1549380323-Statistical Mechanics Theory and Molecular Simulation

186 Canonical ensemble

most situations, when the surroundings are integrated out in this way, the microscopic
equations of motion obeyed by the system are no longer Hamiltonian.In fact, it is
often possible to model the effect of the surroundings by simply positing a set of
non-Hamiltonian equations of motion and then proving that the equations of motion
generate the desired ensemble distribution. Under such a protocol, it is possible to
treat systems interacting with heat and particle reservoirs or systems subject to exter-
nal driving forces. Consequently, it is important to develop an approach that allows
us to predict the phase space distribution function for a given set of non-Hamiltonian
equations of motion.
Let us begin by assuming that a system interacting with its surroundings and pos-
sibly subject to driving forces is described by non-Hamiltonian microscopic equations
of the form
̇x =ξ(x,t). (4.9.1)

We do not restrict the vector functionξ(x,t) except to assume that it is smooth and
at least once differentiable. In particular, the phase space compressibility∇x· ̇x =
∇x·ξ(x,t) need not vanish for a non-Hamiltonian system. If it does not vanish, then
the system is non-Hamiltonian. Note, however, that the converseis not necessarily
true. That is, there are dynamical systems for which the phase space compressibility
is zero but which cannot be derived from a Hamiltonian. Recall that the vanishing of
the phase space compressibility was central to the derivation of the Liouville theorem
and Liouville’s equation in Sections 2.4 and 2.5. Thus, in order to understand how these
results change when the dynamics is compressible, we need to revisitthese derivations.

4.9.1 The phase space metric

Recall from Section 2.4 that a collection of trajectories initially in a volume element
dx 0 about the point x 0 will evolve to dxtabout the point xt, and the transformation
x 0 →xtis a unique one with a JacobianJ(xt; x 0 ) satisfying the equation of motion

d

dt

J(xt; x 0 ) =J(xt; x 0 )∇xt· ̇xt. (4.9.2)

Since the compressibility will occur many times in our discussion of non-Hamiltonian
systems, we introduce the notationκ(xt,t), to represent this quantity

κ(xt,t) =∇xt· ̇xt=∇xt·ξ(xt,t). (4.9.3)

Sinceκ(xt,t) cannot be assumed to be zero, the Jacobian is not unity for all time, and
the Liouville theorem dxt= dx 0 no longer holds.
The Jacobian can be determined by solving eqn. (4.9.2) using the method of char-
acteristics subject to the initial conditionJ(x 0 ; x 0 ) = 1 yielding

J(xt; x 0 ) = exp

[∫t

0

ds κ(xs,s)

]

. (4.9.4)

However, eqn. (4.9.2) implies that there exists a functionw(xt,t) such that

κ(xt,t) =

d

dt

w(xt,t) (4.9.5)