1549380323-Statistical Mechanics Theory and Molecular Simulation

216 Canonical ensemble

u(r) =

A

rn

,

wherenis an integer andA >0. In the low density limit, compute the

pressure of such a system as a function ofn. Explain why a system described

by such a potential cannot exist stably forn≤3.

Hint: You may express the answer in terms of the Γ-function

Γ(x) =

∫∞

0

dt tx−^1 e−t.

Also, the following properties of the Γ-function may be useful:

Γ(x)> 0 x > 0 ,

Γ(0) =∞,

Γ(n) =∞ for integern < 0

Γ(− 1 /2) =− 2

√

π.

4.14 Often a pair potential is insufficient to describe accurately the behavior of many
real liquids and gases. One then often includesthree-bodyterms in the potential,
which appear as follows:

U(r 1 ,...,rN) =

∑

i>j

u(|ri−rj|) +

∑

i>j>k

v(|ri−rj|,|rj−rk|,|ri−rk|),

where the first term is the usual pair interaction term and the second contains

the three-body contributions.

a. Derive an expression for the average energy in terms ofg(2)(r 1 ,r 2 ) and

g(3)(r 1 ,r 2 ,r 3 ). What is the expression forg(3)?

b. Explain whyg(3)(r 1 ,r 2 ,r 3 ) should only depend onr 1 −r 2 ,r 3 −r 2 andr 3 −r 1.

By making the following coordinate transformation:

R=

1

3

(r 1 +r 2 +r 3 )

r=r 1 −r 2

s=r 3 −

1

2

(r 1 +r 2 ),

show thatg(3)really only depends onrands. By integrating over the variable

R, obtain a new distribution function ̃g(3)(r,s).