70 Theoretical foundations
forf(x,t) require input of further information; we will return to this point again as
specific ensembles are considered in subsequent chapters.
Finally, note that if we use the definition ofη(x,t) in eqn. (1.6.24) and apply the
analysis leading up to eqn. (1.6.19), it is clear thatη(x,t)·∇xf(x,t) ={f(x,t),H(x,t)},
where{...,...}is the Poisson bracket. Thus, the Liouville equation can also be written
as
∂
∂t
f(x,t) +{f(x,t),H(x,t)}= 0, (2.5.14)a form we will employ in the next section for deriving general equilibriumsolutions.
2.6 Equilibrium solutions of the Liouville equation
In Section 2.3, we argued that thermodynamic variables can be computed from averages
over an ensemble. Such averages must, therefore, be expressed in terms of the ensemble
distribution function. Ifa(x) is a microscopic phase space function corresponding to a
macroscopic observableA, then a proper generalization of eqn. (2.3.1) is
A=〈a(x)〉=∫
dxf(x,t)a(x). (2.6.1)Iff(x,t) has an explicit time dependence, then so will the observableA, in general.
However, we also remarked earlier that a system in thermodynamic equilibrium has a
fixed thermodynamic state. This means that the thermodynamic variables characteriz-
ing the equilibrium state do not change in time. Thus, ifAis an equilibrium observable,
the ensemble average in eqn. (2.6.1) must yield a time-independent result, which is
only possible if the ensemble distribution of a system in thermodynamicequilibrium
hasnoexplicit time dependence, i.e.,∂f/∂t= 0. This will be the case, for example,
when no external driving forces act on the system, in which caseH(x,t)→H(x) and
η(x,t)→η(x).
When∂f/∂t= 0, the Liouville equation eqn. (2.5.14) reduces to
{f(x),H(x)}= 0. (2.6.2)The general solution to eqn. (2.6.2) is any function of the HamiltonianH(x):
f(x)∝F(H(x)). (2.6.3)This is as much as we can say from eqn. (2.6.2) without further information about
the ensemble. In order to ensure thatf(x) is properly normalized according to eqn.
(2.5.1), we write the solution as
f(x) =1
Z
F(H(x)), (2.6.4)whereZis defined to be
Z=∫
dxF(H(x)). (2.6.5)The quantityZ, referred to as thepartition function, is one of the central quanti-
ties in equilibrium statistical mechanics. The partition function is a measure of the