcanonical case, we add in the ad hoc quantum corrections to the classical result to give Z 1 Q(N V T ) = N !h3N dxe H (x) ;

The thermodynamic relations are thus, Hemlholtz free energy:

A(N V T ) = ; 1 ln Q(N V T ) To see that this must be the denition of A(N V T ), recall the denition of A:

A = E ; TS = hH (x)i ; TS 5

But we saw that

 @A  S = ; @T NV

Substituting this in gives

A = hH (x)i ; T @A @T

or, noting that

@A = @A @ = ; 1 @A @T @ @T kT 2 @

it follows that

A = hH (x)i +  @A @

This is a simple dierential equation that can be solved for A. We will show that the solution is A = ; 1 ln Q( )



Note that

1 ln Q( ) ; 1 @Q = A ; hH (x)i  @A = @  Q @

Substituting in gives, therefore

A = hH (x)i + A ; hH (x)i = A

so this form of A satises the dierential equation. Other thermodynamics follow: Average energy:

Z 1 E = hH (x)i = Q CN dxH (x)e H (x) ;

@ ln Q(N V T ) = ; @

Pressure:

Entropy:

 @A   @ ln Q(N V T )  P = ; @V = kT @V NT NT @A @ 1 @A S = ; @A @T =; @ @T = kT 2 @  @ ; 1 ln Q(N V T ) = ;k @ ln Q + k ln Q = k 2 @  @ E = kE + k ln Q = k ln Q + T 6

Heat capacity at constant volume:

 @E  @ = k 2 @ ln Q(N V T ) CV = @T = @E @ 2 NV @ @T

C. Relation between canonical and microcanonical ensembles We saw that the E (N V S ) and A(N V T ) could be related by a Legendre transformation. The partition functions (N V E ) and Q(N V T ) can be related by a Laplace transform. Recall that the Laplace transform f~() of a function f (x) is given by

Z

f~() =

1

dxe x f (x) ;

0

Let us compute the Laplace transform of (N V E ) with respect to E : ~ (N V ) = CN

Z

1

Z dEe E dx
(H (x) ; E ) ;

0

Using the
-function to do the integral over E : Z ~(N V ) = CN dxe H (x) ;

By identifying  =  , we see that the Laplace transform of the microcanonical partition function gives the canonical partition function Q(N V T ).

D. Classical Virial Theorem (canonical ensemble derivation) Again, let xi and xj be specic components of the phase space vector x = (p1  ::: p3N  q1  ::: q3N ). Consider the canonical average given by

@H i hxi @x j

@H i = 1 C Z dxx @H e H hxi @x i @x Q N j  j1 @  Z 1 = Q CN dxxi ;  @x e ;

But

(x)

  @xi xi @x@ e H (x) = @x@ xi e H (x) ; e H (x) @x j j j  H (x)  @ = @x xi e ;
ij e H (x) j ;

;

;

;

Thus,

H (x)

;

j

;

@H i = ; 1 C Z dx @ x e H  + 1
C Z dxe hxi @x i Q N Z @x Q ij N j j Z  @  1 C + kT
ij = ; Q N dx dxj @x xi e H j Z  = dx xi e H  + kT
ij ;

0

0

;

(x)

(x)

;

;

1

xj =

;1

Several cases exist for the surface term xi exp(;H (x)): 7

(x)

H (x)

1. xi = pi a momentum variable. Then, since H  p2i , exp(;H ) evaluated at pi = 1 clearly vanishes. 2. xi = qi and U ! 1 as qi ! 1, thus representing a bound system. Then, exp(;H ) also vanishes at qi = 1. 3. xi = qi and U ! 0 as qi ! 1, representing an unbound system. Then the exponential tends to 1 both at qi = 1, hence the surface term vanishes. 4. xi = qi and the system is periodic, as in a solid. Then, the system will be represented by some supercell to which periodic boundary conditions can be applied, and the coordinates will take on the same value at the boundaries. Thus, H and exp(;H ) will take on the same value at the boundaries and the surface term will vanish. 5. xi = qi , and the particles experience elastic collisions with the walls of the container. Then there is an innite potential at the walls so that U ;! 1 at the boundary and exp(;H ) ;! 0 at the bondary. Thus, we have the result

@H i = kT
hxi @x ij j

The above cases cover many but not all situations, in particular, the case of a system conned within a volume V with reecting boundaries. Then, surface contributions actually give rise to an observable pressure (to be discussed in more detail in the next lecture).

8

G25.2651: Statistical Mechanics Notes for Lecture 5

I. TEMPERATURE AND PRESSURE ESTIMATORS From the classical virial theorem

@H i = kT
hx @x i

ij

j

we arrived at the equipartition theorem:

*X

p2 =1 2m

+

N

i

i

i

= 32 NkT

where p1  ::: p are the N Cartesian momenta of the N particles in a system. This says that the microscopic function of the N momenta that corresponds to the temperature, a macroscopic observable of the system, is given by N

K (p1  ::: p ) = N

X p2 N

i

=1 2m

i

i

The ensemble average of K can be related directly to the temperature 2 hK (p  ::: p )i = 2 hK (p  ::: p )i T = 3Nk 1 1 3 3nR K (p1  ::: p ) is known as an estimator (a term taken over from the Monte Carlo literature) for the temperature. An estimator is some function of the phase space coordinates, i.e., a function of microscopic states, whose ensemble average gives rise to a physical observable. An estimator for the pressure can be derived as well, starting from the basic thermodynamic relation:  @ ln Q(N V T )   @A  = kT P =; N

N

@V

with

Q(N V T ) = C

N

Z

@V

N T

Z

dx e; (x) = C
H

N

d p N

N T

Z V

d r e; (p r) N


H

The volume dependence of the partition function is contained in the limits of integration, since the range of integration for the coordinates is determined by the size of the physical container. For example, if the system is conned within a cubic box of volume V = L3 , with L the length of a side, then the range of each q integration will be from 0 to L. If a change of variables is made to s = q =L, then the range of each s integration will be from 0 to 1. The coordinates s are known as scaled coordinates. For containers of a more general shape, a more general transformation is i

i

i

s = V ;1 3 r In order to preserve the phase space volume element, however, we need to ensure that the transformation is a canonical one. Thus, the corresponding momentum transformation is  = V 1 3p With this coordinate/momentum transformation, the phase space volume element transforms as d p d r = d d s =

i

=

i

N

N

i

N

1

i

N

Thus, the volume element remains the same as required. With this transformation, the Hamiltonian becomes

H=

X p2 X V ;2 3  2 13 13 + U ( r  ::: r ) = 1 2m 2m + U (V s1 ::: V s ) N

N

i

=1

N

=1

i

i

=

i

=

=

N

i

i

and the canonical partition function becomes

Z

Q(N V T ) = C

d  N

N

( "X ;2 3 2 #) V  1 3 1 3 d s exp ; 2m + U (V s1  ::: V s )

Z

N

=

N

=1

i

=

=

N

i

i

Thus, the pressure can now be calculated by explicit di erentiation with respect to the volume, V : P = kT 1 @Q

Q @V

# " Z Z X  2  ;2 3 X kT 2 @U ; 5 3 = QC d  d s 3 V ; 3V s
@ (V 1 3 s ) e; 2 m " X 2 =1 X # =1 Z Z  p ;  @U e; (p r) = kT C d p d r r
Q 3V =1 m 3V =1 @ r *X  2 +  p 1 = 3V +r
F =1 m = h; @H @V i N

N

N

N

N

=

i

=

i

N

=

i

N

N


H

i

i

i

N

N

i

i


H

i

i

i

i

N

i

i

i

i

i

Thus, the pressure estimator is

X  p2 + r
F ( r ) (p1  ::: p  r1  ::: r ) = (x) = 31V =1 2m N

N

i

N

i

i

i

i

and the pressure is given by

P = h(x)i For periodic systems, such as solids and currently usedPmodels of liquids, an absolute Cartesian coordinate q is ill-dened. Thus, the virial part of the pressure estimator q F must be rewritten in a form appropriate for periodic systems. This can be done by recognizing that the force F is obtained as a sum of contributions F , which is the force on particle i due to particle j . Then, the classical virial becomes i

i

i

i

i

X N

=1

r
F = i

X N

i

i

i

=1

2

r


X

i

j

6=

F

ij

ij

3

i

X X X X = 12 4 r
F + r
F 5 6= 6= i

2

i

ij

j

j

ji

j

i

i

3

j

X X X X = 12 4 r
F ; r
F 5 6= 6= X X 1 1 =2 (r ; r )
F = 2 r
F 6= 6= i

ij

i

j

i

i j i

j

j

i

j

j

ij

i

j

ij

ij

i j i

ij

j

where r is now a relative coordinate. r must be computed consistent with periodic boundary conditions, i.e., the relative coordinate is dened with respect to the closest periodic image of particle j with respect to particle i. This gives rise to surface contributions, which lead to a nonzero pressure, as expected. ij

ij

2

II. ENERGY FLUCTUATIONS IN THE CANONICAL ENSEMBLE In the canonical ensemble, the total energy is not conserved. (H (x) 6= const). What are the uctuations in the energy? The energy uctuations are given by the root mean square deviation of the Hamiltonian from its average hH i:

q

p

E = h(H ; hH i)2 i = hH 2 i ; hH i2

@ ln Q(N V T ) hH i = ; @ Z hH 2 i = Q1 C dxH 2 (x)e; 1 Z @2


H

N

= QC

(x)

dx @ 2 e; (x)
H

N

@2 Q = Q1 @ 2

  @ 2 ln Q + 1 @Q 2 = @ 2 Q2 @ 



@ 2 ln Q + 1 @Q 2 = @ 2 Q @  2 @ ln Q + @ ln Q 2 = @ 2 @ Therefore

2

@ ln Q hH 2 i ; hH i2 = @ 2 But

@ 2 ln Q = kT 2C @ 2

V

Thus,

p

E = kT 2C

V

Therefore, the relative energy uctuation E=E is given by

p

E = kT 2C E E

V

Now consider what happens when the system is taken to be very large. In fact, we will dene a formal limit called the thermodynamic limit, in which N ;! 1 and V ;! 1 such that N=V remains constant. Since C and E are both extensive variables, C  N and E  N , E  p1 ;! 0 as N ! 1 V

V

E

N

But E=E would be exactly 0 in the microcanonical ensemble. Thus, in the thermodynamic limit, the canonical and microcanonical ensembles are equivalent, since the energy uctuations become vanishingly small.

3

III. ISOTHERMAL-ISOBARIC ENSEMBLE A. Basic Thermodynamics The Helmholtz free energy A(N V T ) is a natural function of N , V and T . The isothermal-isobaric ensemble is generated by transforming the volume V in favor of the pressure P so that the natural variables are N , P and T (which are conditions under which many experiments are performed { \standard temperature and pressure," for example). Performing a Legendre transformation of the Helmholtz free energy

@A A~(N P T ) = A(N V (P ) T ) ; V (P ) @V

But

@A @V = ;P

Thus,

A~(N P T ) = A(N V (P ) T ) + PV  G(N P T ) where G(N P T ) is the Gibbs free energy. The di erential of G is

 @G 

dG = @P

 @G 

N T

dP + @T

 @G 

N P

dT + @N

dN P T

But from G = A + PV , we have

dG = dA + PdV + V dP but dA = ;SdT ; PdV + dN , thus

dG = ;SdT + V dP + dN Equating the two expressions for dG, we see that

 @G 

V = @P  @G  S = ; @T  @G   = @N

N T

N P

P T

B. The partition function and relation to thermodynamics In principle, we should derive the isothermal-isobaric partition function by coupling our system to an innite thermal reservoir as was done for the canonical ensemble and also subject the system to the action of a movable piston under the inuence of an external pressure P . In this case, both the temperature of the system and its pressure will be controlled, and the energy and volume will uctuate accordingly. However, we saw that the transformation from E to T between the microcanonical and canonical ensembles turned into a Laplace transform relation between the partition functions. The same result holds for the transformation from V to T . The relevant \energy" quantity to transform is the work done by the system against the external pressure P in changing its volume from V = 0 to V , which will be PV . Thus, the isothermal-isobaric partition function can be expressed in terms of the canonical partition function by the Laplace transform: 4

Z1 (N P T ) = V1 dV e;


P V

0 0

Q(N V T )

where V0 is a constant that has units of volume. Thus, Z1 Z (N P T ) = V N1!h3 dV dxe; ( (x)+ ) 0 0
H

PV

N

The Gibbs free energy is related to the partition function by G(N P T ) = ; 1 ln (N P T )



This can be shown in a manner similar to that used to prove the A = ;(1= ) ln Q. The di erential equation to start with is

G = A + PV = A + P @G @P

Other thermodynamic relations follow: Volume:



N P T ) V = ;kT @ ln (@P

 N T

Enthalpy:

@ ln (N P T ) H = hH (x) + PV i = ; @ Heat capacity at constant pressure

 @ H 

C = @T P

Entropy:

2

N P

@ ln (N P T ) = k 2 @

 

S = ; @G @T

N P

 = k ln (N P T ) + H T

The uctuations in the enthalpy H are given, in analogy with the canonical ensemble, by p H = kT 2C P

so that

p

H = kT 2C H H p  H  1= N which vanish in the thermodynamic limit. so that, since C and H are both extensive, H= P

5

P

C. Pressure and work virial theorems As noted earlier, the quantity ;@H=@V is a measure of the instantaneous value of the internal pressure Pint . Let us look at the ensemble average of this quantity Z1 Z @H hPint i = ; 1 C dV e; dx @V e; (x) 0 Z1 Z 1 @ e; (x) ; dV e dxkT @V = C Z 10 1 @ Q(N V T ) = dV e; kT @V 0 Doing the volume integration by parts gives @    1 Z 1 ; hPint i = 1 e; kTQ(N V T ) 1 ; dV kT e Q(N V T ) 0  0 @V Z1 dV e; Q(N V T ) = P = P 1 0 Thus,
P V

N


H


P V

N


H


P V


P V


P V


P V

hPint i = P This result is known as the pressure virial theorem. It illustrates that the average of the quantity ;@H=@V gives the xed pressure P that denes the ensemble. Another important result comes from considering the ensemble average hPint V i

hPint V i = 1

Z1 0

dV e;


P V

@ Q(N V T ) kTV @V

Once again, integrating by parts with respect to the volume yields e; kTV Q(N V T )1 ; 1 Z 1 dV kT  @ V e; 1 hPint V i =  0  0 @V  Z Z 1 1 = 1 ;kT dV e; Q(V ) + P dV e; V Q(V ) 0 0 = ;kT + P hV i
P V


P V


P V



Q(N V T )


P V

or

hPint V i + kT = P hV i This result is known as the work virial theorem. It expresses the fact that equipartitioning of energy also applies to the volume degrees of freedom, since the volume is now a uctuating quantity.

6



       
 
  


  


      
    
          
   
  
    

   
  
       
   
  
  


 

    
  
     
        
   

 
  

 
  
      

         
      

    

          





            


 
 
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G25.2651: Statistical Mechanics Notes for Lecture 7

I. THE GRAND CANONICAL ENSEMBLE In the grand canonical ensemble, the control variables are the chemical potential , the volume V and the temperature T . The total particle number N is therefore allowed to uctuate. It is therefore related to the canonical ensemble by a Legendre transformation with respect to the particle number N . Its utility lies in the fact that it closely represents the conditions under which experiments are often performed and, as we shall see, it gives direct access to the equation of state.

A. Thermodynamics In the canonical ensemble, the Helmholtz free energy A(N
V
T ) is a natural function of N , V and T . As usual, we perform a Legendre transformation to eliminate N in favor of = @A=@N :


@A A~(
V
T ) = A(N ( )
V
T ) ; N @N VT = A(N ( )
V
T ) ; N

It turns out that the free energy A~(
V
T ) is the quantity ;PV . We shall derive this result below in the context of the partition function. Thus,

;PV = A(N ( )
V
T ) ; N To motivate the fact that PV is the proper free energy of the grand canonical ensemble from thermodynamic considerations, we need to introduce a mathematical theorem, known as Euler's theorem: Euler's Theorem: Let f (x1
:::
xN ) be a function such that

f (x1
:::
xN ) = n f (x1
:::
xN ) Then f is said to be a homogeneous function of degree n. For example, the function f (x) = 3x2 is a homogeneous function of degree 2, f (x
y
z ) = xy2 + z 3 is a homogeneous function of degree 3, however, f (x
y) = exy ; xy is not a homogeneous function. Euler's Theorem states that, for a homogeneous function f , N @f X nf (x1
:::
xN ) = xi @x i i=1

Proof: To prove Euler's theorem, simply di erentiate the the homogeneity condition with respect to lambda: d d n d f (x1
:::
xN ) = d  f (x1
:::
xN ) N X = nn;1 f (x1
:::
xN ) xi @ (@f x ) i i=1

Then, setting  = 1, we have N @f X xi = nf (x1
:::
xN ) i=1 @xi

which is exactly Euler's theorem. 1

Now, in thermodynamics, extensive thermodynamic functions are homogeneous functions of degree 1. Thus, to see how Euler's theorem applies in thermodynamics, consider the familiar example of the Gibbs free energy: G = G(N
P
T ) The extensive dependence of G is on N , so, being a homogeneous function of degree 1, it should satisfy G(N
P
T ) = G(N
P
T ) Applying Euler's theorem, we thus have or, for a multicomponent system,

@G = N G(N
P
T ) = N @N G=

X j

j Nj

But, since

G = E ; TS + PV it can be seen that G = N is consistent with the rst law of thermodynamics. Now, for the Legendre transformed free energy in the grand canonical ensemble, the thermodynamics are dA~ = dA ; dN ; Nd = ;PdV ; SdT ; Nd But, since ~ A~ = A (
V
! T )  !  ! ~ ~ ~ @ A @ A ~ dA = @ d + @V dV + @@TA dT VT

the thermodynamics will be given by


T


V

! ~ @ A N =; @  !V T @ A~ P = ; @V  !
T ~ S = ; @@TA 

V


Since, A~ is a homogeneous function of degree 1, and its extensive argument is V , it should satisfy A~(
V
T ) = A~(
V
T ) Thus, applying Euler's theorem, @ A~ = ;PV A~(
V
T ) = V @V and since

A~ = A ; N = E ; TS ; N

the assignment A~ = ;PV is consistent with the rst law of thermodynamics. It is customary to work with PV , rather than ;PV , so PV is the natural free energy in the grand canonical ensemble, and, unlike the other ensembles, it is not given a special name or symbol! 2

B. Partition function Consider two canonical systems, 1 and 2, with particle numbers N1 and N2 , volumes V1 and V2 and at temperature T . The systems are in chemical contact, meaning that they can exchange particles. Furthermore, we assume that N2 >> N1 and V2 >> V1 so that system 2 is a particle reservoir. The total particle number and volume are V = V1 + V2 N = N1 + N2 The total Hamiltonian H (x
N ) is H (x
N ) = H1 (x1
N1 ) + H2 (x2
N2 ) If the systems could not exchange particles, then the canonical partition function for the whole system would be Z 1 Q(N
V
T ) = N !h3N dxe;(H1 (x1 N1 )+H2 (x2 N2)) = N1N!N! 2 ! Q1(N1
V1
T )Q2(N2
V2
T ) where

Z Q1 (N1
V1
T ) = N !h1 3N1 dxe;H1 (x1 N1 ) 1 Z Q2 (N2
V2
T ) = N !h1 3N2 dxe;H2 (x2 N2 ) 2

However, N1 and N2 are not xed, therefore, in order to sum over all microstates, we need to sum over all values that N1 can take on subject to the constraint N = N1 + N2 . Thus, we can write the canonical partition function for the

whole system as

Q(N
V
T ) =

N X N1 =0

f (N1
N ) N1N!N! 2 ! Q1 (N1
V1
T )Q2(N2
V2
T )

where f (N1
N2 ) is a function that weights each value of N1 for a given N . Thus, f (0
N ) is the number of congurations with 0 particles in V1 and N particles in V2 . f (1
N ) is the number of congurations with 1 particles in V1 and N ; 1 particles in V2 . etc. Determining the values of f (N1
N ) amounts to a problem of counting the number of ways we can put N identical objects into 2 baskets. Thus, f (0
N ) = 1 f (1
N ) = N = N !=1!(N ; 1)! f (2
N ) = N (N ; 1)=2 = N !=2!(N ; 2)! etc. or generally, f (N1
N ) = N !(NN;! N )! = N N!N! ! 1 1 1 2 which is clearly a classical degeneracy factor. If we were doing a purely classical treatment of the grand canonical ensemble, then this factor would appear in the sum for Q(N
V
T ), however, we always include the ad hoc quantum 3

correction 1=N ! in the expression for the canonical partition function, and we see that these quantum factors will exactly cancel the classical degeneracy factor, leading to the following expression:

Q(N
V
T ) =

N X N1 =0

Q1(N1
V1
T )Q2(N2
V2
T )

which expresses the fact that, in reality, the various congurations are not distinguishable from each other, and so each one should count with equal weighting. Now, the distribution function (x) is given by 1 e;H (x N ) (x
N ) = N !hQ3N(N
V
T )

which is chosen so that

Z

dx(x
N ) = 1

However, recognizing that N2
N , we can obtain the distribution for 1 (x1
N1) immediately, by integrating over the phase space of system 2: Z 1 (x1
N1 ) = Q(N
1V
T ) N !h1 3N1 e;H1 (x1 N1) N !h1 3N2 dx2 e;H2(x2 N2) 1 2 where the 1=N1!h3N1 prefactor has been introduced so that N Z X

N1 =0

dx1 (x1
N1 ) = 1

and amounts to the usual ad hoc quantum correction factor that must be multiplied by the distribution function for each ensemble to account for the identical nature of the particles. Thus, we see that the distribution function becomes 1 (x1
N1) = QQ2 ((NN
2
V
V2T
T) ) N !h1 3N1 e;H1 (x1 N1 ) 1 Recall that the Hemlholtz free energy is given by A = ; 1 ln Q



Thus,

or

Q(N
V
T ) = e;A(N V T ) Q2(N2
V2
T ) = e;A(N2 V2 T ) = e;A(N ;N1 V ;V1 T ) Q2 (N2
V2
T ) = e;(A(N ;N1 V ;V1 T );A(N V T )) Q(N
V
T )

But since N >> N1 and V >> V1 , we may expand:

@A N ; @A V +    A(N ; N1
V ; V1
T ) = A(N
V
T ) ; @N 1 @V 1 = A(N
V
T ) ; N1 + PV1 +   

Therefore the distribution function becomes

1 (x1
N1 ) = N !h1 3N1 e
N1 e;PV1 e;H1 (x1 N1 ) 1 1 e
N1 e;H1 (x1 N1 ) = N !h1 3N1 ePV 1 1

4

Dropping the \1" subscript, we have



1 1 
N ;H (x N ) (x
N ) = ePV N !h3N e e We require that (x
N ) be normalized: 1 Z X

1

"1 X

ePV N =0

N =0



dx(x
N ) = 1 #

1 e
N Z dxe;H (x N ) = 1 N !h3N

Now, we dene the grand canonical partition function

Z (
V
T ) =

1 X

N =0

1 e
N Z dxe;H (x N ) N !h3N

Then, the normalization condition clearly requires that Z (
V
T ) = ePV ln Z (
V
T ) = PV kT

Therefore PV is the free energy of the grand canonical ensemble, and the entropy S (
V
T ) is given by

@ @ ( PV ) S (
V
T ) = = k ln Z (
V
T ) ; k ln Z (
V
T )

@T

@


V

We now introduce the fugacity dened to be

= e



V

Then, the grand canonical partition function can be written as Z 1 1 X X 1 N Z (
V
T ) = N !h3N dxe;H (x N ) = N Q(N
V
T ) N =0 N =0 which allows us to view the grand canonical partition function as a function of the thermodynamic variables , V , and T . Other thermodynamic quantities follow straightforwardly: Energy: 1 N Z X E = hH (x
N )i = Z1 dxH (x
N )e;H (x N ) 3N N ! h N =0
@ ln Z (
V
T ) =;

@

Average particle number:

V


@ ln Z (
V
T ) hN i = kT @ VT

This can also be expressed using the fugacity by noting that

@ = @ @ =  @ @ @ @ @

Thus,

hN i = @ @ ln Z (
V
T ) 5

C. Ideal gas Recall the canonical partition function expression for the ideal gas:

"
3=2 #N 1 V 2

m Q(N
V
T ) = N ! h3 

Dene the thermal wavelength ( ) as

2
1=2 ( ) = 2h

m

which has a quantum mechanical meaning as the width of the free particle distribution function. Here it serves as a useful parameter, since the canonical partition can be expressed as


N Q(N
V
T ) = N1 ! V3 The grand canonical partition function follows directly from Q(N
V
T ): 1 1 V
N X Z (
V
T ) = N ! 3 = eV =3 N =0

Thus, the free energy is

PV = ln Z = V kT 3

In order to obtain the equation of state, we rst compute the average particle number hN i

hN i = @ @ ln Z = V3 Thus, eliminating in favor of hN i in the equation of state gives

PV = hN ikT as expected. Similarly, the average energy is given by
E = ; @ ln Z

3V @ = 3 hN ikT = @ V 4 @ 2 where the fugacity has been eliminated in favor of the average particle number. Finally, the entropy (
V
T ) S (
V
T ) = k ln Z (
V
T ) ; k @ ln Z@








3 = 25 hN ik + hN ik ln VhNi
V which is the Sackur-Tetrode equation derived in the context of the canonical and microcanonical ensembles.

D. Particle number
uctuations In the grand canonical ensemble, the particle number N is not constant. It is, therefore, instructive to calculate the uctuation in this quantity. As usual, this is dened to be

p

N = hN 2 i ; hN i2 Note that 6

"1 #2 1 X X N @ 1 1 @ 2 N N Q(N
V
T ) ; Z 2 N Q(N
V
T ) @ @ ln Z (
V
T ) = Z N =0 N =0 = hN 2 i ; h N i 2 Thus,

@ @ ln Z (
V
T ) = (kT )2 @ 2 ln Z (
V
T ) = kTV @ 2 P (N )2 = @ @ @ 2 @ 2

In order to calculate this derivative, it is useful to introduce the Helmholtz free energy per particle dened as follows: a(v
T ) = 1 A(N
V
T )

N

where v = V=N = 1= is the volume per particle. The chemical potential is dened by

@A = a(v
T ) + N @a @v = @N @v @N = a(v
T ) ; v @a @v

Similarly, the pressure is given by

@A = ;N @a @v = ; @a P = ; @V @v @V @v

Also,

@ = ;v @ 2 a @v @v2

Therefore, and

  @P = @P @v = @ 2 a v @ 2 a ;1 = 1 @ @v @ @v2 @v2 v   @ 2 P = @ @P @v = 1 v @ 2 a ;1 = ; 1 @ 2 @v @ @ v2 @v2 v3 @P=@v

But recall the denition of the isothermal compressibility: 1 T = ; V1 @V @P = ; v@p=@v Thus, @2P = 1 

@

and

N = and the relative uctuation is given by

v2 T

2

r

hN ikTT v

r

N = 1 hN ikTT  p 1 ! 0 ashN i ! 1 N hN i v hN i Therefore, in the thermodynamic limit, the particle number uctuations vanish, and the grand canonical ensemble is equivalent to the canonical ensemble. 7

G25.2651: Statistical Mechanics Notes for Lecture 8

I. STRUCTURE AND DISTRIBUTION FUNCTIONS IN CLASSICAL LIQUIDS AND GASES So far, we have developed the classical theory of ensembles and applied it to the ideal gas, for which there was no potential of interaction between the particles: U = 0. We were able to derive all the thermodynamics for this sytem using the various ensembles available to us, and, in particular, we could compute the equation of state. Now, we wish to consider the general case that the potential is not zero. Of course, all of the interesting physics and chemistry of real systems results from the specic interactions between the particles. Real systems can exhibit spatial structure, undergo phase transitions, undergo chemical changes, exhibit interesting dynamics, basically, a wide variety of rich behavior. Consider the two snapshots below. On the left is shown a conguration of an ideal gas, and on the right is shown a conguration of liquid argon. Can you see any inherent structure in the snapshot of liquid argon? While it may not be readily apparent, there is considerable structure in the liquid argon system that is clearly not present in the ideal gas. One way of quantifying spatial structure is through the use of the radial distribution function g(r), which will be discussed in great detail later. For now, it is su cient to know that g(r) is a measure of the probability that a particle will be located a distance r from a another particle in the system. The gure below shows the function g(r) for the ideal gas and for the liquid argon systems. It can be seen that the radial distribution function for the ideal gas is completely featureless signifying that it is equally likely to nd a particle at any distance r from a given particle. (Since the probability is uniform, g(r) 1=r2 for small r. This is the particularly normalization condition on g(r) that gives rise to uniform probability. Hence its rapidly rising behavior for small r.) For the liquid argon system, g(r) exhibits several peaks, indicating that at certain radial values, it is more likely to nd particles than at others. This is a result of the attractive nature of the interaction at such distances. The plot of g(r) also shows that there is essentially zero probability of nding particles at distances less than about 2.5 A from each other. This is due to the presence of very strong repulsive forces at short distances. Sometimes, the structure can be readily seen in snapshots of congurations. Consider the following snapshot of a system of water molecules: The red spheres are oxygen atoms, the grey spheres are hydrogen atoms, the green lines are hydrogen bonds, and the reddish-grey lines are covalent bonds. A good deal of structure can be seen in the form of a complex network of hydrogen bonds. This high degree of structure is characteristic of water and gives rise to the ease which with water can form stable, organized structures around other molecules. In water, one can ask several questions related to structure. For example, what is the probability that an oxygen atom will be found at a distance r away from another oxygen atoms? What is the probability that a hydrogen atom will be located at a distance r from an oxygen atom, etc. The plot below shows the radial distribution functions corresponding to these two scenarios. The peak in the O-O radial distribution function occurs at roughly 2.8 A which is the well known average hydrogen bond length in water. Of course, one could also ask about structure from the point of view of a hydrogen atom and obtain two other representations of structure in water. One can also ask questions pertinent to chemical reactions. The following three snapshots depict a proton transfer reaction in water upon solvation of an excess proton: While the reaction dynamics is expected to be very complicated, involving the breaking of hydrogen bonds and other uctuations away from the rst solvation shell, it might be possible to characterize certain features of the reaction by a single, special coordinate that somehow describes the motion of the proton through the hydrogen bond. Such a coordinate is called a reaction coordinate and can, in general, be a funtion of all of some subset of the cartesian positions of the other atoms in the system:

qreac = q(r1
r2
:::
rN ) There are many things one can do with such a coordinate, such as study its dynamics or obtain its distribution and hence its free energy prole. In the case of the latter, the free energy prole describes the \eective" potential at a given temperature that the proton sees as it moves through the hydrogen bond. Computing this free energy prole, one might obtain a plot that looks as follows: 1

In this case, the free energy prole is obtained with and without quantum eects (which cannot be neglected for protons). For the classical prole, the energy dierence between the top of the barrier and the wells is the eective activation energy. Quantum mechanically, this activation energy is reduced due to zero-point motion (a concept we will discuss in the context of quantum statistical mechanics).

