Probability Distribution for the Freely Jointed Chain

To determine the probability distribution function for the chain end-to-end distance, we first consider a freely jointed, one-dimensional chain having links of length = //VI, which are aU constrained to lie along the x axis. What is the probabihty that the end-to-end distance of this one-dimensional chain is mLl The 

Taking the natural logarithm of both sides of Eq. (10.2.5) and recognizing that 

Equation (10.2.8) represents a discrete probability distribution and is the probability that x lies between ml and (m + 2)1. This is because if n+ increases by 1, has to decrease by 1 and m increases by 2. Simultaneously, the distance between the chain ends goes up by 2/. To obtain the continuous probability distribution p(x) dx, which is the probabdity that the end-to-end distance ranges from to X -I- dx, we merely multiply p ni) by dx/ 2l. Furthermore, because m equals 

In order to extend the one-dimensional results embodied in Eq. (10.2.9) to the three-dimensional case of practical interest, we use the law of joint probability. According to this law, the probability of a number of events happening simultaneously is the product of the probabdities of each of the events occurring individually. Thus, the probability, p r) dr, that the unconstrained end of the freely jointed chain lies in a rectangular parallelepiped defined by X, j, z, x- - ix, y + dy, and z + dz (see Fig. 10.1) is the product p x) dx p y) dy p(z) dz, where p(y) dy and p z) dz are defined in a manner analogous to / (x) dx. Therefore, 

Having obtained the average value of the square of the chain end-to-end distance and the distribution of end-to-end values about this mean, it is worth pausing and again asking if there is any relation between these results and results for real polymer molecules. In other words, how closely do freely jointed chains approximate actual macromolecules If the answer is not very closely, then how do we modify the freely jointed chain results to make them apply to polymers  

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