Physical Chemistry Third Edition

28.7 Polymer Conformation 1195

The difference equation will maintain the normalization

∑∞

x−∞

∑∞

y−∞

∑∞

z−∞

p(n,x,y,z) 1 (28.7-3)

where the summations are over all values ofx,y, andzcorresponding to lattice points

and where we recognize that the lattice is not limited in size (although the probabilities

for locations very far from the origin will be zero).

We define a one-dimensional probability by summingp(n+1,x,y,z) over all val-

ues ofyandz. We omit the limits on the sum and write

p(n+1,x)

∑

y

∑

z

p(n+1,x,y,z) (28.7-4)

which is the probability that the end of link numbern+1isatx, irrespective of the

yandzvalues. Equation (28.7-1) is now summed over all values ofyandzto give

p(n+1,x)

1

6

[p(n,x+a)+ 4 p(n,x)+p(n,x−a)] (28.7-5)

where we have recognized that theyandzdirections are mathematically equivalent so

that four terms are equal to each other after the summation.

A variable that characterizes the width of a distribution is the mean of the square of

the distance, called thesecond momentor thevariance, which is equal to the square

of the standard deviation of the distribution. The variance of thexcoordinate for link

numbern+1 is given by

〈x^2 〉n+ 1 

∑

x

x^2 p(n+1,x)



1

6

∑

x

x^2 p(n,x+a)+

2

3

∑

x

x^2 p(n,x)+

1

6

∑

x

x^2 p(n,x−a) (28.7-6)

In the first term letx+ax′, and in the third sum letx−ax′′. The second sum is

equal to〈x^2 〉n, so that

〈x^2 〉n+ 1 

1

6

∑

x′

(

x′^2 − 2 ax′+a^2

)

p

(

n,x′

)

+

2

3

〈x^2 〉n

+

1

6

∑

x′′

(

x′′^2 + 2 ax′′+a^2

)

p

(

n,x′′

)

(28.7-7)

Since the limits of the sum are−∞to∞, there is no distinction between a sum

overx,x′,orx′′after the summation is done. Therefore, we can replacex′orx′′byx

in the sums. The two sums containing 2axcancel. The two sums containingx^2 give

〈x^2 〉n, and the two sums containinga^2 can be combined:

〈x^2 〉n+ 1 

(

2

3

+

2

6

)

〈x^2 〉n+

2

3

a^2

∑

x

p(n,x)〈x^2 〉n+

a^2

3

(28.7-8)

where we have used the fact that the distribution is normalized as in Eq. (28.7-3).

Equation (28.7-8) is arecursion relation, analogous to the recursion relation used in

the solution of the Schrödinger equation for the harmonic oscillator in Chapter 15.

If the value forn0 is known the value forn1 can be calculated, and from