584 Langevin and generalized Langevin equations
and averaging over a canonical distribution of the initial conditions at temperatureT,
exp(−μv(0)^2 /kT)/Q:
〈v(0)v(t)〉=〈
(v(0))^2〉
e−γ^0 t+∫t0dτ〈v(0)f(τ)〉e−γ^0 (t−τ). (15.3.6)The second term in eqn. (15.3.6) vanishes because〈v(0)f(τ)〉=〈v(0)R(τ)/μ〉= 0.
Thus, interestingly, the velocity autocorrelation function is determined by the transient
term, hence, the short-time dynamics. Performing the average〈(v(0))^2 〉
〈
(v(0))^2〉
=
∫∞
−∞dv v(^2) e−μv^2 / 2 kT
∫∞
−∞dve
−μv^2 / 2 kT =
kT
μ
(15.3.7)
yields the velocity autocorrelation function as
〈v(0)v(t)〉=
kT
μe−γ^0 t. (15.3.8)Finally, from eqn. (13.3.32), we find
D=∫∞
0dt〈v(0)v(t)〉=kT
μ∫∞
0dte−γ^0 t=kT
μγ 0=
kT
ζ 0, (15.3.9)
which has the expected units of length^2 ·(time)−^1. Note that asζ 0 → ∞, the bath
becomes infinitely dissipative and the diffusion constant goes to zero. Note that this
simple picture of diffusion cannot capture the long-time algebraic decay of the velocity
autocorrelation function mentioned in Section 13.3.
15.3.2 Free particle in a bath with memory
If the bath has memory, then the dynamics of the particle is given bythe GLE, which,
for a free particle, reads:
μq ̈=−∫t0dτq ̇(τ)ζ(t−τ) +R(t). (15.3.10)As a concrete example, suppose the dynamic friction kernel is givenby an exponential
function
ζ(t) =λAe−λ|t|, (15.3.11)
which could describe the long-time decay of a realistic friction kernel.Although the
cusp att= 0 is problematic for the short-time behavior of a typical friction kernel, the
exponential friction kernel is, nevertheless, a convenient and simple model that can be
solved analytically and has been studied in some detail in the literature(Berneet al.,
1966). Once again, let us introduce the velocityv= ̇q. For the exponential friction
kernel of eqn. (15.3.11), the GLE then reads
μv ̇=−λA∫t0dτ v(τ)e−λ(t−τ)+R(t), (15.3.12)where we are restricting the time domain tot >0. Let us introduce the quantities
a=A/μandf(t) =R(t)/μ. The Laplace transform can turn this integro-differential