# Difference between revisions of "H-theorem"

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+ | ==Boltzmann's H-theorem== | ||

Boltzmann's '''H-theorem''' states that the [[entropy]] of a closed system can only increase in the course of time, and must | Boltzmann's '''H-theorem''' states that the [[entropy]] of a closed system can only increase in the course of time, and must | ||

approach a limit as time tends to infinity. | approach a limit as time tends to infinity. | ||

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where the function C() represents binary collisions. | where the function C() represents binary collisions. | ||

At equilibrium, <math>\sigma = 0</math>. | At equilibrium, <math>\sigma = 0</math>. | ||

− | ==H-function== | + | ==Boltzmann's H-function== |

Boltzmann's ''H-function'' is defined by (Eq. 5.66 Ref. 3): | Boltzmann's ''H-function'' is defined by (Eq. 5.66 Ref. 3): | ||

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:<math>\frac{dH}{dt} \leq 0</math> | :<math>\frac{dH}{dt} \leq 0</math> | ||

+ | ==Gibbs's H-function== | ||

==See also== | ==See also== | ||

*[[Boltzmann equation]] | *[[Boltzmann equation]] | ||

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#[http://store.doverpublications.com/0486647412.html Sybren R. De Groot and Peter Mazur "Non-Equilibrium Thermodynamics", Dover Publications] | #[http://store.doverpublications.com/0486647412.html Sybren R. De Groot and Peter Mazur "Non-Equilibrium Thermodynamics", Dover Publications] | ||

#[http://www.oup.com/uk/catalogue/?ci=9780195140187 Robert Zwanzig "Nonequilibrium Statistical Mechanics", Oxford University Press (2001)] | #[http://www.oup.com/uk/catalogue/?ci=9780195140187 Robert Zwanzig "Nonequilibrium Statistical Mechanics", Oxford University Press (2001)] | ||

+ | '''Related reading''' | ||

+ | *[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)] | ||

+ | *[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]] |

## Latest revision as of 17:01, 13 January 2012

## Contents

## Boltzmann's H-theorem[edit]

Boltzmann's **H-theorem** states that the entropy of a closed system can only increase in the course of time, and must
approach a limit as time tends to infinity.

where is the *entropy source strength*, given by (Eq 36 Chap IX Ref. 2)

where the function C() represents binary collisions. At equilibrium, .

## Boltzmann's H-function[edit]

Boltzmann's *H-function* is defined by (Eq. 5.66 Ref. 3):

where is the molecular velocity. A restatement of the H-theorem is

## Gibbs's H-function[edit]

## See also[edit]

## References[edit]

- L. Boltzmann "", Wiener Ber.
**63**pp. 275- (1872) - Sybren R. De Groot and Peter Mazur "Non-Equilibrium Thermodynamics", Dover Publications
- Robert Zwanzig "Nonequilibrium Statistical Mechanics", Oxford University Press (2001)

**Related reading**

- 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) - 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)