Difference between revisions of "H-theorem"

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{{Stub-general}}
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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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:<math>\sigma = -k \sum_{i,j} \int C(f_i,f_j) \ln f_i d {\mathbf u}_i</math>
 
:<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.
 
At equilibrium, <math>\sigma = 0</math>.
 
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==
 
==See also==
 
==See also==
 
*[[Boltzmann equation]]
 
*[[Boltzmann equation]]
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# 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]
 
#[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]]

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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.

\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.

Boltzmann's H-function[edit]

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

Gibbs's H-function[edit]

See also[edit]

References[edit]

  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