1549380323-Statistical Mechanics Theory and Molecular Simulation

Problems 607

c. Now consider a simple continuous model for the friction kernelγ(t) =

λAe−λt. In what time range would you expect this model to break down

physically and why?

d. Solve the Volterra equation for the velocity autocorrelation function using

the simple exponential friction kernel model in part c. Discuss the influ-

ence of the parametersAandλon your solution. In addition, examine

the free particle case by taking the limitω→0.

e. Finally, discretize your velocity autocorrelation function into timesteps

of size ∆tfor a given choice ofAandλ, and use this discretizedCvv(t) to

test the algorithm you developed in part b. How well can you recoverthe

exponential friction kernel? How robust is your algorithm to the addition

of a little random noise to your discretizedCvv(t)?

15.6. For a particle obeying eqn. (15.3.21), show that the density ofvibrational
states is related to the friction kernel by

I(ω) =

ω^2 γ′(ω)

[ω^2 −ω ̃^2 −ωγ′′(ω)]^2 + [ωγ′(ω)]^2

,

whereγ′(ω) andγ′′(ω) are the real and imaginary parts of ̃γ(iω) andI(ω) is

the Fourier transform of the velocity autocorrelation function

I(ω) =

1

2 π

∫∞

−∞

dt〈q ̇(0) ̇q(t)〉e−iωt.

In general, the connection between the autocorrelation functionof a ran-

dom process and a spectral density is known as theWiener–Khintchine theo-

rem(Wiener, 1930; Khintchine, 1934; Kuboet al., 1985). Using eqn. (15.2.12),

derive the spectral density for a harmonic bath corresponding to〈R(0)R(t)〉.

15.7. Write a program to integrate the Langevin equation with a Gaussian random
force for a harmonic oscillator with massm= 1, frequencyω= 1, tem-
peraturekT = 1, and frictionγ= 1, using the integrator of Section 15.5.
Verify that the correct momentum and position distribution functions of the
canonical ensemble are obtained (see Problem 4.9).

15.8. Consider the adiabatic free energy dynamics approach of Section 8.10.
a. Reformulate this technique using a set of coupled Langevin equations
with two different temperaturesTqandTfor the firstncoordinates and
remaining 3N−ncoordinates, respectively.

b. Write a program to integrate your equations for the example in eqn.

(8.10.21) using the parameters given following eqn. (8.10.22) and verify

that you are able to generate the analytical free energy profile in eqn.

(8.10.22).