1549380323-Statistical Mechanics Theory and Molecular Simulation

262 Isobaric ensembles

5.1. Show that the distribution function for the isothermal-isobaricensemble can

be derived starting from a microcanonical description of a system coupled to

both a thermal reservoir and a mechanical piston.

5.2. Calculate the volume fluctuations ∆Vgiven by

∆V=

√

〈V^2 〉−〈V〉^2

in the isothermal-isobaric ensemble. Express the answer in terms ofthe isother-

mal compressibilityκdefined to be

κ=−

1

〈V〉

(

∂〈V〉

∂P

)

N,T

.

Show that ∆V/〈V〉∼ 1 /

√

Nand hence vanish in the thermodynamic limit.

5.3. Prove the tensorial version of the work virial theorem in eqn. (5.6.15).

∗5.4. a. For the ideal gas in Problem 4.6 of Chapter 4, calculate the isothermal-

isobaric partition function assuming that only the length of the cylinder

can vary.

Hint: You might find the binomial theorem helpful in this problem.

b. Derive an expression for the average length of the cylinder.

5.5. Prove that the isotropicNPTequations of motion in eqns. (5.9.5) generate

the correct ensemble distribution function using the techniques ofSection 4.9

for the following cases:

a.

∑N

i=1Fi^6 = 0,

b.

∑N

i=1Fi= 0, for which there is an additional conservation law

K=Pexp

[(

1 +

d

Nf

)

ǫ+η 1

]

,

whereP=

∑N

i=1piis the center-of-mass momentum;

c. “Massive” thermostatting is used on the particles.

5.6. Prove that eqns. (5.10.2) for generating anisotropic volume fluctuations gen-

erate the correct ensemble distribution when

∑N

i=1Fi= 0.

5.7. One of the first algorithms proposed for generating the isotropicNPTen-

semble via molecular dynamics is given by the equations of motion

r ̇i=

pi

mi

+

pǫ

W

ri