602 Langevin and generalized Langevin equations
=F(t) +∫t0dτ〈iLF(τ)A†〉〈AA†〉−^1 eiL(t−τ)A(0)=F(t) +∫t0dτ〈iLF(τ)A†〉〈AA†〉−^1 A(t−τ). (15.7.18)SinceF(t) is orthogonal toA(t), it follows that
QF(t) =F(t). (15.7.19)Eqn. (15.7.19) can be used to simplify the ensemble average appearing in eqn. (15.7.18).
We first express the ensemble average ofiLF(τ)A†as
〈iLF(τ)A†〉=〈iLQF(τ)A†〉. (15.7.20)We next transfer the operatoriLtoA†by taking the Hermitian conjugate ofiL.
Recalling thatL, itself, is Hermitian, we only need to changeito−iso that
〈iLF(τ)A†〉=−〈QF(τ)(iLA)†〉. (15.7.21)Using eqn. (15.7.19) and the fact thatQis Hermitian allows us to write eqn. (15.7.21)
as
〈iLF(τ)A†〉=−〈Q^2 F(τ)(iLA)†〉=−〈QF(τ)(QiLA)†〉=−〈F(τ)F†(0)〉. (15.7.22)Therefore, eqn. (15.7.18) becomes
eiLtQiLA=F(t)−∫t0dτ〈F(τ)F†(0)〉〈AA†〉−^1 A(t−τ). (15.7.23)Finally, combining eqn. (15.7.23) with eqns. (15.7.10) and (15.7.8) givesan equation
of motion forAin which bath degrees of freedom have been effectively eliminated:
dA
dt=iΩA−∫t0dτ〈F(τ)F†(0)〉〈AA†〉−^1 A(t−τ) +F(t). (15.7.24)Eqn. (15.7.24) takes the form of a GLE for a harmonic potential of mean force if
the autocorrelation function appearing in the integral is identified with the dynamic
friction kernel
K(t) =〈F(t)F†(0)〉〈AA†〉−^1. (15.7.25)
The quantityK(t) is called thememory functionormemory kernel. Note thatK(t) is
a matrix. Substituting eqn. (15.7.25) into eqn. (15.7.24) gives the generalized Langevin
equation for a general bath
dA
dt=iΩA(t)−∫t0dτK(τ)A(t−τ) +F(t). (15.7.26)Although eqn. (15.7.26) is formally exact, the problem of determiningF(t) andK(t)
is generally more difficult than simply simulating the full system becauseof the need