A. General distribution functions and correlation functions We begin by considering a general N -particle system with Hamiltonian 3N X

p2i + U (r
:::
r ) 1 N 2m i=1

H=

For simplicity, we consider the case that all the particles are of the same type. Having established the equivalence of the ensembles in the thermodynamic limit, we are free to choose the ensemble that is the most convenient on in which to work. Thus, we choose to work in the canonical ensemble, for which the partition function is Z

Q(N
V
T ) = N !h1 3N

d3N p d3N re;

P3N p2i i=1 2m e; U (r1
:::
rN )

The 3N integrations over momentum variables can be done straighforwardly, giving Z 1 Q(N
V
T ) = N !3N dr1


drN e; U (r1
:::
rN ) = NZ!N3N where  = function

p

h2 =2m is the thermal wavelength and the quantity ZN is known as the congurational partition ZN =

Z

dr1


drN e;

U (r1
:::
rN )

The quantity

e;

U (r1
:::
rN )

ZN

dr1


drN  P (N ) (r1
:::
rN )dr1


drN

represents the probability that particle 1 will be found in a volume element dr1 at the point r1 , particle 2 will be found in a volume element dr2 at the point r2 ,..., particle N will be found in a volume element drN at the point rN . To obtain the probability associated with some number n < N of the particles, irrespective of the locations of the remaining n + 1
:::
N particles, we simply integrate this expression over the particles with indices n + 1
:::
N :
Z 1 (n) ; U (r1
:::
rN ) P (r
:::
r )dr


dr = dr


dr e dr


dr 1

n

1

n

n+1

ZN

N

1

n

The probability that any particle will be found in the volume element dr1 at the point r1 and any particle will be found in the volume element dr2 at the point r2 ,...,any particle will be found in the volume element drn at the point rn is dened to be (n) (r1
:::
rn )dr1


drn = (N N;!n)! P (n)(r1
:::
rn )dr1


drn which comes about since the rst particle can be chosen in N ways, the second chosen in N ; 1 ways, etc. Consider the specal case of n = 1. Then, by the above formula, Z (1) (r1 ) = Z1 (N N;! 1)! dr2


drN e; U (r1
:::
rN ) N Z N =Z dr2


drN e; U (r1
:::
rN ) N

2

Thus, if we integrate over all r1 , we nd that 1

V

Z

dr1 (1) (r1 ) = N V =

Thus, (1) actually counts the number of particles likely to be found, on average, in the volume element dr1 at the point r1 . Thus, integrating over the available volume, one nds, not surprisingly, all the particles in the system.

3

G25.2651: Statistical Mechanics Notes for Lecture 9

I. DISTRIBUTION FUNCTIONS IN CLASSICAL LIQUIDS AND GASES (CONT'D) A. General correlation functions A general correlation function can be de ned in terms of the probability distribution function (n) (r1
:::
rn ) according to g(n) (r
:::
r ) = 1 (n) (r
:::
r ) 1

n

n

n

1

Z nN ! V = Z N n (N ; n)! drn+1


drN e; U (r1
:::
rN ) N

Another useful way to write the correlation function is Z nN ! V (n) g (r1
:::
rn ) = Z N n (N ; n)! dr01


dr0N e; U (r1
:::
rN ) (r1 ; r01 )


(rn ; r0n ) N *Y + n nN ! V 0 = Z N n (N ; n)! (ri ; ri ) N i=1 r1
:::
rN i.e., the general n-particle correlation function can be expressed as an ensemble average of the product of -functions, with the integration being taken over the variables r01
:::
r0N . 0

0

0

0

B. The pair correlation function Of particular importance is the case n = 2, or the correlation function g(2) (r1
r2 ) known as the pair function. The explicit expression for g (2) (r1
r2 ) is 2 g(2) (r1
r2 ) = N 2V(NN;! 2)! h(r1 ; r01 )(r2 ; r02 )i Z 2 V ( N ; 1) = dr3


drN e; U (r1
:::
rN )

correlation

NZN = N (N 2; 1) h(r1 ; r01 )(r2 ; r02 )ir1
:::
rN 0

0

In general, for homogeneous systems in equilibrium, there are no special points in space, so that g(2) should depend only on the relative position of the particles or the di erence r1 ; r2 . In this case, it proves useful to introduce the change of variables r = r1 ; r 2
R = 12 (r1 + r2 ) r1 = R + 12 r
r2 = R ; 21 r

Then, we obtain a new function g~(2) , a function of r and R: Z 2 g~(2)(r
R) = V (N ; 1) dr3


drN e; U (R+ 21 r
R; 12 r
r3
:::
rN )

NZN      N ( = N ; 1)  R + 1 r ; r0  R ; 1 r ; r0 2

2

1

1

2

2

r1
:::
rN 0

0

In general, we are only interested in the dependence on r. Thus, we integrate this expression over R and obtain a new correlation function g~(r) de ned by Z 1 g~(r) = dRg~(2)(r
R)

V N ; 1) Z dRdr


dr e; U (R+ 12 r
R; 21 r
r3
:::
rN ) = V (NZ 3 N N Z = (N Z; 1) dRdr3


drN e; U (R+ 21 r
R; 12 r
r3
:::
rN ) N

For an isotropic system such as a liquid or gas, where there is no preferred direction in space, only the maginitude or r, jrj  r is of relevance. Thus, we seek a choice of coordinates that involves r explicitly. The spherical-polar coordinates of the vector r is the most natural choice. If r = (x
y
z ) then the spherical polar coordinates are

x = r sin  cos  y = r sin  sin  x = r cos  dr = r2 sin drdd where  and  are the polar and azimuthal angles, respectively. Also, note that

r = rn where

n = (sin  cos 
sin  sin 
cos ) Thus, the function g(r) that depends only on the distance r between two particles is de ned to be Z g(r) = 41 sinddg~(r) Z = (4N ;Z1) sindddRdr3


drN e; U (R+ 21 rn
R; 12 rn
r3
:::
rN ) N  0)   ( r ; r ( N ; 1) = 4 rr0 r


R
r3
:::
rN 0

0

0

0

0

0

Integrating g(r) over the radial dependence, one nds that 4

Z1 0

drr2 g(r) = N ; 1  N

The function g(r) is important for many reasons. It tells us about the structure of complex, isotropic systems, as we will see below, it determines the thermodynamic quantities at the level of the pair potential approximation, and it can be measured in neutron and X-ray di raction experiments. In such experiments, one observes the scattering of neutrons or X-rays from a particular sample. If a detector is placed at an angle  from the wave-vector direction of an incident beam of particles, then the intensity I () that one observes is proportional to the structure factor

+ *X  eikrm    *Xm + 1 ik rm ;rn 2

I ()  N1 =N

 S (k)

m
n

e

(

)

where k is the vector di erence in the wave vector between the incident and scattered neutrons or X-rays (since neutrons and X-rays are quantum mechanical particles, they must be represented by plane waves of the form exp(ik
r)). 2

By computing the ensemble average (see problem 4 of problem set #5), one nds that S (k) = S (k) and S (k) is given by Z1 4  drr sin(kr)g(r) S (k) = 1 +

k

0

Thus, if one can measure S (k), g(r) can be determined by Fourier transformation.

C. Thermodynamic quantities in terms of g(r) In the canonical ensemble, the average energy is given by

@ ln Q(N
V
 ) E = ; @ ln Q(N
V
 ) = ln ZN ; 3N ln ( ) ; ln N ! Therefore,

@ ; 1 @ZN E = 3 N @ ZN @ Since

 h  = 2

= 2m @ 1 @ = 2 Thus,

1 2

Z 1 3 dr1


drN U (r1
:::
rN )e U (r1
:::
rN ) E = 2 NkT + Z N = 32 NkT + hU i

In order to compute the average energy, therefore, one needs to be able to compute the average of the potential hU i. In general, this is a nontrivial task, however, let us work out the average for the case of a pairwise-additive potential of the form X u(jri ; rj j)  Upair (r1
:::
rN ) U (r1
:::
rN ) = 12 i
j
i6=j i.e., U is a sum of terms that depend only the distance between two particles at a time. This form turns out to be an excellent approximation in many cases. U therefore contains N (N ; 1) total terms, and hU i becomes XZ 1 hU i = 2Z dr1


drN u(jri ; rj j)e; Upair (r1
:::
rN ) N i
j
i6=j Z = N (2NZ ; 1) dr1


drN u(jr1 ; r2 j)e; Upair (r1
:::
rN ) N The second line follows from the fact that all terms in the rst line are the exact same integral, just with the labels changed. Thus,

3

 Z Z hU i = 12 dr dr u(jr ; r j) N (NZ ; 1) dr


drN e; N Z 1 = 2 dr dr u(jr ; r j) (r
r ) Z N dr dr u(jr ; r j)g (r
r ) = 2V 1

2

1

2

1

2

1

2

3

(2)

2

2

1

2

1

2

1

(2)

Upair (r1
:::
rN )



2

1

2

Once again, we change variables to r = r1 ; r2 and R = (r1 + r2 )=2. Thus, we nd that

Z 2 N hU i = 2V 2 drdRu(r)~g(2) (r
R) Z Z 2 = 2NV 2 dru(r) dRg~(2)(r
R) 2 Z =N 2V dru(r)~g (r) Z1 2 =N dr4r2 u(r)g(r) 2V 0

Therefore, the average energy becomes

Z1 3 N E = 2 NkT + 2 4 drr2 u(r)g(r) 0 Thus, we have an expression for E in terms of a simple integral over the pair potential form and the radial distribution function. It also makes explicit the deviation from \ideal gas" behavior, where E = 3NkT=2. By a similar procedure, we can develop an equation for the pressure P in terms of g(r). Recall that the pressure is given by P = 1 @ ln Q  @V N = Z1 @Z N @

The volume dependence can be made explicit by changing variables of integration in ZN to

si = V ;1=3 ri

Using these variables, ZN becomes

Z

ZN = V N ds1


dsN e; U (V 1=3 s1
:::
V 1=3 sN ) Carrying out the volume derivative gives

N @ZN = N Z ; V N Z ds


ds 1 X ri
@U e; U (V 1=3 s1
:::
V 1=3 sN ) 1 N 3V @V V N @ r i i=1 Z X N 1 = V ZN +  dr1


drN 3V ri
Fi e; U (r1
:::
rN ) i=1

Thus,

*

N 1 @ZN = N +  X ZN @V V 3V i=1 ri
Fi

+

Let us consider, once again, a pair potential. We showed in an earlier lecture that 4

N X i=1

ri
Fi =

N X X i=1 j =1
j 6=i

ri
Fij

where Fij is the force on particle i due to particle j . By interchaning the i and j summations in the above expression, we obtain

2

N X

3

X X ri
Fi = 12 4 ri
Fij + rj
Fji 5 i=1 i
j
i6=j i
j
i6=j

However, by Newton's third law, the force on particle i due to particle j is equal and opposite to the force on particle j due to particle i:

Fij = ;Fji Thus,

2

N X

3

X X X X ri
Fi = 12 4 ri
Fij ; rj
Fij 5 = 21 (ri ; rj )
Fij  12 rij
Fij i=1 i
j
i6=j i
j
i6=j i
j
i6=j i
j
i6=j

The ensemble average of this quantity is

*

+

*

+

Z N X X  X   r
F = r
F = d r rij
Fij e; Upair (r1
:::
rN ) i i ij ij 1


drN 3V i=1 6V i
j
i6=j 6V ZN i
j
i6=j

As before, all integrals are exactly the same, so that

*

+

N  X N (N ; 1) Z dr


r r
F e; Upair (r1
:::
rN ) r
F = 1 N 12 12 3V i=1 i i 6V ZN   Z Z = 6V dr1 dr2 r12
F12 N (NZ ; 1) dr3


drN e; Upair (r1
:::
rN ) N Z  = 6V dr1 dr2 r12
F12 (2) (r1
r2 ) Z 2 N = 6V 3 dr1 dr2 r12
F12 g(2)(r1
r2 )

Then, for a pair potential, we have pair = ;u0 (jr ; r j) (r1 ; r2 ) = ;u0 (r ) r12 F12 = ; @U 1 2 12 @r jr ; r j r 12

1

2

12

where u0 (r) = du=dr, and r12 = jr12 j. Substituting this into the ensemble average gives

*

+

Z N 2  X N 0 (2) 3V i=1 ri
Fi = ; 6V 3 dr1 dr2 u (r12 )r12 g (r1
r2 )

As in the case of the average energy, we change variables at this point to r = r1 ; r2 and R = (r1 + r2 )=2. This gives

*

+

N  X N 2 Z drdRu0 (r)rg~(2) (r
R) r
F = ; 3V i=1 i i 6V 3 Z 2 0 = ; N 6V 2 dru (r)rg~(r) Z1 2 3 0 = ; N 6V 2 0 dr4r u (r)g(r)

5

Therefore, the pressure becomes

P = ; 2 Z 1 dr4r3 u0 (r)g(r) kT 6kT 0

which again gives a simple expression for the pressure in terms only of the derivative of the pair potential form and the radial distribution function. It also shows explicitly how the equation of state di ers from the that of the ideal gas P=kT = . From the de nition of g(r) it can be seen that it depends on the density and temperature T : g(r) = g(r
T ). Note, however, that the equation of state, derived above, has the general form

P kT = + B

2

which looks like the rst few terms in an expansion about ideal gas behavior. This suggests that it may be possible to develop a general expansion in all powers of the density about ideal gas behavior. Consider representing g(r
T ) as such a power series:

g (r 
T ) =

1 X j =0

j gj (r T )

Substituting this into the equation of state derived above, we obtain

1 P = +X Bj+2 (T ) j+2 kT j =0 This is known as the virial equation of state, and the coecients Bj+2 (T ) are given by

1 Bj+2 (T ) = ; 6kT

Z1 0

dr 4r3 u0(r)gj (r T )

are known as the virial coe
cients. The coecient B2 (T ) is of particular interest, as it gives the leading order deviation from ideal gas behavior. It is known as the second virial coecient. In the low density limit, g(r
T )  g0 (r T ) and B2 (T ) is directly related to the radial distribution function.

6

G25.2651: Statistical Mechanics Notes for Lecture 10

I. DISTRIBUTION FUNCTIONS AND PERTURBATION THEORY A. General formulation Recall the expression for the con gurational partition function:

Z ZN = dr1

drN e; U (r1
:::
rN )

Suppose that the potential U can be written as a sum of two contributions U (r1 ::: rN ) = U0 (r1 ::: rN ) + U1 (r1 ::: rN ) where U1 is, in some sense, small compared to U0 . An extra bonus can be had if the partition function for U0 can be evaluated analytically. Let

ZN (0) =

Z

drN e; U0(r1
:::
rN )

dr1

Then, we may express ZN as

(0) Z Z N ZN = Z (0) dr1 drN e; U0(r1
:::
rN ) e; U1 (r1
:::
rN ) N = ZN (0) e; U1 (r1
:::
rN ) 0 0 means average with respect to U0 only. If U1 is small, then the average can be expanded in powers of U1 : 2 3 e; U1 = 1  U +  U 2  U 3 + h

where

h

i

i

h

i0

;

= The free energy is given by

h

1 i0

1 (  )k X U1k k ! k=0 ;

h

2!

h

1 i0

;

3!

h

1 i0

i0







(0) 1 ; U1 A(N V T ) = 1 ln NZ!N3N = 1 ln NZN!3N  ln e Separating A into two contributions, we have A(N V T ) = A(0) (N V T ) + A(1) (N V T ) where A(0) is independent of U1 and is given by (0)
1 Z N (0) A (N V T ) =  ln N !3N and A(1) (N V T ) = 1 ln e; U1 ;

;

;

;

;



h

i0

1 (  )k X = 1 ln U1k k ! k=0 ;

h

1

;

h

i0

h

i0

We wish to develop an expansion for A(1) of the general form

A(1) =

1 (  )k;1 X !k k=1 k ! ;

where !kPare a set of expansion coe cients that are determined by the condition that such an expansion be consistent k k with ln 1 k=0 (  ) U1 0 =k!. Using the fact that h

;

h

i

ln(1 + x) = we have that ;

1 X

k=1

k

( 1)k;1 xk ;

1 !  1 ! X (  )k k (  )k U k 1 1 ln X  k=0 k! 1 0 =  ln 1 + k=0 k! U1 0 ;

h

i

;

;

h



i

1 1 (  )l X X = 1 ( 1)k;1 k1 U1l l ! k=1 l=1 ;

;

;

h

!k i0

Equating this expansion to the proposed expansion for A(1) , we obtain 1 X k=1

1 ! 1 X (  )l l k X U = (  )k !kk! k l=1 l! 1 0 k=1

(;1)k;1 1

;

h

i

;

This must be solved for each of the undetermined parameters !k , which can be done by equating like powers of  on both sides of the equation. Thus, from the  1 term, we nd, from the right side: Right Side :

;

!1 1!

and from the left side, the l = 1 and k = 1 term contributes: Left Side :

;

 U1 0 h

from which it can be easily seen that

1!

i

!1 = U1 0 h

i

Likewise, from the  2 term,

2 !

Right Side :

2! 2 and from the left side, we see that the l = 1 k = 2 and l = 2 k = 1 terms contribute: Left Side : Thus,

2 ; U 2 2

1 i0

h

;h

U1 20 i



!2 = U12 0 U1 20 h

i h

i

For  3 , the right sides gives: Right Side :

3 !

3! 3 the left side contributes the l = 1 k = 3, k = 2 l = 2 and l = 3 k = 1 terms: 2

;

3 Left Side : 6 U13 + ( 1)2 31 (  U1 0 )3 21 h

;

Thus,

i

;

;

h

i

 U1 0 + 12  2 U12

;

;

h

i

h


2 i

!3 = U13 0 + 2 U1 30 3 U1 0 U12 0 h

i

h

i

;

h

i h

i

Now, the free energy, up to the third order term is given by

2 A = A(0) + !1 2 !2 + 6 !3 (0)
;  2;  = 1 ln NZN!3N + U1 0 2 U12 0 U1 20 + 6 U13 0 3 U1 0 U12 0 + 2 U1 30 + In order to evaluate U1 0 , suppose that U1 is given by a pair potential X U1 (r1 ::: rN ) = 12 u1 ( ri rj ) ;

h

;

h

i

;

h

i

;h

i

h

i

;

h

i h

i

h

i

i

j

i6=j

Then, h

U1 0 = Z 1(0) i

Z

N

dr1

drN 12

X i6=j

u1 ( ri j

;

;

rj j)e;

Z Z = N (N (0)1) dr1 dr2 u1 ( r1 r2 ) dr3 2ZN 2 Z N = 2V 2 dr1 dr2 u1( r1 r2 )g0(2) (r1 r2 ) 2V Z 1  = 4r2 u (r)g (r)dr ;

j

j

;

;

j

j

U0 (r1
:::
rN )

drN e; U0(r1
:::
rN )

j

1 0 2 0 The free energy is therefore given by Z1 (0)
A(N V T ) = 1 ln NZN!3N + 21 2 V 4r2 u1 (r)g0 (r)dr 0 ;

;

 ; U2 2

h

1 i0

;h

U1 20 i



B. Derivation of the Van der Waals equation As a speci c example of the application of perturbation theory, we consider the Van der Waals equation of state. Let U0 be given by a pair potential: X U0 (r1 ::: rN ) = 12 u0 ( ri rj ) i6=j j

with



;

j

r> r

u0 (r) = 0

1



This potential is known as the hard sphere potential. In the low-density limit, the radial distribution function can be shown to be given correctly by g0 (r) exp( u0 (r)) or  r> g0 (r) = 10 r 

;



= (r ) u1 (r) is taken to be some arbitrary attractive potential, whose speci c form is not particularly important. Then, the full potential u(r) = u0 (r) + u1 (r) might look like: ;

3

u(r)

r

σ

FIG. 1.

Now, the rst term in A(1) is

A

(1)



Z1 1 2 = 2 V 4r2 u1 (r)g0 (r) 0 Z1 = 12 2 V 4r2 u1 (r) (r ) 0 ;

= 22 V where

a = 2 ;

Z1 

Z1 

r2 u1 (r)dr

aN

 ;

drr2 u1 (r) > 0

is a number that depends on and the speci c form of u1 (r). Since the potential u0 (r) is a hard sphere potential, ZN (0) can be determined analytically. If were 0, then u0 would describe an ideal gas and

ZN (0) = V N However, because two particles may not approach each other closer than a distance between their centers, there is some excluded volume: 4

d d/2

d/2

FIG. 2.

If we consider two hard spheres at closest contact and draw the smallest imaginary sphere that contains both particles, then we nd this latter sphere has a radius :

d

FIG. 3.

Hence the excluded volume for these two particles is 4  3 3 and hence the excluded volume per particle is just half of this: 2  3 b 3 Therefore Nb is the total excluded volume, and we nd that, in the low density limit, the partition function is given approximately by 

ZN (0) = (V Thus, the free energy is

A(N V T )



 ;

;

1 ln (V



5

Nb)N

Nb)N  aN 2 N !3N V ;

;

If we now use this free energy to compute the pressure from

P=

;

@A
@V N
T

we nd that

P = N aN 2 kT V Nb kTV 2 2 = 1  b a kT ;

;

;

;

This is the well know Van der Waals equation of state. In the very low density limit, we may assume that b << 1, hence 1 1 b 1 + b Thus, ;

P kT





a

 + 2 b kT ;

from which we can approximate the second virial coe cient: a = 2  3 + 2 Z 1 drr2 u (r) B2 (T ) b kT 1 3 kT  A plot of the isotherms of the Van der Waals equation of state is shown below: 

;

6

critical pt.

Critical isotherm

Volume discontinuity

FIG. 4.

The red and blue isotherms appear similar to those of an ideal gas, i.e., there is a monotonic decrease of pressure with increasing volume. The black isotherm exhibits an unusual feature not present in any of the ideal-gas isotherms - a small region where the curve is essentially horizontal (at) with no curvature. At this point, there is no change in pressure as the volume changes. Below this isotherm, the Van der Waals starts to exhibit unphysical behavior. The green isotherm has a region where the pressure decreases with decreasing volume, behavior that is not expected on physical grounds. What is observed experimentally, in fact, is that at a certain pressure, there is a dramatic, discontinuous change in the volume. This dramatic jump in volume signi es that a phase transition has occurred, in this case, a change from a gaseous to a liquid state. The dotted line shows this jump in the volume. Thus, the small 7

at neighborhood along the black isotherm becomes larger on isotherms below this one. The black isotherm just represents a boundary between those isotherms along which no such phase transition occurs and those that exhibit phase transitions in the form of discontinuous changes in the volume. For this reason, the black isotherm is called the critical isotherm, and the point at which the isotherm is at and has zero curvature is called a critical point. A critical point is a point at which

@P @V = 0

@2P = 0 @V 2 Using these two conditions, we can solve for the critical volume Vc and critical temperature Tc : 8a Vc = 3Nb kTc = 27 b

and the critical pressure is therefore

Pc = 27ab2

Using these values for the critical pressure, temperature and volume, we can show that the isothermal compressibility, given by

= 1 @V

V @P

T

diverges as the critical point is approached. To see this, note that


2 NkT 2 aN kT 2 a 1 8 a @P  @V V =Vc = 2N 2b2 + 27N 3b3 = 4Nb2 + 27Nb3 = 4Nb2 27b kT (T Tc) ;

;

;



;

Thus,

Tc);1 It is observed that at a critical point, T diverges, generally, as T Tc ; . To determine the heat capacity, note that )N e aN 2=V Q(N V T ) = e; A(N
V
T ) = (VN !Nb 3N

T



(T

;

j

;

j

;

so that

@ ln Q(N V T ) E = @  2 @ aN = @ N ln(V Nb) ln N ! 3N ln  + V @ + aN 2 = 3N @ V 2 3 N  aN =  2 + V 2 = 32 NkT + aN V ;

;

Then, since

it follows that

;

;

;


@E CV = @T V

CV = 23 Nk

T Tc 0

 j

8

;

j

The heat capacity is observed to diverge as T Tc ; . Exponents such as  and  are known as critial exponents. Finally, one other exponent we can easily determine is related to how the pressure depends on density near the critical point. The exponent is called , and it is observed that j

;

P kT

j



const + C ( c ) ;

What does our theory predict for ? To determine  we expand the equation of state about the critical density and temperature:

   P = Pc + 1 @P  (  ) + 1 @ 2 P  (  )2 + 1 @ 3 P  (  )3 + c 2kT @2  c c kT kTc kTc @ =c 6kTc @3 =c c =c " 2 #  
2 2V  P 1 @P @V 1 @ P @V @P @ 1 @ 3 P  (  )3 + c = kT + kT @V @  ( c ) + 2kT @V 2 @ + @V @2  ( c)2 + 6kT c c c c c @3 =c =c =c ;

;

;

;

;

;

The second and third terms vanish by the conditions of the critical point. The third derivative term can be worked out straightforwardly and does not vanish. Rather

 243 2 kTc @3 =c = 8 b = 0 1 @ 3 P 

6

Thus, we see that, by the above expansion,  = 3. The behavior of these quantities near the critical temperature determine three critical exponents. To summarize the results, the Van der Waals theory predicts that

=0

=1

=3

The determination of critical exponents such as these is an active area in statistical mechanical research. The reason for this is that such exponents can be grouped into universality classes { groups of systems with exactly the same sets of critical exponents. The existence of universality classes means that very dierent kinds of systems exhibit essentially the same behavior at a critical point, a fact that makes the characterization of phase transitions via critical exponents quite general. The values obtained above for ,  and  are known as the mean-
eld exponents and shows that the Van der Waals theory is really a mean eld theory. These exponents do not agree terribly well with experimental values ( = 0:1  = 1:35,  = 4:2). However, the simplicty of mean- eld theory and its ability to give, at least, qualitative results, makes it, nevertheless, useful. To illustrate universality classes, it can be shown that, within mean eld theory, the Van der Waals gas/liquid and a magnetic system composed of spins at particular lattice sites, which composes the so called Ising model, have exactly the same mean eld theory exponents, despite the completely dierent nature of these two systems. Another problem with the Van der Waals theory that is readily apparent is the fact that it predicts @P=@V > 0 for certain values of the density . Such behavior is unstable. Possible ways of improving the approximations used are the following: 1. Improve the approximation to g0 (r). 2. Choose a better zeroth order potential U0 (r1 ::: rN ). 3. Go to a higher order in perturbation theory. Barker and Henderson have shown that going to second order in perturbation theory (see discussion in McQuarrie's book, chapter 14), yields well converged results for a square well uid, compared to \exact" results from a molecular dynamics simulation.

