1549380323-Statistical Mechanics Theory and Molecular Simulation

Problems 263

p ̇i=−

∂U

∂ri

−

pǫ

W

pi−

pη

Q

pi

V ̇=dV pǫ

W

p ̇ǫ=dV(P(int)−P)−

pη

Q

pǫ

η ̇=

pη

Q

p ̇η=

∑N

i=1

p^2 i

mi

+

p^2 ǫ

W

−(Nf+ 1)kT,

(Hoover, 1985), whereP(int)is the pressure estimator of eqn. (5.7.28). These

equations have the conserved energy

H′=H(r,p) +

p^2 ǫ

2 W

+

p^2 η

2 Q

+ (Nf+ 1)kTη+PV.

Determine the ensemble distribution functionf(r,p,V) generated by these

equations for

a.

∑N

i=1Fi^6 = 0. Would the distribution be expected to approach the correct

isothermal-isobaric ensemble distribution in the thermodynamic limit?

∗b. ∑N

i=1F= 0, in which case, there is an additional conservation law

K=Peǫ+η,

wherePis the center-of-mass momentum. Be sure to integrate overall

nonphysical variables.

5.8. Prove that eqns. (5.11.6) generate the correct isobaric-isoenthalpic ensemble
distribution when the pressure is determined using a molecular virial.

5.9. A simple model for the motion of particles through a nanowire consists of a
one-dimensional ideal gas ofNparticles moving in a periodic potential. Let
the Hamiltonian for one particle with coordinateqand momentumpbe

h(q,p) =

p^2

2 m

+

kL^2

4 π^2

[

1 −cos

(

2 πq

L

)]

,

wheremis the mass of the particle,kis a constant, andLis the length of

the one-dimensional “box” or unit cell.

a. Calculate the change in the Helmholtz free energyper particlerequired

to change the length of the “box” fromL 1 toL 2. Express your answer in

terms of the zeroth-order modified Bessel function