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Difference between revisions of "Boltzmann H-theorem"

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One of the consequences of the kinetic [[Boltzmann equation|Boltzmann equation]], according to which the function
 
One of the consequences of the kinetic [[Boltzmann equation|Boltzmann equation]], according to which the function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016840/b0168402.png" /></td> </tr></table>
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$$H(t)\equiv\int h(t,\mathbf r)d\mathbf r=\int f\ln fd\mathbf pd\mathbf r,$$
  
is a non-increasing function of the time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016840/b0168403.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016840/b0168404.png" /> is the dimensionless classical single-particle distribution function of the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016840/b0168405.png" /> and the momenta <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016840/b0168406.png" />, which satisfies the Boltzmann equation. The evolution of the density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016840/b0168407.png" /> with time is determined by the relaxational nature of the evolution of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016840/b0168408.png" /> towards the local [[Maxwell distribution|Maxwell distribution]], while the limit value of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016840/b0168409.png" />-function as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016840/b01684010.png" /> is equal to the entropy of the ideal gas, calculated according to Gibbs, with the opposite sign. If the increments of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016840/b01684011.png" /> under consideration are much larger than the time needed for the local Maxwell distribution to be established, the quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016840/b01684012.png" /> may be identified with the entropy density, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016840/b01684013.png" /> may be identified with the non-equilibrium entropy of an ideal gas.
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is a non-increasing function of the time $t$. Here $f=f(t,\mathbf r,\mathbf p)$ is the dimensionless classical single-particle distribution function of the coordinates $\mathbf r$ and the momenta $\mathbf p$, which satisfies the Boltzmann equation. The evolution of the density $h(t,\mathbf r)$ with time is determined by the relaxational nature of the evolution of the function $f$ towards the local [[Maxwell distribution|Maxwell distribution]], while the limit value of the $H$-function as $t\to\infty$ is equal to the entropy of the ideal gas, calculated according to Gibbs, with the opposite sign. If the increments of $t$ under consideration are much larger than the time needed for the local Maxwell distribution to be established, the quantity $(-h(t,\mathbf r))$ may be identified with the entropy density, while $(-H(t))$ may be identified with the non-equilibrium entropy of an ideal gas.
  
From the point of view of statistical mechanics, the principal significance of the Boltzmann <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016840/b01684014.png" />-theorem consists in the mathematical expression of the fundamental assumptions of macroscopic thermodynamics, according to which, for example, an isolated system spontaneously tends to the state of thermodynamic equilibrium, the process being accompanied by an increase in entropy.
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From the point of view of statistical mechanics, the principal significance of the Boltzmann $H$-theorem consists in the mathematical expression of the fundamental assumptions of macroscopic thermodynamics, according to which, for example, an isolated system spontaneously tends to the state of thermodynamic equilibrium, the process being accompanied by an increase in entropy.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016840/b01684016.png" />-theorems are statements analogous to the original Boltzmann <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016840/b01684017.png" />-theorem, but formulated for statistical systems of a different or a more general type, including the case of non-ideal and quantified systems.
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$H$-theorems are statements analogous to the original Boltzmann $H$-theorem, but formulated for statistical systems of a different or a more general type, including the case of non-ideal and quantified systems.
  
 
The theory was presented by L. Boltzmann in 1872.
 
The theory was presented by L. Boltzmann in 1872.

Revision as of 13:57, 20 April 2014

One of the consequences of the kinetic Boltzmann equation, according to which the function

$$H(t)\equiv\int h(t,\mathbf r)d\mathbf r=\int f\ln fd\mathbf pd\mathbf r,$$

is a non-increasing function of the time $t$. Here $f=f(t,\mathbf r,\mathbf p)$ is the dimensionless classical single-particle distribution function of the coordinates $\mathbf r$ and the momenta $\mathbf p$, which satisfies the Boltzmann equation. The evolution of the density $h(t,\mathbf r)$ with time is determined by the relaxational nature of the evolution of the function $f$ towards the local Maxwell distribution, while the limit value of the $H$-function as $t\to\infty$ is equal to the entropy of the ideal gas, calculated according to Gibbs, with the opposite sign. If the increments of $t$ under consideration are much larger than the time needed for the local Maxwell distribution to be established, the quantity $(-h(t,\mathbf r))$ may be identified with the entropy density, while $(-H(t))$ may be identified with the non-equilibrium entropy of an ideal gas.

From the point of view of statistical mechanics, the principal significance of the Boltzmann $H$-theorem consists in the mathematical expression of the fundamental assumptions of macroscopic thermodynamics, according to which, for example, an isolated system spontaneously tends to the state of thermodynamic equilibrium, the process being accompanied by an increase in entropy.

$H$-theorems are statements analogous to the original Boltzmann $H$-theorem, but formulated for statistical systems of a different or a more general type, including the case of non-ideal and quantified systems.

The theory was presented by L. Boltzmann in 1872.

References

[1] A. Sommerfeld, "Thermodynamics and statistical mechanics" , Acad. Press (1956) (Translated from German)
[2] G.E. Uhlenbeck, G.V. Ford, "Lectures in statistical mechanics" , Amer. Math. Soc. (1963)
How to Cite This Entry:
Boltzmann H-theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boltzmann_H-theorem&oldid=31887
This article was adapted from an original article by I.A. Kvasnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article