1549380323-Statistical Mechanics Theory and Molecular Simulation

Problems 361

configuration, average the 1s and 0s, and record the average value asp(k). 5) Repeat
for all of the configurations sampled in step 2 until the full set of averaged probabilities
p(1),...,p(M)is generated. 6) Plot a histogram of the probabilitiesp(1),...,p(M). If the
histogram from step 6 peaks sharply at 1/2, thenq(r) is a good reaction coordinate.
However, if the histogram is broad over the entire range (0,1), thenq(r) is a poor reac-
tion coordinate. Illustrations of good and poor reaction coordinates obtained from the
histogram test are shown in Fig. 8.11. Although the histogram test can be expensive
to carry out, it is, nevertheless, an important evaluation of the quality of a reaction
coordinate and its associated free energy profile. Once the investment in the histogram
test is made, the payoff can be considerable, regardless of whether the reaction coor-
dinate passes the test. If it does pass the test, then the same coordinate can be used
in subsequent studies of similar systems. If it does not pass the test, then it is clear
that the coordinateq(r) should be avoided for the present and similar systems.

8.13 Problems

8.1. Derive eqn. (8.3.6).

8.2. Write a program to compute the free energy profile in eqn. (8.3.6) using

thermodynamic integration. How manyλpoints do you need to compute

the integral accurately enough to obtain the correct free energy difference

A(1)−A(0)?

8.3. Write a program to compute the free energy differenceA(1)−A(0) from

eqn. (8.3.6) using the free energy perturbation approach. Can you obtain an

accurate answer using a one-step perturbation, or do you need intermediate

states?

8.4. Derive eqn. (8.7.25).

8.5. Derive eqn. (8.10.22).

8.6. Consider a classical system with two degrees of freedomxandydescribed by

a potential energy

U(x,y) =

U 0

a^4

(

x^2 −a^2

) 2

+

1

2

ky^2 +λxy,

and consider a process in whichxis moved from the positionx=−ato the

positionx= 0.

a. Calculate the Helmholtz free energy difference ∆Afor this process in a

canonical ensemble.

b. Consider now an irreversible process in which the ensemble is frozen in

time and, in each member of the ensemble,xis moved instantaneously