9

G25.2651: Statistical Mechanics Notes for Lecture 11

I. REACTION COORDINATES Often simple chemical processes can be described in terms of a one-dimensional coordinate, which is not one of the Cartesian coordinates in a system but a generalized coordinate, q. Such a coordinate is, in general, a function of the Cartesian coordinates: q = q(r1 ::: rN ) In some cases, a set of reaction coordinates is required. Examples 1: Dissociation reactions. Consider a dissociation reaction of the form AB ;! A + B A reaction coordinate for such a process is the distance r between the two atoms in the diatomic. If r1 and r2 are the Cartesian positions of atoms A and atom B , respectively, then r = jr1 ; r2 j What set of generalized coordinates contains r explicitly? Let us transform to center-of-mass and relative coordinate: B r2 R = mAmr1 ++ m A mB r = r1 ; r2 Then, let r = (x y z ) be transformed to spherical polar coordinates (x y z ) ;! (r  ) x = r sin  cos  y = r sin  sin  z = r cos  Example 2: Proton transfer reactions. Consider proton transfer through a hydrogen bond according to: A ; HA * ) AH ; A which is illustrated schematically in the cartoon below:

rp

H d1

d2 s

O

O

r1

r2 FIG. 1.

1

The two heavy atoms are assumed to be of the same type (e.g. oxygen atoms). A reaction coordinate describing this process is the di erence in the two distances d1 and d2 :

 = d1 ; d2 = jrp ; r1 j ; jrp ; r2 j What generalized coordinate system contains  explicitly?

To see what coordinate system this is, consider transforming to center of mass and two relative coordinates as follows: R = mO(r21m+ r+2 )m+ mHrp O H r = r1 ; r2 s = rp ; 12 (r1 + r2 )

Now the six degrees of freedom (rx ry rz sx sy sz ) are transformed to 6 new degrees of freedom, which are the spherical polar coordinates of r and the confocal elliptic coordinates for the position of the proton: q

r = rx2 + ry2 + rz2 q rx2 + ry2  = tan;1 r z r y ; 1  = tan r z

 = d1 + d2 = js + 21 rj + js + 12 rj

 = d1 + d2 = js + 21 rj ; js + 12 rj = tilt angle of plane containing three atoms Then, the coordinate  is the reaction coordinate .

II. FREE ENERGY PROFILES For a reaction coordinate q, we can de ne a probability distribution function according to

P (q) = h(q(Zr1 ::: rN ) ; q) = CQN dN pdN r e; H (p
r)(q(r1
:::
rN );q) Then, we de ne the free energy pro le, A(q), to be A(q ) = ;kT ln P (q) Apart from the normalization of P (q) by the partition function, Q, P (q ), itself is a kind of partition function corresponding to a xed value of the reaction coordinate. Thus, de ning the free energy pro le in terms of the log of P (q) is analogous to de ning the global free energy A = ;kT ln Q.

A. Physical meaning of A q

( )

Consider the free energy di erence between two values of the reaction coordinate q and q0 , which can be written as

A(q) ; A(q0 ) =

Z q

Z q 1 dP dq0 ddA = ; kT dq0 P ( 0 q  q0 ) dq0 q0 q0

The integrand can be written as 2

R

1 dP = CQN dN pdN re; H (p
r) @@q (q(r1 ::: rN ) ; q0 ) P (q0 ) dq0 h(q ; q0 )i Now, we introduce a coordinate transformation from Cartesian coordinates to a set of generalized coordinates that contains q: 0

(r1 ::: rN ) ;! (u1 ::: un q) where n = 3N ; 1. In addition, a corresponding transformation is made on the conjugate momenta according to: (p1 ::: pN ) ;! (pu1 ::: pun pq ) so that, in the measure, no overall Jacobian appears:

dN pdN r = dn pu dpq dn udq Thus, we have

CN R n u Q

d p dpq dn udqe; H (pu
pq
u
q) @@q (q ; q0 ) h(q ; q0 )i Next, the derivative with respect to q0 is changed to a derivative with respect to q: R 1 dP = ; CQN dn pu dpq dn udqe; H (pu
pq
u
q) @q@ (q ; q0 ) P (q0 ) dq0 h(q ; q0 )i 1 dP P (q0 ) dq0 =

0

Then, an integration by parts is performed, which yields CN R n u Q

h

i

e; H (pu
pq
u
q) (q ; q0 ) 1 dP = P (q0 ) dq0 h(q ; q0 )i R ; CN dn pu dpq dn udq @H e; H (pu
pq
u
q) (q ; q0 ) @q = Q h(q ; q0 )i h @H (q ; q0 )i = ; @q h(q ; q0 )i cond  ; h @H @q iq d p dpq dn udq

@ @q

0

where the nal average is an ensemble average conditional upon the restriction that the reaction coordinate q is equal to q0 . Generally, h(  )(q ; q0 )i h  icond = q h(q ; q0 )i Thus, the free energy di erence becomes: 0

A(q) ; A(q0 ) =

Z q q0

cond dq0 h @H @q iq 0

Note that the average in the above expression can also be performed with respect to Cartesian positions and momenta, in which case the derivative can be carried out via the chain rule. If

H=

N X i=1

p2i + U (r ::: r ) 1 N 2m i

then 3

N @H  @ pi + @H  @ ri @H = X @q i=1 @ pi @q @ ri @q




= =

pi  @ pi + @U  @ ri
mi @q @ ri @q pi  @ pi ; F  @ ri


N X i=1

N X

mi @q

i=1

i

@q

The quantity ;@H=@q is the generalized force on the generalized coordinate q. Thus, let the conditional average of this force be Fq . Then, 0

cond Fq = ;h @H @q iq 0

0

and

A(q ) ; A(q0 ) = ;

Z q q0

dq0 Fq

0

Thus, the free energy di erence can be seen to be equal to the work performed against the averaged generalized force in bringing the system from q0 to q, irrespective of the values of the other degrees of freedom. Such an integration is called thermodynamic integration. Another useful expression for the free energy derivative can be obtained by integrating out the momenta before performing a transformation. We begin with R 1 dP = CQN dN pdN re; H (p
r) @@q (q(r1 ::: rN ) ; q0 ) P (q0 ) dq0 h(q ; q0 )i Now, noting that the -function condition is independent of momenta, we can integrate out the Cartesian momenta to yield: R 1 dP = QCN3N dN re; U (r1
:::
rN ) @@q (q(r1 ::: rN ) ; q0 ) P (q0 ) dq0 h(q ; q0 )i Next, the coordinate transformation to generalized coordinates is performed: (r1 ::: rN ) ;! (u1 ::: un q) associated with which there is a Jacobian given by J (u1 ::: un q) = @@(u(r1 :::::: urN q) ) 1 n Introducing the coordinate transformation, we obtain R 1 dP = QCN3N dn udqJ (u1 ::: un q)e; U @@q (q ; q0 ) P (q0 ) dq0 h(q ; q0 )i CN R dn udqJ (u ::: u q )e; U @  (q ; q0 ) 1 n 3N @q = ; Q h  (q ; q0 )i i h CN R dn udq @ J (u ::: u q )e; U  (q ; q0 ) 1 n 3 N @q = Q h(q ; q0 )i i h CN R n @ ; (U ;kT ln J )  (q ; q0 ) Q3N d udq @q e = h(q ; q0 )i   R ; C3NN dn udq @U ; kT @ ln J e; (U ;kT ln J ) (q ; q0 ) Q @q @q = h(q ; q0 )i 0

0

0

4

=

; C3NN R dn udqJ (u1 ::: un q) Q 





; kT @ @qln J e; U (q ; q0 ) h(q ; q0 )i



@U @q

@ ln J icond = ; h @U ; kT @q @q q 0

or





dA @U @ ln J cond dq0 = h @q ; kT @q iq 0

III. EXAMPLE OF A DISSOCIATION REACTION Let us again look at the dissociation reaction AB ;! A + B for which the reaction coordinate r = jr1 ; r2 j is appropriate. The transformation to center of mass and relative coordinate is: R = mAr1 + mB r2

M

r = r1 ; r2

The inverse of this transformation is

r1 = R + mM2 r r2 = R ; mM1 r

Next, the vector r = (x y z ) is transformed to spherical polar coordinates according to

x = r sin  cos  y = r sin  sin  z = r cos  and the derivative of the potential with respect to r can be easily worked out using the chain rule: @U = @U  @ r @r @ r @r @U = @U m2 ; @U m1 @ r @ r1 M @ r2 M = ; M1 m2 F1 ; m1 F2 ] @ r =  x y z   n^  (sin  cos  sin  sin  cos ) @r r r r @U = ; 1 m F ; m F ]  n^ @r M 2 1 1 2 The Jacobian is clearly

J = r2 sin  so that Thus, the free energy derivative is

@ ln J = 2 @r r 5

dA = h; 1 m F ; m F ]  n^ ; 2kT icond dr0 M 2 1 1 2 jr1 ; r2 j r 0

which is expressed completely in terms of Cartesian quantities and can be calculated straightforwardly. Once dA=dr0 is known, it can be integrated to obtain the full free energy pro le A(r0 ). This is another use of thermodynamic integration. It is always interesting to see what A(r) looks like compared to the bare potential. Suppose the dissociation reaction is governed by a pair potential describing the interaction of the dissociating molecule with a solvent:

U (r1 ::: rN ) = u0 (jr1 ; r2 j) +

X

j 6=1
2

~u(jr1 ; rj j) + u~(jr2 ; rj j)] +

X

i
j
i6=j
i
j 6=1
2

v(jri ; rj j)

The potential u0 (r) might look like:

u 0(r)

r FIG. 2.

If, at a given temperature T , the solvent assists in the dissociation process, then we might expect A(r) to have a lower dissociation threshold and perhaps a slightly longer e ective minimum bond length:

6

A(r)

u 0(r)

r FIG. 3.

If, on the other hand, at a given temperature, T , the solvent hinders dissociation, by causing the molecule to bury itself in a cavity, for example, then we might expect A(r) to appear as follows:

7

A(r)

u 0(r)

r

FIG. 4.

with a higher dissociation threshold energy and slightly shorter e ective minimum bond length. Such curves will, of course, be temperature dependent and depend on the speci c nature of the interactions with the solvent.

8

G25.2651: Statistical Mechanics Notes for Lecture 12

I. THE FUNDAMENTAL POSTULATES OF QUANTUM MECHANICS The fundamental postulates of quantum mechanics concern the following questions: 1. How is the physical state of a system described? 2. How are physical observables represented? 3. What are the results of measurements on quantum mechanical systems? 4. How does the physical state of a system evolve in time? 5. The uncertainty principle.

A. The physical state of a quantum system The physical state of a quantum system is represented by a vector denoted (t) which is a column vector, whose components are probability amplitudes for di erent states in which the system might be found if a measurement were made on it. A probability amplitude is a complex number, the square modulus of which gives the corresponding probability j

i

P

P =

2

j

j

The number of components of (t) is equal to the number of possible states in which the system might observed. The space that contains (t) is called a Hilbert space . The dimension of is also equal to the number of states in which the system might be observed. It could be nite or innite (countable or not). (t) must be a unit vector. This means that the inner product: (t) (t) = 1 In the above, if the vector (t) , known as a Dirac \ket" vector, is given by the column j

j

j

i

i

H

H

i

h

j

i

j

i

0
1 1 B
2 CC (t) = B B@ CA

j

i

then the vector (t) , known as a Dirac \bra" vector, is given by ) (t) = (
1
2 so that the inner product becomes h

j

h

j

(t) (t) =

h

j

i

X i


i 2 = 1

j

j

We can understand the meaning of this by noting that
i , the components of the state vector, are probability amplitudes, and
i 2 are the corresponding probabilities. The above condition then implies that the sum of all the probabilities of being in the various possible states is 1, which we know must be true for probabilities. j

j

1

B. Physical Observables Physical observables are represented by linear, hermitian operators that act on the vectors of the Hilbert space. If

A is such an operator, and  is an arbitrary vector in the Hilbert space, then A might act on  to produce a vector 0 , which we express as A  = 0 Since  is representable as a column vector, A is representable as a matrix with components  A11 A12 A13 ! A = A21 A22 A23 j

j

i

j

i

i

j

j

i

j

i

i

The condition that A must be hermitian means that

Ay = A or

Aij = Aji

C. Measurement The result of a measurement of the observable A must yield one of the eigenvalues of A. Thus, we see why A is required to be a hermitian operator: Hermitian operators have real eigenvalues. If we denote the set of eigenvalues of A by ai , then each of the eigenvalues ai satises an eigenvalue equation f

g

A ai = ai ai where ai is the corresponding eigenvector. Since the operator A is hermitian and ai is therefore real, we have also j

j

i

j

i

i

the left eigenvalue equation

ai A = ai ai The probability amplitude that a measurement of A will yield the eigenvalue ai is obtained by taking the inner product of the corresponding eigenvector ai with the state vector (t) , ai (t) . Thus, the probability that the value ai is obtained is given by Pa = ai (t) 2 h

j

j

h

j

i

j

jh

i

j

i

h

j

i

ij

Another useful and important property of hermitian operators is that their eigenvectors form a complete orthonormal basis of the Hilbert space, when the eigenvalue spectrum is non-degenerate. That is, they are linearly independent, span the space, satisfy the orthonormality condition h

ai aj = ij j

i

and thus any arbitrary vector  can be expanded as a linear combination of these vectors: j

i

 =

j

i

X i

ci ai j

i

By multiplying both sides of this equation by aj and using the orthonormality condition, it can be seen that the expansion coecients are h

j

ci = ai  h

The eigenvectors also satisfy a closure relation: 2

j

i

X

I=

i

j

ai ai ih

j

where I is the identity operator. Averaging over many individual measurements of A gives rise to an average value or expectation value for the observable A, which we denote A and is given by h

i

A = (t) A (t) That this is true can be seen by expanding the state vector (t) in the eigenvectors of A: h

i

h

j

j

i

j

(t) =

j

i

X i

i

i (t)jai i

where i are the amplitudes for obtaining the eigenvalue ai upon measuring A, i.e., expansion into the expectation value expression gives

A (t) =

h

i

= =

X i
j

X i
j

X i

i

= ai (t) . Introducing this h

j

i

i (t) j (t)hai jAjai i i (t) j ai (t)ij

ai i (t) 2 j

j

The interpretation of the above result is that the expectation value of A is the sum over possible outcomes of a measurement of A weighted by the probability that each result is obtained. Since i 2 = ai (t) 2 is this probability, the equivalence of the expressions can be seen. Two observables are said to be compatible if AB = BA. If this is true, then the observables can be diagonalized simultaneously to yield the same set of eigenvectors. To see this, consider the action of BA on an eigenvector ai of A. BA ai = ai B ai . But if this must equal AB ai , then the only way this can be true is if B ai yields a vector proportional to ai which means it must also be an eigenvector of B . The condition AB = BA can be expressed as j

j

jh

j

ij

j

j

i

j

j

i

j

i

j

i

i

i

AB BA = 0 A B ] = 0 where, in the second line, the quantity A B ] AB BA is know as the commutator between A and B . If A B ] = 0, then A and B are said to commute with each other. That they can be simultaneously diagonalized implies that one can simultaneously predict the observables A and B with the same measurement. As we have seen, classical observables are functions of position x and momentum p (for a one-particle system). Quantum analogs of classical observables are, therefore, functions of the operators X and P corresponding to position and momentum. Like other observables X and P are linear hermitian operators. The corresponding eigenvalues x and p and eigenvectors x and p satisfy the equations X x =xx P p =pp which, in general, could constitute a continuous spectrum of eigenvalues and eigenvectors. The operators X and P ;



j

i

;

j i

j

i

j

j i

i

j i

are not compatible. In accordance with the Heisenberg uncertainty principle (to be discussed below), the commutator between X and P is given by X P ] = ihI

and that the inner product between eigenvectors of X and P is x p = 1 eipx=h 2h Since, in general, the eigenvalues and eigenvectors of X and P form a continuous spectrum, we write the orthonormality and closure relations for the eigenvectors as: h

j i

p

3

x x0 = Z(x x0 )  = dx x x 

h

j

j

i

p p0 = Z(p p0 )  = dp p p 

;

i

Z

j

ih

j

I = dx x x j

ih

h j

i

i

j

;

i

Z

j ih j

i

I = dp p p

j

j ih j

The probability that a measurement of the operator X will yield an eigenvalue x in a region dx about some point is

P (x t)dx = x (t) 2 dx The object x (t) is best represented by a continuous function (x t) often referred to as the wave function. It is a representation of the inner product between eigenvectors of X with the state vector. To determine the action of the operator X on the state vector in the basis set of the operator X , we compute x X (t) = x(x t) The action of P on the state vector in the basis of the X operator is consequential of the incompatibility of X and P jh

h

j

j

ij

i

h

j

j

i

and is given by

@ (x t) x P (t) = hi @x

h

j

j

i

Thus, in general, for any observable A(X P ), its action on the state vector represented in the basis of X is





@ (x t) x A(X P ) (t) = A x hi @x

h

j

j

i

D. Time evolution of the state vector The time evolution of the state vector is prescribed by the Schrodinger equation

@ (t) = H (t) ih @t j

i

j

i

where H is the Hamiltonian operator. This equation can be solved, in principle, yielding (t) = e;iHt=h (0)

j

i

j

i

where (0) is the initial state vector. The operator j

i

U (t) = e;iHth is the time evolution operator or quantum propagator. Let us introduce the eigenvalues and eigenvectors of the Hamiltonian H that satisfy

H Ei = Ei Ei j

i

j

i

The eigenvectors for an orthonormal basis on the Hilbert space and therefore, the state vector can be expanded in them according to (t) =

j

i

X i

ci (t) Ei j

i

where, of course, ci (t) = Ei (t) , which is the amplitude for obtaining the value Ei at time t if a measurement of H is performed. Using this expansion, it is straightforward to show that the time evolution of the state vector can be written as an expansion: h

j

i

4

(t) = e;iHth (0) X = e;iHt=h Ei Ei (0)

j

i

j

=

X i

i

j

ih

j

i

i ;iEi t=h  e jE

i ihEi j(0)i

Thus, we need to compute all the initial amplitudes for obtaining the di erent eigenvalues Ei of H , apply to each the factor exp( iEi t=h) Ei and then sum over all the eigenstates to obtain the state vector at time t. If the Hamiltonian is obtained from a classical Hamiltonian H (x p), then, using the formula from the previous section for the action of an arbitrary operator A(X P ) on the state vector in the coordinate basis, we can recast the Schrodiner equation as a partial di erential equation. By multiplying both sides of the Schrodinger equation by x , we obtain ;

j

i

h

j

@ x (t) x H (X P ) (t) = ih @t  h @  @ (x t) H x i @x (x t) = ih @t h

j

j

i

h

j

i

If the classical Hamiltonian takes the form 2 H (x p) = 2pm + U (x)

then the Schrodinger equation becomes

 h2 @ 2  @ (x t) + U ( x ) ( x t ) = i h  2 2m @x @t ;

which is known as the Schrodinger wave equation or the time-dependent Schrodinger equation. In a similar manner, the eigenvalue equation for H can be expressed as a di erential equation by projecting it into the X basis:

 h @ xH Ei = Ei x Ei H x i @x
i (x) = Ei
i (x)   h2 @ 2 h

;

j

j

i

h

j

i

2m @x2 + U (x)
i (x) = Ei
i (x)

where
i (x) = x Ei is an eigenfunction of the Hamiltonian. h

j

i

E. The Heisenberg uncertainty principle Because the operators X and P are not compatible, X P ] = 0, there is no measurement that can precisely determine both X and P simultaneously. Hence, there must be an uncertainty relation between them that species how uncertain we are about one quantity given a denite precision in the measurement of the other. Presumably, if one can be determined with innite precision, then there will be an innite uncertainty in the other. Recall that we had dened the uncertainty in a quantity by 6

A = Thus, for X and P , we have

p

A2

h

p

x = X 2 p p = P 2

A2

i ;h

i

X2 P2

h

i ;h

h

i; h

i

i

These quantities can be expressed explicitly in terms of the wave function (x t) using the fact that 5

Z

X = (t) X (t) = dx (t) x x X (t) =

h

i

h

j

j

i

h

j

ih

j

and

X 2 = (t) X 2 (t) =

h

i

h

Similarly,

P = (t) P (t) =

h

i

h

j

j

j

Z

j

h

j

i

h

j

j

i

Z

ih

j

Z

dx (x t)x(x t)

 (x t)x2 (x t)

dx (t) x x P (t) =

i

P 2 = (t) P 2 (t) = i

Z

i

and h

j

j

Z

i

dx (x t)



@ (x t) dx (x t) hi @x



@ (x t) h 2 @x 2 2

;

Then, the Heisenberg uncertainty principle states that xp

>



h

which essentially states that the greater certainty with which a measurement of X or P can be made, the greater will be the uncertainty in the other.

F. The Heisenberg picture In all of the above, notice that we have formulated the postulates of quantum mechanics such that the state vector (t) evolves in time but the operators corresponding to observables are taken to be stationary. This formulation of quantum mechanics is known as the Schr
odinger picture. However, there is another, completely equivalent, picture in which the state vector remains stationary and the operators evolve in time. This picture is known as the Heisenberg picture. This particular picture will prove particularly useful to us when we consider quantum time correlation functions. The Heisenberg picture species an evolution equation for any operator A, known as the Heisenberg equation. It states that the time evolution of A is given by dA = 1 A H ] dt ih While this evolution equation must be regarded as a postulate, it has a very immediate connection to classical mechanics. Recall that any function of the phase space variables A(x p) evolves according to j

i

dA = A H dt f

g

where ::: ::: is the Poisson bracket. The suggestion is that in the classical limit (h small), the commutator goes over to the Poisson bracket. The Heisenberg equation can be solved in principle giving f

g

A(t) = eiHt=h Ae;iHt=h = U y (t)AU (t) where A is the corresponding operator in the Schrodinger picture. Thus, the expectation value of A at any time t is computed from

A(t) =  A(t) 

h

i

h

j

j

i

where  is the stationary state vector. Let's look at the Heisenberg equations for the operators X and P . If H is given by j

i

2 H = 2Pm + U (X )

6

then Heisenberg's equations for X and P are

dX = 1 X H ] = P dt ih m dP = 1 P H ] = @U dt ih @X Thus, Heisenberg's equations for the operators X and P are just Hamilton's equations cast in operator form. Despite ;

their innocent appearance, the solution of such equations, even for a one-particle system, is highly nontrivial and has been the subject of a considerable amount of research in physics and mathematics. Note that any operator that satises A(t) H ] = 0 will not evolve in time. Such operators are known as constants of the motion. The Heisenberg picture shows explicitly that such operators do not evolve in time. However, there is an analog with the Schrodinger picture: Operators that commute with the Hamiltonian will have associated probabilities for obtaining di erent eigenvalues that do not evolve in time. For example, consider the Hamiltonian, itself, which it trivially a constant of the motion. According to the evolution equation of the state vector in the Schrodinger picture, (t) =

j

i

X i

e;iE t=h Ei Ei (0) i

j

ih

j

i

the amplitude for obtaining an energy eigenvalue Ej at time t upon measuring H will be

Ej (t) =

h

j

i

=

X X i i

e;iE t=h Ej Ei Ei (0) i

h

j

ih

j

i

e;iE t=h ij Ei (0) i

h

= e;iE t=h Ej (0) j

h

j

j

i

i

Thus, the squared modulus of both sides yields the probability for obtaining Ej , which is jh

Ej (t) 2 = Ej (0) j

ij

jh

j

ij

2

Thus, the probabilities do not evolve in time. Since any operator that commutes with H can be diagonalized simultaneously with H and will have the same set of eigenvectors, the above arguments will hold for any such operator.

7

G25.2651: Statistical Mechanics Notes for Lecture 13

I. PRINCIPLES OF QUANTUM STATISTICAL MECHANICS The problem of quantum statistical mechanics is the quantum mechanical treatment of an N -particle system. Suppose the corresponding N -particle classical system has Cartesian coordinates

q1 ::: q3N and momenta

p1 ::: p3N and Hamiltonian

H=

3N X

p2i + U (q ::: q ) 1 3N i=1 2mi

Then, as we have seen, the quantum mechanical problem consists of determining the state vector (t) from the Schr odinger equation j

i

@ (t) H (t) = i h @t Denoting the corresponding operators, Q1 ::: Q3N and P1 ::: P3N , we note that these operators satisfy the commuj

i

j

i

tation relations:

Qi Qj ] = Pi Pj ] = 0 Qi Pj ] = i hIij and the many-particle coordinate eigenstate q1 :::q3N is a tensor product of the individual eigenstate q1 ::: q3N : q1 :::q3N = q1 q3N j

i

j

j

i

j

i

j

i

j

i

i

The Schr odinger equation can be cast as a partial dierential equation by multiplying both sides by q1 :::q3N : h

@ q :::q (t) q1 :::q3N H (t) = i h @t 1 3N # 3N X

h2 @ 2 + U (q ::: q ) (q ::: q t) = i h @ (q ::: q t) 1 3N 1 3N 3N 2 @t 1 i=1 2mi @qi j

h

" ;

j

j

i

h

j

i

where the many-particle wave function is (q1 :::: q3N t) = q1 :::q3N (t) . Similarly, the expectation value of an operator A = A(Q1 ::: Q3N P1 ::: P3N ) is given by h

Z

A = dq1

h

i



j

i



dq3N  (q1 ::: q3N )A q1 ::: q3N hi @q@ ::: hi @q@ (q1 ::: q3N ) 1 3N

1

A. The density matrix and density operator In general, the many-body wave function (q1 ::: q3N t) is far too large to calculate for a macroscopic system. If we wish to represent it on a grid with just 10 points along each coordinate direction, then for N = 1023, we would need 23 1010 total points, which is clearly enormous. We wish, therefore, to use the concept of ensembles in order to express expectation values of observables A without requiring direct computation of the wavefunction. Let us, therefore, introduce an ensemble of systems, with a total of Z members, and each having a state vector ( ) ,  = 1 ::: Z . Furthermore, introduce an orthonormal set of vectors k ( k j = ij ) and expand the state vector for each member of the ensemble in this orthonormal set: h

j

j

i

h

j

i

i

i

(

j

)

i

=

X

k

Ck( ) k j

i

The expectation value of an observable, averaged over the ensemble of systems is given by the average of the expectation value of the observable computed with respect to each member of the ensemble:

A = Z1

h

Z X

( ) A (

h

i

=1

j

)

j

Substituting in the expansion for ( ) , we obtain X A = Z1 Ck( ) Cl( ) k A l j

i

i

h

i

=

k
l  Z X 1X

Z

k
l

=1

Let us dene a matrix

lk = and a similar matrix

h

j

j

!

