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1.2 Equilibrium properties 11

of the H

function is deﬁned as a functional of the one-particle distribution function

f (r, p, t) as

(t) =



dp f (p, t) ln f (p, t). (1.1.41)

Note that it is consistent to assume that the distribution function is independent of r since

there is no external ﬁeld present. On taking the time derivative of eqn. (1.1.41), we obtain

(t)



∂ f (p, t)

∂t

1 + ln f (p, t)

. (1.1.42)

It is straightforward to establish (Huang, 1987) using the Boltzmann equation that

(t)

≤ 0, (1.1.43)

showing that the H function always decreases. This is known as the Boltzmann H theorem.

For f (p, t) equal to the Maxwell–Boltzmann distribution function

( p) =



2πm



3/2

exp



−

β p



(1.1.44)

the H function H

(t) is independent of time and is equal to zero. The function φ

( p) is

normalized and the constant β is determined from the average kinetic energy as

p

/(2m)=3/(2β). Hence β is identiﬁed with the temperature T of the equilibrium ﬂuid

β = 1/(k

T ).

In general the nature of the interaction potential plays an important role in evaluating the

collision integral on the RHS of eqn. (1.1.39). The irreversible kinetic equation obtained

from closing the BBGKY hierarchy is used to compute the basic transport equations for

ﬂuid with dissipative dynamics. The most widely studied system in this regard is a system

of hard spheres. We discuss this further in Section 5.2.2 below.

1.2 Equilibrium properties

By deﬁnition for thermodynamic equilibrium, the distribution function f

(N )

depends on

the coordinates r

(t) and p

(t) but is independent of time t. The equilibrium state of a

ﬂuid corresponds to the stationary solutions of the distribution functions,

∂

∂t

(N )

, p

, t) = 0. (1.2.1)

Therefore from eqn. (1.1.20) it follows that the N-particle equilibrium distribution func-

tion f

(N )

must satisfy the condition

(N )

, H

=0. Hence f

(N )

must be expressed as a

functional of the Hamiltonian H or of function {A

} (say), all of which have vanishing

Poisson bracket with the Hamiltonian. These stationary distribution functions determine

the Gibbsian ensembles. In this regard it is particularly useful to consider the following

theorem for the equilibrium state.

12 Statistical physics of liquids

1.2.1 The Gibbs H-theorem

The H function in terms of the N -particle distribution function f

(N )

is deﬁned as



(N )

, p

) ln f

(N )

, p

). (1.2.2)

In some cases the ensemble may include variation of the total number of particles N or the

total volume of the system. We denote by Tr an averaging over the whole phase space. If

the different members of the ensemble have only energy as the variable with both the total

number of particles N and the volume V ﬁxed (a canonical ensemble) then Tr stands for

Tr ≡

N !

α=1

, (1.2.3)

where the factor of h

is included to take into account the quantum-mechanical counting

of possible states and the factor of N ! to account for the permutation among the N identical

particles. If the ensemble is one in which, for example, variation of the total number of

particles as well as of the energy is allowed (a grand-canonical ensemble) then we deﬁne

Tr ≡

∞



N =0

N !



. (1.2.4)

We now deﬁne the H function H

= Tr



(N )

, p

)ln f

(N )

, p

)

. (1.2.5)

Let us consider the problem of minimizing the above H function subject to the following

constraints in terms of a speciﬁc set of functions A

, p

), i = 1,...,m, given by {

}

such that

= Tr



(N )

, p

)

. (1.2.6)

In equilibrium the time-independent f

(N )

, p

) describes the probability for the dif-

ferent members of the corresponding ensemble. The condition (1.2.6) implies that the

ensemble averages {

} characterize the thermodynamic equilibrium state. The H theo-

rem at the N -particle level in this case states that, subject to the constraints (1.2.6),the

normalized distribution function f

(N )

corresponding to which the value of H

reaches a

minimum is given by

(N )

, p

) = W

−1

exp



−



, p

)



. (1.2.7)

The set of parameters {α

} for i = 1,...,m deﬁnes the thermodynamic state characterized

by optimum f

(0)

. W

is the normalization constant

= Tr



exp



−



, p

)



(1.2.8)

1.2 Equilibrium properties 13

which ensures the condition

Tr f

(N )

, p

) = 1. (1.2.9)

In order to prove the above theorem we ﬁrst consider an arbitrary choice of the distri-

bution function f

(N )

= f

(N )

. Since both the distribution functions represent probabilities,

both of them are real positive numbers and x = f

(N )

/ f

(N )

is also a positive number. Since

both the distributions are normalized, we must have

Tr f

(N )

, p

) = Tr f

(N )

, p

) = 1. (1.2.10)

Now let us consider the difference of the function H

as evaluated with the arbitrary

functional f

(N )

from that corresponding to f

(N )

[ f ]−H [ f

]=Tr



x ln{xf

, p

, t)}− f

ln f

, (1.2.11)

where we have dropped the superscript N on f to keep the notation simple. Using the

normalization condition (1.2.10),theRHSofeqn. (1.2.11) reduces to

[ f ]−H [ f

]=Tr

{

x ln x − x + 1

}

(

f − f

)

ln f

= I

+ I

. (1.2.12)

The ﬁrst integral, I

,ontheRHSofeqn. (1.2.12) is always positive since, according to the

Gibbs inequality (see Appendix A1.1), the quantity (x ln x −x +1) is positive deﬁnite for

positive x. The second integral, I

,ontheRHSofeqn. (1.2.12) vanishes. To demonstrate

this we use the expression (1.2.7) for evaluating ln f

=−Tr



( f − f

)

ln W



, p

)

'

= 0. (1.2.13)

The above result for I

follows easily from the normalization condition (1.2.10) and the

constraints (1.2.6) for the ensemble averages of the set of observable {A

}. Note that W

and the parameters {α

} are not dependent on the phase-space variables. We have thus

proved that H

for all arbitrary choices of f

(N )

increases from its value corresponding to

the function f

(N )

. The latter, as given by the RHS of eqn. (1.2.7), therefore minimizes the

function H

. If we identify −k

with the thermodynamic entropy of the equilibrium

state, the other parameters are also readily identiﬁed from the relation

−

=−ln W

−



. (1.2.14)

Since for the equilibrium state the entropy is a maximum, we obtain, on taking the deriva-

tive of eqn. (1.2.14) with respect to α

, the following result for the thermodynamic average:

=−

∂

∂α

ln W

. (1.2.15)

14 Statistical physics of liquids

On taking a second derivative we obtain

∂

∂α

∂

∂α

ln W

. (1.2.16)

Using the expression (1.2.8) for ln W

,theRHSofeqn. (1.2.16) is evaluated as the average

of the mean-square ﬂuctuation of A

∂

∂α

ln W

= Tr



−{Tr[A

]}

= Tr[{A

−

}

]=A

, (1.2.17)

giving the relative ﬂuctuation of A

A

=−

∂

∂α

. (1.2.18)

The result (1.2.18) implies that the relative ﬂuctuation of A

is of the order of the square

root of the size S of the system,

A

1/2

∼ O





. (1.2.19)

The RHS approaches zero in the thermodynamic limit. Different representative statisti-

cal ensembles for the equilibrium state of the system are obtained with corresponding

speciﬁc choices of the set of extensive variables {A

}. The ensemble-averaged quantities

{

} remain ﬁxed at their experimentally measured values in the corresponding thermody-

namic state which is characterized by the corresponding set of intensive thermodynamic

variables {α

}. The above result (1.2.19) therefore demonstrates the equivalence of the dif-

ferent ensembles since the ﬂuctuation of a speciﬁc property A

around its average value is

negligible in the thermodynamic limit of a large system, e.g., lim N →∞, lim V →∞

but N/ V remaining ﬁnite.

1.2.2 The equilibrium ensembles

Let us consider some of the speciﬁc ensembles in the following. For a completely isolated

system we have the micro-canonical ensemble in which the assumption of equal a-priori

probability for each state is applied.

