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However, we can give a simple heuristic argument for the dependence of the critical restitution coefficient rc on N for large N . Consider a test particle with speed 1 that is incident on an array of static, equally-spaced particles when the restitution coefficient is nearly 1. After the first collision, the target particle moves with a slightly smaller speed, while the incident particle comes nearly to rest. It is now helpful to exchange the identities of the two particles when they collide, so that the worldline of particle 1 is merely deflected slightly as it “passes through” particle 2.
The transformed contour is also a small circle about the origin since y and x are proportional to each other near the origin. Let’s apply this inversion formula to f (y) = y e−y = x. From Eq. 39) and using we have I I 1 y 1 − y ny 1 −y (1 − y) e dy = e dy An = 2πi (y e−y )n+1 2πi yn dx dy = (1 − y) e−y , To find the residue we simply expand the exponential in a power series and then read off the coefficient of y1 in the integral. Thus An = 1 2πi I X ∞ ” nn−1 nk “ k−n nn−2 nn−1 y − y k+1−n dy = − = , k!
F (ǫ) = 3 ǫ−ǫ3 −ǫ1/2 1+ 5 ǫ 1+ 2 ( 4 ) ǫ(1−ǫ2 ) 1/2 . 5. Assume that the restitution coefficient is randomly chosen according to the distribution f (ǫ). Generalize Eq. (refroot). 1. Heuristic derivation of extremal statistics. Evaluate the limiting behaviors of the scaling function Φ(z) in Eq. 35) when z → 0 and z → ∞. 32) with the scaling function (problem 2) Φ(z) = az µ exp [−bz µ ] . 38) 2. Evaluate the cluster size distribution P (v, t) and the cumulative distribution Qm (v, t) for the special case P0 (v) = e−v .