Linear response 503shorthand notation for a time correlation function that stands in for the left side of
the equation. Using this notation, eqn. (13.2.27) can be written compactly as
A(t) =〈a〉t=〈a〉−β∫t0dsFe(s)〈a(t−s)j(0)〉. (13.2.29)Second, as eqn. (13.2.27) suggests, in linear response theory, the observableA(t)
obtained by averaging the phase space functiona(x) over the nonequilibrium ensemble
is expressible solely in terms of averages over the equilibrium ensemblecharacterized
byf 0 (H(x)). All information concerning the response of the system to the external
perturbation is embodied in the equilibrium time correlation function. This remarkable
result indicates that, within linear response theory, a nonequilibriumaverage can be
generated entirely within an equilibrium calculation. We will study several applications
of linear response theory in Section 13.3.
13.2.1 Properties of equilibrium time correlation functions
Before we apply the linear response theory to specific examples, wefirst explore some of
the properties of time correlation functions. We define the equilibrium time correlation
functionCAB(t) between two observablesAandB, corresponding to phase space
functionsa(x) andb(x), with respect to a normalized equilibrium distribution function
f(x) and dynamics generated by a Liouville operatoriLas
CAB(t) =〈a(0)b(t)〉=∫
dxf(x)a(x)eiLtb(x)=
∫
dxf(x)a(x)b(xt(x)). (13.2.30)Since the propagator exp(iLt) can be taken to act either to the right as a forward
propagator or to the left as a backward propagator, the time correlation function
satisfies the property
〈a(0)b(t)〉=〈a(−t)b(0)〉. (13.2.31)
Att= 0,
CAB(0) =〈ab〉=∫
dxf(x)a(x)b(x), (13.2.32)which is a simple equilibrium average ofa(x)b(x). The long time (t→ ∞) limit,
by contrast, is a little more subtle. In complex many-body systems characterized by
highly nonlinear forces, the influence of each initial condition on the resultant trajec-
tories generated by exp(iLt) rapidly becomes negligible as time proceeds (recall the
rapid decay of the transient component of the forced-damped harmonic oscillator in
Section 13.1). This loss of memory of the initial condition means that there is a charac-
teristic time, called thecorrelation time, over which the trajectory xt(x) appears to be
particular to a given choice of x and beyond which xt(x) is essentially indistinguishable
from any other trajectory. This is merely a conceptual device, asevery trajectory is
uniquely determined for all time by its initial conditions. However, in complex many-
body systems, nearly all trajectories will exhibit the same type of chaotic behavior