The canonical ensemble

For the canonical ensemble the system is connected to an external heat bath allowing

exchange of energy. The average energy is ﬁxed and is determined by the equilibrium

temperature T of the thermodynamic state. In the present case we have A ≡ H and the cor-

responding parameter is α

≡ β. The number of particles N is ﬁxed in this case. Evaluation

of the H function gives



ln f

=−ln Z

−

H, (1.2.20)

1.2 Equilibrium properties 15

where Z

≡ W

. Upon identifying the average energy

H as the thermodynamic internal

energy U , the above relation (1.2.14) reduces to

−

=−ln Z

− βU. (1.2.21)

We identify β = 1/(k

T ) and the Helmholtz free energy of the system F as

− k

T ln Z

= U − TS = F. (1.2.22)

The equilibrium probability distribution is therefore given by

(N )



, p



N !

exp

[

−β H

]

, (1.2.23)

with the partition function Z(N, V , T ) deﬁned as

Z(N, V, T ) ≡ Z

N !



exp



−β H(r

, p

)

. (1.2.24)

The key relation connecting the thermodynamic property (the Helmholtz free energy F )

with the statistical-mechanical quantity (the canonical partition function Z

) is given by

F =−k

T ln Z

. (1.2.25)

For a noninteracting system (ideal gas) the Hamiltonian H has only a kinetic part and it

is straightforward to obtain the partition function Z

by evaluating the Gaussian integrals

involving the p

, α = 1,...,N :

N !

, (1.2.26)

with the thermal de Broglie wavelength 

=h/

√

2πmk

T . The free energy for the

noninteracting system is obtained as

= Nk



(

n

)

− 1

. (1.2.27)

By using the relation P =−(∂ F/∂ V )

we obtain from eqn. (1.2.27) the ideal-gas equation

of state PV = Nk

T . The Gibbs free energy is given by

G = F + PV = Nk

T ln

(

n

)

, (1.2.28)

and the chemical potential for the ideal gas is

= k

T ln

(

n

)

. (1.2.29)

The average energy, which is identiﬁed as the thermodynamic internal energy U ,is

obtained as

U =

H =−

∂

∂β

ln Z

. (1.2.30)

16 Statistical physics of liquids

The mean-square ﬂuctuation of the internal energy is obtained from the general formula

(1.2.18) as

H

=−



∂U

∂β



V,N

= Nk

, (1.2.31)

where c

is the speciﬁc heat per particle at constant volume. The relative root-mean-square

ﬂuctuation per particle is of O



√



, making it negligible in the thermodynamic limit.

The grand-canonical ensemble

The grand-canonical ensemble describes the system connected to an external bath allowing

exchange of both energy and particles. The ensemble average of the total energy and the

total particle numbers of the system are ﬁxed. In this case we have A ≡{H, N } and the

corresponding α

is denoted by {β, −βμ}. Equation (1.2.14) reduces to the form

−

=−ln  −

H + βμ

N , (1.2.32)

where  ≡ W

in this case. The average energy

H and

N , respectively, correspond to the

thermodynamic internal energy U and the average number of particles at temperature T .

If we identify β = 1/(k

T ) and μ as the chemical potential, eqn. (1.2.32) reduces to the

thermodynamic relation

− k

T ln  = U − TS− μ

N = F − G =−, (1.2.33)

where G is the Gibbs free energy of the system and  =−PV is the thermodynamic

potential. The equilibrium probability distribution in the grand-canonical ensemble is

therefore given by

(N )

, p

) =

N !

exp

[

−β(H − μN )

]



, (1.2.34)

where the grand-canonical partition function (μ, V, T ) is obtained as

(μ, V, T ) =

∞



N =0

N !



exp



−β(H (r

, p

) − μN )

. (1.2.35)

The average energy and the average number of particles are respectively obtained as

U =−

∂

ln 

μ,V

N =

∂

∂(βμ)

ln 

T,V

= z

∂

∂z

ln 

T,V

. (1.2.36)

where z = exp(βμ) is the fugacity. Note that the averaging in the grand-canonical ensem-

ble can be interpreted as a weighted average with respect to the canonical-ensemble

1.2 Equilibrium properties 17

N -particle partition function Z

with all possible values of N. Thus we can express the

grand-canonical partition function as

(μ, V, T ) =

∞



N =0

N !

