1549380323-Statistical Mechanics Theory and Molecular Simulation

606 Langevin and generalized Langevin equations

b. Compute the autocorrelation function〈ˆσy(0)ˆσy(t)〉assuming an initial

canonical distribution.

c. In order to mimic the effect of an environment, the above two-level system

is often coupled to a bath of quantum-mechanical harmonic oscillators for

which the Hamiltonian is given by

Hˆ=ε

2

ˆσz+

∆

2

ˆσx+

∑

α

̄hωα

[

ˆa†αˆaα+

1

2

]

+

̄h

2

σˆz

∑

α

gα

(

ˆa†α+ ˆaα

)

,

whereαis an index that runs over all of the bath modes,ωαare the bath

frequencies, ˆaαand ˆa†αare the bath annihilation and creation (lowering

and raising) operators, respectively, andgαare coupling constants. For

this Hamiltonian, write down the Heisenberg equations of motion forall

operators, including the Pauli matrices of the system and the creation

and annihilation operators of the bath.

d. By solving the Heisenberg equations for the bath operators, develop gen-

eralized Langevin type equations for the spin operators ˆσxand ˆσy.

15.4. a. Derive eqn. (15.4.4).

∗b. Derive eqn. (15.4.6).

∗15.5. 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. Supposeq(0) = 0.

a. Show that the velocity autocorrelation functionCvv(t) obeys the following

integro-differential equation:

d

dt

Cvv(t) =−

∫t

0

dτ K(t−τ)Cvv(τ),

whereK(t) =ω^2 +γ(t). This equation is one in a class of integro-

differential equations known asVolterraequations.

b. Devise a numerical algorithm for extracting the memory functionand

friction kernel given a velocity autocorrelation function obtained from a

molecular dynamics calculation. Note that this requires inversion of the

Volterra equation. Discuss any numerical difficulties you expect to arise

in the implementation of your algorithm.