Comptes Rendus
Fourier and the science of today / Fourier et la science d'aujourd'hui
A microscopic view of the Fourier law
[Une vision microscopique de la loi de Fourier]
Comptes Rendus. Physique, Volume 20 (2019) no. 5, pp. 402-418.

La loi de Fourier permet de décrire la diffusion de la chaleur dans des systèmes physiques à l'échelle macroscopique. Cette loi est très bien vérifiée expérimentalement, et une question naturelle est de la justifier à partir de modèles microscopiques en utilisant les principes fondamentaux de la mécanique.

The Fourier law of heat conduction describes heat diffusion in macroscopic systems. This physical law has been experimentally tested for a large class of physical systems. A natural question is to know whether it can be derived from the microscopic models using the fundamental laws of mechanics.

Publié le :
DOI : 10.1016/j.crhy.2019.08.002
Keywords: Boltzmann equation, Fourier law, Diffusion equation, Hard-sphere gas, Low-density limit, Relative entropy method
Mot clés : Équation de Boltzmann, Loi de Fourier, Équation de diffusion, Gaz de sphères dures, Limite de basse densité, Méthode de l'entropie relative
Thierry Bodineau 1 ; Isabelle Gallagher 2 ; Laure Saint-Raymond 3

1 CMAP, École polytechnique, France
2 DMA, École normale supérieure, France
3 UMPA, École normale supérieure de Lyon, France
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Thierry Bodineau; Isabelle Gallagher; Laure Saint-Raymond. A microscopic view of the Fourier law. Comptes Rendus. Physique, Volume 20 (2019) no. 5, pp. 402-418. doi : 10.1016/j.crhy.2019.08.002. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2019.08.002/

[1] F. Bonetto; J. Lebowitz; L. Rey-Bellet Fourier's law: a challenge to theorists, Math. Phys. (2000), pp. 128-150

[2] L. Boltzmann Further studies on the thermal equilibrium of gas molecules (1872), the kinetic theory of gases (S.G. Brush; N.S. Hall, eds.), History of Modern Physical Sciences, vol. 1, 2003, pp. 262-349

[3] H. Grad Asymptotic theory of the Boltzmann equation II, Proceedings of the Third International Symposium on Rarefied Gas Dynamics, Palais de l'Unesco, Paris, vol. I, 1962, pp. 26-59

[4] C. Bardos; F. Golse; C.D. Levermore Fluid dynamic limits of the Boltzmann equation I, J. Stat. Phys., Volume 63 (1991), pp. 323-344

[5] C. Bardos; F. Golse; C.D. Levermore Fluid dynamic limits of kinetic equations II: convergence proofs for the Boltzmann equation, Commun. Pure Appl. Math., Volume 46 (1993), pp. 667-753

[6] P.-L. Lions; N. Masmoudi From Boltzmann equation to the Navier–Stokes and Euler equations II, Arch. Ration. Mech. Anal., Volume 158 (2001), pp. 195-211

[7] F. Golse; L. Saint-Raymond The incompressible Navier–Stokes limit of the Boltzmann equation for hard cutoff potentials, J. Math. Pures Appl., Volume 91 (2009), pp. 508-552

[8] N. Masmoudi; L. Saint-Raymond From the Boltzmann equation to the Stokes–Fourier system in a bounded domain, Commun. Pure Appl. Math., Volume 56 (2003), pp. 1263-1293

[9] L. Saint-Raymond Hydrodynamic Limits of the Boltzmann Equation, Lecture Notes in Mathematics, vol. 1971, Springer-Verlag, 2009

[10] C. Bardos; S. Ukai The classical incompressible Navier–Stokes limit of the Boltzmann equation, Int. J. Math. Models Methods Appl. Sci., Volume 1 (1991), pp. 235-257

[11] T.-P. Liu; T. Yang; S.-H. Yu Energy method for Boltzmann equation, Physica D, Volume 188 (2004), pp. 178-192

