The Freely Jointed Chain in One Dimension

The algebraic average of the component of r in any arbitrarily specified direction must likewise be zero. 

The average square of the projection of a bond vector on the x-axis is not zero. It may be calculated in a similar manner as follows  

Thus the root-mean-square value of the projection is given by 

According to Eq. (5) the most probable value of x occurs at x = 0. As X increases in magnitude, W(x) decreases monotonically from its maximum at x = 0, the decrease being the more rapid the smaller n. Eq. (5) obviously becomes invalid at values of x approaching nl W x) remains finite, though very small, according to Eq. (5) even for x nlj where it necessarily is zero. 

The foregoing derivation may appear artificial in view of the assumptions involved. The contribution of a given bond to x is by no means restricted to the two unique values, + as has been assumed. On the contrary, one may show that all values of h from 0 to Z occur with equal probability for freely jointed connections between links. A more detailed study of the problem shows that the final result is unaffected by this assumption so long as n is large. The freely jointed chain model under consideration is an artifice also, but the form of the results obtained will be shown to apply also to real polymer chains. 

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