1549380323-Statistical Mechanics Theory and Molecular Simulation

Problems 529

a. Write a program to integrate the driven equations of motion under con-

ditions of constant energy, and use the program to show that thesystem

never reaches a steady state by computing the average〈q ̇〉tin the long

time limit.

b. Next try coupling the system to a Nos ́e–Hoover chain thermostat (see

Section 4.10) at temperaturekT= 1. Does the system now reach a steady

state? If so, calculate the average〈q ̇〉tfor different values offand find

a regime for which the dependence of the average onfis linear, and

estimate the value of the diffusion constantD. Takem= 1,k= 1, and

V 0 = 1.

∗∗c. Finally, couple the system to the thermostat of Problem 4.2, usingM= 2.

Algorithms for integrating these equations are given by Liu and Tuck-

erman (2000) and by Ezra (2007). Repeat the analysis of part b, and

estimate the diffusion constant for different values ofV 0.

13.6. Verify that the cell matrix evolution in eqn. (13.5.20) reproduces eqn. (13.5.1)
for planar Couette flow starting withh(0) = diag(L,L,L).

13.7. Consider a fluid confined between corrugated plates as in Fig. 13.8 but with
the fluid particles obeying the equations of motion

r ̇i=

pi

m

p ̇i=Fi+Feˆex,

whereFeis a constant andFiis the force on particleidue to the other fluid

particles and the corrugated wall. Describe the velocity profile you would

expect to develop in a steady state, and give the mathematical form of this

profile.

13.8. Prove that〈Pxy〉= 0 in the canonical ensemble.