Statistics of SAWs in continuous space

Expressions (8.13)-(8.18) represent the SAW statistics at /-dimensional lattice. In a metric space, incorporating the variable of shifting the x,-semi-axis of conformational ellipsoid, with the states Mp(s) belonging to its surface 

Substitution of (8.19) induces an essential distinction between w(x) and ( 1 w(x) determines the probability (jc) that the SAW trajectory at given parameters TV and Oi will end in an elementary volume dx, laying on an ellipsoid surface with semi-axes jc i= i,d. In the other case, all the surface of ellipsoid is a geometrical place of points or chain end states with the same corresponding distribution density w(x). 

The maximum of w(jc) at given N and oj corresponds to the most probable or equilibrium state of a polymer chain. The semi-axes x of the equilibrium conformational ellipsoid can be found firom the condition 

We will continue to consider the situation when all the directions of chain steps are equiprobable, i.e  

Substitution of (8.27) into (8.25) makes the semi-axes of equilibrium ellipsoid equal, and equal to the radius of the Flory ball 

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