This is the proper Liouville equation, which is very important in statistical

mechanics. It states that for a Hamiltonian system the (probability) density

in phase space does not change with time. This also means that a volume in

phase space does not change with time: if one follows a bundle of trajectories

that start in an initial region of phase space, then at a later time these

trajectories will occupy a region of phase space with the same volume as

the initial region.29 This is also expressed by saying that the Hamiltonian

probability ¬‚ow in phase space is incompressible.

If the volume does not change, neither will a volume element used for

integration over phase space. We have to be careful what we call a volume

element. Normally we write the volume element somewhat loosely by a prod-

uct dz or d2n z or dz1 . . . dz2n or Π2n dzi , while we really mean the volume

i=1

spanned by the local displacement vectors corresponding to the increments

dzi . These displacement vectors are proportional, but not necessarily equal

to zi . The volume spanned by the local displacement vectors only equals

their product when these vectors are orthogonal; in general the volume is

equal to the determinant of the matrix formed by the set of displacement

vectors. The proper name for such a volume is the wedge product, but we

29 We skip the intricate discussion on the possibility to de¬ne such regions, which relates to the

fact that di¬erent trajectories can never cross each other. Even more intricate is the discussion

on the possible chaotic behavior of Hamiltonian dynamical systems that destroys the notion

of conserved volume.

17.8 Liouville equations in phase space 495

shall not use the corresponding wedge notation. The volume element is

written as

√

dV = gdz1 · · · dz2n , (17.159)

where g is the determinant of the metric tensor gij , which de¬nes the metric

of the space: the square of the length of the displacement ds caused by

dz1 , . . . , dz2n is determined by (ds)2 = i,j gij dzi dzj .30 .

When coordinates are transformed from z(0) to z(t), the transformation

is characterized by a transformation matrix J and the volume element trans-

forms with the Jacobian J of the transformation, which is the determinant

of J:

g(t) dz1 (t) · · · dz2n (t) = g(0) dz1 (0) · · · dz2n (0), (17.160)

dz1 (t) · · · dz2n (t) = J dz1 (0) · · · dz2n (0), (17.161)

dz(t) = J dz(0), (17.162)

J = det J, (17.163)

J= g(t)/g(o). (17.164)

The Liouville equation (17.158) implies that the Jacobian of the transfor-

mation from z(0) to z(t) equals 1 for Hamiltonian systems. Hence the

volume element in phase space, which is denoted by the product dz =

dz1 (t) · · · dz2n (t), is invariant under a Hamiltonian or canonical transforma-

tion. A canonical transformation is symplectic, meaning that the so-called

two-form dq § dp = i dqi § dpi of any two vectors dq and dp, spanning

a two-dimensional surface S, is invariant under the transformation.31 This

property is important in order to retain the notion of probability density in

phase space f (z) dz.

In practice, time evolutions are not always Hamiltonian and the probabil-

ity ¬‚ow could well loose its incompressibility. The question how the Jacobian

(or the metric tensor) develops in such cases and in¬‚uences the distribution

functions has been addressed by Tuckerman et al. (1999). We™ll summarize

their results. First de¬ne the phase space compressibility κ:

‚ zi

™

def

κ(z) = ∇ · z =

™ , (17.165)

‚zi

i

30 Consider polar coordinates (r, θ, φ) of a point in 3D space: changing θ to θ + dθ causes a

displacement vector of length r dθ. Changing φ to φ + dφ causes a displacement vector of

length r sin θ dφ. What is the metric tensor for polar coordinates and what is the square root

of its determinant and hence the proper volume element?

31 The two-form is the sum of areas of projections of the two-dimensional surface S onto the

qi ’ pi planes. See, e.g., Arnold (1975).

496 Review of statistical mechanics

which is the essential factor on the right-hand side of the generalized Liou-

ville equation (17.157). As we have seen, κ = 0 for incompressible Hamil-

tonian ¬‚ow. Tuckerman et al. then proceed to show that the Jacobian J of

the transformation from z(0) to z(t) obeys the di¬erential equation

dJ

= J κ(z), (17.166)

dt

with solution

t

J(t) = exp κ[z(„ )] d„ . (17.167)

0

If a function w(z) is de¬ned for which w = κ, then

™

J(t) = ew(z(t))’w(z(0) , (17.168)

and

e’w(z(t)) dz1 (t) · · · dz2n (t) = e’w(z(0)) dz1 (0) · · · dz2n (0). (17.169)

√

Hence this modi¬ed volume element, with g = e’w(z) , is invariant under

the non-Hamiltonian equations of motion. This enables us to compute equi-

librium distribution functions generated by the non-Hamiltonian dynamics.

Examples are given in Section 6.5 on page 194.

Let us return to Hamiltonian systems for which the Liouville equation

(17.158) is valid. The time derivative of f , measured at a stationary point

in phase space, is

‚f ˆ

= ’z · ∇f = ’iLf,

™ (17.170)

‚t

where the Liouville operator is de¬ned as32

2n 2n

‚ ‚H ‚

def

ˆ

iL z·∇=

™

= zi

™ = L0ij

‚zi ‚zj ‚zi

i=1 i,j=1

n

‚H ‚ ‚H ‚

’

= . (17.171)

‚pj ‚qi ‚qj ‚pi

i,j=1

This sum is called a Poisson bracket; if applied to a function f , it is written

as {H, f }. We shall not use this notation. Assuming a Hamiltonian that

does not explicitly depend on time, the formal solution is

f (z, t) = e’itL f (z, 0).

ˆ

(17.172)

The convention to write the operator as iL and not simply L is that there is a corresponding

ˆ ˆ

32

operator in the quantum-mechanical evolution of the density matrix and the operator now is

hermitian.

17.9 Canonical distribution functions 497

Note that the time-di¬erential operator (17.170) for the space phase density

has a sign opposite to that of the time-di¬erential operator (17.152) for a

point in phase space.

17.9 Canonical distribution functions

In this section we shall consider the classical distribution functions for the

most important ensemble, the canonical ensemble, for various cases. The

cases concern the distributions in phase space and in con¬guration space,

both for cartesian and generalized coordinates and we shall consider what

happens if internal constraints are applied.

In general phase space the canonical (NVT) equilibrium ensemble of a

Hamiltonian system of n degrees of freedom (= 3N for a system without