1549380323-Statistical Mechanics Theory and Molecular Simulation

Problems 493

∗12.6. Consider two distinguishable particles in one dimension with respective coor-

dinatesxandyand conjugate momentapxandpywith a Hamiltonian

Hˆ= pˆ

2

x

2 m

+

pˆ^2 y

2 M

+U(ˆx) +

1

2

Mω^2 yˆ^2 −λˆxy.ˆ

a. Show that the density matrixρ(x,y,x′,y′;β) can be written in the form

ρ(x,y,x′,y′;β)

=

∫x(β ̄h)=x′

x(0)=x

Dx(τ) exp

[

−

1

̄h

∫β ̄h

0

dτ

(

1

2

mx ̇^2 (τ) +U(x(τ))

)]

T[x;y,y′],

whereT[x;y,y′] is known as theinfluence functional. What is the func-

tional integral expression forT[x;y,y′], and of what function isT[x;y,y′]

a functional?

b. Using the method of expansion about the classical path, derive aclosed

form expression forT(x(τ),y,y′) by evaluating the functional integral.

12.7. A fourth-order Trotter formula valid for traces (Takahashi and Imada, 1984)

is

Tr

[

e−λ(

Aˆ+Bˆ)]

≈Tr

{[

e−λ

A/Pˆ

e−λ

C/Pˆ ]P

}

+O

(

λ^5 P−^4

)

,

when [A,ˆBˆ] 6 = 0.

Cˆ=Bˆ+^1

24

(

λ

P

) (^2) [
B,ˆ

[

A,ˆBˆ

]]

.

Derive the discrete path integral expression for the canonical partition func-

tionQ(N,V,T) forNBoltzmann particles in three dimensions that results

from applying this approximation. In particlar, show that theN-particle po-

tentialU(r 1 ,...,rN) is replaced by a new effective potentialU ̃(r 1 ,...,rN) and

derive the expression for this new potential.

12.8. Consider a system of two distinguishable degrees of freedom with position

operators ˆxandXˆand corresponding momenta ˆpandPˆ, respectively, with

Hamiltonian

Hˆ= pˆ

2

2 m

+

Pˆ^2

2 M

+U(ˆx,Xˆ).

Assume that the massesMandmare such thatM≫m, meaning that the

two degrees of freedom are adiabatically decoupled.

a. Show that the partition function of the system can be approximated as

Q(β) =

∑

n

∮

DX(τ) exp

{

−

1

̄h

∫β ̄h

0

dτ

[

1

2

MX ̇^2 (τ) +εn(X(τ))

]}

,