H-theorem
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[edit] 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 approach a limit as time tends to infinity.
\[\sigma \geq 0\]
where \(\sigma\) is the entropy source strength, given by (Eq 36 Chap IX Ref. 2)
\[\sigma = -k \sum_{i,j} \int C(f_i,f_j) \ln f_i d {\mathbf u}_i\]
where the function C() represents binary collisions. At equilibrium, \(\sigma = 0\).
[edit] Boltzmann's H-function
Boltzmann's H-function is defined by (Eq. 5.66 Ref. 3):
\[H=\iint f({\mathbf V}, {\mathbf r}, t) \ln f({\mathbf V}, {\mathbf r}, t) ~ d {\mathbf r} d{\mathbf V}\]
where \({\mathbf V}\) is the molecular velocity. A restatement of the H-theorem is
\[\frac{dH}{dt} \leq 0\]
[edit] Gibbs's H-function
[edit] See also
[edit] References
- 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)
