Entropy maximisation problem for quantum relativistic particles
Bulletin de la Société Mathématique de France, Volume 133 (2005) no. 1, pp. 87-120.

The entropy of an ideal gas, both in the case of classical and quantum particles, is maximised when the number particle density, linear momentum and energy are fixed. The dispersion law energy to momentum is chosen as linear or quadratic, corresponding to non-relativistic or relativistic behaviour.

L'entropie d'un gaz idéal de particules, classiques ou quantiques, est maximisée lorsque la densité du nombre de particules, l'impulsion et l'énergie sont fixées. La loi de dispersion qui relie l'impulsion et l'énergie est linéaire ou quadratique, selon que le comportement des particules est non relativiste ou relativiste.

DOI: 10.24033/bsmf.2480
Classification: 82B40, 82C40, 83-02
Keywords: entropy, maximisation problem, moments, bosons, fermions
Mot clés : entropie, maximisation, moments, bosons, fermions
@article{BSMF_2005__133_1_87_0,
     author = {Escobedo, Miguel and Mischler, St\'ephane and Valle, Manuel A.},
     title = {Entropy maximisation problem for quantum relativistic particles},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {87--120},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {133},
     number = {1},
     year = {2005},
     doi = {10.24033/bsmf.2480},
     mrnumber = {2145021},
     zbl = {1074.82011},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/bsmf.2480/}
}
TY  - JOUR
AU  - Escobedo, Miguel
AU  - Mischler, Stéphane
AU  - Valle, Manuel A.
TI  - Entropy maximisation problem for quantum relativistic particles
JO  - Bulletin de la Société Mathématique de France
PY  - 2005
SP  - 87
EP  - 120
VL  - 133
IS  - 1
PB  - Société mathématique de France
UR  - http://www.numdam.org/articles/10.24033/bsmf.2480/
DO  - 10.24033/bsmf.2480
LA  - en
ID  - BSMF_2005__133_1_87_0
ER  - 
%0 Journal Article
%A Escobedo, Miguel
%A Mischler, Stéphane
%A Valle, Manuel A.
%T Entropy maximisation problem for quantum relativistic particles
%J Bulletin de la Société Mathématique de France
%D 2005
%P 87-120
%V 133
%N 1
%I Société mathématique de France
%U http://www.numdam.org/articles/10.24033/bsmf.2480/
%R 10.24033/bsmf.2480
%G en
%F BSMF_2005__133_1_87_0
Escobedo, Miguel; Mischler, Stéphane; Valle, Manuel A. Entropy maximisation problem for quantum relativistic particles. Bulletin de la Société Mathématique de France, Volume 133 (2005) no. 1, pp. 87-120. doi : 10.24033/bsmf.2480. http://www.numdam.org/articles/10.24033/bsmf.2480/

[1] H. Andréasson - « Regularity of the gain term and strong L 1 convergence to equilibrium for the relativistic Boltzmann equation », SIAM J. Math. Anal. 27 (1996), no. 5, p. 1386-1405. | MR | Zbl

[2] S. Bose - « Plancks Gesetz und Lichtquantenhypothese », Z. Phys. 26 (1924), p. 178-181. | JFM

[3] R. Caflisch & C. Levermore - « Equilibrium for radiation in a homogeneous plasma », Phys. Fluids 29 (1986), p. 748-752. | MR

[4] C. Cercignani - The Boltzmann equation and its applications, Applied Math. Sciences, vol. 67, Springer Verlag, 1988. | MR | Zbl

[5] N. Chernikov - « Equilibrium distribution of the relativistic gas », Acta Phys. Polon. 26 (1964), p. 1069-1092. | MR

[6] F. Demengel & R. Temam - « Convex functions of a measure and applications », Indiana Univ. Math. J. 33 (984), no. 5, p. 673-709. | MR | Zbl

[7] J. Dolbeault - « Kinetic models and quantum effects, a modified Boltzmann equation for Fermi-Dirac particles », Arch. Rat. Mech. Anal. 127, p. 101-131. | MR | Zbl

[8] M. Dudyński & M. Ekiel-Jeżewska - « Global existence proof for relativistic Boltzmann equation », J. Statist. Phys. 66 (1992), no. 2,3, p. 991-1001. | MR | Zbl

[9] A. Einstein - « Quantentheorie des einatomingen idealen Gases », Stiz. Presussische Akademie der Wissenschaften Phys-math. Klasse, Sitzungsberichte 23 (1925), p. 3-14. | JFM

[10] -, « Zur Quantentheorie des idealen Gases », Stiz. Presussische Akademie der Wissenschaften, Phys-math. Klasse, Sitzungsberichte 23 (1925), p. 18-25.

[11] M. Escobedo & S. Mischler - « On a quantum Boltzmann equation for a gas of photons », J. Math. Pures Appl. 80 (2001), no. 55, p. 471-515. | MR | Zbl

[12] M. Escobedo, S. Mischler & M. Valle - Homogeneous Boltzmann equation for quantum and relativistic particles, Electronic J. Diff. Eqns. Monographs, vol. 4, 2003, http://ejde.math.swt.edu. | Zbl

[13] R. Glassey - The Cauchy Problem in Kinetic Theory, SIAM, Philadelphia, 1996. | MR | Zbl

[14] R. Glassey & W. Strauss - « Asymptotic stability of the relativistic Maxwellian », Publ. Res. Inst. Math. Sciences, Kyoto University 29 (1993), no. 2. | MR | Zbl

[15] -, « Asymptotic stability of the relativistic Maxwellian via fourteen moments », Transport Theory Stat. Physics 24 (1995), p. 657-678. | MR | Zbl

[16] S. Groot, W. Van Leeuwen & C. Van Weert - Relativistic kinetic theory, North Holland Publishing Company, 1980. | MR

[17] X. Lu - « A modified Boltzmann equation for Bose-Einstein particles: isotropic solutions and long time behavior », J. Statist. Phys. 98 (2000), no. 5,6, p. 1335-1394. | MR | Zbl

[18] -, « On spatially homogeneous solutions of a modified Boltzmann equation for Fermi-Dirac particles », J. Statist. Phys. 105 (2001), no. 1,2, p. 353-388. | MR | Zbl

[19] X. Lu & B. Wennberg - « On stability and strong convergence for the spatially homogeneous Boltzmann equation for Fermi-Dirac particles », Arch. Rat. Mech. Anal. 168 (2003), no. 1, p. 1-34. | MR | Zbl

[20] G. Naber - The geometry of Minkowski spacetime, Springer Verlag, 1992. | MR | Zbl

[21] R. Pathria - Statistical Mechanics, Pergamon Press, 1972. | Zbl

[22] C. Villani - « A review of mathematical topics on collisional kinetic theory », Handbook of Mathematical Fluid Mechanics, Vol. I (S. Friedlander & D. Serre, éds.), North Holland, Amsterdam, 2002. | MR | Zbl

Cited by Sources: