1549380323-Statistical Mechanics Theory and Molecular Simulation

608 Langevin and generalized Langevin equations

15.9. a. Derive eqns. (15.7.27), (15.7.30), and (15.7.31).

∗b. Derive eqn. (15.7.34).

∗15.10. A solution contains a very low concentration of solute molecules (denoted A)

and solvent molecules (denoted B). Let the number of solute molecules be

NAwith positionsr 1 (t),...,rNA(t). We can introduce a phase space function

for the solute concentrationc(r,t) at any pointrin space as

c(r,t) =

∑NA

i=1

δ(r−ri(t)).

a. Assuming the solution is in a cubic periodic box of lengthL, show that

the spatial Fourier transform ̃ck(t) ofc(r,t) is

̃ck(t) =

∑NA

i=1

e−ik·ri(t),

wherek= 2πn/L, wherenis a vector of integers.

b. Use the Mori–Zwanzig theory to derive a generalized Langevin equation

for ̃ck(t) and give the explicit expressions for all terms in the equation.

c. Now consider the correlation function

C(k,t) =

〈 ̃c−k(0) ̃ck(t)〉

〈 ̃c−k ̃ck〉

.

Starting with your generalized Langevin equation of part b, derive an

integro-differential equation satisfied byC(k,t).

d. Show that the memory kernel in your equation is at least second order in

the wave vectork.

e. Show that exp(QiLt)→exp(iLt) as|k|→∞.

f. Suppose the memory kernel decays rapidly in time. In this limit, show

that the correlation function satisfies an equation of the form

∂

∂t

C(k,t) =−k·D·kC(k,t),

whereDis the diffusion tensor. Give an expression forDin terms of a

velocity autocorrelation function.