188 Canonical ensemble
the volume element, when
√
ghas no explicit time dependence, so that the average
can be performed at any instant in time.
Imbuing phase space with a metric is not as strange as it might at firstseem. After
all, phase space is a fictitious mathematical construction, a background space on which
a dynamical system evolves. There is no particular reason that we need to attach the
same fixed, Euclidean space toeverydynamical system. In fact, it is more natural to
allow the properties of a given dynamical system dictate the geometry of the phase
space on which it lives. Thus, if imbuing a phase space with a metric thatis particular
to a given dynamical system leads to a volume conservation law, thensuch a phase
space is the most natural choice for that dynamical system. Oncethe geometry of the
phase space is chosen, the form of the Liouville equation and its equilibrium solution
are determined, as we will now show.
4.9.2 Generalizing the Liouville equation
In order to generalize the Liouville equation for the phase space distributionf(xt,t)
for a non-Hamiltonian system, it is necessary to recast the derivation of Section 2.5 on
a space with a nontrivial metric. The mathematics required to do thisare beyond the
scope of the general discussion we wish to present here but are discussed elsewhere by
Tuckermanet al.(1999, 2001), and we simply quote the final result,
∂
∂t(
f(x,t)√
g(x,t))
+∇x·(
̇x√
g(x,t)f(x,t))
= 0. (4.9.12)
Now, combining eqns. (4.9.10) and (4.9.2), we find that the phase spa√ ce metric factor
g(x,t) satisfies
d
dt
√
g(xt,t) =−κ(xt,t)√
g(xt,t), (4.9.13)which, by virtue of eqn. (4.9.12), leads to an equation forf(x,t) alone:
∂
∂t
f(x,t) +ξ(x,t)·∇xf(x,t) = 0 (4.9.14)or simply
d
dt
f(xt,t) = 0. (4.9.15)That is, when the non-Euclidean nature of the non-Hamiltonian phase space is properly
accounted for, the ensemble distribution functionf(xt,t) is conserved just as it is in
the Hamiltonian case, but it is conserved on a different phase space,namely, one with
a nontrivial metric. Consequently, eqn. (2.5.11) generalizes to
f(xt,t)√
g(xt,t)dxt=f(x 0 ,0)√
g(x 0 ,0)dx 0. (4.9.16)Eqn. (4.9.12) assumes smoothness both of the metric factor
√
g(x,t) and of the distri-
bution functionf(x,t), which places some restrictions on the class of non-Hamiltonian
systems for which it is valid. This and related issues have been discussed by oth-
ers (Ramshaw, 2002; Ezra, 2004) and are beyond the scope of this book.