Cl Ck ( )

Z X =1

i

( )

Cl( ) Ck(

h

k A l j

j

i

)

Z X ~lk = Z1 Cl( ) Ck( =1

)

Thus, lk is a sum over the ensemble members of a product of expansion coecients, while ~lk is an average over the ensemble of this product. Also, let Akl = k A l . Then, the expectation value can be written as follows: X X A = 1  A = 1 (A) = 1 Tr(A) = Tr(~A) h

j

j

i

kk Z Z k where  and A represent the matrices with elements lk and Akl in the basis of vectors k . The matrix lk is h

i

Z k
l lk kl

fj

ig

known as the density matrix. There is an abstract operator corresponding to this matrix that is basis-independent. It can be seen that the operator

=

Z X

(

j

=1

)

(

ih

)

j

and similarly Z X ~ = Z1 ( =1 j

2

)

ih

(

)

j

have matrix elements lk when evaluated in the basis set of vectors k . fj

l  k =

h

j j

Z X

i

l (

h

=1

)

j

Z X

( ) k =

ih

j

ig

i

=1

Cl( ) Ck(

)

= lk

Note that  is a hermitian operator

y =  so that its eigenvectors form a complete orthonormal set of vectors that span the Hilbert space. If wk and wk represent the eigenvalues and eigenvectors of the operator ~, respectively, then several important properties they must satisfy can be deduced. Firstly, let A be the identity operator I . Then, since I = 1, it follows that X 1 = 1 Tr() = Tr(~) = w j

i

j 

Pk .

h i

Z

k

k

Thus, the eigenvalues of ~ must sum to 1. Next, let A be a projector onto an eigenstate of ~, A = wk wk Then Pk = Tr(~ wk wk ) But, since ~ can be expressed as j

h

i

j

X

~ =

ih

j

wk wk wk j

k

ih

ih

j

and the trace, being basis set independent, can be therefore be evaluated in the basis of eigenvectors of ~, the expectation value becomes

Pk =

h

i

=

X

wj

h

j X

i
j = wk

X

j

wi wi wi wk wk wj j

i

ih

j

ih

j

i

wi ij ik kj

However,

Pk = Z1

h

Z X

i

( ) wk wk (

h

=1

j

ih

Z X ( ) wk = Z1 jh

j

=1

j

ij

2



)

i

0

P

Thus, wk 0. Combining these two results, we see that, since k wk = 1 and wk 0, 0 wk 1, so that wk satisfy the properties of probabilities. With this in mind, we can develop a physical meaning for the density matrix. Let us now consider the expectation value of a projector ai ai ai onto one of the eigenstates of the operator A. The expectation value of this operator is given by 

j

ih

i





j  P

Z X 1 ( ai = Z

hP



h

=1

)

jP

ai j

( )

i

Z X 1 =Z ( ) ai ai ( h

j

ih

=1

j

)

i

Z X 1 =Z ai ( jh

j

) 2 ij

=1

But ai ( ) 2 Pa(i ) is just probability that a measurement of the operator A in the th member of the ensemble will yield the result ai . Thus, jh

j

ij



P 1X ( ) = ai Z Pai

hP

i

=1

3

or the expectation value of ai is just the ensemble averaged probability of obtaining the value ai in each member of the ensemble. However, note that the expectation value of ai can also be written as P

P

X

Pai ) = Tr( ai i = Tr(~

hP

= =

X

k
l X k

k

j

wk kl wk ai ai wl h

wk ai wk jh

j

X

wk wk wk ai ai ) =

ih

ih

j

ih

j

wl wk wk wk ai ai wl

h

k
l

j

j

ih

j

ih

j

i

i

2

ij

Equating the two expressions gives Z X 1X ( ) P = wk ai wk a i Z h

i

jh

k

=1

j

2

ij

The interpretation of this equation is that the ensemble averaged probability of obtaining the value ai if A is measured is equal to the probability of obtaining the value ai in a measurement of A if the state of the system under consideration were the state wk , weighted by the average probability wk that the system in the ensemble is in that state. Therefore, the density operator  (or ~) plays the same role in quantum systems that the phase space distribution function f (;) plays in classical systems. j

i

B. Time evolution of the density operator The time evolution of the operator  can be predicted directly from the Schr odinger equation. Since (t) is given by

(t) =

Z X j

=1

( ) (t) ( ) (t) ih

j

the time derivative is given by Z @ = X @t =1



@ () @t  (t) j

 i

h

j

Z h X = i1 h H ( ) (t) ( ) (t) j

i



@ ( ) (t)  (t) +  (t) @t ( )

h

( )

j

i

( ) (t)

j ;j

=1

h

 h

 j

( ) (t) H

h

i

j

= i1h (H H ) = i1 h H ] ;

@ = 1 H ] @t i h

where the second line follows from the fact that the Schr odinger equation for the bra state vector ( ) (t) is h

;

@ ( ) (t) = ( ) (t) H i h @t h

j

h

j

j

Note that the equation of motion for (t) diers from the usual Heisenberg equation by a minus sign! Since (t) is constructed from state vectors, it is not an observable like other hermitian operators, so there is no reason to expect that its time evolution will be the same. The general solution to its equation of motion is (t) = e;iHt=h (0)eiHt=h = U (t)(0)U y (t) The equation of motion for (t) can be cast into a quantum Liouville equation by introducing an operator 4

iL = i1 h ::: H ]

In term of iL, it can be seen that (t) satises

@ = iL @t (t) = e;iLt (0) ;

What kind of operator is iL? It acts on an operator and returns another operator. Thus, it is not an operator in the ordinary sense, but is known as a superoperator or tetradic operator (see S. Mukamel, Principles of Nonlinear Optical Spectroscopy, Oxford University Press, New York (1995)). Dening the evolution equation for  this way, we have a perfect analogy between the density matrix and the state vector. The two equations of motion are

@ (t) = i H (t) @t

h @ (t) = iL(t) @t j

i

;

j

i

;

We also have an analogy with the evolution of the classical phase space distribution f (; t), which satises

@f = iLf @t ;

with iL = ::: H being the classical Liouville operator. Again, we see that the limit of a commutator is the classical Poisson bracket. f

g

C. The quantum equilibrium ensembles At equilibrium, the density operator does not evolve in time thus, @=@t = 0. Thus, from the equation of motion, if this holds, then H ] = 0, and (t) is a constant of the motion. This means that it can be simultaneously diagonalized with the Hamiltonian and can be expressed as a pure function of the Hamiltonian

 = f (H ) Therefore, the eigenstates of , the vectors, we called wk are the eigenvectors Ei of the Hamiltonian, and we can write H and  as j

H= =

i

X

j

Ei Ei Ei j

i X

ih

j

f (Ei ) Ei Ei j

i

i

ih

j

The choice of the function f determines the ensemble. 1. The microcanonical ensemble

Although we will have practically no occasion to use the quantum microcanonical ensemble (we relied on it more heavily in classical statistical mechanics), for completeness, we dene it here. The function f , for this ensemble, is

f (Ei )E = (Ei (E + E )) (Ei E ) where (x) is the Heaviside step function. This says that f (Ei )E is 1 if E < Ei < (E + E ) and 0 otherwise. The partition function for the ensemble is Tr(), since the trace of  is the number of members in the ensemble: ;

(N V E ) = Tr() =

X

i

;

;

(Ei (E + E )) (Ei E )] ;

5

;

;

The thermodynamics that are derived from this partition function are exactly the same as they are in the classical case:

S (N V E ) = k ln (N V E )  1 @ ln  T = k @E ; ;

N
V

etc. 2. The canonical ensemble

In analogy to the classical canonical ensemble, the quantum canonical ensemble is dened by

 = e;H f (Ei ) = e;Ei Thus, the quantum canonical partition function is given by

Q(N V T ) = Tr(e;H ) =

X

i

e;Ei

and the thermodynamics derived from it are the same as in the classical case: A(N V T ) = 1 ln Q(N V T )

@ ln Q(N V T ) E (N V T ) = @

@ ln Q(N V T ) P (N V T ) = 1 @V ;

;

etc. Note that the expectation value of an observable A is A = 1 Tr(Ae;H ) h

i

Q

Evaluating the trace in the basis of eigenvectors of H (and of ), we obtain X X A = Q1 Ei Ae;H Ei = Q1 e;Ei Ei A Ei h

i

i

h

j

j

i

h

i

j

j

i

The quantum canonical ensemble will be particularly useful to us in many things to come. 3. Isothermal-isobaric and grand canonical ensembles

Also useful are the isothermal-isobaric and grand canonical ensembles, which are dened just as they are for the classical cases: (N P T ) = Z

( V T ) =

Z

1

0 1 X

N =0

dV e;PV Q(N V T ) = eN Q(N

V T) =

6

1 X

N =0

Z

1 0

dV Tr(e;(H +PV ) )

Tr(e;(H ;N ) )

D. A simple example { the quantum harmonic oscillator As a simple example of the trace procedure, let us consider the quantum harmonic oscillator. The Hamiltonian is given by 2 H = 2Pm + 12 m!2 X 2 and the eigenvalues of H are   1 En = n + 2 h! n = 0 1 2 ::: Thus, the canonical partition function is

Q( ) =

1 X

n=0

e;(n+1=2)h! = e;h !=2

1 X ; ;h ! n e n=0

This is a geometric series, which can be summed analytically, giving ;h  !=2 Q( ) = 1e e;h ! = eh !=2 1e;h !=2 = 21 csch( h!=2) The thermodynamics derived from it as as follows: 1. Free energy: The free energy is ;  A = 1 ln Q( ) = h 2! + 1 ln 1 e;h ! ;

;

;

;

2. Average energy: The average energy E = H is h

i





@ ln Q( ) = h! + h!e;h! = 1 + n h! E = @

2 1 e;h ! 2 ;

3. Entropy The entropy is given by

;

h

i

;h ! ;  S = k ln Q( ) + ET = k ln 1 e;h ! + hT! 1 e e;h ! ;

;

;

Now consider the classical expressions. Recall that the partition function is given by Z ; Q( ) = h1 dpdxe

Thus, the classical free energy is



p2 1 2m + 2

m!2 x2



 1=2   2 1=2 = 2 = 1 = h1 2 m

m!2

!h h!

Acl = 1 ln( h!)

In the classical limit, we may take h to be small. Thus, the quantum expression for A becomes, approximately, in this limit: h ! + 1 ln( h!) AQ 2

and we see that

h! AQ Acl 2 The residual h !=2 (which truly vanishes when h 0) is known as the quantum zero point energy. It is a pure quantum eect and is present because the lowest energy quantum mechanically is not E = 0 but the ground state energy E = h !=2. ;!

;

;!

!

7

G25.2651: Statistical Mechanics Notes for Lecture 14

I. DERIVATION OF THE DISCRETIZED PATH INTEGRAL We begin our discussion of the Feynman path integral with the canonical ensemble. The expressions for the partition function and expectation value of an observable A are, respectively Q(N V T ) = Tr(e; H ) hAi = Q1 Tr(Ae; H ) It is clear that we need to be able to evaluate traces of the type appearing in these expressions. We have already derived expressions for these in the basis of eigenvectors of H . However, since the trace is basis independent, let us explore carrying out these traces in the coordinate basis. We will begin with the partition function and treat expectation values later. Consider the ensemble of a one-particle system. The partition function evaluated as a trace in the coordinate basis is

Q(
) =

Z

dxhxje; H jxi

We see that the trace involves the diagonal density matrix element hxje; H jxi. Let us solve the more general problem of any density matrix element hxje; H jx0 i. If the Hamiltonian takes the form

2 H = 2Pm + U (X ) K + U then we cannot evaluate the operator exp(;
H ) explicitly because the operators for kinetic (T ) and potential energies (U ) do not commute with each other, being, respectively, functions of momentum and position, i.e., K U ] 6= 0 In this instance, we will make use of the Trotter theorem, which states that given two operators A and B , such that A B ] 6= 0, then for any number , h
B=2P
A=P
B=2P iP e
(A+B) = Plim e e e !1

Thus, for the Boltzmann operator,

h ; U=2P ; K=P ; U=2P iP e e; (K +U ) = Plim e e !1

and the partition function becomes

Z h ; U=2P ; K=P ; U=2P iP Q(
) = Plim dx h x j e e e jxi !1

De ne the operator in brackets to be : Then,

= e; U=2P e; K=P e; U=2P

Z Q(
) = Plim dxhxj P jxi !1 1

In between each of the P factors of , the coordinate space identity operator

I=

Z

dxjxihxj

is inserted. Since there are P factors, there will be P ; 1 such insertions. the integration variables will be labeled x2 ::: xP . Thus, the expression for the matrix element becomes

Z

hxj jx0 i = dx2    dxP hxj jx2 ihx2 j jx3 ihx3 j    jxP ihxP j jx0 i =

Z

dx2    dxP

P Y i=1

hxi j jxi+1 ijx1 =xxP +1 =x

0

The next step clearly involves evaluating the matrix elementx

hxi j jxi+1 i = hxi je;

U (X )=2P e; P 2 =2mP e; U (X )=2P jxi+1 i

Note that in the above expression, the operators involving the potential U (X ) act on their eigenvectors and can thus be replaced by the corresponding eigenvalues:

hxi j jxi+1 i = e;

(U (xi )+U (xi+1 )=2

hxi je;

P 2 =2mP jxi+1 i

In order to evaluate the remaining matrix element, we introduce the momentum space identity operator

I=

Z

dpjpihpj

Letting K = P 2 =2m, the matrix remaining matrix element becomes

hxi je; K=P jxi+1 i = =

Z Z

dphxi jpihpje; P 2 =2mP jxi+1 i dphxi jpihpjxi+1 ie; p2 =2mP

Using the fact that

hxjpi = p 1 eipx=h 2h

it follows that

hxi je; K=P jxi+1 i = 1

Z

ip(xi ;xi+1 )=h ; p2 =2mP

e 2h dpe The remaining integral over p can be performed by completing the square, leading to the result



mP 1=2 exp ; mP (x ; x )2 2
h2 2
h2 i+1 i Collecting the pieces together, and introducing the P ! 1 limit, we have for the density matrix " P
#
P=2 Z X mP mP
 2 hxje; H jx0 i = Plim dx2    dxP exp ; 2 (xi+1 ; xi ) + 2P (U (xi ) + U (xi+1 )) x =xx =x !1 2
h2 i=1 2
h 1 P +1

hxi je;

K=P jxi+1 i =

The partition function is obtained by setting x = x0 , which is equivalent to setting x1 = xP +1 and integrating over x, or equivalently x1 . Thus, the expression for Q(
) becomes

" P
#
P=2 Z X 1 2 mP 1 2 Q(
) = Plim m!P (xi+1 ; xi ) + P U (xi )  dx1    dxP exp ;
!1 2
h2 2 i=1 xP +1 =x1 2

0

where we have introduced a \frequency"

p

!P =
Ph When expressed in this way, the partition function, for a nite value of P , is isomorphic to a classical con guration integral for a P -particle system, that is a cyclic chain of particles, with harmonic nearest neighbor interactions and interacting with an external potential U (x)=P . That is, the partition function becomes

Q(
) 

Z

where

Ue (x1 ::: xP ) =

dx1    dxP e; Ueff (x1 :::xP ) P 1 X i=1

1 2 2 2 m!P (xi+1 ; xi ) + P U (xi )



Thus, for nite (if large) P the partition function in the discretized path integral representation can be treated as any ordinary classical con guration integral. Consider the integrand of Q(
) in the limit that all P points on the cyclic chain are at the same location P x. Then the harmonic nearest neighbor coupling (which is due to the quantum kinetic energy) vanishes and (1=P ) Pi=1 U (xi ) ! U (x), and the integrand becomes

e; U (x) which is just the true classical canonical position space distribution function. Therefore, the greater the spatial spread in the cyclic chain, the more \quantum" the system is, since this indicates a greater contribution from the quantum kinetic energy. The spatially localized it is, the more the system behaves like a classical system. It remains formally to take the limit that P ! 1. There we will see an elegant formulation for the density matrix and partition function emerges.

3

G25.2651: Statistical Mechanics Notes for Lecture 15

I. THE FUNCTIONAL INTEGRAL REPRESENTATION OF THE PATH INTEGRAL A. The continuous limit In taking the limit P so that P

! 1, it will prove useful to de
ne a parameter " =
Ph

! 1 implies " ! 0. In terms of ", the partition function becomes m
Q(
) = !1 lim !0 2" h

P=

P

2

"

Z

"

 

X m x +1 ; x dx1  dx exp ; h" 2 " =1 P

i

i

2

P

i

!# + U (x )  P +1 = 1 i

x

x

We can think of the points x1  ::: x as speci
c points of a continuous functions x( ), where x = x( = (k ; 1)") such that x(0) = x( = P") = x( =
h ): P

k

x

x4 x2

x P+1

x1

0

2 ε 4ε

Pε FIG. 1.

1

τ

Note that lim !0

 x +1 ; x  k

"

"

and that the limit

k

 x(k") ; x((k ; 1)")  dx = lim = !0

" 

" X m x +1 ; x lim !1 !0 h =1 2 " P

P

"

"

"

i

i

2

d

#

+ U (x ) i

i

is just a Riemann sum representation of the continuous integral

"  2

1 j Z  d m dx h 0 2 d h

Finally, the measure P

lim !0 !1 "

#

+ U (x( ))

 m  2 dx1  dx 2"h2 P=

P

represents an integral overa all values that the function x( ) can take on between  = 0 and  =
h such that x(0) = x(
h). We write this symbolically as Dx(). Therefore, the P ! 1 limit of the partition function can be written as

Q(
) = =

Z I

Z

"

Z  m
1 dx D x() exp ; h d 2 x_ 2 + U (x( )) (0)= 0 " Z 
# 1 m 2 Dx() exp ; h 0 d 2 x_ + U (x( )) (  )=x

x h

x

h

#

x

h

The above expression in known as a functional integral. It says that we must integrate over all functions (i.e., all values that an arbitrary function x( ) may take on) between the values  = 0 and  =
h. It must really be viewed as the limit of the discretized integral introduced in the last lecture. The integral is also referred to as a path integral because it implies an integration over all paths that a particle might take between  = 0 and  =
h such that x(0) = x(
h , where the paths are paramterized by the variable  (which is not time!). The second line in the above expression, which is equivalent to the
rst, indicates that the integration is taken over all paths that begin and end at the same point, plus a
nal integration over that point. The above expression makes it clear how to represent a general density matrix element hxj exp(;
H )jx0 i:

hxje; jx0 i = H

Z

(  )=x0

x h

(0)=x

x

"

Z  m
1 Dx() exp ; h 0 d 2 x_ 2 + U (x( )) h

#

which indicates that we must integrate over all functions x( ) that begin at x at  = 0 and end at x0 at  =
h:

2

x

x’

x

βh

0

τ

FIG. 2.

Similarly, diagonal elements of the density matrix, used to compute the partition function, are calculated by integrating over all periodic paths that satisfy x(0) = x(
h) = x:

3

x

x’

x τ

βh

0 FIG. 3.

Note that if we let
= it=h, then the density matrix becomes

(x x0  it=h) = hxje;



iH t=h

jx0 i = U (x x0  t)

which are the coordinate space matrix elements of the quantum time evolution operator. If we make a change of variables  = is in the path integral expression for the density matrix, we
nd that the quantum propagator can also be expressed as a path integral:

U (x x0  t) = hxje;



iHt=h

jx0 i =

Z

( )=x0

x t

Dx() exp

i Z

t


ds m2 x_ (s) ; U (x(s))



h 0 Such a variable transformation is known as a Wick rotation. This nomenclature comes about by viewing time as a complex quantity. The propagator involves real time, while the density matrix involves a transformation t = ;i
h to the imaginary time axis. It is because of this that the density matrix is sometimes referred to as an imaginary time path integral. (0)=x

x

B. Dominant paths in the propagator and density matrix Let us
rst consider the real time quantum propagator. The quantity appearing in the exponential is an integral of 1 mx_ 2 ; U (x)  L(x x_ ) 2 which is known as the Lagrangian in classical mechanics. We can ask, which paths will contribute most to the integral

Z

t

0

i Z h ds m2 x_ 2 (s) ; U (x(s)) = dsL(x(s) x_ (s)) = S x] t

0

4

known as the action integral. Since we are integrating over a complex exponential exp(iS=h), which is oscillatory, those paths away from which small deviations cause no change in S (at least to
rst order) will give rise to the dominant contribution. Other paths that cause exp(iS=h) to oscillate rapidly as we change from one path to another will give rise to phase decoherence and will ultimately cancel when integrated over. Thus, we consider two paths x(s) and a nearby one constructed from it x(s) + x(s) and demand that the change in S between these paths be 0 S x + x] ; S x] = 0 Note that, since x(0) = x and x(t) = x0 , x(0) = x(t) = 0, since all paths must begin at x and end at x0 . The change in S is

S = S x + x] ; S x] =

Z

0

t

dsL(x + x x_ + x_ ) ;

Z

t

0

dsL(x x_ )

Expanding the
rst term to
rst order in x, we obtain

@L x ; Z L(x x_ ) = Z ds @L x_ + @L x

ds L(x x_ ) + @L  x _ + @ x_ @x @ x_ @x 0 0 0 The term proportional to x_ can be handled by an integration by parts:  Z d @L Z @L Z @L d  ds @ x_ x_ = ds @ x_ dt x = @L x @ x_ 0 ; 0 ds dt @ x_ x 0 0 because x vanishes at 0 and t, the surface term is 0, leaving us with Z d @L @L

S = ds ; dt @ x_ + @x x = 0 0 S =

Z

t

t

t

t

t

t

t

t

Since the variation itself is arbitrary, the only way the integral can vanish, in general, is if the term in brackets vanishes:

d @L @L dt @ x_ ; @x = 0

This is known as the Euler-Lagrange equation in classical mechanics. For the case that L = mx= _ 2 ; U (x), they give

d (mx_ ) + @U = 0 dt @x mx = ; @U @x

which is just Newton's equation of motion, subject to the conditions that x(0) = x, x(t) = x0 . Thus, the classical path and those near it contribute the most to the path integral. The classical path condition was derived by requiring that S = 0 to
rst order. This is known as an action stationarity principle. However, it turns out that there is also a principle of least action, which states that the classical path minimizes the action as well. This is an important consideration when deriving the dominant paths for the density matrix, which takes the form

(x x0 
) =

Z

(  =x0

x h

(0)=x

x

The action appearing in this expression is

S x] = E

Z



h

0

"

Z  m
1 Dx() exp ; h 0 d 2 x_ ( ) + U (x( )) h

h i Z d m2 x_ 2 + U (x( )) =



h

0

#

dH (x x_ )

which is known as the Euclidean action and is just the integral over a path of the total energy or Euclidean Lagrangian H (x x_ ). Here, we see that a minimum action principle is needed, since the smallest values of S will contribute most to the integral. Again, we require that to
rst order S x + x] ; S x] = 0. Applying the same logic as before, we obtain the condition E

E

E

5

d @H ; @H = 0 d @ x_ @x @ U (x ) mx = @x

which is just Newton's equation of motion on the inverted potential surface ;U (x), subject to the conditions x(0) = x, x(
h ) = x0 . For the partition function Q(
), the same equation of motion must be solved, but subject to the conditions that x(0) = x(
h ), i.e., periodic paths.

II. DOING THE PATH INTEGRAL: THE FREE PARTICLE The density matrix for the free particle 2 H = 2Pm

will be calculated by doing the discrete path integral explicitly and taking the limit P The density matrix expression is

(x x0 
) =

 mP  2 Z lim dx2  dx !1 2
h2

"

P=

P

P

! 1 at the end.

X exp ; mP2 (x +1 ; x )2 2
h =1 P

i

i

i

#   1= x

P +1 =x0

x x

Let us make a change of variables to

u1 = x1 u = x ; x~ k

k

k

x~ = (k ; 1)xk +1 + x1 k

k

The inverse of this transformation can be worked out explicitly, giving

x1 = u1 x = k

+1 X k;1

P

l

The Jacobian of the transformation is simply 01 B0 J = det B B@ 0 0



=1

P ;k+1 l ; 1 u + P u1

;1=2 1 0 0



l

0 ;2=3 1 0



 1  CC  CA = 1    

0 0 ;3=4 1

Let us see what the eect of this transformation is for the case P = 3. For P = 3, one must evaluate (x1 ; x2 )2 + (x2 ; x3 )2 + (x3 ; x4 )2 = (x ; x2 )2 + (x2 ; x3 )2 + (x3 ; x0 )2

According to the inverse formula,

x1 = u 1 x2 = u2 + 21 u3 + 13 x0 + 32 x

Thus, the sum of squares becomes

x3 = u3 + 23 x0 + 13 x 6

(x ; x2 )2 + (x2 ; x3 )2 + (x3 ; x0 )2 = 2u22 + 32 u23 + 31 (x ; x0 )2 = 2 ;2 1 u22 + 3 ;3 1 u23 + 31 (x ; x0 )2 From this simple exmple, the general formula can be deduced:

X P

(x +1 ; x )2 = i

=1

i

i

X k 2 1 02 k ; 1 u + P (x ; x ) P

k

k

=2

Thus, substituting this transformation into the integral gives

(x x0 
) =

" X #

m P m (x ; x0 )2 2 exp ; u exp ; 2
h2 2
h2

 m 1 2 Y  m P 1 2 Z du2  du 2
h2 2
h2 P

=

P

=

k

k

P

k

k

=2

where

k

=2

m = k ;k 1 m k

and the overall prefactor has been written as

 mP 

2

P=

2
h

2

 m 1 2 Y  m P 1 2 = =

P

=

k

2
h

2

k

=2

2
h

2

Now each of the integrals over the u variables can be integrated over independently, yielding the
nal result

(x x0 
) =

 m 1 2 m

0 2 exp ; 2 (x ; x ) 2
h2 2
h =

In order to make connection with classical statistical mechanics, we note that the prefactor is just 1= , where

 2
h2 1 2 
h2 1 2

= = =

=

m

2m is the kinetic prefactor that showed up also in the classical free particle case. In terms of , the free particle density matrix can be written as (x x0 
) = 1 e; ( ; )2 2



 x

0

x

=

Thus, we see that represents the spatial width of a free particle at
nite temperature, and is called the \thermal de Broglie wavelength."

7

G25.2651: Statistical Mechanics Notes for Lecture 16

I. THE HARMONIC OSCILLATOR { EXPANSION ABOUT THE CLASSICAL PATH It will be shown how to compute the density matrix for the harmonic oscillator: 2 H = 2Pm + 12 m!2 X 2 using the functional integral representation. The density matrix is given by


(x x   ) = 0

Z x( h )=x

0

x(0)=x

"

# Z h
1 1 1 2 2 2 Dx( ) exp ; h d 2 mx_ + 2 m! x 0

As we saw in the last lecture, paths in the vicinity of the classical path on the inverted potential give rise to the dominant contribution to the functional integral. Thus, it proves useful to expand the path x( ) about the classical path. We introduce a change of path variables from x( ) to y( ), where

x( ) = xcl ( ) + y( ) where xcl ( ) satis es

mx cl = m!2xcl subject to the conditions

xcl ( h) = x

xcl(0) = x

0

so that y(0) = y( h) = 0. Substituting this change of variables into the action integral yields

S=

Z h 0

Z h



d 12 mx_ 2 + 21 m!2 x2







= d 12 m(x_ cl + y_ )2 + 21 m!2 (xcl + y)2 0  Z h   Z h  = d 12 mx_ 2cl + 12 m!2 x2cl + d 21 my_ 2 + 21 m!2 y2 0 0 +

Z h 0



d mx_ cl y_ + m!2 xcl y



An integration by parts makes the cross terms vanish: Z h 0





d mx_ cly_ + m!2xcl y = mx_ cl yj0 h +

Z h 0





d ;mx cl + m!2xcl y = 0

where the surface term vanishes becuase y(0) = y( h) = 0 and the second term vanishes because xcl satis es the classical equation of motion. The rst term in the expression for S is the classical action, which we have seen is given by Z h





m! h(x2 + x 2 )cosh( h!) ; 2xx i d 21 mx_ 2cl + 21 m!2 x2cl = 2sinh(  h!) 0 Therefore, the density matrix for the harmonic oscillator becomes 0

1

0





m! (x2 + x 2 )cosh( h!) ; 2xx
(x x   ) = I y] exp ; 2sinh(  h!) 0

0

where I y] is the path integral

I y] =

Z y( h )=0

y(0)=0

0

# Z h
2  m m! 1 2 2 Dy( ) exp ; h 2 y_ + 2 y 0 "

Note that I y] does not depend on the points x and x and therefore can only contribute an overall (temperature dependent) constant to the density matrix. This will aect the thermodynamics but not any averages of physical observables. Nevertheless, it is important to see how such a path integral is done. In order to compute I y] we note that it is a functional integral over functions y( ) that vanish at  = 0 and  =  h. Thus, they are a special class of periodic functions and can be expanded in a Fourier sine series: 0

X 1

y ( ) =

n=1

cn sin(!n  )

where

!n = n  h Thus, we wish to change from an integral over the functions y( ) to an integral over the Fourier expansion coecients cn . The two integrations should be equivalent, as the coecients uniquely determine the functions y( ). Note that y_ ( ) =

X 1

n=1

!n cn cos(!n  )

Thus, terms in the action are:

Z h 0

d m1 y_ 2 = m2

XX 1

1

n=1 n =1

cn cn !n !n 0

Z h

0

0

d cos(!n  ) cos(!n  ) 0

0

Since the cosines are orthogonal between  = 0 and  =  h, the integral becomes Z h 0

d m1 y_ 2 = m2

X 1

n=1

Z h

c2n !n2

0

Similarly,

d cos2 (!n  ) = m2

X 1

n=1

c2n !n2

Z h 1 0

Z h 0





X d 21 + 12 cos(2!n ) = m4 h c2n !n2 n=1 1

m h !2 X c2 2 2 m! y = n 2 4 1

n=1

The measure becomes

Dy(t) !

