[Article] A Thermodynamic Study of Electrolytic Solutions by Hitchcock F.L.

By Hitchcock F.L.

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This is again a special case of the L´evy distribution obtained when we set α = 1. Consider N independent and identically distributed L´evy random variables, with the common distribution denoted by L(x; α). The sum YN is again a L´evy distribution denoted by LN (x; α), obtained from L(x; α) by replacing D by N D. Let us scale YN by N 1/α and consider the random variable YN /N 1/α . Its distribution is L(x; α). Thus N 1/α provides a natural scaling for Levy distributions. We have, LN (x; α) = N −1/α LX x N 1/α ;α .

Metropolis algorithm is widely used in Monte Carlo simulation of models in statistical physics. Here I shall illustrate the technique for sampling from an arbitrary discrete distribution f (i), where the random variable takes discrete integer values i between 1 and N . We call 1, 2, · · · , N as states and denote the set of all states by Ω = {1, 2, · · · N }. 05 0 −5 −4 −3 −2 −1 0 1 2 3 4 5 x Figure 13: Sampling from a Gaussian of mean zero and variance unity employing the technique proposed by Ferna´ ndez and Criado [39].

2 (108) Repeat the above several times. After initial warm up time of say 4N iterations or so, the velocities of the pair of particles you pick up in all subsequent iterations are the desired pairs of independent Gaussian random numbers with mean zero and variance unity. Gaussian of desired mean, say µ and standard deviation, say σ can be obtained 2 by the transformation x = σv + µ. Note that N i=1 vi = N at all stages of iteration. This algorithm is found to be ten times faster than the Box-Muller algorithm.

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