1549380323-Statistical Mechanics Theory and Molecular Simulation

Examples 147

the integration ofxover all space with no significant loss of accuracy. Therefore, the
partition function becomes

Q(β) =

1

h

∫∞

−∞

dpe−βp

(^2) / 2 m

∫∞

−∞

dxe−βmω

(^2) x (^2) / 2

=

1

h

(

2 πm

β

) 1 / 2 (

2 π

mω^2

) 1 / 2

=

2 π

βhω

=

1

β ̄hω

, (4.5.19)

where ̄h=h/ 2 π. From eqn. (4.5.19), it follows that the energy isE=kT, the pressure
isP= 0 (which is expected for a bound system), and the heat capacity isCV=k.
If we now consider a collection ofNuncoupled harmonic oscillators with different
masses and frequencies with a Hamiltonian

H=

∑N

i=1

[

p^2 i

2 mi

+

1

2

miω^2 ix^2 i

]

. (4.5.20)

Since the oscillators are not identical, the 1/N! factor is not needed, and the partition
function is just a product of single particle partition functions for theNoscillators:

Q(N,β) =

∏N

i=1

1

β ̄hωi

. (4.5.21)

For this system, the energy isE=NkT, and the heat capacity is simplyCv=Nk.

4.5.3 The harmonic bead-spring model

Another important class of harmonic models is a simple model of a polymer chain
based on harmonic nearest-neighbor interactions. Consider a polymer with endpoints
at positionsrandr′havingNrepeat units in between, each of which will be treated
as a single ’particle’. The particles are indexed from 0 toN+ 1, and the Hamiltonian
takes the form

H=

N∑+1

i=0

p^2 i

2 m

+

1

2

mω^2

∑N

i=0

(ri−ri+1)^2 , (4.5.22)

wherer 0 ,...,rN+1andp 0 ,...,pN+1are the positions and momenta of the particles
with the additional identificationr 0 =randrN+1=r′andp 0 =pandpN+1=p′as
the positions and momenta of the endpoint particles, andmω^2 is the force constant.
The polymer is placed in a cubic container of volumeV =L^3 such thatLis much
larger than the average distance between neighboring particles|rk−rk+1|.
Let us first consider the case in which the endpoints are fixed at given positionsr
andr′so thatp=p′= 0. We seek to calculate the partition functionQ(N,V,T,r,r′)
given by