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Angular momentum by D M Brink; G R Satchler

By D M Brink; G R Satchler

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J. Gumbel (1958); R. Rammal (1985)]. For example, the low-temperature thermodynamics of a system is often governed by the statistics of low-energy states. The problem arises generally in the following way. Suppose that {X,} is an ensemble of N independent random variables and YN is the random variable whose value takes the minimum value in the ensemble {Xj} : YN=Minj{Xj} . 36 Central Limit Theorem and Stable Laws The cumulative distribution of YN is : TV -n/ Piob[YN>y} = Tl fXj(x)dx I since if y denotes the minimum of the values of Xj, then all values of Xj must be larger than y.

36 Central Limit Theorem and Stable Laws The cumulative distribution of YN is : TV -n/ Piob[YN>y} = Tl fXj(x)dx I since if y denotes the minimum of the values of Xj, then all values of Xj must be larger than y. The probability density of the variable Yjv is just the opposite of the derivative of Prob[Yjv > y] with respect to y. Suppose now that all random variables X have the same probability distribution fx- In this case one obtains : fYN(y) = Nfy(y)(l- f fx(x)dx\ . 42) To see whether this probability distribution leads to some definite limiting distribution when the number N of variables tends to infinity, first one has to search for the most probable value of Yff [E.

Zolotarev (1986)] : ^Ao(z) = V l £ e X P ( 2 ^ ) W ~WI* ( 2 ^ ) where Wp,q is the Whittaker function [M. Abramowitz k. I. A. Stegun (1964)], Stable Laws for Sum of Uncorrelated Variables 23 • The distribution p, = 1/2,/? = 0 [E. W. Montroll k J. T. Bendler (1984)] : Pi/2,o(x) = y/2^ + I2 2 \/2dd S (y/2Z\x\ ) ) 008(1/41*1) sin(l/4|d,) with C and S the sine and cosine Fresnel integrals, respectively. 18) corresponds to the Gaussian law with a variance equal 2 and not exactly to the normal law. 14).

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