1549380323-Statistical Mechanics Theory and Molecular Simulation

Non-Hamiltonian statistical mechanics 189

4.9.3 Equilibrium solutions

In equilibrium, bothf(xt,t) and

√

g(xt,t) have no explicit time dependence, and eqn.
(4.9.16) reduces to

f(xt)

√

g(xt)dxt=f(x 0 )

√

g(x 0 )dx 0 , (4.9.17)

which means that equilibrium averages can be performed at any instant in time, just
as in the Hamiltonian case.
Although the equilibrium Liouville equation takes the same form as it does in the
Hamiltonian case
ξ(x)·∇xf(x) = 0, (4.9.18)

we cannot express this in terms of a Poisson bracket with the Hamiltonian because
there is no Hamiltonian to generate the equations of motion ̇x =ξ(x). In cases for which
we can determine the full metric tensorgij(x), a non-Hamiltonian generalization of
the Poisson bracket is possible (Sergi, 2003; Tarasov, 2004; Ezra, 2004); however, no
general theory of this metric tensor yet exists. Nevertheless, the fact that df/dt= 0
allows us to construct a general equilibrium solution that is suitable for our purposes
in this book. The non-Hamiltonian systems we will be studying in subsequent chapters
are assumed to be complete in the sense that they represent the physical system plus
some additional variables that grossly represent the surroundings. Thus, in order to
construct a distribution functionf(x) that satisfies df/dt= 0, it is sufficient to know
all of the conservation laws satisfied by the equations of motion. Let there beNc
conservation laws of the form

Λk(xt)−Ck= 0,

d

dt

Λk(xt) = 0, (4.9.19)

wherek= 1,...,Nc. If we can identify these, then a general “microcanonical” solution
forf(x) can be constructed from these conservation laws in the form

f(x) =

∏Nc

k=1

δ(Λk(x)−Ck). (4.9.20)

This solution simply states that the distribution generated by the dynamics is one that
samples the intersection of the hypersurfaces represented by all of the conservation laws
in eqn. (4.9.19). Under the usual assumptions of ergodicity, the system will sample all
of the points on this intersection surface in an infinite time. Consequently, the non-
Hamiltonian system has an associated “microcanonical” partition function obtained
by integrating the distribution in eqn. (4.9.20):

Z=

∫

dx

√

g(x)f(x) =

∫

dx

√

g(x)

∏Nc

k=1

δ(Λk(x)−Ck). (4.9.21)

The appearance of the metric determinant in the phase space integral conforms to the
requirement of eqn. (4.9.17), which states that the number of microstates available to
the system is determined byf(x) when it is integrated with respect to theconserved
volume element

√

g(x)dx. Eqns. (4.9.20) and (4.9.21) lie at the heart of our theory of
non-Hamiltonian phase spaces and will be used to analyze a variety ofnon-Hamiltonian
systems in this and subsequent chapters.