Examples 58315.3 Analytically solvable examples based on the GLE
In the next few subsections, a number of simple yet illustrative examples of both
Langevin and generalized Langevin dynamics will be examined in detail. In particular,
we will study the free Brownian and compute its diffusion constant and then consider
the free particle in a more general bath with memory. Finally, we will consider the
harmonic oscillator and derive well-known relations for the vibrational and energy
relaxation times.
15.3.1 The free Brownian particle
A particle diffusing in a dissipative bath with no external forces is known as afree
Brownian particle. The dynamics is described by eqn. (15.2.23) withW(q) = 0:
μq ̈=−ζ 0 q ̇+R(t). (15.3.1)Since only ̈qand ̇qappear in the equation of motion, we can rewrite eqn. (15.3.1) in
terms of the velocityv= ̇q
μv ̇=−ζ 0 v+R(t). (15.3.2)Eqn. (15.3.2) can be treated as an inhomogeneous first-order equation that can be
solved in terms ofR(t). In order to derive the solution for a given initial valuev(0),
we take the Laplace transform of both sides, which yields
μ(s ̃v(s)−v(0)) =−ζ 0 v ̃(s) +R ̃(s). (15.3.3)Definingγ 0 =ζ 0 /μandf(t) =R(t)/μand solving for ̃v(s) gives
̃v(s) =v(0)
s+γ 0+
f ̃(s)
s+γ 0. (15.3.4)
The function 1/(s+γ 0 ) has a single pole ats=−γ 0. Hence, the inverse Laplace
transform (see Appendix D) yields the solution forv(t) as
v(t) =v(0)e−γ^0 t+∫t0dτ f(τ)e−γ^0 (t−τ). (15.3.5)From eqn. (15.3.5), it is clear that the solution for a free Brownian particle has two
components: a transient component dependent onv(0) that decays at larget, and a
steady-state term involving a convolution of the random force withexp(−γ 0 t). Thus,
the system quickly loses memory of its initial condition, and the dynamics for long
times is determined by the bath, as we would expect for a random walkprocess such
as Brownian motion.
If we wish to compute the diffusion constant of the Brownian particle, we can use
eqn. (13.3.32) and calculate the the velocity autocorrelation function〈v(0)v(t)〉. From
eqn. (15.3.5), the velocity correlation is obtained by multiplying both sides byv(0)