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: This equation is readily applied to discuss² an idealized case of random fragmentation of area. Consider a rectangle of sides l ₁ and l ₂ (area Σ = l ₁ l ₂) and imagine it to be divided into subrectangles by drawing at random N ₁and 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): 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, For S≫S ₀ we obtain the approximate relation: