1549380323-Statistical Mechanics Theory and Molecular Simulation

210 Canonical ensemble

Show that these equations obey the following two conservation laws:

C=

p^2

2 m

+

p^2 η

2 Q

+kTη≡H′

K=peη.

b. Show, therefore, that the distribution function in the physicalmomentum

pgenerated by the equations is

f(p) =

√

2 Q

√

p^2 (C−(p^2 / 2 m) +kTln(p/K))

rather than the expected Maxwell-Boltzmann distribution

f(p) =

1

√

2 πmkT

exp(−p^2 / 2 mkT).

c. Plot the distributionf(p) and show that it matches the distribution shown

in Fig. 4.9.

d. Write a program that integrates the equations of motion using the algo-

rithm of Section 4.11 and verify that the numerical distribution matches

that of part c.

e. Next, consider the Nos ́e–Hoover chain equations withM= 2 for the same

free particle:

p ̇=−

pη 1

Q

p, η ̇k=

pηk

Q

, p ̇η 1 =

p^2

m

−kT−

pη 2

Q

pη 1 , p ̇η 2 =

p^2 η 1

Q

−kT.

Here,k= 1,2. Show that these equations of motion generate the correct

Maxwell-Boltzmann distribution inp.

∗f. Will these equations yield the correct Maxwell-Boltzmann distribution

in practice if implemented using the Liouville-based integrator of Sec-

tion 4.11?

Hint: Consider how an initial momentump(0)>0 evolves under the

action of the integrator. What happens ifp(0)<0?

∗g. Derive the general distribution generated by eqns. (4.8.19) when no exter-

nal forces are present and the conservation law in eqn. (4.9.27) is obeyed.

Hint: Since the conservation law involves the center of mass momentum

P, it is useful to introduce a canonical transformation to center-of-mass

momentum and position (R,P) and thed(N−1) corresponding relative

coordinatesr′ 1 ,r′ 2 ,...and momentap′ 1 ,p′ 2 ,....

∗h. Show that when∑N

i=1Fi= 0, the Nos ́e–Hoover chain equations generate

the correct canonical distribution in all variables except the center-of-

mass momentum.