Y 1

n=1

p

dcn 4=m!n2

which, is not an equivalent measure (since it is not derived from a determination of the Jacobian), but is chosen to q give the correct free-particle (! = 0) limit, which can ultimately be corrected by attaching an overall factor of m=2 h2 . With this change of variables, I y] becomes

I y] =

YZ 1

1

n=1

;1

 Y 1=2  2 ! dc m n n 2 2 2 p exp ; 4 (! + !n )cn = 2 2 4=m!n2 n=1 ! + !n 1

The in nite product can be written as 2

Y 1

n=1

"






2 n2 = 2 h2 = Y 1 +  2 h2 !2 2 n2 !2 + 2 n2 = 2 h2 n=1 1

#

;

1

the product in the square brackets is just the in nite product formula for sinh( h!)=( h!), so that I y] is just s

 h ! I y] = sinh(  h!)

q

Finally, attaching the free-particle factor m=2 h2 , the harmonic oscillator density matrix becomes: 



r m! m! (x2 + x 2 )cosh( h!) ; 2xx
(x x   ) = 2 hsinh( exp ;  h!) 2sinh( h!) Notice that in the free-particle limit (! ! 0), sinh( h!)   h! and cosh( h!)  1, so that   r m m 2
(x x   ) ! exp ; 2 (x ; x ) 2 h2 2 h 0

0

0

0

0

which is the expected free-particle density matrix.

II. THE STATIONARY PHASE APPROXIMATION Consider the simple integral:

I = lim

Z

1

dx e f (x) ;

!1

;1

Assume f (x) has a global minimum at x = x0 , such that f (x0 ) = 0. If this minimum is well separated from other minima of f (x) and the value of f (x) at the global minimum is signi cantly lower than it is at other minima, then the dominant contributions to the above integral, as  ! 1 will come from the integration region around x0 . Thus, we may expand f (x) about this point: f (x) = f (x0 ) + f (x0 )(x ; x0 ) + 21 f (x0 )(x ; x0 )2 +    Since f (x0 ) = 0, this becomes: f (x)  f (x0 ) + 12 f (x0 )(x ; x0 )2 Inserting the expansion into the expression for I gives 0

0

00

0

00

I = lim e f (x0 )

Z

1

;

!1

dx e 2 f (x0 )(x x0)2 ;

;1



00

;



1=2 = lim f 2(x ) e f (x0 ) 0 Corrections can be obtained by further expansion of higher order terms. For example, consider the expansion of f (x) up to fourth order: 1 f (iv) (x )(x ; x )4 f (x)  f (x0 ) + 21 f (x0 )(x ; x0 )2 + 16 f (x0 )(x ; x0 )3 + 24 0 0 Substituting this into the integrand and further expanding the exponential would give, as the lowest order nonvanishing correction: ;

00

!1

00

I = lim e

f (x0 )

000

Z

1

;

!1

;1

dx e

;

2

 f (x0 )(x x0 )2 1 ; 00

;

3

 f (iv) (x )(x ; x )4 0 0 24



This approximation is known as the stationary phase or saddle point approximation. The former may seem a little out-of-place, since there is no phase in the problem, but that is because we formulated it in such a way as to anticipate its application to the path integral. But this is only if  is taken to be a real instead of an imaginary quantity. The application to the path integral follows via a similar argument. Consider the path integral expression for the density matrix:


(x x   ) = 0

Z x( h )=x

0

x(0)=x

We showed that the classical path satisfying

SE x]=h

Dx]e

;





mx cl = @U @x x=xcl x(0) = x x( h) = x is a stationary point of the Euclidean action SE x], i.e., SExcl ] = 0. Thus, we can develop a stationary phase or 0

saddle point approximation for the density matrix by introducing an expansion about the classical path according to

x( ) = xcl ( ) + y( ) = xcl ( ) +

X

cn n ( )

n

where the correction y( ), satisfying y(0) = y( h) = 0 has been expanded in a complete set of orthonormal functions f n ( )g, which are orthonormal on the interval 0  h] andsatisfy n (0) = n ( h) = 0 as well as the orthogonality condition: Z h 0

d n ( ) m ( ) = mn

Setting all the expansion coecients to 0 recovers the classical path. Thus, we may expand the action S x] (the \E" subscript will henceforth be dropped from this discussion) with respect to the expansion coecients:



@S

1 X @ 2 S

S x] = S xcl ] + @c c + j 2 jk @cj ck c =0 cj ck +    j c =0 j X

f g

Since

f g





Z h

S x] = d 12 mx_ 2 + U (x( )) 0 the expansion can be worked out straightforwardly by substitution and subsequent dierentiation: S x] = @S = @cj

Z h 0

Z h 0

2

X d 4 12 m x_ cl + cn _ n n "

d m(x_ cl +

X

!2

+ U (xcl +

X

n

0

n

n

0

= mx_ cl j j0 h + =0

Z h 0

" @ 2 S = Z h m _ _ + U j k @cj @ck 0

d ;mx cl + U (xcl )] j 0

xcl +

00

4

cn n )5 !

X cn _ n ) _ j + U xcl + cn n j

Z h h i @S

_ = d m x _

+ U ( x )

cl j cl j @cj c =0 0 f g

3

X

n

!

cn n ] j k

#

#



Z h h i @ 2 S

_ j _ k + U (xcl ) j k = d m



@cj @ck c =0 0 00

f g

= =

Z h 0

Z h 0

h

d ;m j

k + U (xcl ( )) j k 00



i 

2 d j ( ) ;m dd 2 + U (xcl ( )) k ( ) 00

where the fourth and eighth lines are obtained from an integration by parts. Let us write the integral in the last line in the suggestive form:

@ 2 S

@ 2 + U (x ( ))j i =  = h

j ; m j cl k jk

@cj @ck c =0 @ 2 00

f g

which emphasizes the fact that we have matrix elements of the operator ;md2=d 2 + U (xcl ( )) with respect to the basis functions. Thus, the expansion for S can be written as X S x] = S xcl ] + 12 cj jk ck +    jk 00

and the density matrix becomes


(x x   ) = N 0

Z Y

j

pdcj e 2 h

Scl (xx 

;

0

)

e

;

1 2

P

 j
k cj jk ck =h

where Scl(x x   ) = S xcl ]. N is an overall normalization constant. The integral over the coecients becomes a p generalized Gaussian integral, which brings down a factor of 1= det:
(x x   ) = N e Scl (xx  ) p 1 det 1 S (xx  ) q cl = Ne  ; det ;m dd22 + U (xcl ( )) 0

0

0

;

0

;

00

where the last line is the abstract representation of the determinant. The determinant is called the Van Vleck-Paulideterminant. If we choose the basis functions n ( ) to be eigenfunctions of the operator appearing in the above expression, so that they satisfy Morette





;m dd 2 + U (xcl ( )) n ( ) = n n ( ) 2

00

Then, jk = j jk = j (x x   ) jk and the determinant can be expressed as a product of the eigenvalues. Thus, Y
(x x   ) = N e Scl (xx  ) p 1 j (x x   ) j 0

0

0

;

0

The product must exclude any 0-eigenvalues. Incidentally, by performing a Wick rotation back to real time according to  = ;it= h, the saddle point or stationary phase approximation to the real-time propagator can be derived. The derivation is somewhat tedious and will not be given in detail here, but the result is 1 i=2 U (x x  t) = e hi Scl (xx t) q ; e 2 d det ;m dt2 ; U (xcl (t)) 0

0

;

00

5

where xcl (t) satis es





mx cl = ; @U @x x=xcl xcl (0) = x xcl (t) = x

0

and  is an integer that increases by 1 each time the determinant vanishes along the classical path.  is called the Maslov index. It is important to note that because the classical paths satisfy an endpoint problem, rather than an initial value problem, there can be more than one solution. In this case, one must sum the result over classical paths: X 1 U (x x  t) = e hi Scl (xx t) i=2 q ;  2 d det ;m dt2 ; U (xcl (t)) classical paths 0

0

;

00

with a similar sum for the density matrix.

6

G25.2651: Statistical Mechanics Notes for Lecture 17

I. EXPECTATION VALUES OF OBSERVABLES Recall the basic formula for the expectation value of an observable A: A = Q(1 ) Tr(Ae; H ) Two important cases pertaining to the evaluation of the trace in the coordinate basis for expectation values will be considered below: h

i

A. Case 1: Functions only of position If A = A(X ), i.e., a function of the operator X only, then the trace can be easily evaluated in the coordinate basis: Z 1 A = dx x A(X )e; H x h

Q

i

h

j

j

Since A(X ) acts to the left on one of its eigenstates, we have Z A = 1 dxA(x) x e; h

Q

i

h

j

Hj

i

x

i

which only involves a diagonal element of the density matrix. This can, therefore, be written as a path integral:

" P

# X 1 2 mP P=2 Z dx dx A(x ) exp 1 2 1 P 1 2 m!P (xi+1 xi ) + P U (xi ) 2
h2 i=1 However, since all points x1  :: xP are equivalent, due to the fact that they are all integrated over, we can make P equivalent cyclic renaming of the coordinates x1 x2 , x2 x3 , etc. and generate P equivalent integrals. In each, the function A(x1 ) or A(x2 ), etc. will appear. If we sum these P equivalent integrals and divide by P , we get an h

A = Q1 Plim !1 i

;

!

expression:

A = Q1 Plim !1

h

i

mP 2
h2


P=2 Z

dx1

;

!

P X 1 dxP P A(xi ) exp

" ;

i=1


P X 1 m!2 (x 1 2 2 P i+1 xi ) + P U (xi )

#

;

i=1

This allows us to de ne an estimator for the observable A. Recall that an estimator is a function of the P variables x1  ::: xP whose average over the ensemble yields the expectation value of A: P X 1 aP (x1  ::: xP ) = P A(xi ) i=1 Then

A = Plim a !1 p x1 :::xP

h

i

h

i

where the average on the right is taken over many con gurations of the P variables x1  ::: xP (we will discuss, in the nex lecture, a way to generate these con gurations). The limit P can be taken in the same way that we did in the previous lecture, yielding a functional integral expression for the expectation value: ! 1

h

I A = Q1 i

"

D

#

Z h x( ) 1h dA(x( )) exp 0 1


#

"

;

1 Z h d 1 mx_ 2 + U (x( )) h 0 2

B. Case 1: Functions only of momentum Suppose that A = A(P ), i.e., a function of the momentum operator. Then, the trace can still be evaluated in the coordinate basis: Z A = 1 dx x A(P )e; H x h

i

Q

h

j

j

i

R

However, A(P ) acting to the left does not act on an eigenvector. Let us insert a coordinate space identity I = dx x x between A and exp( H ): Z 1 dxdx0 x A(P ) x0 x e; H x A = j

ih

j

;

h

Q

i

h

j

j

ih

j

j

i

Now, we see that the expectation value can be obtained by evaluating all the coordinate space matrix elements of the operator and all the coordinate space matrix elements of the density matrix. A particularly useful form for the expectation value can be obtained if a momentum space identity is inserted: Z A = 1 dxdx0 dp x A(P ) p p x0 x0 e; H x h

i

Q

h

j

j ih j

ih

j

j

i

Now, we see that A(P ) acts on an eigenstate (at the price of introducing another integral). Thus, we have Z Z A = 1 dpA(p) dxdx0 x p p x0 x0 e; H x

Q Using the fact that x p = (1=2
h) exp(ipx=h), we nd that Z Z 1 A = 2
hQ dpA(p) dxdx0 eip(x;x )=h x0 e; h

h

i

h

j ih j

ih

j

j

i

j i

x

Hj

0

h

i

h

j

i

In the above expression, we introduce the change of variables 0 r = x +2 x s = x x0 Then Z Z A = 2
1h Q dpA(p) dr dseips=h r 2s e; H r + 2s De ne a distribution function Z W (r p) = 2
1h dseips=h r 2s e; H r + 2s Then, the expectation value can be written as Z 1 A = drdpA(p) (r p) ;

h

i

h

h

h

i

Q

;

;

j

j

j

j

i

i

W

which looks just like a classical phase space average using the \phase space" distribution function W (r p). The distribution function W (r p) is known as the Wigner density matrix and it has many interesting features. For one thing, its classical limit is

W (r p) = exp



;

p2 + U (r) 2m




which is the true classical phase space distribution function. There are various examples, in which the exact Wigner distribution function is the classical phase space distribution function, in particularly for quadratic Hamiltonians. Despite its compelling appearance, the evaluation of expectation values of functions of momentum are considerably more dicult than functions of position, due to the fact that the entire density matrix is required. However, there are a few quantities of interest, that are functions of momentum, that can be evaluated without resorting to the entire density matrix. These are thermodynamic quantities which will be discussed in the next section. 2

II. THERMODYNAMICS FROM PATH INTEGRALS Although general functions of momentum are dicult (though not intractable) to evaluate by path integration, certain functions of momentum (and position) can be evaluated straightforwardly. These are thermodynamic quantities such as the energy and pressure, given respectively by

E = @@ ln Q(  V ) @ ln Q(  V ) P = 1 @V ;

We shall derive estimators for these two quantities directly from the path integral expression for the partition function. However, let us work with the partition function for an ensemble of 1-particle systems in three dimensions, which is given by

mP Q(  V ) = Plim !1 2
h2


3P=2 Z

dr1

drP exp

"

;

P X 1 m!2 (r 2 P i+1

;

i=1


#

1 ri ) + U (ri ) P 2

Using the above thermodynamic relation, the energy becomes E = 1 @Q ;

Q@

" P
X 1 2 mP 3P=2 Z dr dr exp 1 P 2 2 m!P (ri+1 2
h i=1 # P P X X 1 m!2 (r 1 2 2 P i+1 ri ) + P U (ri )

= Q1 Plim !1

"



3P 2

;

;

;


#

1 ri ) + U (ri ) P 2

;

i=1

i=1

= Plim " (r  ::: rP ) !1 P 1 h

i

where

"P (r1  ::: rP ) = 32P

;

P X 1 i=1

2 2 m!P (ri+1

;

ri )2

P X + P1 U (ri ) i=1

is the thermodynamic estimator for the total energy. Similarly, an estimator for the internal pressure can be derived using P = kT@ ln Q=@V . As we have done in the past for classical systems, the volume dependence can be made explicity by introducing the change of variables: rk

= V 1=3 sk

In terms of the scaled variables sk , the partition function expression reads:

mP Q(  V ) = Plim !1 2
h2


3P=2

VP

Z

ds1

dsP exp

"

;

P X 1 2 2=3 2 m!P V (si+1 i=1


#

;

1 si )2 + U (V 1=3 si ) P

Evaluating the derivative with respect to volume gives the internal pressure: P = 1 @Q

Q @V

= Q1 Plim !1

"



P V

;


3P=2

"

Z

P X 1 m!2 V 2=3 (s V ds1 dsP exp i+1 P 2 i=1 # P P X @U 1 m!2 V ;1=3 X 1 1 2 ; 2=3 (si+1 si ) P 3 P @ (V 1=3 s ) 3 V si

mP 2
h2

P

;

;

i=1

;

i=1

3

i

;


#

1 si ) + U (V 1=3 si ) P 2

= Q1 Plim !1

"



P V

;

"


3P=2 Z

P X 1 m!2 (r dr1 drP exp 2 P i+1 i=1 # P P X 1 m!2 X 1 @U 2 3V P i=1 (ri+1 ri ) 3V P i=1 @ ri si

mP 2
h2

;

;

;


#

1 ri ) + U (ri ) P 2

;

= Plim p (r  ::: rP ) !1 P 1 h

i

where

pP (r1  ::: rP ) = PV

;

P  1 X 2 3V i=1 m!P (ri+1

;

1 @U ri ) + ri P @r 2



i

is the thermodynamic estimator for the pressure. Clearly, both the energy and pressure will be functions of the particle momenta, however, because they are related to the partition function by thermodynamic dierentiation, estimators can be derived for them that do not require the o-diagonal elements of the density matrix.

III. PATH INTEGRAL MOLECULAR DYNAMICS (OPTIONAL READING) Consider once again the path integral expression for the one-dimensional canonical partition function (for a nite but large value of P ):

Q( ) = mP 2 2
h


P=2 Z

dx1

dxP exp

"

;


# P X 1 1 m!2 (x 2 2 P i+1 xi ) + P U (xi )

(1)

;

i=1

(the condition xP +1 = x1 is understood). Recall that, according to the classical isomorphism, the path integral expression for the canonical partition function is isomorphic to the classical con guration integral for a certain P particle system. We can carry this analogy one step further by introducing into the above expression a set of P momentum integrations:

Z

Q( ) = dp1

dpP dx1

dxP exp

"

;


P X p2i + 1 m!2 (x 1 2 2m0 2 P i+1 xi ) + P U (xi ) i=1 ;

#

(2)

Note that these momentum integrations are completely uncoupled from the position integrations, and if we were to carry out these momentum integrations, we would reproduce Eq. (1) apart from trivial constants. Written in the form Eq. (2), however, the path integral looks exactly like a phase space integral for a P -particle system. We know from our work in classical statistical mechanics that dynamical equations of motion can be constructed that will generate this partition function. In principle, one would start with the classical Hamiltonian

 P  2 X 1 1 p i 2 2 H= 2m0 + 2 m!P (xi+1 xi ) + P U (xi ) i=1 ;

derive the corresponding classical equations of motion and then couple in thermostats. Such an approach has certainly been attempted with only limited success. The diculty with this straightforward approach is that the more \quantum" a system is, the large the paramester P must be chosen in order to converge the path integral. However, if P is large, the above Hamiltonian describes a system with extremely sti nearest-neighbor harmonic bonds interacting with a very weak potential U=P . It is, therefore, almost impossible for the system to deviate far harmonic oscillator solutions and explore the entire available phase space. The use of thermostats can help this problem, however, it is also exacerbated by the fact that all the harmonic interactions are coupled, leading to a wide variety of time scales associated with the motion of each variable in the Hamiltonian. In order to separate out all these time scales, one must somehow diagonalize this harmonic interaction. One way to do this is to use normal mode variables, and this is 4

a perfectly valid approach. However, we will explore another, simpler approach here. It involves the use of a variable transformation of the formed used in previous lectures to do the path integral for the free-particle density matrix. Consider a change of variables: u1 = x1 uk = xk x~k k = 2 ::: P where x~ = (k 1)xk+1 + x1 ;

;

i

k

The inverse of this transformation can be worked out in closed form: x1 = u1 P X 1) u xk = u1 + ((kl 1) l l=k ;

;

and can also be expressed as a recursive inverse: x1 = u1 xk = uk + k k 1 xk+1 + k1 x1 ;

The term k = P here can be used to start the recursion. We have already seen that this transformation diagonalized the harmonic interaction. Thus, substituting the transformation into the path integral gives:

Z

Q( ) = dp1

dpP du1

duP exp

"

;


# P X p2i + 1 m !2 u2 + 1 U (x (u  ::: u )) P 2m0i 2 i P i P i 1 i=1

The parameters mi are given by

m1 = 0 mi = i i 1 m ;

Note also that the momentum integrations have been changed slightly to involve a set of parameters m0i . Introducing these parameters, again, only changes the partition function by trivial constant factors. How these should be chosen will become clear later in the discussion. The notation xi (u1  ::: uP ) indicates that each variable xi is a generally a function of all the new variables u1  ::: uP . A dynamics scheme can now be derived using as an eective Hamiltonian:

 P  2 X 1 1 p i 2 2 H= 2m0i + 2 mi !P ui + P U (xi (u1  ::: uP )) i=1

which, when coupled to thermostats, yields a set of equations of motion m0 u = m !2 u 1 @U _ u_ i i

;

i P i

Q i = mi u_ 2i 1

;

P @ui

;

i i

;

(3)

These equations have a conserved energy (which is not a Hamiltonian):

H0 =

 P  X 1 m0 u_ 2 + 1 m !2 u2 + 1 U (x (u  ::: u )) + 1 Q _ 2 + 1

i i 1 P i 2 i i 2 P i P 2 i i=1

Notice that each variable is given its own thermostat. This is done to produce maximum ergodicity in the trajectories. In fact, in practice, the chain thermostats you have used in the computer labs are employed. Notice also that the 5

time scale of each variable is now clear. It is just determined by the parameters mi . Since the object of using such dynamical equations is not to produce real dynamics but to sample the phase space, we would really like each variable to move on the same time scale, so that there are no slow beads trailing behind the fast ones. This eect can be produced by choosing each parameter m0i to be proportional to mi : m0i = cmi . Finally, the forces on the u variables can be determined easily from the chain rule and the recursive inverse given above. The result is f

g

P 1 @U = 1 X @U P @u1 P i=1 @xi   1 @U = 1 (k 2) @U + @U P @ui P (k 1) @uk;1 @xk ; ;

where the rst (i = 1) of these expressions starts the recursion in the second equation. Later on, when we discuss applications of path integrals, we will see why a formulation such as this for evaluating path integrals is advantageous.

IV. PATH INTEGRALS FOR N -PARTICLE SYSTEMS If particle spin statistics must be treated in a given problem, the formulation of the path integral is more complicated, and we will not treat this subject here. The extension of path integrals to N -particle systems in which spin statistics can safely be ignored, however, is straightforward, and we will give the expressions below. The partition function for an N -particle system in the canonical ensemble without spin statistics can be formulated essentially by analogy to the one-particle case. The partition function that one obtains is

"N Y mI P
3P=2 Z (1) Q(N V T ) = Plim drI !1 I =1 2
h2

drI

(P )

#

exp

"

;

N P X X 1 2 (i+1) 2 mI !P (rI i=1

;

I =1

) + P1 U (r(1i)  ::: r(Ni) )

(i) 2

rI

!#

Thus, it can be seen that the N -particle potential must be evaluated for each imaginary time discretization, however, there is no coupling between separate imaginary time slices due arising from the potential. Thus, interactions occur only between particles in the same time slice. From a computational point of view, this is advantageous, as it allows for easily parallelization over imaginary time slices. The corresponding energy and pressure estimators for the N -particle path integral are given by

P ( pP (

f

r(1)  ::: r(P ) ) = (1)

r

f

 ::: r

(P )

3NP 2

) = NP V

;

P X N X 1 i=1 I =1



(i) (i+1) 2 rI ; rI +

 1 X X m !2 r(i) 3V i=1 I =1 I P I P

;

2 2 mI !P

N

6

;



(i+1) 2

rI

P 1X (i) (i) P U (r1  ::: rN ) i=1

+ P1 r(Ii)

r

(i)

rI

U





       
 
   


   


 
         
           

             
    
     


    

 
   
 
 

      
   

 
   
     
                  
  
       
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C



G25.2651: Statistical Mechanics Notes for Lecture 21

I. CLASSICAL LINEAR RESPONSE THEORY Consider Hamilton's equations in the form

q_i = @H @pi p_i = ; @H @qi We noted early in the course that an ensemble of systems evolving according to these equations of motion would generate an equilibrium ensemble (in this case, microcanonical). Recall that the phase space distribution function f (;
t) satised a Liouville equation:

@f + iLf = 0 @t where iL = f:::
H g. We noted that if @f=@t = 0, then f = f (H ) is a pure function of the Hamiltonian which dened

the general class of distribution functions valid for equilibrium ensembles. What does it mean, however, if @f=@t 6= 0? To answer this, consider the problem of a simple harmonic oscillator. In an equilibrium ensemble of simple harmonic oscillators at temperature T , the members of the ensemble will undergo oscillatory motion about the potential minimum, with the amplitude of this motion determined by the temperature. Now, however, consider driving each oscillator with a time-dependent driving force F (t). Depending on how complicated the forcing function F (t) is, the motion of each member of the ensemble will, no longer, be simple oscillatory motion about the potential minimum, but could be a very complex kind of motion that explores large regions of the potential energy surface. In other words, the ensemble of harmonic oscillators has been driven away from equilibrium by the time-dependent force F (t). Because of this nonequilibrium behavior of the ensemble, averages over the ensemble could become time-dependent quantities rather than static quantities. Indeed, the distribution function f (;
t), itself, could be time-dependent. This can most easily be seen by considering the equation of motion for a forced oscillator

mx = ;m!2x + F (t) The solution now depends on the entire history of the forcing function F (t), which can introduce explicit timedependence into the ensemble distribution function.

A. Generalized equations of motion The most general way a system can be driven away from equilibrium by a forcing function Fe (t) is according to the equations of motion:

q_i = @H @pi + Ci (;)Fe (t) p_i = ; @H @pi + Di (;)Fe (t) where the 3N functions Ci and Di are required to satisfy the incompressibility condition 3N X

i=1




@Ci + @Di = 0 @qi @pi 1

in order to insure that the Liouville equation for f (;
t) is still valid. These equations of motion will give rise to a distribution function f (;
t) satisfying

@f + iLf = 0 @t R with @f=@t 6= 0. (We assume that f is normalized so that d;f (;
t) = 1.) What does the Liouville equation say about the nature of f (;
t) in the limit that Ci and Di are small, so that

the displacement away from equilibrium is, itself, small? To examine this question, we propose to solve the Liouville equation perturbatively. Thus, let us assume a solution of the form f (;
t) = f0(H (;)) + f (;
t) Note, also, that the equations of motion ;_ take a perturbative form ;_ (t) = ;_ 0 + ;_ (t) and as a result, the Liouville operator contains two pieces: iL = ;_  r; = ;_ 0  r; + ;_  r; = iL0 + iL where iL0 = f:::
H g and f0 (H ) is assumed to satisfy iL0f0 (H (;)) = 0 ;_ 0 means the Hamiltonian part of the equations of motion

q_i = @H @pi p_i = ; @H @qi For an observable A(;), the ensemble average of A is a time-dependent quantity: Z

hA(t)i = d;A(;)f (;
t) which, when the assumed form for f (;
t) is substituted in, gives Z

Z

Z

hA(t)i = d;A(;)f0 (;) + d;A(;)f (;
t) = hAi0 + d;A(;)f (;
t) where hi0 means average with respect to f0 (;).

