1549380323-Statistical Mechanics Theory and Molecular Simulation

528 Classical time-dependent statistical mechanics

b. Comparing the two diffusion constants, what two physical situations do

these two models describe?

13.2. The classical isotropic isothermal-isobaric (NPT) ensemble is particularly use-

ful for determining thebulk viscosityof a substance via Green–Kubo theory.

a. Show that the linear response formula does not change if the initial dis-

tribution is chosen to bef 0 (H(r,p),V), i.e., the isothermal-isobaric dis-

tribution function exp(−β(H(r,p) +PV))/∆(N,P,T).

b. Next, consider coupling a system to an external compression field de-

scribed by the equations of motion

r ̇i,α=

pi,α

mi

+

∑

β

ri,βMβα

p ̇i,α=Fi,α−

∑

β

pi,βMβα,

whereαandβindex the three spatial directionsx,y, andz. Show that

the equations of motion satisfy the incompressibility condition.

c. Consider the specific choice

Mαβ=

1

3

γδαβ,

whereγis the compression rate. The coefficient of bulk viscosityηVis

given by a generalization of Newton’s law of viscosity:

〈V〉ηV=−lim

t→∞

〈P(t)V(t)〉

γ

,

where〈···〉on the right represents an average over the equilibrium isotropic

NPT distribution function and〈···〉ηVon the left is the full nonequilib-

rium average. Using the linear response formula to evaluate〈P(t)V(t)〉,

derive the appropriate Green–Kubo expression forηV.

∗13.3. Show that the Einstein relation in eqn. (13.3.34) can be derived from the

Green–Kubo relation in eqn. (13.3.33).

13.4. Verify that eqn. (13.5.23) is conserved by eqn. (13.5.22) in theabsence of

time-dependent boundary conditions.

∗13.5. In Problem 5.9, we considered a particle moving through a periodic potential.

Now suppose we add a linear potential−fqto this system, so that the particle

is driven by a constant forcef. The total potential that results is known as

Galton’s staircase

U(q) =V 0 cos(kq)−fq.