1549380323-Statistical Mechanics Theory and Molecular Simulation

Problems 363

∗8.12. Develop a weighted histogram procedure to obtain the free energy derivative

dA/dqiat a set of integration pointsqistarting with eqn. (8.8.22). Describe

the difference between your algorithm and that corresponding to the original

WHAM procedure for obtainingAk.

∗8.13. In this problem, we will illustrate how a simple change of integration vari-

ables in the partition function can be used to create an enhanced sampling

method. The approach was originally introduced by Zhuet al.(2002) and

later enhanced by Minaryet al.(2007). Consider the double-well potential

U(x) =

U 0

a^4

(

x^2 −a^2

) 2

.

The configurational partition function is

Z(β) =

∫

dxe−βU(x).

a. Consider the change of variablesq=f(x). Assume that the inversex=

f−^1 (q)≡g(q) exists. Show that the partition function can be expressed

as an integral of the form

Z(β) =

∫

dqe−βφ(q),

and give an explicit form for the potentialφ(q).

b. Now consider the transformation

q=f(x) =

∫x

−a

dye−β

U ̃(y)

for−a≤x≤aandq=xfor|x|> aandU ̃(x) a continuous poten-

tial energy function. This transformation is known as aspatial-warping

transformation(Zhuet al., 2002; Minaryet al., 2007). Show thatf(x) is a

monotonically increasing function ofxand, therefore, thatf−^1 (q) exists.

Write down the partition function that results from this transformation.

c. If the functionU ̃(x) is chosen to beU ̃(x) =U(x) for−a≤x≤aand

U ̃(x) = 0 for|x|> a, then the functionφ(x) is a single-well potential

energy function. Sketch a plot ofqvs.x, and compare the shape ofφ(x)

as a function ofxtoφ(q) as a function ofq.

d. Argue, therefore, that a Monte Carlo calculation carried out based on

φ(q), or molecular dynamics calculation performed using the Hamiltonian

H(q,p) =p^2 / 2 m+φ(q), leads to an enhanced sampling algorithm for

high barriers, and show that the same equilibrium and thermodynamic

properties will result.

Hint: From the plot ofqvs.x, argue that a small change inqleads to a

change inxlarge enough to move it from one well ofU(x) to the other.