Linear response 497ing feature of Fig. 13.2(a) is the fact that many more phase space points are visited,
even on the short time scale of the simulation used to generate the figure, than in
Fig. 1.3. Recalling that the equations of motion generate a distribution of accessible
microscopic states, the number of accessible states is larger for the driven oscillator.
Thus, the ensembles generated by the driven and undriven oscillators are not equiva-
lent. Indeed, the undriven oscillator generates a microcanonical ensemble, while eqns.
(13.1.1) generate a more complex ensemble, which is an example of anonequilibrium
ensemble. Moreover, ifFe(t) were more complicated than the simple periodic function
considered, this complexity would be reflected in the phase space distribution.
Suppose, next, that the oscillator is subject to a frictional force−γp/min addition
to the driving forceFe(t). The equations of motion now read
x ̇=p
m, p ̇=−mω^2 x−γp
m+Fe(t). (13.1.2)A typical position trajectoryx(t), forFe(t) =F 0 cos Ωtwith Ω/ω=
√
2, is shown in
Fig. 13.2(b). The combination of damping and driving forces generates motion with
two components. There is an initialtransientphase that disappears after a certain
length of time, leaving a regular component known as asteady state. The steady
state persists in the long time limit. In general, the phase space distribution of an
ensemble that is allowed to reach a steady state is significantly different from that
of the corresponding equilibrium distribution and, therefore, has different properties.
Moreover, it is clear that the phase space distribution functionf(x,p,t) can have an
explicit time dependence due to the presence of the time-dependent driving force and
the transient component of the motion. From our analysis of this simple system, we
can conclude that ensembles of driven systems are more complex than equilibrium
ensembles. They generally contain many more accessible microscopicstates due to
the presence of the driving forces, they are described by time-dependent phase space
distribution functions, and they often exhibit steady-state behavior.
We now proceed to make these simple arguments more formal by deriving an
approximation to the phase space distribution of a system weakly driven away from
equilibrium by a time-dependent perturbation. Interested readers are also referred
to the detailed article by B. J. Berne (1971) on time-dependent properties in the
condensed phase.
13.2 Driven systems and linear response theory
In this section, we will consider a general class of driven classical systems and their
corresponding phase space distributions. Consider a classical system described by 3N
generalized coordinatesq 1 ,...,q 3 N≡q, 3Ngeneralized momentap 1 ,...,p 3 N≡p, and
a HamiltonianH(q,p), which, in the absence of driving forces, satisfies Hamilton’s
equations of motion, eqns. (1.6.11). We now wish to include the effectof a weak driving
force that is assumed to perturb the system only slightly away fromequilibrium. To
this end, we introduce the following equations of motion: