600 Langevin and generalized Langevin equations
and orthogonal toA(0). This is done by inserting the identity operatorIinto eqn.
(15.7.7) and using the fact thatP+Q=I, which yields
dA
dt= eiLt(P+Q)iLA(0) = eiLtPiLA(0) + eiLtQiLA(0). (15.7.8)The first term can be evaluated by introducing eqn. (15.7.3) into eqn. (15.7.8):
eiLtPiLA(0) = eiLt〈iLAA†〉〈AA†〉−^1 A(0). (15.7.9)The integrations implied by the angular brackets in eqn. (15.7.9) are performed over
an ensemble distribution of initial conditionsA(0). The propagator exp(iLt) can be
pulled across the ensemble averages, since the quantity〈iLAA†〉〈AA†〉−^1 is a matrix
independent of the phase space variables. Thus,
eiLtPiLA(0) =〈iLAA†〉〈AA†〉−^1 eiLtA(0)=〈iLAA†〉〈AA†〉−^1 A(t)≡iΩA(t),whereΩis a force-constant matrix given by
Ω=〈LAA†〉〈AA†〉−^1. (15.7.10)Note that because of the application of the operators exp(iLt) andP, the first term
in eqn. (15.7.8) is effectively linear inA(t).
In order to evaluate the second term, we start with the trivial identity
eiLt= eQiLt+ eiLt−eQiLt. (15.7.11)The operator difference exp(iLt)−exp(QiLt) appearing in this identity can be evalu-
ated as follows. We first take the Laplace transform of the exp(iLt)−exp(QiLt):
∫∞0dte−st[
eiLt−eQiLt]
= (s−iL)−^1 −(s−QiL)−^1. (15.7.12)Eqn. (15.7.12) involves the difference of operator inverses. In general, given a generic
operator difference of the formO− 11 −O− 21 , we multiply the first term by the iden-
tity operator expressed asI=O 2 O− 21 and the second term by the identity operator
expressed asI=O− 11 O 1 to yield
O− 11 −O 2 −^1 =O− 11 (O 2 −O 1 )O− 21. (15.7.13)Applying eqn. (15.7.13) to the difference in eqn. (15.7.12) gives