1549380323-Statistical Mechanics Theory and Molecular Simulation

314 Monte Carlo

Can this algorithm be used in conjunction with hybrid Monte Carlo? Whyor

why not?

7.6. Write a replica exchange Monte Carlo algorithm to sample the Boltzmann

distribution corresponding to a double-well potentialU(x) of the form

U(x) =D 0

(

x^2 −a^2

) 2

fora= 1 andD 0 values of 5 and 10. For each case, optimize the temperature

ladderT 1 ,...,TM, the number of replicasM, and the frequency of exchange

attempts. Use separate Nos ́e-Hoover chains to control the temperatures on

each of theMreplicas. For each simulation, plot the following measure of

convergence:

ζk=

1

Nbins

N∑bins

i=1

|Pk(xi)−Pexact(xi)|,

wherePexact(x) is the exact probability distribution,Pk(x) is the probability

at thekth step of the simulation, andNbinsis the number of bins used in the

calculation of the histogram of the system with temperatureT 1.

∗7.7 Suppose a non-Hamiltonian molecular dynamics algorithm, such as the Nos ́e-

Hoover chain method of Section 4.10, is used to generate paths in the transi-

tion path sampling algorithm. Assuming symmetry of the rule for generating

shooting points, i.e.,τ(yj∆t|xj∆t)/τ(xj∆t|yj∆t), show that the acceptance rule

in eqn. (7.7.15) must be modified to read

Λ[X(T)|Y(T)] =hA(x 0 )hB(xn∆t)min

[

1 ,

f(x 0 )

f(y 0 )

J(yj∆t; y 0 )

J(xj∆t; x 0 )

]

,

whereJ(xt; x 0 ) is the Jacobian of the transformation from x 0 to xt(x 0 ) de-

termined by eqn. (4.9.2).