. (1.2.37)

The average number of particles as given by eqn. (1.2.36) is expressed as

N =

∞



N =0

NP(N ), (1.2.38)

where P(N ) is the probability for the N -particle state given by

P(N ) =

N !



. (1.2.39)

The key relation in the case of the grand-canonical ensemble connecting the thermody-

namic property (the thermodynamic potential  ≡−PV) with the statistical-mechanical

quantity (the grand-canonical partition function )isgivenby

 =−k

T ln . (1.2.40)

The mean-square ﬂuctuation of the number of particles is obtained from the general for-

mula (1.2.18) as

N

∂

∂(βμ)

T,V

= k



∂

∂μ



. (1.2.41)

The relative root-mean-square ﬂuctuation per particle is of O



√



, making it negligi-

ble in the thermodynamic limit. The RHS of eqn. (1.2.41) is further simpliﬁed by using the

Gibbs–Duhem relation (McQuarie, 2000)

Ndμ = VdP− SdT, (1.2.42)

and hence



∂μ

∂





∂ P

∂





∂ P

∂n



, (1.2.43)

since n =

N /V . However, (∂ P/∂n)

can also be written as



∂ P

∂n





∂ P

∂V





∂V

∂n



=−



∂ P

∂V



nκ

, (1.2.44)

where

=−



∂

∂ P



(1.2.45)

18 Statistical physics of liquids

is the isothermal compressibility of the system. Using the result (1.2.44) in eqn. (1.2.41),

we obtain

−

. (1.2.46)

The root-mean-square ﬂuctuation N is therefore of O(1/

√

N ) and negligible in the ther-

modynamic limit – except near the critical point, where the compressibility is diverging

and hence the role of ﬂuctuations becomes important there.

The isobaric–isothermal ensemble

The isobaric ensemble describes the system connected to an external bath allowing both

exchange of energy and variation of the volume V of the system. The number of particles

remains constant in this case. The ensemble average of the total energy and the volume

of the system are ﬁxed. In this case we have A

≡{H, V } and the corresponding α

denoted by {β, βα}. Evaluation of the H function gives



ln f

=−ln  − β





− βα





, (1.2.47)

where  ≡ W

. Upon identifying the average energy





as the thermodynamic internal

energy U the above relation (1.2.14) reduces to

−

=−ln  − β(U + α

V ), (1.2.48)

where

V is the volume in the thermodynamic equilibrium state at temperature T . We iden-

tify β = 1/(k

T ) and α as the thermodynamic pressure P so that eqn. (1.2.48) reduces to

the thermodynamic relation

− k

T ln  = U − TS+ α

V = U − TS+ PV = G. (1.2.49)

The equilibrium probability distribution in the isobaric ensemble is therefore given by

(N )



, p



= 

−1

exp

[

−β{H + PV}

]

, (1.2.50)

where (N, P, T ) is the isobaric partition function. The latter is obtained as a Laplace

transform (see eqn. (1.3.10) for its deﬁnition) at “frequency” β P with respect to the

volume V ,

(N, P, T ) =



N !



∞



exp

[

−β{H + PV}

]

, (1.2.51)

where 

is a constant with the dimension of volume introduced so as to make the par-

tition function  dimensionless. The average volume and the mean square of the volume

ﬂuctuations are, respectively, obtained as

V =−k

∂

∂ P

ln 

T,N

, (1.2.52)

1.2 Equilibrium properties 19

Table 1.1 Different ensembles, the corresponding partition functions,

and how they relate to the appropriate thermodynamic function.

