H-theorem: Difference between revisions
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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]] | ||
Revision as of 10:44, 4 September 2007
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.
- 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 \geq 0}
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 \sigma} is the entropy source strength, given by (Eq 36 Chap IX Ref. 2)
- 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 = -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, .
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}
Gibbs's H-function
See also
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)