1549380323-Statistical Mechanics Theory and Molecular Simulation

Problems 605

15.2. Consider a single harmonic degree of freedomq, having a massm, that obeys
the generalized Langevin equation

q ̈=−ω^2 q−

∫t

0

dτq ̇(τ)γ(t−τ) +f(t).

Hereωis the frequency associated with the motion ofq,γ(t) =ζ(t)/m, and

f(t) =R(t)/m, whereR(t) andζ(t) are the random force and friction kernel,

respectively. As a crude model of a rapidly decaying friction kernel,consider

aγ(t) that is constant over a very short timet 0 and then drops suddenly to

0:

γ(t) =γ 0 θ(t 0 −t).

Hereγ 0 is a constant,θ(x) is the Heaviside step function,t≥0, andt 0 ≥0.

a. Show that ift 0 is small enough that exp(−st 0 )≈ 1 −st 0 for all relevant

values ofs, then the presence of memory in this system causes the normal-

ized velocity autocorrelation functionCvv(t) to oscillate with a frequency

less thanωand to decay slowly in time. Determine the decay constant

and oscillation frequency ofCvv(t).

b. Suppose that we now lett 0 become very large. Show that the velocity

autocorrelation no longer decays but oscillates with a frequency different

fromω. What is the oscillation frequency ofCvv(t)?

c. Explain the physical origin of the behavior of the velocity correlation

function in the two limits considered in parts a and b in terms of the

response of the bath to the system.

15.3. A simple model for electron transfer is defined by a quantum Hamiltonian of
the form
Hˆ=ε
2

σˆz+

∆

2

ˆσx,

whereǫand ∆ are constants and ˆσx, ˆσy, and ˆσzare the Pauli matrices given

by

σˆx=

(

0 1

1 0

)

σˆy=

(

0 −i

i 0

)

ˆσz=

(

1 0

0 − 1

)

.

The model assumes that there are only two states for the electron and is thus

a simplification of the true electron-transfer problem.

a. In the Heisenberg picture, the operators that describe the observables of

a system evolve in time according to the equation of motion

dAˆ

dt

=

1

i ̄h

[A,ˆHˆ],

whereAˆis an arbitrary operator. Write down Heisenberg’s equations for

ˆσx, ˆσy, and ˆσz.