Derivation of the GLE 577Taking the Laplace transform of both sides of the second line in eqn.(15.2.2) yieldss^2 ̃xα(s)−x ̇α(0)−sxα(0) +ωα^2 x ̃α(s) =−gα
mα̃q(s). (15.2.6)The use of the Laplace transform has the effect of turning a differential equation into
an algebraic equation for ̃xα(s). Solving this equation for ̃xα(s) gives
̃xα(s) =s
s^2 +ω^2 αxα(0) +1
s^2 +ωα^2x ̇α(0)−gα
mαq ̃(s)
s^2 +ω^2 α. (15.2.7)
We now obtain the solution to the differential equation by computing the inverse
transform ̃xα(s) in eqn. (15.2.7). Applying the inverse Laplace transform relations in
Appendix D (eqn. (D.2)), and recognizing that the last term in eqn. (15.2.7) is the
product of two Laplace transforms and can be inverted using the convolution theorem,
we find that the solution forxα(t) is
xα(t) =xα(0) cosωαt+1
ωαx ̇α(0) sinωαt−
gα
mαωα∫t0dτ sinωα(t−τ)q(τ). (15.2.8)For reasons that will be clear shortly, we integrate the convolutionterm by parts to
express it in the form
∫t0dτ sinωα(t−τ)q(τ) =1
ωα[q(t)−q(0) cosωαt]−
1
ωα∫t0dτ cosωα(t−τ) ̇q(τ). (15.2.9)Substituting eqn. (15.2.9) and eqn. (15.2.8) into the first line of eqn.(15.2.2) yields
the equation of motion forq:
μq ̈=−dV
dq−
∑
αgαxα(t)=−
dV
dq−
∑
αgα[
xα(0) cosωαt+pα(0)
mαωαsinωαt+gα
mαω^2 αq(0) cosωαt]
−
∑
αgα^2
mαωα^2∫t0dτq ̇(τ) cosωα(t−τ) +∑
αg^2 α
mαωα^2q(t). (15.2.10)Eqn. (15.2.10) is in the form of an integro-differential equation for the system coor-
dinate that depends explicitly on the bath dynamics. Although the dynamics of each
bath coordinate are relatively simple, the collective effect of the bath on the system
coordinate can be nontrivial, particularly if the initial conditions of the bath are ran-
domly chosen, the distribution of frequencies is broad, and the frequencies are not
all commensurate. Indeed, the bath might appear to affect the system coordinate in a
random and unpredictable manner, especially if the number of bath degrees of freedom