500 Classical time-dependent statistical mechanics
this approximation. Interestingly, however, the approximation oflinear response the-
ory proves to be remarkably robust (Bianucciet al., 1996). Within linear response
theory, eqn. (13.2.12) reduces to
(
∂
∂t
+iL 0)
∆f(x,t) =−i∆L(t)f 0 (H(x)), (13.2.13)which follows from the facts that∂f 0 /∂t= 0 andiL 0 f 0 (H(x)) = 0. In order to solve
eqn. (13.2.13), we take the driving force to be 0 fort <0, so that att= 0, the ensemble
is described byf(x,0) =f 0 (H(x)), and ∆f(x,0) = 0. Eqn. (13.2.13) is a simple first-
order inhomogeneous differential equation that can be solved usingthe unperturbed
classical propagator exp(iL 0 t) as an integrating factor. The solution that satisfies the
initial condition is
∆f(x,t) =−∫t0dse−iL^0 (t−s)i∆L(s)f 0 (H(x)). (13.2.14)In order to simplify eqn. (13.2.14), we note that
i∆L(s)f 0 (H(x)) = (iL(s)−iL 0 )f 0 (H(x))=iL(s)f 0 (H(x))= ̇x(s)·∇xf 0 (H(x)) (13.2.15)sinceiL 0 f 0 (H(x)) = 0. However,
̇x(s)·∇xf 0 (H(x)) = ̇x(s)·
∂f 0
∂H∂H
∂x=
∂f 0
∂H∑^3 N
i=1[
p ̇i(s)∂H
∂pi+ ̇qi(s)∂H
∂qi]
=
∂f 0
∂H∑^3 N
i=1[
∂H
∂pi(
−
∂H
∂qi+Di(x)Fe(s))
+
∂H
∂qi(
∂H
∂pi+Ci(x)Fe(s))]
=
∂f 0
∂H∑^3 N
i=1[
Di(x)∂H
∂pi+Ci(x)∂H
∂qi]
Fe(s). (13.2.16)The quantity
j(x) =−∑^3 N
i=1[
Di(x)∂H
∂pi+Ci(x)∂H
∂qi]
(13.2.17)
appearing in eqn. (13.2.16) is known as thedissipative flux. In terms of this quantity,
we have