1549380323-Statistical Mechanics Theory and Molecular Simulation

502 Classical time-dependent statistical mechanics

a∗(xt) =a∗(x 0 )e−iL^0 t

a(xt) =a(x 0 )e−iL^0 t, (13.2.24)

where the last line follows from the fact that physical observables are real. In eqn.
(13.2.24), the propagator acts to the left ona(x 0 ). Thus, we see that the action of
exp(−iL 0 t) on the left evolvesa(x 0 )forwardin time just as exp(iL 0 t) produces for-
ward evolution when it acts to the right. According to this result, the propagator
exp[−iL 0 (t−s)] in eqn. (13.2.22) can be taken to act to the left ona(x) to pro-
ducea(xt−s), assuming x is an initial condition to the undriven equations of motion.
Consequently, we could also write eqn. (13.2.22) as

A(t) =〈a〉−β

∫t

0

ds

∫

dxf 0 (H(x))j(x)eiL^0 (t−s)a(x)Fe(s). (13.2.25)

Both eqns. (13.2.25) and (13.2.22) indicate that the nonequilibrium ensemble average
can be expressed as

A(t) =〈a〉−β

∫t

0

ds Fe(s)

∫

dxf 0 (H(x))a(xt−s)j(x). (13.2.26)

Since every solution to Hamilton’s equations is a unique function of theinitial condi-
tions, i.e., xt−s= xt−s(x) is a unique function of the initial condition x, we can write
eqn. (13.2.26) more explicitly as

A(t) =〈a〉−β

∫t

0

ds Fe(s)

∫

dxf 0 (H(x))a(xt−s(x))j(x). (13.2.27)

Eqn. (13.2.27) is an expression for the ensemble average of a phasespace functiona(x)
for the driven system described by eqns. (13.2.1) valid within linear response theory.
At this point, several comments are in order. First, we interpret the second term in
eqn. (13.2.27) as follows: We take each point x in phase space and useit as an initial
condition for Hamilton’s equations of motion, evolving each initial condition up to time
t−s. This evolution yields a new phase space point xt−s, which depends uniquely on
x. We evaluate the phase space functiona(x) at the point xt−s, givinga(xt−s(x)).
We then take an average ofa(xt−s(x))j(x) over the phase space with respect to the
unperturbed distribution functionf 0 (H(x)) of all possible initial conditions. Finally,
we integrate the result multiplied byFe(s) oversfrom 0 tot. The quantity

∫

dxf 0 (H(x))a(xt−s(x))j(x)≡〈a(t−s)j(0)〉 (13.2.28)

has a form we have not previously encountered. Specifically, it is known as anequi-
librium time correlation function; these functions play a fundamental role in time-
dependent statistical mechanics. The right side of eqn. (13.2.28) isa commonly used