B. Perturbative solution of the Liouville equation Substituting the perturbative form for f (;
t) into the Liouville equation, one obtains

@ @t (f0 (;) + f (;
t)) + (iL0 + iL(t))(f0 (;) + f (;
t)) = 0

Recall @f0 =@t = 0. Thus, working to linear order in small quantities, one obtains the following equation for f (;
t): 



@ + iL f (;
t) = ;iLf (;) 0 0 @t

which is just a rst-order inhomogeneous dierential equation. This can easily be solved using an integrating factor, and one obtains the result f (;
t) = ;

Z t 0

dse;iL0 (t;s) iL(s)f0(;) 2

Note that

iLf0(;) = iLf0(;) ; iL0f0 (;) = iLf0(;) = ;_  r; f0 (;)

But, using the chain rule, we have

0 @H ;_  r; f0 (;) = ;_  @f @H @ ; 0 = @f @H

N X

p_ i  @H + q_ i @H @ pi @ qi




i=1  
N p X 0 i  (F + D F (t)) ; F  pi + C F (t) = @f i m i e @H i=1 mi i i e i


N X 0 = @f Di (;)  pi ; Ci (;)  Fi Fe (t) @H i=1 mi

where Fi (q1
:::
qN ) is the force on the ith particle. Dene

j (;) = called the dissipative ux. Then

N X i=1

pi

Ci (;)  Fi ; Di (;)  m




i

;_  r; f0 (;) = ; @f0 j (;)Fe (t)

@H

Now, suppose f0 (;) is a canonical distribution function

f0 (H (;)) = Q(N
1V
T ) e; H (;) then

@f0 = ;f (H ) 0 @H

so that

;_  r; f0 (;) = f0 (;)j (;)Fe (t)

Thus, the solution for f (;
t) is f (;
t) = ;

Z t 0

dse;iL0 (t;s) f0 (;)j (;)Fe (s)

The ensemble average of the observable A(;) now becomes Z

hA(t)i = hAi0 ;  d;A(;) = hAi0 ;  = hAi0 ; 

Z t 0

Z t 0

Z

Z t 0

dse;iL0 (t;s) f0 (;)j (;)Fe (s)

ds d;A(;)e;iL0 (t;s) f0 (;)j (;)Fe (s) Z

ds d;f0 (;)A(;)e;iL0 (t;s) j (;)Fe (s)

Recall that the classical propagator is exp(iLt). Thus the operator appearing in the above expression is a classical propagator of the unperturbed system for propagating backwards in time to ;(t ; s). An observable A(;) evolves in time according to 3

dA = iLA dt A(t) = eiLtA(0) A(;t) = e;iLtA(0) Now, if we take the complex conjugate of both sides, we nd

A
(t) = A
(0)e;iLt where now the operator acts to the left on A
(0). However, since observables are real, we have

A(t) = A(0)e;iLt which implies that forward evolution in time can be achieved by acting to the left on an observable with the time reversed classical propagator. Thus, the ensemble average of A becomes

hA(t)i = hAi0 ;  = hAi0 ; 

Z t 0

Z t 0

Z

dsFe (s) d;0 f0 (;0 )A(;t;s (;0 ))j (;0 ) dsFe (s)hj (0)A(t ; s)i0

where the quantity on the last line is an object we have not encountered yet before. It is known as an equilibrium time correlation function. An equilibrium time correlation function is an ensemble average over the unperturbed (canonical) ensemble of the product of the dissipative ux at t = 0 with an observable A evolved to a time t ; s. Several things are worth noting: 1. The nonequilibrium average hA(t)i, in the linear response regime, can be expressed solely in terms of equilibrium averages. 2. The propagator used to evolve A(;) to A(;
t ; s) is the operator exp(iL0 (t ; s)), which is the propagator for the unperturbed, Hamiltonian dynamics with Ci = Di = 0. That is, it is just the dynamics determined by H . 3. Since A(;
t ; s) = A(;(t ; s)) is a function of the phase spaceRvariables evolved to a time t ; s, we must now specify over which set of phase space variables the integration d; is taken. The choice is actually arbitrary, and for convenience, we choose the initial conditions. Since ;(t) is a function of the initial conditions ;(0), we can write the time correlation function as Z 1 hj (0)A(t ; s)i = d; e; H (;0 ) j (; )A(; (; )) 0

Q

0

0

t;s

C. General properties of time correlation functions Dene a time correlation function between two quantities A(;) and B (;) by

CAB (t) = hZA(0)B (t)i = d;f (;)A(;)eiLt B (;) The following properties follow immediately from the above denition:

hA(0)B (t)i = hA(;t)B (0)i 1.

CAB (0) = hA(;)B (;)i 2. Thus, if A = B , then 4

0

CAA (t) = hA(0)A(t)i known as the autocorrelation function of A, and

CAA (0) = hA2 i If we dene A = A ; hAi, then

C
A
A (0) = h(A)2 i = h(A ; hAi)2 i = hA2 i ; hAi2 which just measures the uctuations in the quantity A. 3. A time correlation function may be evaluated as a time average, assuming the system is ergodic. In this case, the phase space average may be equated to a time average, and we have 1

CAB (t) = Tlim !1 T ; t

Z T ;t 0

dsA(;(s))B (;(t + s))

which is valid for t << T . In molecular dynamics simulations, where the phase space trajectory is determined at discrete time steps, the integral is expressed as a sum

CAB (kt) = N 1; k

NX ;k j =1

A(;k )B (;k+j )

k = 0
1
2
:::
Nc

where N is the total number of time steps, t is the time step and Nc << N . 4. Onsager regression hypothesis: In the long time limit, A and B eventually become uncorrelated from each other so that the time correlation function becomes

CAB (t) = hA(0)B (t)i ! hAihB i For the autocorrelation function of A, this becomes

CAA (t) ! hAi2 Thus, CAA (t) decays from hA2 i at t = 0 to hAi2 as t ! 1. An example of a signal and its time correlation function appears in the gure below. In this case, the signal is the magnitude of the velocity along the bond of a diatomic molecule interacting with a Lennard-Jones bath. Its time correlation function is shown beneath the signal:

5

FIG. 1.

Over time, it can be seen that the property being autocorrelated eventually becomes uncorrelated with itself.

6

II. TIME CORRELATION FUNCTIONS AND TRANSPORT COEFFICIENTS A. The shear viscosity The shear viscosity of a system measures is resistance to ow. A simple ow eld can be established in a system by placing it between two plates and then pulling the plates apart in opposite directions. Such a force is called a shear force, and the rate at which the plates are pulled apart is the shear rate. A set of microscopic equations of motion for generating shear ow is q_ = pi + y x^ i

mi

i

p_ i = Fi ; py x^ i

where  is a parameter known as the shear rate. These equations have the conserved quantity

H0 =

N X i=1

(pi + mi yi x^ )2 + U (q1
::
qN )

The physical picture of this dynamical system corresponds to the presence of a velocity ow eld v(y) = yx^ shown in the gure. The ow eld points in the x^ direction and increases with increasing y-value. Thus, layers of a uid, for example, will slow past each other, creating an anisotropy in the system. From the conserved quantity, one can see that the momentum of a particle is the value of pi plus the contribution from the eld evaluated at the position of the particle pi ! pi + mi v(yi )

7

FIG. 2.

Such an applied external shearing force will create an asymmetry in the internal pressure. In order to describe this asymmetry, we need an analog of the internal pressure that contains a dependence on specic spatial directions. Such a quantity is known as the pressure tensor and can be dened analogously to the isotropic pressure P that we encountered earlier in the course. Recall that an estimator for the pressure was
N p2 X i + qi  Fi p = 31V i=1 mi

and P = hpi in equilibrium. Here, V is the volume of the system. By analogy, one can write down an estimator for the pressure tensor p :
N (p  e^ )(p  e^ ) X 1 i  i + (qi  e^ )(Fi  e^ ) p = V m

and

i=1

i

P  = hp  i where e^ is a unit vector in the direction, = x
y
z . This (nine-component) pressure tensor gives information

about spatial anisotropies in the system that give rise to o-diagonal pressure tensor components. The isotropic pressure can be recovered from 8

P = 13 P  X

which is just 1/3 of the trace of the pressure tensor. While most systems have diagonal pressure tensors due to spatial isotropy, the application of a shear force according to the above scheme gives rise to a nonzero value for the xy component of the pressure tensor Pxy . In fact, Pxy is related to the velocity ow eld by a relation of the form x Pxy = ; @v @y = ; 

where the coecient is known as the shear viscosity and is an example of a transport coe
cient. Solving for we nd

= ; Pxy = ; lim hpxy (t)i





t!1

where hpxy (t)i is the nonequilibrium average of the pressure tensor estimator using the above dynamical equations of motion. Let us apply the linear response formula to the calculation of the nonequilibrium average of the xy component of the pressure tensor. We make the following identications:

Fe (t) = 1

Di (;) = ;py x^

Ci (;) = yi x^

i

Thus, the dissipative ux j (;) becomes

j (;) =

N X i=1 N X

pi Ci  Fi ; Di  m




i

pi  x^




yi (Fi  x^ ) + pyi m i i=1
N (p  y^ )(p  x^ ) X i i ^ ^ = + ( q  y )( F  x ) i i mi i=1

=

= V pxy According to the linear response formula,

hpxy (t)i = hpxy i0 ; V

Z t 0

dshpxy (0)pxy (t ; s)i0

so that the shear viscosity becomes




hpxy i0 + V Z t dshp (0)p (t)i

= tlim ; xy xy 0 !1  0 Recall that h  i0 means average of a canonical distribution with  = 0. It is straightforward to show that hpxy i0 = 0 for an equilibrium canonical distribution function. Finally, taking the limit that t ! 1 in the above expression gives the result

V Z 1 dthp (0)p (t)i

= kT xy xy 0 0

which is a relation between a transport coecient, in this case, the shear viscosity coecient, and the integral of an equilibrium time correlation function. Relations of this type are known as Green-Kubo relations. Thus, we have expressed a new kind of thermodynamic quantity to an equilibrium time correlation function, which, in this case, is an autocorrelation function of the xy component of the pressure tensor. 9

B. The di
usion constant The diusive ow of particles can be studied by applying a constant force f to a system using the microscopic equations of motion q_ = pi i

mi

p_ i = Fi (q1
::
qN ) + f x^

which have the conserved energy

H0 =

N X

N X p2i + U ( q1
:::
qN ) ; f xi i=1 2mi i=1

Since the force is applied in the x^ direction, there will be a net ow of particles in this direction, i.e., a current Jx . Since this current is a thermodynamic quantity, there is an estimator for it:

ux = ;

N X i=1

x_ i

and Jx = hux i. The constant force can be considered as arising from a potential eld

(x) = ;xf The potential gradient @=@x will give rise to a concentration gradient @c=@x which is opposite to the potential gradient and related to it by @c = ; 1 @

@x

kT @x However, Fick's law tells how to relate the particle current Jx to the concentration gradient @c = D @ = ; D f Jx = ;D @x kT @x kT where D is the diusion constant. Solving for D gives hux (t)i D = ;kT Jfx = ;kT tlim !1 f

Let us apply the linear response formula again to the above nonequilibrium average. Again, we make the identication: Fe (t) = 1 Di = f x^ Ci = 0 Thus,

hux (t)i = hux i0 ; 

Z t 0

= hux i0 ; f

dsf h

Z t 0

ds

N X

i=1

x_ i (0)

! N X

i=1

!

x_ i (t ; s) i0

X

ij

hx_ i (0)x_ j (t ; s)i0

In equilibrium, it can be shown that there are no cross correlations between dierent particles. Consider the initial value of the correlation function. From the virial theorem, we have which vanishes for i 6= j . In general,

hx_ i x_ j i0 = ij hx_ 2i i0 hx_ i (0)x_ j (t)i0 = ij hx_ i (0)x_ i (t ; s)i0 10

Thus,

hux (t)i = hux i0 ; f

Z t 0

ds

N X i=1

x_ i (0)x_ i (t ; s)i0

In equilibrium, hux i0 = 0 being linear in the velocities (hence momenta). Thus, the diusion constant is given by, when the limit t ! 1 is taken,

D=

Z 1X N 0

i=1

hx_ i (0)x_ i (t)i0

However, since no spatial direction is preferred, we could also choose to apply the external force in the y or z directions and average the result over the these three. This would give a diusion constant Z 1 X N 1 D=3 dt hq_ i (0)  q_ i (t)i0 0 i=1

The quantity N X i=1

hq_ i (0)  q_ i (t)i0

is known as the velocity autocorrelation function, a quantity we will encounter again in other contexts.

11



       
 
  


            
 


        



        
   


  
       
   


 
     

 
   

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@

G25.2651: Statistical Mechanics Notes for Lecture 24

I. THE HARMONIC BATH HAMILTONIAN In the theory of chemical reactions, it is often possible to isolate a small number or even a single degree of freedom in the system that can be used to characterize the reaction. This degree of freedom is coupled to other degrees of freedom (for example, reactions often take place in solution). Isomerization or dissociation of a diatomic molecule in solution is an excellent example of this type of system. The degree of freedom of paramount interest is the distance between the two atoms of the molecule { this is the degree of freedom whose detailed dynamics we would like to elucidate. The dynamics of the \bath" or environment to which is couples is less interesting, but still must be accounted for in some manner. A model that has maintained a certain level of both popularity and success is the so called \harmonic bath" model, in which the environment to which the special degree(s) of freedom couple is replaced by an eective set of harmonic oscillators. We will examine this model for the case of a single degree of freedom of interest, which we will designate q. For the case of the isomerizing or dissociating diatomic, q could be the coordinate r ; hri, where r is the distance between the atoms. The particular denition of q ensures that hqi = 0. The degree of freedom q is assumed to couple to the bath linearly, giving a Hamiltonian of the form "




2 X p2 1 g 2 H = 2pm + (q) + 2m + 2 m ! x + m !2 q

2 #

where the index  runs over all the bath degrees of freedom, ! are the harmonic bath frequencies, m are the harmonic bath masses, and g are the coupling constants between the bath and the coordinate q. p is a momentum conjugate to q, and m is the mass associated with this degree of freedom (e.g., the reduced mass  in the case of a diatomic). The coordinate q is assumed to be subject to a potential (q) as well (e.g., an internal bond potential). The form of the coupling between the system (q) and the bath (x ) is known as bilinear. Below, using a completely classical treatment of this Hamiltonian, we will derive an equation for the detailed dynamics of q alone. This equation is known as the generalized Langevin equation (GLE).

II. DERIVATION OF THE GLE The GLE can be derived from the harmonic bath Hamiltonian by simply solving Hamilton's equations of motion, which take the form q_ = mp X X 2 p_ = ; @@q ; g x ; mg !2 q x_ = mp p_ = ;m !2 x ; g q

This set of equations can also be written as second order dierential equation: X X 2 mq = ; @@q ; g x ; mg !2 q

m x = ;m !2 x ; g q

In order to derive an equation for q, we solve explicitly for the dynamics of the bath variables and then substitute into the equation for q. The equation for x is a second order inhomogeneous dierential equation, which can be solved by Laplace transforms. We simply take the Laplace transform of both sides. Denote the Laplace transforms of q and x as 1

q~(s) = x~ = and recognizing that

Z

1

0

Z

1

Z0 1 0

dt e;st q(t) dt e;st x (t)

dt e;st x (t) = s2 x~ (s) ; sx (0) ; x_ (0)

we obtain the following equation for x~ (s):

(s2 + !2 )~x (s) = sx (0) + x_ (0) ; mg q~(s)

or

x~ (s) = s2 +s !2 x (0) + s2 +1 !2 x_ (0) ; mg s2q~+(s!) 2

x (t) can be obtained by inverse Laplace transformation, which is equivalent to a contour integral in the complex s-plane around a contour that encloses all the poles of the integrand. This contour is known as the Bromwich contour. To see how this works, consider the rst term in the above expression. The inverse Laplace transform is 1 I ds sest = 1 I ds sest 2 2 2i s +! 2i (s + i! )(s ; i! ) The integrand has two poles on the imaginary s-axis at
i! . Integration over the contour that encloses these poles picks up both residues from these poles. Since the poles are simple poles, then, from the residue theorem: 
 1 I ds sest 1 2i i! ei! t + ;i! e;i! t = cos ! t = 2i (s + i! )(s ; i! ) 2i 2i! ;2i! By the same method, the second term will give (sin ! t)=! . The last term is the inverse Laplace transform of a product of q~(s) and 1=(s2 + !2 ). From the convolution theorem of Laplace transforms, the Laplace transform of a convolution gives the product of Laplace transforms:

1

Z

0

dt e;st

Zt 0

d f ( )g(t ;  ) = f~(s)~g(s)

Thus, the last term will be the convolution of q(t) with (sin ! t)=! . Putting these results together, gives, as the solution for x (t): Zt x (t) = x (0) cos ! t + x_ !(0) sin ! t ; mg ! dq( ) sin ! (t ;  ) 0

The convolution term can be expressed in terms of q_ rather than q by integrating it by parts:

g m!

Zt 0

Zt d q( ) sin ! (t ;  ) = mg !2 q(t) ; q(0) cos ! t] ; mg !2 d q_( ) cos ! (t ;  ) 0

The reasons for preferring this form will be made clear shortly. The bath variables can now be seen to evolve according to Zt x (t) = x (0) cos ! t + x_ (0) sin ! t + g d q_( ) cos ! (t ;  ) ; g q(t) ; q(0) cos ! t]

!

m !2 0 m !2 Substituting this into the equation of motion for q, we nd  X  p (0) sin ! t + g q(0) cos ! t ;X g2 Z t d q_( ) cos ! (t; )+X g2 q(t);X mq = ; @@q ; g x (0) cos ! t + m ! m !2 m !2 0 m !2 We now introduce the following notation for the sums over bath modes appearing in this equation: 2

1. Dene a dynamic friction kernel

 (t) = 2. Dene a random force

R(t) = ;

X

g

X




g cos ! t m !2 



p (0) sin ! t x (0) + mg !2 q(0) cos ! t + m !

Using these denitions, the equation of motion for q reads

Zt mq = ; @@q ; d q_( ) (t ;  ) + R(t) 0

(1)

Eq. (1) is known as the generalized Langevin equation. Note that it takes the form of a one-dimensional R particle subject to a potential (q), driven by a forcing function R(t) and with a nonlocal (in time) damping term ; 0t d q_( ) (t ;  ), which depends, in general, on the entire history of the evolution of q. The GLE is often taken as a phenomenological equation of motion for a coordinate q coupled to a general bath. In this spirit, it is worth taking a moment to discuss the physical meaning of the terms appearing in the equation.

III. PROPERTIES OF THE GLE Below we discuss the physical meaning of the terms appearing the GLE

A. The random force term Within the context of a harmonic bath, the term \random force" is something of a misnomer, since R(t) is completely deterministic and not random at all!!! We will return to this point momentarily, however, let us examine particular features of R(t) from its explicit expression from the harmonic bath dynamics. Note, rst of all, that it does not depend on the dynamics of the system coordinate q (except for the appearance of q(0)). In this sense, it is independent or \orthogonal" to q within a phase space picture. From the explicit form of R(t), it is straightforward to see that the correlation function hq_(0)R(t)i = 0 i.e., the correlation function of the system velocity q_ with the random force is 0. This can be seen by substituting in the expression for R(t) and integrating over initial conditions with a canonical distribution weighting. For certain potentials (q) that are even in q (such as a harmonic oscillator), one can also show that hq(0)R(t)i = 0 Thus, R(t) is completely uncorrelated from both q and q_, which is a property we might expect from a truly random process. In fact, R(t) is determined by the detailed dynamics of the bath. However, we are not particularly interested or able to follow these detailed dynamics for a large number of bath degrees of freedom. Thus, we could just as well model R(t) by a completely random process (satisfying certain desirable features that are characteristic of a more general bath), and, in fact, this is often done. One could, for example, postulate that R(t) act over a maximum time tmax at discrete points in time kt, giving N = tmax=t values of Rk = R(kt), and assume that Rk takes the form of a gaussian random process:

Rk =

N h X j =1

aj e2ijk=N + bj e;2ijk=N

i

where the coecients faj g and fbj g are chosen at random from a gaussian distribution function. This might be expected to be suitable for a bath of high density, where strong collisions between the system and a bath particle are essentially nonexistent, but where the system only sees feels the relatively \soft" uctuations of the less mobile bath. For a low density bath, one might try modeling R(t) as a Poisson process of very strong collisions. Whatever model is chosen for R(t), if it is a truly random process that can only act at discrete points in time, then the GLE takes the form of a stochastic (based on random numbers) integro-dierential equation. There is a whole body of mathematics devoted to the properties of such equations, where heavy use of an It^o calculus is made. 3

B. The dynamic friction kernel The convolution integral term

Zt 0

d q_( ) (t ;  )

is called the memory integral because it depends, in general, on the entire history of the evolution of q. Physically it expresses the fact that the bath requires a nite time to respond to any uctuation in the motion of the system (q). This, in turn, aects how the bath acts back on the system. Thus, the force that the bath exerts on the system presently depends on what the system coordinate q did in the past. However, we have seen previously the regression of uctuations (their decay to 0) over time. Thus, we expect that what the system did very far in the past will no longer the force it feels presently, i.e., that the lower limit of the memory integral (which is rigorously 0) could be replaced by t ; tmem, where tmem is the maximum time over which memory of what the system coordinate did in the past is important. This can be interpreted as a indicating a certain decay time for the friction kernel  (t). In fact,  (t) often does decay to 0 in a relatively short time. Often this decay takes the form of a rapid initial decay followed by a slow nal decay, as shown in the gure below: Consider the extreme case that the bath is capable of responding innitely quickly to changes in the system coordinate q. This would be the case, for example, if there were a large mass disparity between the system and the bath (m >> m ). Then, the bath retains no memory of what the system did in the past, and we could take  (t) to be a

-function in time:

 (t) = 20 (t) Then

Zt 0

d q_( ) (t ;  ) =

Zt

and the GLE becomes

0

d q_(t ;  ) ( ) = 20

Zt 0

d ( )q_(t ;  ) = 0 q_(t)

mq = ; @@q ; 0 q_ + R(t)

This simpler equation of motion is known as the Langevin equation and it is clearly a special case of the more generalized equation of motion. It is often invoked to describe brownian motion where clearly such a mass disparity is present. The constant 0 is known as the static friction and is given by

0 =

1

Z

0

dt  (t)

In fact, this is a general relation for determining the static friction constant. The other extreme is a very sluggish bath that responds slowly to changes in the system coordinate. In this case, we may take  (t) to be a constant    (0), at least, for times short compared to the response time of the bath. Then, the memory integral becomes Zt 0

and the GLE becomes

d q_( ) (t ;  )   (q(t) ; q(0))

@ mq = ; @q






(q) + 21  (q ; q0 )2 + R(t)

where the friction term now manifests itself as an extra harmonic term added to the potential. Such a term has the eect of trapping the system in certain regions of conguration space, an eect known as dynamic caging. An example of this is a dilute mixture of small, light particles in a bath of heavy, large particles. The light particles can get trapped in regions of space where many bath particles are in a sort of spatial \cage." Only the rare uctuations in the bath that open up larger holes in conguration space allow the light particles to escape the cage, occasionally, after which, they often get trapped again in a new cage for a similar time interval. 4

C. Relation between the dynamic friction kernel and the random force From the denitions of R(t) and  (t), it is straightforward to show that there is a relation between them of the form

hR(0)R(t)i = kT (t) This relation is known as the second uctuation dissipation theorem. The fact that it involves a simple autocorrelation function of the random force is particular to the harmonic bath model. We will see later that a more general form of this relation exists, valid for a general bath. This relation must be kept in mind when introducing models for R(t) and  (t). In eect, it acts as a constraint on the possible ways in which one can model the random force and friction kernel.

IV. MORI-ZWANZIG THEORY: A MORE GENERAL DERIVATION OF THE GLE A derivation of the GLE valid for a general bath can be worked out. The details of the derivation are given in the book by Berne and Pecora called Dynamic Light Scattering. The system coordinate q and its conjugate momentum p are introduced as a column vector:


A = pq

and, in addition, one introduces statistical projection operators P and Q that project onto subspaces in phase space parallel and orthogonal to A. These operators take the form

P = h:::AT ihAAT i;1 Q=I ;P

These operators are Hermitian and satisfy the property of idempotency:

P2 = P Q2 = Q Also, note that

PA = A QA = 0

The time evolution of A is given by application of the classical propagator:

A(t) = eiLtA(0)

Note that the evolution of A is unitary, i.e., it preserves the norm of A:

jA(t)j2 = jA(0)j2

Dierentiating both sides of the time evolution equation for A gives:

dA = eiLt iLA(0) dt

Then, an identity operator is inserted in the above expression in the form I = P + Q:

dA = eiLt (P + Q)iLA(0) = eiLtPiLA(0) + eiLtQiLA(0) dt

The rst term in this expression denes a frequency matrix acting on A: 5

eiLt PiLA(0) = eiLt hiLAAT ihAAT i;1 A = hiLAAT ihAAT i;1 eiLt A = hiLAAT ihAAT i;1 A(t)

 i A(t) where

= hLAAT ihAAT i;1

In order to evaluate the second term, another identity operator is inserted directly into the propagator:

eiLt = ei(P +Q)Lt Consider the dierence between the two propagators:

eiLt ; eiQLt If this dierence is Laplace transformed, it becomes

(s ; iL);1 ; (s ; iQL);1

which can be simplied via the general operator identity:

A;1 ; B;1 = A;1 (B ; A)B;1

Letting A = (s ; iL) B = (s ; iQL) we have

or

(s ; iL);1 ; (s ; iQL);1 = (s ; iL);1 (s ; iQL ; s + iL)(s ; iQL);1 = (s ; iL);1 iPL(s ; iQL);1 (s ; iL);1 = (s ; iQL);1 + (s ; iL);1(s ; iQL ; s + iL)(s ; iQL);1

Now, inverse Laplace transforming both sides gives

eiLt = eiQLt + Thus, multiplying fromthe right by QiLA gives

Zt 0

eiLt QiLA = eiQLt QiLA + Dene a vector

d eiL(t; )iPLeiQL Zt 0

d eiL(t; )iPLeiQL QiLA

F(t) = eiQLt QiLA(0)

so that

eiLt QiLA = F(t) +

Zt 0

d hiLF( )AT ihAAT i;1 A(t ;  ) 6

Because F(t) is completely orthogonal to A(t), it is straightforward to show that QF(t) = F(t) Then, hiLF( )AT ihAAT i;1 A = hiLQF( )AT ihAAT i;1 A = ;hQF( )(iLA)T ihAAT i;1 A = ;hQ2 F( )(iLA)T ihAAT i;1 A = ;hQF( )(QiLA)T ihAAT i;1 A = ;hF( )FT (0)ihAAT i;1 A

Thus,

A = F(t) ;

eiLt QiL

Finally, we dene a memory kernel matrix:

Zt 0

d hF( )FT (0)ihAAT i;1 A(t ;  )

K(t) = hF( )FT (0)ihAAT i;1 Then, combining all results, we nd, for dA=dt: dA = i (t)A ; Z t d K( )A(t ;  ) + F(t) dt 0

which equivalent to a generalized Langevin equation for a particle subject to a harmonic potential, but coupled to a general bath. For most systems, the quantities appearing in this form of the generalized Langevin equation are
 0 1=m i = ;m! 2 0


K(t) = 00  (t)0=m






F(t) = R0(t) It is easy to derive these expressions for the case of the harmonic bath Hamiltonian when (q) = m!2 q2 =2. For the case of a harmonic bath Hamiltonian, we had shown that the friction kernel was related to the random force by the uctuation dissipation theorem: hR(0)R(t)i = hR(0)eiLt R(0)i = kT (t) For a general bath, the relation is not as simple, owing to the fact that F(t) is evolved using a modied propagator exp(iQLt). Thus, the more general form of the uctuation dissipation theorem is hR(0)eiQLt R(0)i = kT (t) so that the dynamics of R(t) is prescribed by the propagator exp(iQLt). This more general relation illustrates the diculty of dening a friction kernel for a general bath. However, for the special case of a sti harmonic diatomic molecule interacting with a bath for which all the modes are soft compared to the frequency of the diatomic, a very useful approximation results. One can show that hR(0)eiQLt R(0)i  hR(0)eiLcons t R(0)i where iLcons is the Liouville operator for a system in which the diatomic is held rigidly xed at some particular bond length (i.e., a constrained dynamics). Since the friction kernel is not sensitive to the details of the internal potential of the diatomic, this approximation can also be used for diatomics with sti, anharmonic potentials. This approximation is referred to as the rigid bond approximation (see Berne, et al, J. Chem. Phys. 93, 5084 (1990)). 7

V. EXAMPLE: VIBRATIONAL DEPHASING AND ENERGY RELAXATION Recall that the Fourier transform of a time correlation function can be related to some kind of frequency spectrum. For example, the Fourier transform of the velocity autocorrelation function of a particular degree of freedom q of interest q_(t)i Cvv (t) = hq_(0) hq_2 i where v = q_, gives the relevant frequencies contributing to the dynamics of q, but does not give amplitudes. This \frequency" spectrum I (!) is simply given by

I (!) =

1

Z

0

dt ei!t Cvv (t)

