1549380323-Statistical Mechanics Theory and Molecular Simulation

Linear response 501

iL(s)f 0 (H(x)) =−

∂f 0

∂H

j(x)Fe(s). (13.2.18)

Now, suppose thatf 0 (H(x)) is given by a canonical distribution

f 0 (H(x)) =

exp[−βH(x)]

Q(N,V,T)

. (13.2.19)

Then, since∂f 0 /∂H=−βf 0 , eqn. (13.2.18) becomes

iL(s)f 0 (H(x)) =βf 0 (H(x))j(x)Fe(s). (13.2.20)

Note that eqn. (13.2.16) also holds whenf 0 (H(x)) is an isothermal-isobaric or grand
canonical distribution. When we substitute eqn. (13.2.20) into eqn.(13.2.14), we obtain

∆f(x,t) =−β

∫t

0

dse−iL^0 (t−s)f 0 (H(x))j(x)Fe(s). (13.2.21)

We now substitute eqn. (13.2.21) into eqn. (13.2.10) to express theaverage ofa(x) in
the nonequilibrium ensemble as

A(t) =〈a〉t=〈a〉−β

∫

dxa(x)

∫t

0

dse−iL^0 (t−s)f 0 (H(x))j(x)Fe(s)

=〈a〉−β

∫t

0

ds

∫

dxa(x)e−iL^0 (t−s)f 0 (H(x))j(x)Fe(s)

=〈a〉−β

∫t

0

ds

∫

dxf 0 (H(x))a(x)e−iL^0 (t−s)j(x)Fe(s). (13.2.22)

The second term in eqn. (13.2.22) involves a classical propagator exp[−iL 0 (t−s)].
However, since the propagator contains onlyiL 0 , its action generates the evolution
of the undriven system obtained by solving Hamilton’s equations in theabsence of
any perturbation. If we write the propagator as exp[iL 0 (−(t−s))], it is clear that
this propagator evolves a system backwards in time. Recall from Section 3.10 that if
the phase space vector evolves in time from an initial condition x 0 to xtaccording to
Hamilton’s equations (under the action of exp[iL 0 t]), the evolution of any phase space
functiona(x) is determined by

da

dt

=iL 0 a

a(xt) = eiL^0 ta(x 0 ). (13.2.23)

However, since it can be shown thatL 0 is a Hermitian operator, if we take the adjoint
of both sides, we obtain