498 Classical time-dependent statistical mechanics
q ̇i=∂H
∂pi+Ci(q,p)Fe(t)p ̇i=−∂H
∂qi+Di(q,p)Fe(t), (13.2.1)whereFe(t) is a time-dependent driving function andCi(q,p) andDi(q,p) are phase
space functions whose forms are determined by the particular external perturbation. In
our treatment, we will not consider forces that give rise to a nonzero phase space com-
pressibility, such as frictional forces. Rather, we will impose the requirement that eqns.
(13.2.1), although possibly non-Hamiltonian, nevertheless satisfy an incompressibility
condition:
∑^3 N
i=1[
∂q ̇i
∂qi+
∂p ̇i
∂pi]
= 0. (13.2.2)
We note, however, that if eqns. (13.2.1) can be generated by a time-dependent Hamil-
tonianH(x,t), where x = (q,p) is the phase space vector, then the usual conservation
law dH/dt= 0 is replaced by
dH
dt
=
∂H
∂t, (13.2.3)
which states that a nonzero total time derivative of the Hamiltonianarises solely from
the explicit time dependence ofH.
Substituting eqns. (13.2.1) into eqn. (13.2.2) leads to a restriction on the choice of
the phase space functionsCi(q,p) andDi(q,p)
∑^3 Ni=1[
∂^2 H
∂pi∂qi+
∂Ci
∂qi
Fe(t)−∂^2 H
∂qi∂pi+
∂Di
∂pi
Fe(t)]
= 0 (13.2.4)
or simply
∑^3 Ni=1[
∂Ci
∂qi+
∂Di
∂pi]
= 0. (13.2.5)
As we showed in Section 2.5, when the equations of motion have zero phase space
compressibility, the phase space distribution functionf(x,t) satisfies the Liouville
equation
∂
∂t
f(x,t) +iLf(x,t) = 0, (13.2.6)whereiL= ̇x·∇xis the Liouville operator.
Solving the Liouville equation for an ensemble of systems described byeqns.
(13.2.1) is nontrivial, in general, especially for largeN. However, if we assume that the
external driving forces constitute only a small perturbation to Hamilton’s equations,
so that the ensemble remains relatively close to its equilibrium distribution, then we