That is, we take the Laplace transform of Cvv (t) using s = ;i!. Since Cvv (t) carries information about the relevant frequencies of the system, the decay of Cvv (t) in time is a measure of how strongly coupled the motion of q is to the rest of the bath, i.e., how much of an overlap there is between the relevant frequencies of the bath and those of q. The more of an overlap there is, the more mixing there will be between the system and the bath, and hence, the more rapidly the motion of the system will become vibrationally \out of phase" or decorrelated with itself. Thus, the decay time of Cvv (t), which is denoted T2 is called the vibrational dephasing time. Another measure of the strength of the coupling between the system and the bath is the time required for the system to dissipate energy into the bath when it is excited away from equilibrium. This time can be obtained by studying the decay of the energy autocorrelation function: C = h"(0)"(t)i ""

h"2 i

where "(t) is dened to be

"(t) = 21 mq_2 + (q) ; kT

The decay time of this correlation function is denoted T1. The question then becomes: what are these characteristic decay times and how are they related? To answer this, we will take a phenomenological approach. We will assume the validity of the GLE for q: Zt mq = ; @@q ; d q_( ) (t ;  ) + R(t) 0

and use it to calculate T1 and T2 . Suppose the potential (q) is harmonic and takes the form (q) = 12 m!2 q2 Substituting into the GLE and dividing through by m gives

q = ;!2q ; where

Zt 0

d q_(t ;  ) ( ) + f (t)

(t)

(t) = m

f (t) = Rm(t)

An equation of motion for Cvv (t) can be obtained directly by multiplying both sides of the GLE by q_(0) and averaging over a canonical ensemble:

hq_(0)q(t)i = ;!2hq_(0)q(t)i ;

Zt 0

d hq_(0)q_(t ;  )i ( ) + hq_(0)f (t)i 8

Recall that

hq_(0)f (t)i = m1 hq_(0)R(t)i = 0

and note that also

hq_(0)q(t)i = dtd hq_(0)q_(t)i = dCdtvv Zt 0

d hq_(0)q_( )i = hq_(0)q(t)i ; hq_(0)q(0)i = hq_(0)q(t)i

Thus,

hq_(0)q(t)i =

Zt 0

d Cvv ( )

Combining these results gives an equation for Cvv (t)

d C (t) = ; Z t d ;!2 + (t ;  ) C ( ) vv dt vv 0 d C (t) = ; Z t d K (t ;  )C ( ) vv dt vv 0

which is known as the memory function equation and the kernel K (t) is known as the memory function or memory kernel. This type of integro-dierential equation is called a Volterra equation and it can be solved by Laplace transforms. Taking the Laplace transform of both sides gives sC~vv (s) ; Cvv (0) = ;C~vv (s)K~ (s) However, it is clear that Cvv (0) = 1 and also 2 K~ (s) = !s + ~(s)

Thus, it follows that






2 sC~vv (s) ; 1 = !s + ~(s) C~vv (s)

C~vv (s) = s2 + s ~s(s) + !2

In order to perform the inverse Laplace transform, we need the poles of the integrand, which will be determined by the solutions of

s2 + s ~(s) + !2 = 0 which we could solve directly if we knew the explicit form of ~(s). However, if ! is suciently larger than ~(0), then it is possible to develop a perturbation solution to this equation. Let us assume the solutions for s can be written as s = s0 + s1 + s2 +    Substituting in this ansatz gives (s0 + s1 + s2 +   )2 + (s0 + s1 + s2 +   )~ (s0 + s1 + s2 +   ) + !2 = 0 Since we are assuming ~ is small, then to lowest order, we have 9

s20 + !2 = 0

so that s0 =
#!. The rst order equation then becomes 2s0 s1 + s0 ~(s0 ) = 0 or s1 = ; ~(2s0 ) = ; ~(
2i!) Note, however, that

~(
i!) =

Z

1

Z0

1

dt (t)ei!t

dt  (t) cos !t
i (t) sin !t]  0 (!)
i 00(!) =

0

Thus, stopping the rst order result, the poles of the integrand occur at 0 0 s 
i (! + 00 (!)) ; (2!) 
i$ ; (2!) Dene 0 s+ = i$ ; (2!) 0 s; = ;i$ ; (2!) Then

C~vv (s)  (s ; s )(s s ; s ) + ;

and Cvv (t) is then given by the contour integral

Cvv (t) = 21i Taking the residue at each pole, we nd

which can be simplied to give

I

sest ds (s ; s+ )(s ; s; )

s+ t s; t Cvv (t) = (ss+;e s ) + (ss;;e s ) + ; ; +

  0(!) 0 (!)t=2

;  Cvv (t) = e cos $t ; 2$ sin $t

Thus, we see that the GLE predicts Cvv (t) oscillates with a frequency $ and decays exponentially. From the exponential decay, we can directly read o the time T2 : 1 = 0 (!) =  0 (!) T2 2 2m That is, the value of the real part of the Fourier (Laplace) transform of the friction kernel evaluated at the renormalized frequency divided by 2m gives the vibrational dephasing time! By a similar scheme, one can easily show that the position autocorrelation function Cqq (t) = hq(0)q(t)i decays with the same dephasing time. It's explicit form is   0 (! ) 0 (!)t=2

;  Cqq (t) = e cos $t + 2$ sin $t 10

The energy autocorrelation function C"" (t) can be expressed in terms of the more primitive correlation functions Cqq (t) and Cvv (t). It is a straightforward, although extremely tedious, matter to show that the relation, valid for the

harmonic potential of mean force, is

2 (t) + 1 C 2 (t) + 1 C_ 2 (t) C"" (t) = 12 Cvv 2 qq !2 qq

Substituting in the expressions for Cqq (t) and Cvv (t) gives

C"" (t) = e; 0(!)t  (oscillatory functions of t)

so that the decay time T1 can be seen to be

1 = 0 (!) =  0 (!) T m 1

and therefore, the relation between T1 and T2 can be seen immediately to be 1 1 T2 = 2T1 The incredible fact is that this result is also true quantum mechanically. That is, by doing a simple, purely classical treatment of the problem, we obtained a result that turns out to be the correct quantum mechanical result! Just how big are these times? If ! is very large compared to any typical frequency relevant to the bath, then the friction kernel evaluated at this frequency will be extremely small, giving rise to a long decay time. This result is expect, since, if ! is large compared to the bath, there are very few ways in which the system can dissipate energy into the bath. The situation changes dramatically, however, if a small amount of anharmonicity is added to the potential of mean force. The gure below illustrates the point for a harmonic diatomic molecule interacting with a Lennard-Jones bath. The top gure shows the velocity autocorrelation function for an oscillator whose frequency is approximately 3 times the characteristic frequency of the bath, while the bottom one shows the velocity autocorrelation function for the case that the frequency disparity is a factor of 6.

11

FIG. 1.

12

G25.2651: Statistical Mechanics Notes for Lecture 25

I. OVERVIEW OF CRITICAL PHENOMENA Consider the phase diagram of a typical substance:

P Melting curve

Solid

Supercritical fluid region

Liquid

.

Pc

Critical point

.

Triple point

Boiling curve

Gas

Sublimation curve

Tc

T

FIG. 1.

The boundary lines between phases are called the coexistence lines. Crossing a coexistence line leads to a rst order phase transition, which is characterized by a discontinuous change in some thermodynamic quantity such as volume, enthalpy, magnetization. Notice that, while the melting curve, in principle, can be extended to innity, the gas-liquid/boiling curve terminates at a point, beyond which the two phases cannot be distinguished. This point is called a critical point and beyond the critical point, the system is in a supercritical
uid state. The temperature at which this point occurs is called the critical temperature Tc. At a critical point, thermodynamic quantities such as given above remain continuous, but derivatives of these functions may diverge. Examples of such divergent quantities are the heat capacity, isothermal compressibility, magnetic susceptibility, etc. These divergences occur in a universal way for large classes of systems (such systems are said to belong to the same universality class, a concept to be dened more precisely in the next lecture). Recall that the equation of state of a system can be expressed in the form

P = g(
T ) i.e., pressure is a function of density an temperature. A set of isotherms of the equation of state for a typical substance might appear as follows: 1

T > Tc

P

T = Tc T < Tc

Inflection pt.

.

Pc

Linear regime

ρG

ρ

ρ

ρ

L

c

FIG. 2.

The dashed curve corresponds to the gas-liquid coexistence curve. Below the critical isotherm, the gas-liquid coexistence curve describes how large a discontinuous change in the density occurs during rst-order gas-liquid phase transition. At the in ection point, which corresponds to the critical point, the discontinuity goes to 0. As noted above, the divergences in thermodynamic derivative quantities occur in the same way for systems belonging to the same universality class. These divergences behave as power laws, and hence can be described by the exponents in the power laws. These exponents are known as the critical exponents. Thus, the critical exponents will be the same for all systems belonging to the same universality class. The critical exponents are dened as follows for the gas-liquid critical point: 1. The heat capacity at constant volume dened by

CV = @E @T




V

2 = ;T @@TA2




V

diverges with temperature as the critical temperature is approached according to

CV
jT ; Tc j;

2. The isothermal compressibility dened by

T = ; V1 @V @P


T

@ = 1 @T


T

diverges with temperature as the critical temperature is approached according to

T
jT ; Tc j;
2

3. On the critical isotherm, the shape of the curve near the in ection point at ( c
Pc ) is given by P ; Pc
j ; c j sign( ; c ) >0 4. The shape of the coexistence curve in the -T plane near the critical point for T < Tc is given by  L ; G
(Tc ; T ) These are the primary critical exponents.

A. Review of the Van der Waals theory Recall that the Van der Waals equation of state was derived earlier by perturbation theory. The unperturbed Hamiltonian describes a system of hard spheres and is given by

H0 =

N 2 X X pi + u0 (jri ; rj j) i=1 2m i<j

and a perturbation of the form

X

H1 = where u0 (r) is the hard-sphere potential given by

i<j

u1 (jri ; rj j)



r> r

0 u 0 (r ) = 1

and u1 (r) is an arbitrary attractive potential. In the low density limit, we had

g(r)  e;u0 (r) = (r ; ) and the free energy was determined to be

  )N ; aN 2 A(N
V
T )  ; 1 ln (VN;! Nb 3N V

with

b = 23 3 Z1 a = ;2 dr r2 u1 (r) 

The equation of state takes the form The critical point is dened by the conditions:

P = 1 ;kT b ; a

@P @V = 0

2

@2P = 0 @V 2

which leads to the following values for the critical pressure, temperature and density: 8a 1 Pc = 27ab2 Tc = 27 c= b 3b The critical exponents predicted by the theory are as follows: 3

1. The internal energy is given by

2 E = 32 NkT + aN V

from which it can be seen that the heat capacity is


@E CV = @T = 32 Nk
jT ; Tcj0 V

) = 0.

2. The isothermal compressibility can be expressed as

1 T = ; V (@P=@V )

and


1 8 a @P  @V V =Vc = 4Nb2 27b ; kT
(T ; Tc )

so that

T
(T ; Tc);1

)=1 3. By Taylor expanding the equation of state about the critical pressure and density, it is easy to show that 1 (P ; P )
const + ( ; )3 +    c

kT

)=3

c

4. The exponent can be computed using the Maxwell construction (see problem set 11), which attempts to x the fact that the Van der Waals equation has an unphysical region where @P=@V > 0. The Maxwell construction is illustrated below:

P

Van der Waals eqn

a2

Tie line

a1

VG

VL FIG. 3.

4

V

When the Maxwell construction is carried out, it can be shown that L

;


(Tc ; T )1=2

G

) = 1=2. The following table compares the Van der Waals exponents to the experimental critical exponents:

α

β

γ

δ

VdW

0

1/2

1

3

Exp.

0.1

0.34

1.35

4.2

FIG. 4.

Thus, one sees that the Van der Waals theory is only qualitatively correct, but not quantitatively. It is an example of a mean eld theory.

II. MAGNETIC SYSTEMS AND THE ISING MODEL Imagine a cubic lattice in which particles carrying spin S are placed on the lattice sites with jSj = h =2 as shown below:

FIG. 5.

5

Such a model describes ferromagnetic materials, which can be magnetized by applying an external magnetic eld . In the absence of a eld, the unperturbed Hamiltonian takes the form X H0 = ; 12 i  Jij  j ij

h

where J is a tensor and  is a spin vector such that jj = 1. Quantum mechanically,  would be the vector of Pauli matrices. In general, the spins can point in any spatial direction, a fact that makes the problem dicult to solve. A simplication introduced by Ising was to allow the spins to point in only one of two possible directions, up or down, e.g., along the z -axis only. In addition, the summation is restricted to nearest neighbor interactions only. In this model, the Hamiltonian becomes X H0 = ; 12 Jij i j hiji where hi
j iindicates restriction of the sum to nearest neighbor pairs only. The variables i now can take on the values 1 only. The couplings Jij are the spin-spin \J " couplings. In the presence of a magnetic eld, the full Hamiltonian becomes

H = ; 12

X hiji

Jij i j ; h

N X i=1

i

which describes a uniaxial ferromagnetic system in a magnetic eld. The parameters T and h are experimental control parameters. Dene the magnetization per spin as N X m = N1 i i=1

Then the phase diagram looks like:

h

Spin up phase

T Spin down phase

Tc

FIG. 6.

6

where the colored lines indicate a nonzero magnetization at h = 0 below a critical temperature T . The persistence of a nonzero magnetization in ferromagnetic systems at h = 0 below T = Tc indicates a transition from a disordered to an ordered phase. In the latter, the spins are aligned in the direction of the applied eld before it is switched o. If h ;! 0+ , then the spins will point in one direction and if h ;! 0; , it will be in the opposite direction. A plot of the isotherms of m vs. h yields:

m T < TC T = TC T>TC h

FIG. 7.

Notice an in ection point along the isotherm T = Tc, at h = 0, where @m=@h ;! 1. The thermodynamics of the magnetic system can be dened in analogy with the liquid-gas system. The analogy is shown in the table below: Gas-Liquid

Magnetic

P

h

V

;M = ;Nm

T = ; V1 @V @P

 = @m @h

A = A(N
V
T )

A = A(N
M
T )

G = A + PV (P )

G = A ; hM (h)

@A P = ; @V

@A h = @M

V = @G @P
2 CV = ;T @@TA2 V

M = ; @G @h 2 CM = ;T @@TA2

7


M

2 CP = ;T @@TG2




2 Ch = ;T @@TG2

P

where  is the magnetic susceptibility. The magnetic exponents are then given by

Ch
jT ; Tcj; h = 0 limit ;

jT ; Tcj h = 0 limit h
jmj sign(m) at T = Tc m
(Tc ; T ) T < Tc

8


h

G25.2651: Statistical Mechanics Notes for Lecture 26

I. MEAN FIELD THEORY CALCULATION OF MAGNETIC EXPONENTS The calculation of critical exponents is nontrivial, even for simple models such as the Ising model. Here, we will introduce an approximate technique known as mean eld theory. The approximation that is made in the mean eld theory (MFT) is that uctuations can be neglected. Clearly, this is a severe approximation, the consequences of which we will see in the nal results. Consider the Hamiltonian for the Ising model: X X H = 21 Jij i j + h i i hi ji The partition function is given by (N
h
T ) =

X

XX


1
2




e



h P 1 2

hi j i



+h i

j

P i




i

i

N

Notice that we have written the partition function as an isothermal-isomagnetic partition function in analogy with the isothermal-isobaric ensemble. (Most books use the notation Q for the partition function and A for the free energy, which is misleading). This sum is nontrivial to carry out. In the MFT approximation, one introduces the magnetization N X m = N1 i i=1 h

i

explicitly into the partition function by using the identity i j = ( i m + m)( j m + m) = m2 + m( i m) + m( j m) + ( i m)( j m) The last term is quadratic in the spins and is of the form ( i )( j ), the average of which measures the spin uctuations. Thus, this term is neglected in the MFT. If this term is dropped, then the spin-spin interaction term in the Hamiltonian becomes: 1 XJ 1 XJ m2 + m( + ) 2m2
i j ij i j 2 hi j i 2 hi ji ij X
= 12 Jij m2 + m( i + j ) hi j i ;

;

;

;

;h

;

i

;h



;

i

;

;

P

We will restrict ourselves to isotropic magnetic systems, for which j Jij is independent of i (all sites are equivalent). P ~ , where z is the number of nearest neighbors of each spin. This number will depend on the number Dene j Jij Jz of spatial dimensions. Since this dependence on spatial dimension is a trivial one, we can absorb the z factor into the ~ . Then, coupling constant and redene J = Jz 1 X J = 1 NJ 2 i 2 



where N is the total number of spins. Finally, 1 X J m( + ) = Jm X i 2 hi ji ij i j i 1

and the Hamiltonian now takes the form X 1 XJ + h ij i j i 2 hi ji i

;! ;

1 NJm2 + (Jm + h) X i 2 i

and the partition function becomes (N
h
T ) = e;NJm =2 2

X



= e;NJm

2


1 X

X




e(Jm+h)

N

e(Jm+h)


P

!N


=
1 2 =2 ; NJm =e 2cosh (Jm + h)]N

The free energy per spin g(h
T ) = G(N
h
T )=N is then given by g(h
T ) = 1 ln (N
h
T ) ;

N

= 12 Jm2 The magnetization per spin can be computed from

;

1 ln 2cosh (Jm + h)]







@g m = @h h = tanh (Jm + h) 0, one nds a transcendental equation for m m = tanh(mJ ) ;

Allowing h

;!

which can be solved graphically as shown below:

2

i




i

tanh(βJm)

f(m)=m

− m0 m m0

FIG. 1.

Note that for J > 1, there are three solutions. One is at m = 0 and the other two are at nite values of m, which we will call m0. For J < 1, there is only one solution at m = 0. Thus, for J > 1, MFT predicts a nonzero magnetization at h = 0. The three solutions coalesce onto a single solution at J = 1. The condition J = 1 thus denes a critical temperature below which (J > 1) there is a nite magnetization at h = 0. The condition J = 1 denes the critical temperature, which leads to 

kTc = J To see the physical meaning of the various cases, consider expanding the free energy about m = 0 at zero-eld. The expansion gives

g(0
m) = const + J (1 J )m2 + cm4 where c is a (possibly temperature dependent) constant with c > 0. For J > 1, the sign of the quadratic term is negative and the free energy as a function of m looks like: ;

3

g(0,m)

m m0

−m 0 FIG. 2.

Thus, there are two stable minima at m0 , corresponding to the two possible states of magnetization. Since a large portion of the spins will be aligned below the critical temperature, the magnetic phase is called an ordered phase. For J > 1, the sign of the quadratic term is positive and the free energy plot looks like: 

4

g(0,m)

m FIG. 3.

i.e., a single minimum function at m = 0, indicating no net magnetization above the critical temperature at h = 0. The exponent  can be obtained directly from this expression for the free energy. For T < Tc, the value of the magnetization is given by 

@g  @m m=m0 = 0 which gives

2J (T T )m + 4cm3 = 0 c 0 0 T 1 = m0 (Tc T ) 2 ;



;

Thus,  = 1=2. From the equation for the magnetization at nonzero eld, the exponent  is obtained as follows: m = tanh (Jm + h)  (Jm + h) = tanh;1 m 

3



Jm h kT m + m3 + = mk(T T ) + kT m3 ;



c

3 where the second line is obtained by expanding the inverse hyperbolic tangent about m = 0. At the critical temperature, this becomes ;

h



m3

so that  = 3. For the exponent , we need to compute the heat capacity at zero-eld, which is either Ch or Cm . In either case, we have, for T > Tc, where m = 0, 5

G = NkT ln 2 ;

so





2 Ch = T @@TG2 = 0 from which is it clear that  = 0. For T < Tc, Ch approaches a dierent value as T on T Tc is the same, so that  = 0 is still obtained. ;

j

;

!

Tc , however, the dependence

j

Finally, the susceptibility, which is given by

1 = @m @h = @h=@m but, near m = 0, 3 h = mk(T Tc ) + kT 3m @h 2 @m = k(T Tc) + kTm As the critical temperature is approached, m 0 and T Tc ;1 which implies = 1. ;

;

!

 j

;

j

The MFT exponents for the Ising model are, therefore

=0

 = 1=2

=1

=3

which are exactly the same exponents that the Van der Waals theory predict for the uid system. The fact that two (or more) dissimilar systems have the same set of critical exponents (at least at the MFT level) is a consequence of a more general phenomenon known as universality, which was alluded to in the previous lecture. Systems belonging to the same universality class will exhibit the same behavior about their critical points, as manifested by their having the same set of critical exponents. A universality class is characterized by two parameters: 1. The spatial dimension d. 2. The dimension, n, of the order parameter. An order parameter is dened as follows: Suppose the Hamiltonian H0 of a system is invariant under all the transformations of a group . If two phases can be distinguished by the appearance of a thermodynamic average  , which is not invariant under , then  is an order parameter for the system. G

h

h

i

G

i

The Ising system, for which H0 is given by

X H0 = 21 Jij i j ;

hi ji

is invariant under the group Z2 , which is the group that contains only two elements, an identity element and a spin reection transformation: Z2 = 1
1. Thus, under Z2 , the spins transform as ;

i

From the form of H0 is can be seen that H0 However, the magnetization

!

!

i

i

! ;

i

H0 under both transformations of Z2 , so that it is invariant under Z2 . X = 1N i m = N1 i h

6

i

is not invariant under a spin reection for T < Tc, when the system is magnetized. In a completely ordered state, with all spins aligned, under a spin reection m m. Thus, m is an order parameter for the Ising model, and, since it is a scalar quantity, its dimension is 1. Thus, the Ising model denes a universality class known as the Ising universality class, characterized by d = 3, n = 1 in three dimensions. Note that the uid system, which has the same MFT critical exponents as the Ising system, belongs to the same universality class. The order parameter for this system, by the analogy table dened in the last lecture, is the volume dierence between the gas an liquid phases, VL VG , or equivalently, the density dierence, L G . Although the solid phase is the truly ordered phase, while the gas phase is disordered, the liquid phase is somewhere in between, i.e., it is a partially ordered phase. The Hamiltonian of a uid is invariant under rotations of the coordinate system. Ordered and partially ordered phases break this symmetry. Note also that a true magnetic system, in which the spins can point in any spatial direction, need an order parameter that is the vector generalization of the magnetization: ! ;

;

;

N X 1 m= N i h

i=1

i

Since the dimension of the vector magnetization is 3, the true magnetic system belongs to the d = 3, n = 3 universality class.

II. EXACT SOLUTIONS OF THE ISING MODEL IN 1 AND 2 DIMENSIONS Exact solutions of the Ising model are possible in 1 and 2 dimensions and can be used to calculate the exact critical exponents for the two corresponding universality classes. In one dimension, the Ising Hamiltonian becomes:

H=

N X

;

i=1

Ji i+1 i i+1 h ;

N X i=1

i

which corresponds to N spins on a line. We will impose periodic boundary conditions on the spins so that N +1 = 1 . Thus, the topology of the spin space is that of a circle. Finally, let all sites be equivalent, so that Ji i+1 J . Then, 

N X

H= J ;

The partition function is then

i=1

(N
h
T ) =

X

X

i i+1 h ;




i=1

i


P P 1 e J

+1 + 2 h =1 (
+
+1 ) i


1

N X

i

N

i

i

i

i

N

In order to carry out the spin sum, let us dene a matrix P with matrix elements: P 0 = e J

+h(
+
)=2] 1 P 1 = e(J +h) 1 P 1 = e(J ;h) 1 P 1 = 1 P 1 = e;J Thus, the matrix P becomes is a 2 2 matrix given by 0

h

j

h j

j

0

i

j i

h; j

j;

i

h j

j;

i

h; j

j i







1 jPj 2 ih 2 jPj 3 i

h

 (J +h) ;J P = e e;J ee(J ;h)

so that the partition function becomes X (N
h
T ) = =

X


1




X


1

h

N

h

1 jPN j 1 i

;

= Tr PN

7

N ;1 jPj N ih N jPj 1 i

A simple way to carry out the trace is diagonalize the matrix, P. From det(P I) = 0 ;

the eigenvalues can be seen to be 

= eJ cosh(h) 

q

sinh2 (h) + e;4J

 




where + corresponds to the choice of + in the eigenvalue expression, etc. The trace of the PN is then ;

Tr PN = N+ + N;

We will be interested in the thermodynamic limit. Note that + > ; for any h, so that as N over N; . Thus, in this limit, the partition function has the single term: (N
h
T )

;!

! 1

, N+ dominates

N+

Thus, the free energy per spin becomes

g(h
T ) = kT ln +   q = J kT ln cosh(h) + sinh2 (h) + e;4J ;

;

;

and the magnetization becomes 



@g m = @h = @ (@h) ln + h)cosh(h) sinh(h) + sinh( 2 (h)+e 4 sinh q = cosh(h) + sinh2 (h) + e;4J ;

p

;
J

which, as h 0, since cosh(h) 1 and sinh(h) 0, itself vanishes. Thus, there is no magnetization at any nite temperature in one dimension, hence no nontrivial critical point. While the one-dimensional Ising model is a relatively simple problem to solve, the two-dimensional Ising model is highly nontrivial. It was only the pure mathematical genius of Lars Onsager that was able to nd an analytical solution to the two-dimensional Ising model. This, then, gives an exact set of critical exponents for the d = 2, n = 1 universality class. To date, the three-dimensional Ising model remains unsolved. Here, the Onsager solution will be outlined only and the results stated. Consider a two-dimension spin-lattice as shown below: !

!

!

8

n+1 . . . .

2

1

. . . . . . . . . . . . . . . . . . . . . . . . 1

2

......

. n+1

FIG. 4.

The Hamiltonian can be written as X X H = J ( i j i+1 j + i j+1 i j ) h ij ;

;

ij

ij

where the spins are now indexed by two indices corresponding to a point on the 2-dimensional lattice. Introduce a shorthand notation for H :

H=

n X j =1

E (j
j+1 ) + E (j )]

where

n X

E (j
k )

 ;

E (j )

 ;

i=1 n X

J

ij ik

i=1

ij i+1 j ; h

and j is dened to be a set of spins in a particular column:

j

 f

1j
:::
nj g

Then, dene a transfer matrix P, with matrix elements: 9

X

ij

j

j P k = e; E(  )+E( )]

h

j

j

j

i

j

k

which is a 2n 2n matrix. The partition function will be given by 

 = Tr (Pn ) and, like, in the one-dimensional case, the largest eigenvalue of P is sought. This is the nontrivial problem that is worked out in 20 pages in Huang's book. In the thermodynamic limit, the nal result at zero eld is:   Z  q kT 1 2 2 g(T ) = kT ln 2cosh(2J )] 2 d ln 2 1 + 1 K sin  0 where 2 K = cosh(2J )coth(2 J ) The energy per spin is ;

;

;

Z  sin2  "(T ) = 2J tanh(2J ) + 2K dK d d 0 (1 + ) ;

where

q

 = 1 K 2 sin2  ;

The magnetization, then, becomes n

m = 1 sinh(2J )];4 ;

o1=8

for T < Tc and 0 for T > Tc , indicating the presence of an order-disorder phase transition at zero eld. The condition for determining the critical temperature at which this phase transition occurs turns out to be 2tanh2 (2J ) = 1 kTc 2:269185J 

Near T = Tc, the heat capacity per spin is given by        C (t) = 2 2J 2 ln 1 T + ln kTc k  kTc Tc 2J Thus, the heat capacity can be seen to diverge logarithmically as T Tc. ;

;

The critical exponents computed from the Onsager solution are

!

 = 0 (log divergence)  = 18

= 74  = 15 which are a set of exact exponents for the d = 2, n = 1 universality class.

10

1 + 4

 ;



G25.2651: Statistical Mechanics Notes for Lecture 27

I. THE EXPONENTS AND
Consider a spin-spin correlation function at zero
eld of the form 1 X X e;
H i j = i j h

Q

i

N

1

If i and j occupy lattice sites at positions ri and rj , respectively, then at large spatial separation, with r = the correlation function depends only r and decays exponentially according to

G(r)

i j i ; h i ih j i

 h



ri rj ,

j

;

j

e;r= rd;2+

for T < Tc. The quantity  is called the correlation length. Since, as a critical point is approached from above, long range order sets in, we expect  to diverge as T Tc+ . The divergence is characterized by an exponent  such that !



T Tc ;

 j

;

j

At T = Tc, the exponential dependence of G(r) becomes 1, and G(r) decays in a manner expected for a system with long range order, i.e., as some small inverse power of r. The exponent  appearing in the expression for G(r) characterizes this decay at T = Tc. The exponents,  and  cannot be determined from MFT, as MFT neglects all correlations. In order to calculate these exponents, a theory is needed that restores uctuations at some level. One such theory is the so called LandauGinzberg theory. Although we will not discuss this theory in great detail, it is worth giving a brief introduction to it.