Ensemble type Partition function Thermodynamic function

Helmholtz free energy, F

Canonical Z

(N , V, T ) F =−k

T ln Z

Grand-canonical (μ, V , T ) Thermodynamic potential, 

 =−PV =−k

T ln 

Isothermal–isobaric (N , P, T ) Gibbs free energy, G

G =−k

T ln 

(V )

=−

∂

∂(βP)

T,N

. (1.2.53)

The above results for the different thermodynamic properties corresponding to the canoni-

cal, grand-canonical, and isobaric ensembles are shown in Table 1.1.

1.2.3 The static structure factor

In the present section we discuss the correlation functions of ﬂuctuations of dynamic

variables at different spatial points but at the same time. These represent the static or

the structural properties of the liquid. For a high-density liquid the most basic quantity

characteristic of its structural or thermodynamic properties is the pair correlation function.

The pair correlation function g(r)

For an N -particle system the equilibrium solution to the Liouville equation (1.1.22) is

given by the Gibbsian distributions discussed in the previous section. Let us consider

the reduced probability distributions f

, p

) of s particles as deﬁned in eqn. (1.1.32),

speciﬁcally for the equilibrium ﬂuid. We use the short-hand notation r

≡{r

, r

,...,r

etc. here. The momentum dependence of the distribution function f

in this case of a

thermodynamic equilibrium state is Maxwellian,

(s)

) =

s=1

)

(s)

), (1.2.54)

where φ

) is the normalized Maxwell distribution function as deﬁned in eqn. (1.1.44).

The conditional probability of there being s particles simultaneously at {r

,..., r

} irre-

spective of their momenta is obtained from the corresponding f

(s)

by integrating out the

momentum variables {p

(s)

) =



(s)

, p

). (1.2.55)

The simplest example of the above distribution function is the s = 2 case, i.e., g

, r

which is termed the radial distribution function. It represents the conditional probability

20 Statistical physics of liquids

of a particle being at r

while another is at r

. In general a distribution function at the

s-particle level is deﬁned from n

(s)

, r

,...,r

) =

(s)

, r

,...,r

)

, (1.2.56)

where we deﬁne n

= N/V , the number of particles per unit volume in the N -particle

system. In the grand-canonical-ensemble description, we average the reduced distribution

function n

(s)

) over the total number of particles N in the system to obtain the averaged

quantity

(s)

) ≡

(s)

)

∞



N =s

P(N )n

). (1.2.57)

The simplest and most important is the s = 2 case. The thermodynamic quantity g(r

, r

)

termed the pair correlation function is deﬁned as

(2)

, r

) = n

g(r

, r

), (1.2.58)

where the factor n

on the LHS is for normalization of g(r); n is the average number of

particles per unit volume of the liquid, n =

N /V =n

. Some useful properties of

this function follow directly from simple physical considerations. Thus, in the equilibrium

state translational invariance holds and for an isotropic system g(r

, r

) is a function only

of the distance between the two points and can be denoted as g(r

). For a small sep-

aration r between two points g(r) vanishes due to the repulsive core of the interaction

potential, whereas, for large r , g(r) → 1 so that the probability of two particles being at

the two points goes to the trivial limit n

. The radial distribution function g(r) is the most

fundamental quantity in describing the thermodynamic properties of the equilibrium ﬂuid.

In this respect it is important to note two very basic relations linking the thermodynamic

properties of a liquid to the pair correlation function.

The virial equation

For a liquid in which the interaction between the particles involves only two-body poten-

tials as introduced in the expression (1.1.16) for the Hamiltonian, the thermodynamic

pressure is obtained in terms of the pair correlation function only. To demonstrate this, we

begin from the thermodynamic relation P =−(∂ F/∂ V )

with the free energy being given

by F =−k

T ln Z

. The canonical partition function Z

is obtained for the interacting

system as

N !



exp

[

−βU (r

,..., r

)

]

, (1.2.59)

where 

is the thermal de Broglie wavelength as deﬁned with eqn. (1.2.26). The pressure

is obtained as

P =

N !

∂

∂V

. (1.2.60)