1549380323-Statistical Mechanics Theory and Molecular Simulation

Polymer model 25

=





0 1

−1 0





= M, (1.6.34)

showing that the symplectic condition is satisfied.

1.7 A simple classical polymer model

Before moving on to more formal developments, we present a simpleclassical model
for a free polymer chain that can be solved analytically. This example will not only
serve as a basis for more complex models of biological systems presented later but
will also reappear in our discussion of quantum statistical mechanics. The model is
illustrated in Fig. 1.7 and consists of a set ofNpoint particles connected by nearest

Fig. 1.7 The harmonic polymer model.

neighbor harmonic interactions. The Hamiltonian for this system is

H=

∑N

i=1

p^2 i

2 m

+

1

2

N∑− 1

i=1

mω^2 (|ri−ri+1|−bi)^2 , (1.7.1)

wherebiis the equilibrium bond length. For simplicity, all of the particles are assigned
the same mass,m. Consider a one-dimensional analog of eqn. (1.7.1) described by

H=

∑N

i=1

p^2 i

2 m

+

1

2

N∑− 1

i=1

mω^2 (xi−xi+1−bi)^2. (1.7.2)

In order to simplify the problem, we begin by making a change of variables of the form