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Stochastic processes relating to particles distributed in a continuous infinity of states

Published online by Cambridge University Press:  24 October 2008

Alladi Ramakrishnan
Alladi Ramakrishnan
Affiliation:
Department of MathematicsUniversity of Manchester
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Extract

Many stochastic problems arise in physics where we have to deal with a stochastic variable representing the number of particles distributed in a continuous infinity of states characterized by a parameter E, and this distribution varies with another parameter t (which may be continuous or discrete; if t represents time or thickness it is of course continuous). This variation occurs because of transitions characteristic of the stochastic process under consideration. If the E-space were discrete and the states represented by E1, E2, …, then it would be possible to define a function

representing the probability that there are ν1 particles in E1, ν2 particles in E2, …, at t. The variation of π with t is governed by the transitions defined for the process; ν1, ν2, … are thus stochastic variables, and it is possible to study the moments or the distribution function of the sum of such stochastic variables

with the help of the π function which yields also the correlation between the stochastic variables νi.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 46 , Issue 4 , October 1950 , pp. 595 - 602
Copyright
Copyright © Cambridge Philosophical Society 1950

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References

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