H-theorem

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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma = 0} .

Boltzmann's H-function

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H=\iint f({\mathbf V}, {\mathbf r}, t) \ln f({\mathbf V}, {\mathbf r}, t) ~ d {\mathbf r} d{\mathbf V}}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbf V}} is the molecular velocity. A restatement of the H-theorem is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dH}{dt} \leq 0}

See also

• Boltzmann equation
• Second law of thermodynamics

References

1. L. Boltzmann "", Wiener Ber. 63 pp. 275- (1872)
2. Sybren R. De Groot and Peter Mazur "Non-Equilibrium Thermodynamics", Dover Publications
3. 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)