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| {{Stub-general}}
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| ==Boltzmann's H-theorem==
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| Boltzmann's '''H-theorem''' states that the [[entropy]] of a closed system can only increase in the course of time, and must
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| approach a limit as time tends to infinity.
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| :<math>\sigma \geq 0</math>
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|
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| where <math>\sigma</math> is the ''entropy source strength'', given by (Eq 36 Chap IX Ref. 2)
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| :<math>\sigma = -k \sum_{i,j} \int C(f_i,f_j) \ln f_i d {\mathbf u}_i</math>
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| where the function C() represents binary collisions.
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| At equilibrium, <math>\sigma = 0</math>.
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| ==Boltzmann's H-function==
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| Boltzmann's ''H-function'' is defined by (Eq. 5.66 Ref. 3):
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| :<math>H=\iint f({\mathbf V}, {\mathbf r}, t) \ln f({\mathbf V}, {\mathbf r}, t) ~ d {\mathbf r} d{\mathbf V}</math>
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| where <math>{\mathbf V}</math> is the molecular velocity. A restatement of the H-theorem is
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| :<math>\frac{dH}{dt} \leq 0</math>
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| ==Gibbs's H-function==
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| ==See also==
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| *[[Boltzmann equation]]
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| *[[Second law of thermodynamics]]
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| ==References== | | ==References== |
| # L. Boltzmann "", Wiener Ber. '''63''' pp. 275- (1872) | | # L. Boltzmann "", Wiener Ber. '''63''' pp. 275- (1872) |
| #[http://store.doverpublications.com/0486647412.html Sybren R. De Groot and Peter Mazur "Non-Equilibrium Thermodynamics", Dover Publications]
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| #[http://www.oup.com/uk/catalogue/?ci=9780195140187 Robert Zwanzig "Nonequilibrium Statistical Mechanics", Oxford University Press (2001)]
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| '''Related reading'''
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| *[http://dx.doi.org/10.1073/pnas.1001185107 Philip T. Gressman and Robert M. Strain "Global classical solutions of the Boltzmann equation with long-range interactions", Proceedings of the National Academy of Sciences of the United States of America '''107''' pp. 5744-5749 (2010)]
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| *[http://dx.doi.org/10.1063/1.3675847 James C. Reid, Denis J. Evans, and Debra J. Searles "Communication: Beyond Boltzmann's H-theorem: Demonstration of the relaxation theorem for a non-monotonic approach to equilibrium", Journal of Chemical Physics '''136''' 021101 (2012)]
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| [[category: non-equilibrium thermodynamics]] | | [[category: non-equilibrium thermodynamics]] |