By Robert Zwanzig

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The one-dimensional harmonic oscillator provides an easy illustration. The Hamiltonian is the equations of motion are and their solution, valid for all t, is Aside from this special case, one can seldom solve Hamilton's equations exactly. " It may be very hard to give a precise prediction of the future behavior of a given initial state. However, because the present concern is with formalism, and not with practical calculations, this difficulty may be ignored. The Liouville Equation In classical statistical mechanics, averages are determined by the phase space distribution function (or phase space density) f(p, q, t) or f(X, t).

At this point, B can be any function of a. FOKKER-PLANCK EQUATIONS 39 Several comments are in order. The derivation as given here depends explicitly on two assumptions, that the noise is Gaussian and that it is delta-function correlated. Otherwise, the factorization of the average over noise will not work. In particular, the derivation will not work for a non-Markovian Langevin equation. Also, no fluctuation-dissipation theorem has been invoked. Nothing has been said about requiring that

The rate of energy transfer is the energy difference times the transition rate, (Ef - Ei)wfi. Since the system starts out in thermal equilibrium, the initial state is taken from a Boltzmann distribution with probability ;. Further, 54 NONEQUILIBRIUM STATISTICAL MECHANICS we are concerned with only the energy absorption at the frequency , and we do not care what the final state is. So we sum over all initial i with the Boltzmann weight i, and sum over all final f, On using the delta functions in the Golden Rule, this becomes The matrix elements are symmetric in (i,f), so by switching indices in the second term, we can combine the delta functions, Because is the Boltzmann distribution, we can relate f to i, and because of the delta function, the exponent in this formula can be changed to , leading to Now we replace the delta function by its integral representation, so that after some rearrangement, the energy absorption is Next, we recall that the time dependence of any quantum mechanical operator is given in the Heisenberg representation by which has the matrix elements MASTER EQUATIONS 55 The energy absorption becomes The sum over / can be done immediately because the set of all states {|j)} is complete, so that The sum over i produces the thermal equilibrium average, The rate of energy absorption is determined by the Fourier transform of the time correlation function of the perturbation V.