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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 ''λ''. If ''X''(t) = ''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]] |
| | |
| | * <math>q_{i, i-1} = i\, \lambda</math> for ''i'' = 1, 2, ..., ''N'' |
| | * <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]] |
Revision as of 10:31, 26 January 2014
The Ehrenfest model (or dog-flea model[1]) of diffusion was proposed by 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 λ. If X(t) = i is defined to be the number of particles in one container at time t, then it is a birth-death process with transition rates
and equilibrium distribution .
Mark Kac proved in 1947 that if the initial system state is not equilibrium, then the entropy, given by
is monotonically increasing (H-theorem). This is a consequence of the convergence to the equilibrium distribution.
References
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- F.P. Kelly Reversibility and Stochastic Networks (Wiley, Chichester, 1979) ISBN 0-471-27601-4 [1] 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.