Random fragmentation

Abstract

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) ˜ π^1/2 N₀(S/S₀)^1/4 exp{ −2 (S/S₀)^1/2}.