II. INTRODUCTION TO LANDAU-GINZBERG THEORY The Landau-Ginzberg (LG) theory is a phenomenological theory meant to be used only near the critical point. Thus, it is formulated as a macroscopic theory. The basic idea of LG theory is to introduce a spin density
eld variable S (x) de
ned by

S (x) =

N X

i

i=1

Then the total magnetization is given by

M=

Z

(x

;

xi )

dd xS (x)

Since the free energy at zero
eld is A = A(N M T ), there should be a corresponding free energy density A(T S (x)) =

a(x) such that

A(N M T ) =

Z

dd xa(x)

It is assumed that a(x) can be represented as a power series according to

a(x) = a0 + a1 (T )S 2(x) + a2 (T )S 4 (x) + 1

such that a1 and a2 are both positive for T > Tc and a1 vanishes at T = Tc. These conditions are analogous to those exhibited by the total free energy in MFT above and near the critical point (see previous lecture). By symmetry, all odd terms vanish and are, therefore, not explicitly included. In the presence of a magnetic
eld h(x), we have a Gibbs free energy density given by

g(x) = a(x) h(x)S (x) = a0 + a1 (T )S 2(x) + a2 (T )S 4(x) h(x)S (x) In addition, a term a3 ( S )2 with a3 > 0 is added to the free energy in order to damp out local uctuations. If there are signi
cant local uctuations, then ( S )2 becomes large, so these
eld con
gurations contribute negligibly to the ;

;

r

partition function, which is de
ned by

Z= =

Z

D

Z D

S ] exp S ] exp

Z

r

 d xg(x)

;

Z

d




d ;

 d x a0 + a1 (T )S (x) + a2 (T )S (x) + a3 ( S )

;

2

4

r

2

;



h(x)S (x)




Thus, the LG theory is a
eld theory. The form of this
eld theory is well known in quantum
eld theory, and is known as a d-dimensional scalar Klein-Gordon theory. In terms of Z , a correlation function at zero
eld can be de
ned by 

2 ln Z   2 h(x)h(x0 ) h=0 Thus, by studying the behavior of the correlation function, the exponents  and  can be determined. For the choice a2 = 0, the theory can be solved exactly analytically. This is known as the Gaussian model, for 1 0 hS (x)S (x )i =

which

Z=

Z D

S ] exp

Z

d ;

 d x a0 + a1 (T )S (x) + a3 ( S )

;

2

r

2

;

h(x)S (x)






which leads to values of  and  of 0 and 1/2, respectively. These values are known as the \classical" exponents. They are independent of the number of spatial dimensions d. Dependence on d comes in at higher orders in S . Comparing these to the exact exponents from the Onsager solution, which gives  = 1=4 and  = 1, it can be seen that the classical exponents are only qualitatively correct. Going to higher orders in the theory leads to improved results, although the theory cannot be solved exactly analytically. It can either be solved numerically using path integral Monte Carlo or Molecular Dynamics or analytically perturbatively using Feynman diagrams.

III. RENORMALIZATION GROUP AND THE SCALING HYPOTHESIS A. General formulation The renormalization group (RG) has little to do with \group theory" as it is meant mathematically. Also, there is no uniqueness to the renormalization group, so the use of \the" in this context is misleading. Rather, the RG is an idea that exploits the physics of systems near their critical point which leads to a procedure for
nding the critical point. It also oers an explanation of universality, perhaps the closest thing there is to a proof of this concept. Finally, through the scaling hypothesis, it generates relations, called scaling relations satis
ed by the critical exponents. It does not allow actual determination of speci
c exponents. However, given a numerical calculation or some other method of determining a small subset of exponents, the scaling relations can be used to determine the remaining exponents. In order to see how the RG works, we will consider a speci
c example. Consider a square spin lattice:

2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

FIG. 1.

which has been separated into 3 3 blocks as shown. Consider de
ning a new spin lattice from the old by application of a coarse-graining procedure in which each 3 3 block is replaced by a single spin. The new spin is an up spin if the majority of spins in the block is up and down is the majority point down. The new lattice is shown below: 



3

.

.

.

. FIG. 2.

Such a transformation is called a block spin transformation. Near a critical point, the system will exhibit long-range ordering, hence the coarse-graining procedure should yield a new spin lattice that is statistically equivalent to the old spin lattice. If this is so, then the spin lattice is said to possess scale invariance. What will be the Hamiltonian for the new spin lattice? To answer this, consider the partition function at h = 0 for the old spin lattice using the Ising Hamiltonian as the starting point:

Q=

X X 1

N

e;
H0 (

1 :::

N)



Tr e;
H0 (

1 :::

N)

The block spin transformation can be expressed by de
ning, for each block, a transformation function: 

0 P9

i=1 i > 0 0 otherwise When inserted into the expression for the partition function, T acts to project out those con
gurations that are consistent with the block spin transformation, leaving a function of only the new spin variables 10  ::: N0 , in terms of which a new partition function can be de
ned. To see how this works, let the new Hamiltonian be de
ned through:

T ( 0  1  ::: 9 ) = 1

f

e;
H0 (f g) = Tr 0

"

Y

0

blocks

0g

#

T ( 0  1  ::: 9 ) e;
H0 (f g)

That this is a consistent de
nition follows from the fact that X

T ( 0  1  ::: 9 ) = 1

0

Thus, tracing both sides of the projected partition function expression over 0 yields: Tr e;
H0 (f g) = Tr e;
H0 (f g) which states that the partition function is preserved by the block spin transformation, hence the physical properties are also preserved. Transformations of this type should be chosen so as to preserve the functional form of the Hamiltonian, for if this is done, then the transformation can be iterated in exactly the same way for each new lattice produced by the previous iteration. The importance of being able to iterate the procedure is that, in a truly ordered state formed at a critical point, the iteration transformation will produce exactly the same lattice as the previous iteration, thus signifying the existence of a critical point. If the functional form of the Hamiltonian is preserved, then only its parameters are aected by the transformation, so we can think of the transformation as acting on these parameters. If the original Hamiltonian contains parameters K1  K2  ::: K (e.g., the J coupling in Ising model), then the transformation yields a Hamiltonian with a new set of parameters K0 = (K10  K20  :::), such that the new parameters are functions of the old parameters 0

0

0



4

K0 = R(K)

The vector function R characterizes the transformation. These equations are called the renormalization group equations or renormalization group transformations. By iterating the RG equations, it is possible to determine if a system has an ordered phase or not and for what values of the parameters such a phase will occur.

B. Example: The one-dimensional Ising model For the one-dimensional Ising model: N X

H0 = J ;

i=1

i i+1

De
ne: = H K = J

H0

so that H0

= K

N X

;

i=1

i i+1

and the partition function becomes

Q = Tr e;H0 We will consider a simple block spin transformation as illustrated below:

σ1

. σ1

σ2

. σ2

3

2

1

σ3

. σ3

σ1

σ2

.

. σ5

σ4

σ3

. σ6

σ1

. σ7

σ2

. σ8

FIG. 3.

The
gure shows the one-dimension spin lattice numbered in two dierent ways { one a straight numbering and one using blocks of three spins, with spins in each block numbered 1-3. The block spin transformation to be employed here is that the spin of a block will be determined by the value of the spin in the center of the block. Thus, for block 1, it is the value of spin 2, for block 2, it is the value of spin 5, etc. This rather undemocratic choice should be reasonable at low temperature, where local ordering is expected, and spins close to the center spin would be expected to be aligned with it, anyway. The transformation function, T for this case is

T ( 0 1 2 3) = 

The new lattice will look like: 5

0

2

.

.

. σ 3’

σ2 ’

σ’ 1

FIG. 4.

with 10 = 2 , 20 = 5 , etc. The new Hamiltonian is computed from

XXX e;cHz (f g) = 0

X;

0

1

=

2

3

XXXX 1

3

4

N



eK

0 1 2



0

1 1

 K e

0 2 5

eK

0

1 3

eK

1 2

3 4

eK

eK

2 3

eK

3 4

eK

4 5

0

4 2

6

The idea is then to
nd a K 0 such that when the sum over 3 and 4 are performed, the new interaction between 10 and 20 is of the form exp(K 0 10 20 ), which preserves the functional form of the old Hamiltonian. The sum over 3 and 4 is XX K e 3

Note that

3 4

0

1 3

eK

3 4

eK

0

4 2

4

= 1. Then, since 

e = cosh + sinh = cosh 1 + tanh] e; = cosh sinh = cosh 1 tanh] ;

we can express exp(K

3 4

;

) as

eK

3 4

= coshK 1 +

3 4

tanhK ]

Letting x = tanhK , the product of the three exponentials becomes:

eK

0

1 3

eK

3 4

eK

0

4 2

When summed over

= cosh3 K (1 + 10 3 x)(1 + 3 4 x)(1 + 4 20 x) = cosh3 K (1 + 10 3 x + 3 4 x + 4 20 x + 10 32 4 x2 + 10 3

2 0 2 3 4 2

x + 10

2 2 0 3 3 4 2

x)

and 4 , most terms in the above expression will cancel, yielding the following expression:

XX 3

0 x2 +

3 4 2

eK

0

1 3

eK

3 4

eK

0

4 2

 = 2cosh3 K 1 + 10 20 x3



coshK 0 1 + 10 20 x0 ]

4

where the last expression puts the interaction into the original form with a new coupling constant K 0. One of the possible choices for the new coupling constant is tanhK 0 = tanh3 K K 0 = tanh;1 tanh3 K This, then, is the RG equation for this particular block spin transformation. With this identi
cation of K 0, the new Hamiltonian can be shown to be N0

0 0 g) = N 0 g(K ) ; K 0 X 0 0 H0 (f i i+1 i=1

where the spin-independent function g(K ) is given by 6

3
K 2 ln 2 g(K ) = 13 ln cosh coshK 0 3 ;

;

Thus, apart from the additional term, the new Hamiltonian is exactly the same functional form as the original Hamiltonian but with a dierent set of spin variables and a dierent coupling constant. The transformation can now be applied to the new Hamiltonian, yielding the same relation between the new and old coupling constants. This is equivalent to iterating the RG equation. Since the coupling constant K depends on temperature through K = J=kT , the purpose of the iteration would be to
nd out if, for some value of K , there is an ordered phase. In an ordered phase, the transformed lattice would be exactly the same as the old lattice, and hence the same coupling constant and Hamiltonian would result. Such points are called
xed points of the RG equations and are generally given by the condition

K = R(K)

The
xed points correspond to critical points. For the one-dimensional Ising model, the
xed point condition is 

K = tanh;1 tanh3 K



or, in terms of x = tanhK ,

x = x3 Since K is restricted to K 0, the only solutions to this equation are x = 0 and x = 1, which are the
xed points of the RG equation. To see what these solutions mean, consider the RG equation away from the
xed point: 

x0 = x3 Since K = J=kT , at high T , K 0 and x = tanhK RG equation as an iteration or recursion of the form !

!

0+ . At low temperature, K

! 1

and x

!

1;. Viewing the

xn+1 = x3n if we start at x0 = 1, each successive iteration will yield 1. However, for any value of x less than 1, the iteration eventually goes to 0 in some
nite (though perhaps large) number of iterations of the RG equation. This can be illustrated pictorially as shown below:

Stable

Unstable

.

.

x=1

x=0

K=

K=0 FIG. 5.

The iteration of the RG equation produces an RG ow through coupling constant space. The
xed point at x = 1 is called an unstable
xed point because any perturbation away from it, if iterated through the RG equation, ows away from this point to the other
xed point, which is called a stable
xed point. As the stable
xed point is approached, the coupling constant gets smaller and smaller until, at the
xed point, it is 0. The absence of a
xed point for any
nite, nonzero value of temperature tells us that there can be no ordered phase in one dimension, hence no critical point in one dimension. If there were a critical point at a temperature Tc , then, at that point, long range order would set it, and there would be a
xed point of the RG equation at Kc = J=kTc. Note, however, that at T = 0, K = , there is perfect ordering in one dimension. Although this is physically meaningless, it suggests that ordered phases and critical points will be associated with the unstable
xed points of the RG equations. 1

7

Another way to see what the T = 0 unstable
xed point means is to study the correlation length. The correlation is a quantity that has units of length. However, if we choose to measure it in units of the lattice spacing, then the correlation length will be a number that can only depend on the coupling constant K or x = tanhK :

 =  (x) Under an RG transformation, the lattice spacing increases by a factor of 3 as a result of coarse graining. Thus, the correlation length, in units of the lattice spacing, must decrease by a factor of 3 in order for the same physical distance to be maintained:  (x0 ) = 13  (x) In general, for block containing b spins, the correlation length transforms as  (x0 ) = 1  (x)

b

A function satisfying this equation is

 (x )



Since, for arbitrary b, the RG equation is

1 ln x

x0 = xb we have

 (x0 ) =  (xb )

1 ln xb = b ln1 x = 1b  (x) 

so that

1 asT 0 ln tanhK so that at T = 0 the correlation length becomes in
nite, indicating an ordered phase. Note, also, that at very low T , where K is large, motion toward the stable
xed point is initially extremely slow. To see this, rewrite the RG equation as

 (K )



;! 1

;!

tanhK 0 = tanh3 K = tanhK tanh2 K
cosh(2K ) 1 = tanhK cosh(2 K) + 1 ;

Notice that the term in brackets is extremely close to 1 if K is large. To leading order, we can say

K0



K

Since the interactions between blocks are predominantly mediated by interactions between boundary spins, which, for one dimension, involves a single spin pair between blocks, we expect that a block spin transformation in 1 dimension yields a coupling constant of the same order as the original coupling constant when T is low enough that there is alignment between the blocks and the new lattice is similar to the original lattice. This is reected in the above statement. 8

G25.2651: Statistical Mechanics Notes for Lecture 28

I. FIXED POINTS OF THE RG EQUATIONS IN GREATER THAN ONE DIMENSION In the last lecture, we noted that interactions between block of spins in a spin lattice are mediated by boundary spins. In one dimension, where there is only a single pair between blocks, the block spin transformation yields a coupling constant that is approximately equal to the old coupling constant at very low temperature, i.e., K K . Let us now explore the implications of this fact in higher dimensions. Consider a two-dimensional spin lattice as we did in the previous lecture. Now interactions between blocks can be mediated by more than a single pair of spin interactions. For the case of 3
3 blocks, there will be 3 boundary spin pairs mediating the interaction between two blocks: 0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FIG. 1.

1

. . . .

Since 3 boundary spin pair interactions mediate the block-block interaction, the result of a block spin transformation should yield, at low T , a coupling constant K roughly three times as large as the original coupling constant, K : 0

K

3K

0

In a three-dimensional lattice, using 3
3
3 blocks, there would be 32 = 9 spin pairs. Generally, in d dimensions using blocks containing bd spins, the RG equation at low T should behave as

K

0

T !0 K!1

bd 1K ;

The number b is called the length scaling factor. The above RG equation implies that for d > 1, K > K for low T . Thus, iteration of the RG equation at low temperature should now ow toward K = 1 and the xed point at T = 0 is now a stable xed point. However, we know that at high temperature, the system must be in a paramagnetic state, so the xed point at T = 1 must remain a stable xed point. These two facts suggest that, for d > 1, between T = 0 and T = 1, there must be another xed point, which will be an unstable xed point. To the extent that an RG ow in more than one dimension can be considered a one-dimensional ow, the ow diagram would look like: 0

Unstable

Stable

Stable

.

.

x=1 K=

.

K=K c

x=0

T=T c

K=0

FIG. 2.

Any perturbation to the left of the unstable xed point iterates to T = 0, K = 1 and any perturbation to the right iterates to T = 1 and K = 0. The unstable xed point corresponds to a nite, nonzero value of K = Kc and a temperature Tc , and corresponds to a critical point. To see that this is so, consider the correlation length evolution of the correlation length under the RG ow. Recall that for a length scaling factor b, the correlation length transform as  (K ) = 1  (K ) 0

b  (K ) = b (K ) Suppose we start at a point K near Kc and require n(K ) iterations of the RG equations to reach a value K0 between K = 0 and K = Kc under the RG ow: 0

Stable

.

x=1 K=

Unstable

. K=K c T=T c FIG. 3.

2

. K0

Stable

. x=0 K=0

If 0 is the correlation length at K = K0, which can expect to be a nite number of order 1, then, by the above transformation rule for the correlation length, we have

 (K ) = 0 bn(K) Now, recall that, as the starting point, K is chosen closer and closer to Kc, the number of iterations needed to reach K0 gets larger and larger. (Recall that near an unstable xed point, the initial change in the coupling constant is small as the iteration proceeds). Of course, if K = Kc initially, than an in nite number of iterations is needed. This tells us that as K approaches Kc, the correlation length becomes in nite, which is what is expected for an ordered phase. Thus, the new unstable xed point must correspond to a critical point. In fact, we can calculate the exponent  knowing the behavior of the RG equation near the unstable xed point. Since this is a xed point, Kc satis es, quite generally,

Kc = R(Kc) Near the xed point, we can expand the RG equation, giving

K  R(Kc) + (K ; Kc)R (Kc ) +    0

0

De ne an exponent y by

y = ln Rln(bKc ) 0

so that

K  Kc + by (K ; Kc ) 0

Near the critical point,  diverges according to

 jT ; Tcj

;

1 1 K ; Kc

K ; Kc K

;

;

K ; Kc Kc

;

Thus,

jK ; Kcj



;

but

 (K ) = b (K ) 0

which implies

jK ; Kc j

bjK ; Kc j = bjby (K ; Kc)j

;

0

;

;

which is only possible if

 = y1 This result illustrates a more general one, namely, that critical exponents are related to derivatives of the RG transformation.

II. GENERAL LINEARIZED RG THEORY The above discussion illustrates the power of the linearized RG equations. We now generalize this approach to a general Hamiltonian H0 with parameters K1 K2 :::  K. The RG equation

K = R(K) 0

3

can be linearized about an unstable xed point at bK according to 

Ka ; Ka  0



where

X

b

Tab (Kb ; Kb ) 

a Tab = @R @Kb K=K

The matrix T need not be a symmetric matrix. Given this, we de ne a left eigenvalue equation for T according to X

a

ia Tab = i ib

where the eigenvalues fg can be assumed to be real (although it cannot be proved). Finally, de ne a scaling variable, ui by

ui =

X

a

ia (Ka ; Ka ) 

They are called scaling variables because they transform multiplicatively near a xed point under the linearized RG ow:

ui = 0

X

a

ia (Ka ; Ka ) 0

=

XX

=

X

a b



ia Tab (Kb ; Kb ) 

b i i b

  (Kb ; Kb ) 

= i ui Since ui scales with i , it will increase if i > 1 and will decrease if i < 1. Rede ning the eigenvalues i according to

i = by

i

we see that

ui = by ui 0

i

By convention, the quantities fyig are called the RG eigenvalues. These will soon be shown to determine the scaling relations among the critical exponents. For the RG eigenvalues, three cases can be identi ed: 1. yi > 0. The scaling variable ui is called a relevant variable. Repeated RG transformations will drive it away from its xed point value, ui = 0. 2. yi < 0. The scaling variable ui is called an irrelevant variable. Repeated RG transformations will drive it toward 0. 3. yi = 0. The scaling variable ui is called a marginal variable. We cannot tell from the linearized RG equations if ui will iterate toward or away from the xed point. Typically, scaling variables are either relevant or irrelevant. Marginality is rare and will not be considered here. The number of relevant scaling variables corresponds to the number of experimentally `tunable' parameters or `knobs' (such as T and h in the magnetic system, or P and T in a uid system in the case of the former, the relevant variables are called the thermal and magnetic scaling variables, respectively). 4

III. UNDERSTANDING UNIVERSALITY FROM THE LINEARIZED RG THEORY In the linearized RG theory, at a xed point, all scaling variables are 0, whether relevant, irrelevant or marginal. Consider the case where there are no marginal scaling variables. Recall, moreover, that irrelevant scaling variables will iterate to 0 under repeated RG transformations, starting from a point near the unstable xed point, while the relevant variables will be driven away from 0. These facts provide us with a formal procedure for locating the xed point: i. Start with the space spanned by the full set of eigenvectors of T. ii. Project out the relevant subspace by setting all the relevant scaling variables to 0 by hand. iii. The remaining subspace spanned by the irrelevant eigenvectors of T de nes a hypersurface in the full coupling constant space. This is called the critical hypersurface. iv. Any point on the critical hypersurface belongs to the irrelevant subspace and will iterate to 0 under successive RG transformations. This will de ne a trajectory on the hypersurface that leads to the xed point as illustrated below:

K

3

Fixed point

. K1 FIG. 4.

5

K2

This xed point is called the critical xed point. Note that it is stable with respect to irrelevant scaling variables and unstable with respect to relevant scaling variables. What is the importance of the critical xed point? Consider a simple model in which there is one relevant and one irrelevant scaling variable, u1 and u2 , with corresponding couplings K1 and K2 . In an Ising-type model, K1 might represent the reduced nearest neighbor coupling and K2 might represent a next nearest neighbor coupling. Relevant variables also include experimentally tunable parameters such as temperature and magnetic eld. The reason u1 is relevant and u2 is irrelevant is that there must be at least nearest neighbor coupling for the existence of a critical point and ordered phase at h = 0 but this can happen whether or not there is a next nearest neighbor coupling. Thus, the condition u1 (K1 K2) = 0 de nes the critical surface, in this case, a one-dimensional curve in the K1 -K2 plane as illustrated below:

K

2

K1 FIG. 5.

Here, the blue curve represents the critical \surface" (curve), and the point where the arrows meet is the critical

xed point. The full coupling constant space represents the space of all physical systems containing nearest neighbor and next nearest neighbor couplings. If we wish to consider the subset of systems with no next nearest neighbor coupling, i.e., K2 = 0, the point at which the line K2 = 0 intersects the critical surface (curve) de nes the critical value, K1c and corresponding critical temperature and will be an unstable xed point of an RG transformation with K2 = 0. Similarly, if we consider a model for which K2 6= 0, but having a xed nite value, then the point at which this line intersects the critical surface (curve) will give the critical value of K1 for that model. For any of these models, K1c lies on the critical surface and will, under the full RG transformation iterate toward the critical xed point. This, then, de nes a universality class: All models characterized by the same critical xed point belong to the same universality class and will share the same critical properties. This need not include only magnetic systems. Indeed, a uid system, near its critical point can be characterized by a so called lattice gas model. This model is similar to the Ising model, except that the spin variables are replaced by site occupance variables ni , which can take on values 0 or 1, depending on whether a given lattice site is occupied by a particle or not. The grand canonical partition function is 6

Z=

X

n1 =0
1



X

nN =0
1



P i

e

ni 2

P

hi j i

Jij ni nj

and hence belongs to the same universality class as the Ising model. The critical surface, then, contains all physical models that share the same universality properties and have the same critical xed point. In order to see how the relevant scaling variables lead to the scaling relations among the critical exponents, we next need to introduce the scaling hypothesis.

IV. THE SCALING HYPOTHESIS Recall that the RG transformation preserves the partition function: Tr0 e

0 (  0 g
K 0 )

;H0 f

= Tr e

(  g
K)

;H0 f

For the one-dimensional Ising model, we found that the block spin transformation lead to a transformed Hamiltonian of the form

H0 (f g K ) = N g(K ) ; K 0

0

0

0

0

0

N X i=1

i i+1 0

0

Thus, H0 (f g K ) contains a term that is of the same functional form as H0 (f g K ) plus an analytic function of K . De ning the reduced free energy per spin from the spin trace as f (K ), the equality of the partition functions allows us to write generally: 0

0

e

;

Nf (fK g)

= Tr e

(  g
fK g)

;H0 f

=e

;

N 0 g(fK g)

Tr e

(  0 g
fK 0 g)

;H0 f

=e

N 0 g(fK g)

;

e

;

N 0 f (fK 0 g)

which implies that

f (fK g) = g(fK g) + NN f (fK g) 0

0

If b is the size of the spin block, then

N = b dN 0

;

from which

f (fK g) = g(fK g) + b df (fK g) ;

0

Now, g(fK g) is an analytic function and therefore plays no role in determining critical exponents, since it does not lead to divergences. Only the so called singular part of the free energy is important for this. Thus, the singular part of the free energy fs (fK g) can be seen to satisfy a scaling relation:

fs (fK g) = b d fs (fK g) ;

0

This is the basic scaling relation for the free energy. From this simple equation, the scaling relations for the critical exponents can be derived. To see how this works, consider the one-dimensional Ising model again with h 6= 0. The free energy depends on the scaling variables through the dependence on the couplings K . For h = 0, we saw that there was a single relevant scaling variable corresponding to the nearest neighbor coupling K = J=kT . This variable is temperature dependent and is called a thermal scaling variable ut . For h 6= 0 there must also be a magnetic scaling variable, uh. These will transform under the linearized RG as

ut = by ut uh = by uh 0

t

0

h

where yt and yh are the relevant RG eigenvalues. Therefore, the scaling relation for the free energy becomes

fs (ut uh) = b dfs (ut uh ) = b dfs (by ut by uh) ;

0

0

7

;

t

h

Now, after n iterations of the RG equations, the free energy becomes

fs (ut uh) = b

nd

;

fs (bny ut bny uh) t

h

Recall that relevant scaling variables are driven away from the critical xed point. Thus, let us choose an n small enough that the linear approximation is still valid. In order to determine n, we only need to consider one of the scaling variables, so let it be ut . Thus, let ut0 be an arbitrary value of ut obtained after n iterations of the RG equation, such that ut0 is still close enough to the xed point that the linearized RG theory is valid. Then,

ut0 = bny ut t

or

n = y1 logb uut0 = logb uut0 t t t

1=yt

and

fs (ut uh ) = uut t0

d=yt

fs ( ut0 uh jut =ut0j

y =yt

; h

)

Now, let

t = T ;T Tc We know that fs must depend on the physical variables t and h. In the linearized theory, the scaling variables ut and uh will be related linearly to the physical variables: ut = t ut0 t0 uh = hh 0

Here, t0 and h0 are nonuniversal proportionality constants, and we see that ut ! 0 when t ! 0 and the same for uh. Then, we have 0 fs ut0 jt=th=h j0 y =y



fs (t h) = jt=t0j




d=yt

h

t

The left side of this equation does not depend on the nonuniversal constant ut0, hence the right side cannot. This means that the function on the right side depends on a single argument. Thus, we rewrite the free energy equation as 0 fs (t h) = jt=t0jd=y  jt=th=h y =y 0j




t

h

t

The function  is called a scaling function. Note that the dependence on the system-particular variables t0 and h0 is trivial, as this only comes in as a scale factor in t and h. Such a scaling relation is a universal scaling relation, in that it will be the same for all systems in the same universality class. From the above scaling relation come all thermodynamic quantities. Note that the scaling relation depends on only two exponents yt and yh. This suggests that there can only be two independent critical exponents among the six,    , and . There must, therefore, be four relations relating some of the critical exponents to others. These are the scaling relations. To derive these, we use the scaling relation for the free energy to derive the thermodynamic functions: 1. Heat Capacity:

Ch

@2f @t2

but 8

jtjd=y

2

t;

h=0

jtj

Ch

;



Thus,

= 2 ; yd

t

2. Magnetization:

m = @f @h

jt=t0 jd=y jt=t0 jy =y

jtj(d

t

h

h=0

y )=yt

; h

t

but

m jtj Thus,

= d ;y yh t

3. Magnetic susceptibility:

@2f @h2

 = @m @h

jtj(d

2yh )=yt

;

but

 jtj



;

Thus,

 = 2yhy; d t

4. Magnetization vs. magnetic
eld: (d m = @f @h = jt=t0 j

y )=yt

; h




0

h=h0 jt=t0 jy =y h



t

but m should remain nite as t ! 0, which means that  (x) must behave as xd=y 0

m jt=t0 j(d

y )=yt

; h

(h=h0 )

(h=h0)(d

jt=t0j

But

m h1=  = d ;yhy h 9

y )=yt

; h

yh (d;yh )=(yt yh )

(d;yh )=yh

Thus,

1

h;

as x ! 1, since then

From these, it is straightforward to show that the four exponents , ,  , and  are related by

+ 2 +  = 2 + (1 + ) = 2 These are examples of scaling relations among the critical exponents. The other two scaling relations involve  and  and are derived from the scaling relation satis ed by the correlation function:

G(r) = b

;

2(d;yh )

G(r=b by t) t

G(r) = jt=t0 j

2(d;yh )=yt

This leads to the relations






jt=t0 j

 = y1 t  = d + 2 ; 2yh

and the scaling relations:

= 2 ; d  =  (2 ; )

10

r



1=yt

;