1549380323-Statistical Mechanics Theory and Molecular Simulation

492 The Feynman path integral

12.7 Problems

12.1. Derive primitive and virial estimators for the full pressure tensorPαβdefined

by

Pαβ=

kT

det(h)

∑

γ

∂lnQ

∂hαγ

hβγ,

wherehμνis the cell matrix.

∗12.2. Derive a virial form for the heat capacity at constant volume using the ther-

modynamic relation

CV=kβ^2

∂^2 lnQ

∂β^2

.

12.3. Derive eqns. (12.5.6) and (12.5.7) and generalize these equations to the case

ofNparticles in three dimensions.

12.4. Derive eqn. (12.4.24).

12.5. The following problem considers the path-integral theory forthe tunneling of

a particle through a barrier.

a. Show that the path-integral expression for the density matrixcan be

written as:

ρ(x,x′;β) =

∫x(β ̄h/2)=x′

x(−β ̄h/2)=x

D[x] exp

[

−

1

̄h

∫β ̄h/ 2

−βh/ ̄ 2

dτ

(

1

2

mx ̇^2 (τ) +U(x(τ))

)]

.

b. Consider a double-well potential of the form

U(x) =

ω^2

8 a^2

(x^2 −a^2 )^2.

Show that, for a particle of unit mass, the dominant path for the density

matrixρ(−a,a;β) is given by

x(τ) =atanh[(τ−τ 0 )ω/2]

in the low-temperature limit with negligible error in the endpoint con-

ditions. This path is called aninstantonorkinksolution. Discuss the

behavior of this trajectory in imaginary timeτ.

c. Calculate the classical imaginary-time action for the kink solution.