1549380323-Statistical Mechanics Theory and Molecular Simulation

66 Theoretical foundations

d

dt

J(xt; x 0 ) =J(xt; x 0 )

∑

k,l

[

∂ ̇xkt

∂xl 0

∂xl 0

∂xkt

]

. (2.4.9)

The summation overlof the term in square brackets, is just the chain-rule expression
for∂ ̇xkt/∂xkt. Thus, performing this sum yields the equation of motion for the Jacobian:

d

dt

J(xt; x 0 ) =J(xt; x 0 )

∑

k

∂ ̇xkt

∂xkt

. (2.4.10)

The sum in the last line of eqn. (2.4.10) is easily recognized as the phasespace com-
pressibility∇xt· ̇xtdefined in eqn. (1.6.25), where∇xt=∂/∂xt. Eqn. (1.6.25) also
revealed that the phase compressibility is 0 for a system evolving under Hamilton’s
equations. Thus, the sum on the right side of eqn. (2.4.10) vanishes, and the equation
of motion for the Jacobian reduces to

d

dt

J(xt; x 0 ) = 0. (2.4.11)

This equation of motion implies that the Jacobian is a constant for all time. The initial
conditionJ(x 0 ; x 0 ) on the Jacobian is simply 1, since the transformation from x 0 to
x 0 is an identity transformation. Thus, since the Jacobian is initially 1 andremains
constant in time, it follows that

J(xt; x 0 ) = 1. (2.4.12)

Substituting eqn. (2.4.12) into eqn. (2.4.2) yields the volume element transformation
condition
dxt= dx 0. (2.4.13)

Eqn. (2.4.13) is an important result known asLiouville’s theorem(named for the
nineteenth-century French mathematician Joseph Liouville (1809–1882)). Liouville’s
theorem is essential to the claim made earlier that ensemble averages can be performed
at any point in time.
If the motion of the system is driven by highly nonlinear forces, thenan initial
hypercubic volume element dx 0 , for example, will distort due to the chaotic nature of
the dynamics. Because of Liouville’s theorem, the volume element canspread out in
some of the phase space dimensions but must contract in other dimensions by an equal
amount so that, overall, the volume is conserved. That is, there can be no net attractors
or repellors in the phase space. This is illustrated in Fig. 2.3 for a two-dimensional
phase space.

2.5 The ensemble distribution function and the Liouville equation

Phase space consists of all possible microstates available to a system ofNparticles.
However, an ensemble contains only those microstates that are consistent with a given
set of macroscopic observables. Consequently, the microstatesof an ensemble are either
a strict subset of all possible phase space points or are clustered more densely in certain