Perturbative solution of the Liouville equation

Substituting the perturbative form for into the Liouville equation, one obtains

Recall . Thus, working to linear order in small quantities, one obtains the following equation for :

which is just a first-order inhomogeneous differential equation. This can easily be solved using an integrating factor, and one obtains the result

Note that

But, using the chain rule, we have

Define

which is known as the dissipative flux. Thus, for a Cartesian Hamiltonian

where is the force on the ith particle, the dissipative flux becomes:

In general,

Now, suppose is a canonical distribution function

then

so that

Thus, the solution for is

The ensemble average of the observable now becomes

Recall that the classical propagator is . Thus the operator appearing in the above expression is a classical propagator of the unperturbed system for propagating backwards in time to -(t-s). An observable evolves in time according to

Now, if we take the complex conjugate of both sides, we find

where now the operator acts to the left on . However, since observables are real, we have

which implies that forward evolution in time can be achieved by acting to the left on an observable with the time reversed classical propagator. Thus, the ensemble average of A becomes

where the quantity on the last line is an object we have not encountered yet before. It is known as an equilibrium time correlation function. An equilibrium time correlation function is an ensemble average over the unperturbed (canonical) ensemble of the product of the dissipative flux at t=0 with an observable A evolved to a time t-s. Several things are worth noting:

1.

The nonequilibrium average , in the linear response regime, can be expressed solely in terms of equilibrium averages.

2.

The propagator used to evolve to is the operator , which is the propagator for the unperturbed, Hamiltonian dynamics with . That is, it is just the dynamics determined by H.

3.

Since is a function of the phase space variables evolved to a time t-s, we must now specify over which set of phase space variables the integration is taken. The choice is actually arbitrary, and for convenience, we choose the initial conditions. Since is a function of the initial conditions , we can write the time correlation function as