1549380323-Statistical Mechanics Theory and Molecular Simulation

Problems 131

Ω(N,V,E) =MN

∫

dE′

∫

dNpδ

(

∑

i=1

p^2 i

2 m

−E′

)

×

∫

D(V)

dNrδ(U(r 1 ,...,rN)−E+E′),

which provides a way to separate the kinetic and potential contributions

to the partition function.

b. Based on the result of part a, show that the partition function can, there-

fore, be expressed as

Ω(N,V,E) =

E 0

N!Γ

( 3 N

2

)

[(

2 πm

h^2

) 3 / 2 ]N

×

∫

D(V)

dNr[E−U(r 1 ,...,rN)]^3 N/^2 −^1 θ(E−U(r 1 ,...,rN)),

whereθ(x) is the Heaviside step function.

∗3.2. Figure 1.7 illustrates the harmonic polymer model introduced in Section 1.7.

If we take the equilibrium bond lengths all to be zero, then the potential

energy takes the simple form

U(r 1 ,...,rN) =

1

2

mω^2

∑N

k=0

(rk−rk+1)^2 ,

wheremis the mass of each particle, andωis the frequency of the harmonic

couplings. Letrandr′be the positions of the endpoints, with the defini-

tion thatr 0 ≡randrN+1≡r′. Consider making the following change of

coordinates:

rk=uk+

k

k+ 1

rk+1+

1

k+ 1

r, k= 1,...,N.

Using this change of coordinates, calculate the microcanonical partition func-

tion Ω(N,V,E) for this system. Assume the polymer to be in a cubic box of

volumeV.

Hint: Note that the transformation is defined recursively. How should you

start the recursion? It might help to investigate how it works for a small

number of particles, e.g. two or three.

3.3. A water molecule H 2 O is subject to an external potential. Let the positions

of the three atoms be denotedrO,rH 1 ,rH 2 , so that the forces on the three

atoms can be denotedFO,FH 1 , andFH 2. Consider treating the molecule