Boltzmann equation: Difference between revisions
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The '''Boltzmann equation''' is given by ( | The '''Boltzmann equation''' is given by (<ref>[http://store.doverpublications.com/0486647412.html Sybren R. De Groot and Peter Mazur "Non-Equilibrium Thermodynamics", Dover Publications]</ref> Eq 1 Chap. IX) | ||
:<math>\frac{\partial f_i}{\partial t} = - {\mathbf u}_i \cdot \frac{\partial f_i}{\partial {\mathbf r}} - {\mathbf F}_i \cdot \frac{\partial f_i}{\partial {\mathbf u}_i } + \sum_j C(f_i,f_j) </math> | :<math>\frac{\partial f_i}{\partial t} = - {\mathbf u}_i \cdot \frac{\partial f_i}{\partial {\mathbf r}} - {\mathbf F}_i \cdot \frac{\partial f_i}{\partial {\mathbf u}_i } + \sum_j C(f_i,f_j) </math> | ||
where <math></math> is an external force and the function C() represents binary collisions. | where <math></math> is an external force and the function C() represents binary collisions. | ||
==Solution== | |||
Recently Gressman and Strain <ref>[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)]</ref> have provided a proof of global existence and rapid decay to equilibrium of classical solutions to the Boltzmann equation. | |||
==See also== | ==See also== | ||
*[[H-theorem]] | *[[H-theorem]] | ||
==References== | ==References== | ||
<references/> | |||
[[category: non-equilibrium thermodynamics]] | [[category: non-equilibrium thermodynamics]] | ||
Revision as of 11:00, 19 May 2010
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The Boltzmann equation is given by ([1] Eq 1 Chap. IX)
- 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{\partial f_i}{\partial t} = - {\mathbf u}_i \cdot \frac{\partial f_i}{\partial {\mathbf r}} - {\mathbf F}_i \cdot \frac{\partial f_i}{\partial {\mathbf u}_i } + \sum_j C(f_i,f_j) }
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 } is an external force and the function C() represents binary collisions.
Solution
Recently Gressman and Strain [2] have provided a proof of global existence and rapid decay to equilibrium of classical solutions to the Boltzmann equation.
See also
References
- ↑ Sybren R. De Groot and Peter Mazur "Non-Equilibrium Thermodynamics", Dover Publications
- ↑ 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)