Mathematical economics: Difference between revisions

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The '''Ehrenfest model''' (or '''dog-flea model'''<ref>{{cite doi|10.1119/1.1632488}}</ref>) of [[diffusion]] was proposed by [[Tatyana Afanasyeva|Tatiana]] and [[Paul Ehrenfest]] to explain the [[second law of thermodynamics]]. The model considers ''N'' particles in two containers. Particles independently change container at a rate&nbsp;''λ''. If ''X''(t)&nbsp;=&nbsp;''i'' is defined to be the number of particles in one container at time t, then it is a [[birth-death process]] with [[Continuous-time Markov process#Mathematical definitions|transition rates]]
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* <math>q_{i, i-1} = i\, \lambda</math> for ''i'' = 1, 2, ..., ''N''
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* <math>q_{i, i+1} = (N-i\,) \lambda</math> for ''i'' = 0, 1, ..., ''N'' – 1
 
and equilibrium distribution <math>\pi_i = 2^{-N} \tbinom Ni</math>.
 
[[Mark Kac]] proved in 1947 that if the initial system state is not equilibrium, then the [[Entropy (information theory)|entropy]], given by
 
:<math>H(t) = -\sum_{i} P(X(t)=i) \log \left( \frac{P(X(t)=i)}{\pi_i}\right) , </math>
 
is monotonically increasing ([[H-theorem]]). This is a consequence of the convergence to the equilibrium distribution.
 
==References==
{{Reflist}}
* [[F.P. Kelly]] Reversibility and Stochastic Networks (Wiley, Chichester, 1979) ISBN 0-471-27601-4 [http://www.statslab.cam.ac.uk/~frank/BOOKS/kelly_book.html] pp. 17–20
* "Ehrenfest model of diffusion." [[Encyclopædia Britannica]] (2008)
* Paul und Tatjana Ehrenfest. Über zwei bekannte Einwände gegen das Boltzmannsche H-Theorem. Physikalishce Zeitschrift, vol. 8 (1907), pp. 311-314.
 
[[Category:Queueing theory]]
[[Category:Diffusion]]
[[Category:Stochastic processes]]

Latest revision as of 15:37, 19 December 2014

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