Non-Hamiltonian statistical mechanics 187or that there exists a function whose derivative yields the compressibility. Substitution
of eqn. (4.9.5) into eqn. (4.9.4) yields
J(xt; x 0 ) = exp [w(xt,t)−w(x 0 ,0)]. (4.9.6)Since the phase space volume element evolves according to
dxt=J(xt; x 0 )dx 0 , (4.9.7)we have
dxt= exp [w(xt,t)−w(x 0 ,0)] dx 0
exp [−w(xt,t)] dxt= exp [−w(x 0 ,0)] dx 0 (4.9.8)(Tuckermanet al., 1999; Tuckermanet al., 2001). Eqn. (4.9.8) constitutes a gen-
eralization of Liouville’s theorem; it implies that a weightedphase space volume
exp[−w(xt,t)]dxtis conserved rather than simply dxt.
Eqn. (4.9.8) implies that a conservation law exists on a phase space that does not
follow the usual laws of Euclidean geometry. We therefore need to view the phase space
of a non-Hamiltonian system in a more general way as a non-Euclideanor Riemannian
space ormanifold. Riemannian spaces are locally curved spaces and, therefore, it
is necessary to consider local coordinates in each neighborhood ofthe space. The
coordinate transformations needed to move from one neighborhood to another give
rise to a nontrivial metric and a corresponding volume element denoted
√
g(x)dx,
whereg(x) is the determinant of a second-rank tensor gij(x), known as themetric
tensor. Given a coordinate transformation from coordinates x to coordinates y, the
Jacobian is simply the ratio of the metric determinant factors:
J(x; y) =√
g(y)
√
g(x). (4.9.9)
It is clear, then, that eqn. (4.9.6) is nothing more than a statementof this fact for a
coordinate transformation x 0 →xt
J(xt; x 0 ) =√
g(x 0 ,0)
√
g(xt,t), (4.9.10)
where √
g(xt,t) = e−w(xt,t) (4.9.11)
when the metric
√
g(xt,t) is allowed to have an explicit time dependence. Although
such coordinate and parameter-dependent metrics are not standard features in the
theory of Riemannian spaces, they do occasionally arise (Sardanashvily, 2002a; Sar-
danashvily, 2002b). Most of the metric factors we will encounter in our treatments of
non-Hamiltonian systems will not involve explicit time-dependence andwill therefore
obey eqn. (4.9.9). The implication of eqn. (4.9.8) is that any phase space integral that
represents an equilibrium ensemble average should be performed using
√
g(x)dx as