[12] R. Esposito; J. Lebowitz; R. Marra Hydrodynamic limit of the stationary Boltzmann equation in a slab, Commun. Math. Phys., Volume 160 (1994), pp. 49-80

[13] R. Esposito; Y. Guo; C. Kim; R. Marra Non-isothermal boundary in the Boltzmann theory and Fourier law, Commun. Math. Phys., Volume 323 (2013), pp. 177-239

[14] R. Esposito; Y. Guo; C. Kim; R. Marra Stationary solutions to the Boltzmann equation in the hydrodynamic limit, Ann. PDE, Volume 4 (2018) (119 pages)

[15] R. Esposito; J. Lebowitz; R. Marra The Navier–Stokes limit of stationary solutions of the nonlinear Boltzmann equation, J. Stat. Phys., Volume 78 (1995), pp. 389-412

[16] J.-S. Darrozès; J.-P. Guiraud Généralisation formelle du théorème H en présence de parois, C. R. Hebd. Séances Acad. Sci., Ser. A, Volume 262 (1966), pp. 368-371

[17] G. Eyink; J. Lebowitz; H. Spohn Lattice gas models in contact with stochastic reservoirs: local equilibrium and relaxation to the steady state, Commun. Math. Phys., Volume 140 (1991), pp. 119-131

[18] A.-L. Dalibard; L. Saint-Raymond Mathematical study of rotating fluids with resonant surface stress, J. Differ. Equ., Volume 246 (2009), pp. 2304-2354

[19] H.T. Yau Scaling limit of particle systems, incompressible Navier–Stokes equation and Boltzmann equation, Doc. Math., Volume 3 (1998), pp. 193-202 (in: Proceedings of the International Congress of Mathematicians, Berlin, 18–27 August 1998)

[20] C. Kipnis; C. Landim; S. Olla Macroscopic properties of a stationary nonequilibrium distribution for a non-gradient interacting particle system, Ann. Inst. Henri Poincaré, Volume 31 (1995), pp. 191-221

[21] J. Farfan; C. Landim; M. Mourragui Hydrostatics and dynamical large deviations of boundary driven gradient symmetric exclusion processes, Stoch. Process. Appl., Volume 121 (2011) no. 4, pp. 725-758

[22] F. Golse; P.-L. Lions; B. Perthame; R. Sentis Regularity of the moments of the solution of a transport equation, J. Funct. Anal., Volume 76 (1988), pp. 110-125

[23] C. Bardos Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels, théorèmes d'approximation, application à l'équation de transport, Ann. Sci. Éc. Norm. Supér., Volume 3 (1970), pp. 185-233

[24] T. Bodineau; I. Gallagher; L. Saint-Raymond From hard spheres to the linearized Boltzmann equation: an L2 analysis of the Boltzmann–Grad limit, Ann. PDE, Volume 3 (2017) (118 pages)

[25] O.E. Lanford Time evolution of large classical systems, Lect. Notes in Physics, vol. 38, Springer Verlag, 1975, pp. 1-111

[26] I. Gallagher; L. Saint-Raymond; B. Texier From Newton to Boltzmann: The Case of Hard-Spheres and Short-Range Potentials, Zurich Lecture Notes in Advanced Mathematics, vol. 18, EMS Publications, 2014 (150 pages)

[27] M. Pulvirenti; C. Saffirio; S. Simonella On the validity of the Boltzmann equation for short range potentials, Rev. Math. Phys., Volume 26 (2014) no. 2

[28] T. Bodineau; I. Gallagher; L. Saint-Raymond The Brownian motion as the limit of a deterministic system of hard-spheres, Invent. Math., Volume 203 (2016), pp. 493-553

[29] R. Denlinger The propagation of chaos for a rarefied gas of hard spheres in the whole space, Arch. Ration. Mech. Anal., Volume 229 (2018), pp. 885-952

[30] T. Bodineau; I. Gallagher; L. Saint-Raymond; S. Simonella One sided convergence in the Boltzmann–Grad limit, Ann. Fac. Sci. Toulouse (2018) (in press)

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