1549380323-Statistical Mechanics Theory and Molecular Simulation

Problems 211

i. Finally, show the conservation

∑N

i=1Fi= 0 isnotobeyed by eqns. (4.10.7)

and, therefore, that they also generate a correct canonical distribution in

allvariables.

4.4. Consider a modified version of the Nos ́e–Hoover equations fora harmonic
oscillator with unit mass, unit frequency, andkT= 1:

x ̇=p−pηx, p ̇=−x−pηp η ̇=pη, p ̇η=p^2 +x^2 − 2.

a. Show that these equations have the two conservation laws:

C=

1

2

(

p^2 +x^2 +p^2 η

)

+ 2η

K=

1

2

(

p^2 +x^2

)

e^2 η.

b. Determine the distributionf(H) of the physical HamiltonianH(x,p) =

(p^2 +x^2 )/2. Is the distribution the expected canonical distributionf(H)∝

exp(−H)?

Hint: Try using the two conservation laws to eliminate the variablesη

andpη.

∗c. Show that a plot of the physical phase spacepvs.xnecessarily must have

a hole centered at (x,p) = (0,0), and find a condition that determines

the size of the hole.

4.5. Suppose the interactions in anN-particle system are described by a pair
potential of the form

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

∑N

i=1

∑N

j>i

u(|ri−rj|).

In the low density limit, we can assume that each particle interacts withat

mostone other particle.

a. Show that the canonical partition function in this limit can be expressed

as

Q(N,V,T) =

(N−1)!!VN/^2

N!λ^3 N

[

4 π

∫∞

0

dr r^2 e−βu(r)

]N/ 2

.

b. Show that the radial distribution function is proportional to exp[−βu(r)]

in this limit.