1549380323-Statistical Mechanics Theory and Molecular Simulation

Problems 649

Hint: Try redefining the coupling constant byu= eKand show that P′can

be put in the same form as P, i.e., P′(u) =c(u)P(u′), and thatccan be

defined implicitly in terms ofu′.

∗16.5. A simple model of a long polymer chain consists of the following assumptions:

i. The conformational energyEof the chain is determined solely from its

backbone dihedral angles.

ii. Each dihedral angle can assume three possible values denotedtfor “trans”

andg+andg−for the two “gauche” conformations. However, the present

model is discrete in the sense thatt,g+, andg−are the only values the

dihedral angles may assume.

iii. Each conformation has an intrinsic energy and is also influenced by the

conformations of nearest neighbor dihedral angles only. If the polymer has

Natomic sites, then there areN−3 dihedral angles numberedφ 1 ,...,φN− 3

by convention. The total energyE(φ 1 ,...,φN− 3 ) can be written as

E(φ 1 ,...,φN− 3 ) =

N∑− 3

i=1

ε 1 (φi) +

N∑− 3

i=2

ε 2 (φi− 1 ,φi),

where eachφihas valuest,g+, org−.

iv. The two energy functionsε 1 andε 2 are assumed to have the following

values:

ε 1 (t) = 0

ε 1 (g+) =ε(g−) =ε

ε 2 (g+,g−) =ε 2 (g−,g+) =∞

ε 2 (φi− 1 ,φi) = 0 for all other combinations.

a. Calculate the canonical partition function for this system. You may ex-

press your answer in terms ofσ≡exp(−βε).

b. Show that, in the limitN→∞, the partition function behaves as

lim

N→∞

1

N

lnQ= lnχ,

where

χ=

1

2

[

(1 +σ) +

√

1 + 6σ+σ^2

]

.

c. What is the probability, for largeN, that all angles will be in the trans

conformation?