Dimensionality of a random chain

A continuous random chain can be considered as a curve defined by a vector function r(s) with 0 s S. However, it would not be right to conceive such a chain as a unidimensional object. In fact, a close relation exists between the dimensionality of a chain and the exponent v. 

In order to define the dimensionality of an object, we can proceed as follows. We introduce an arbitrarily small length 0 and we look for the minimal number N0 of balls of diameter 0 that are necessary to cover the object completely (see Fig. 2.4). We discover that, when 0 goes to zero, N varies according to a law of the form 

The number D defines the dimension of the object in a Hausdorff sense,2 and we shall call it the dimensionality of the object. Of course, this number has no connection with the dimension d of the space in which the object is embedded, in spite of the fact that the balls which cover the object have the dimension d. For usual objects, one easily verifies that the preceding definition of D corresponds to our intuition. 

Let us now try to find out the dimensionality of a Kuhnian chain. For such a chain, the distance between end points is 

Let us consider the chain as made of N parts with equal causes S/N. The size of each chain segment is approximately 

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