Random fragmentation

Auluck, F. C. ; Kothari, D. S. (1954) Random fragmentation Nature, 174 . pp. 565-566. ISSN 0028-0836

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Official URL: http://www.nature.com/nature/journal/v174/n4429/ab...

Related URL: http://dx.doi.org/10.1038/174565a0


The problem of random fragmentation of a line into a finite number of N parts has received considerable attention, partly because of its application in assessing the randomness of radioactive disintegrations and cosmic ray events. For a line of length l the average number of fragments equal to or greater than x is1: N(x) = N(1-x/l)N-1.(1) This equation is readily applied to discuss2 an idealized case of random fragmentation of area. Consider a rectangle of sides l1 and l2 (area ∑= l 1 l2) and imagine it to be divided into subrectangles by drawing at random N1 and N 2 lines respectively parallel to the two sides of the rectangle. If N(S) be the average number of elements of area equal to or exceeding S, we have, using equation (1): N(S) = N1N2(N1-1)(N2-1)/l1l2∫∫ xy>S(1-x/I1)N-11 dxdy ˜ −2N0(S/S0)1/2K1[20(S/S0)1/2] where K 1(z) is the usual Bessel function of imaginary argument, N 0= N 1 N 2 is the total number of elements and S 0 the average area of an element, S0 = ∑/N0 For S»S 0 we obtain the approximate relation: N(S) ˜ π1/2 N0(S/S0)1/4 exp{ −2 (S/S0)1/2}.

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