# American Institute of Mathematical Sciences

March  2014, 7(1): 133-168. doi: 10.3934/krm.2014.7.133

## Thermodynamical considerations implying wall/particles scattering kernels

 1 IUSTI, 5 rue Enrico Fermi, 13453 Marseille CEDEX 13, France

Received  September 2011 Revised  March 2013 Published  December 2013

In this paper we study the thermodynamics of a rarefied gas contained in a closed vessel at constant volume. By adding axiomatic rules to the usual ones derived by Cercignani, we obtain a new symmetry property in the wall/particle scattering kernel. This new symmetry property enables us to show the first and second law of macroscopic thermodynamics for a rarefied gas having collisions with walls. Then we study the behavior of the rarefied gas when it is in contact with several (moving) thermostats at the same time. We show the existence, uniqueness and long time behavior of the solution to the homogeneous (linear) evolution equation describing the system. Finally we apply our thermodynamical model of rarefied gas to the measurement of heat flux in very low density systems and compare it to the experimental results, shading into light new interpretation of observed behaviors.
Citation: Vincent Pavan. Thermodynamical considerations implying wall/particles scattering kernels. Kinetic and Related Models, 2014, 7 (1) : 133-168. doi: 10.3934/krm.2014.7.133
##### References:
 [1] K. Aoki and F. Golse, On the speed of approach to equilibrium for collisionless gas, Kinet. Relat. Models, 4 (2011), 87-107. doi: 10.3934/krm.2011.4.87. [2] W. Arendt, Positive semigroups of Kernel operators, Positivity, 12 (2008), 25-44. doi: 10.1007/s11117-007-2137-z. [3] L. Arkeryd and A. Nouri, Boltzmann asymptotics with diffuse reflection boundary conditions, Monatsh. Math., 123 (1997), 285-298. doi: 10.1007/BF01326764. [4] V. Bagland, P. Degond and M. Lemou, Moment systems derived from relativistic kinetic equations, J. Stat. Phys., 125 (2006), 621-659. doi: 10.1007/s10955-006-9173-0. [5] J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups With Applications, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2006. [6] C. Cercignani, Slow Rarefied Flows. Theory and Application to Micro-Electro-Mechanical Systems, Progress in Mathematical Physics, 41, Birkhäuser Verlag, Basel, 2006. [7] C. Cercignani, The Boltzmann Equation and its Application, Applied Mathematical Sciences, 67, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9. [8] S. K. Diadze and J. G. Méolans, Temperature jump and slip velocity calculations from an anisotropic scattering kernel, Physica A, 358 (2005), 328-346. doi: 10.1016/j.physa.2005.04.013. [9] V. Keicher, Convergence of Positive $C_0$-Semigroups to Rotation Groups, Ph.D thesis, Eberhard Karls Universität Tübingen, 2008. [10] I. Kuscer, Reciprocity in scattering of gas molecules by surfaces, Surface Science, 25 (1971), 225-237. [11] J. Lebowitz and P. G. Bergmann, Irreversible Gibsian ensembles, Ann. Physics, 1 (1957), 1-23. doi: 10.1016/0003-4916(57)90002-7. [12] C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Statist. Phys., 83 (1996), 1021-1065. doi: 10.1007/BF02179552. [13] C. Mouhot, Quantitative linearized study of the Boltzmann collision operator and applications,, Commun. Math. Sci., 2007 (): 73.  doi: 10.4310/CMS.2007.v5.n5.a6. [14] B.-S. Tam, A cone theoretic approach to the spectral theory of positive linear operators: The finite dimensional case, Tawainese J. Math., 5 (2001), 207-277. [15] H. Yamaguchi, K. Anazawa, Y. Matsuda, T. Niimi, A. Polikarpov and I. Graur, Investigation of heat transfer between two coaxial cylinders for measurement of thermal accomodation coefficient, Physics of Fluids, 24 (2012), 062002. doi: 10.1063/1.4726059.

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##### References:
 [1] K. Aoki and F. Golse, On the speed of approach to equilibrium for collisionless gas, Kinet. Relat. Models, 4 (2011), 87-107. doi: 10.3934/krm.2011.4.87. [2] W. Arendt, Positive semigroups of Kernel operators, Positivity, 12 (2008), 25-44. doi: 10.1007/s11117-007-2137-z. [3] L. Arkeryd and A. Nouri, Boltzmann asymptotics with diffuse reflection boundary conditions, Monatsh. Math., 123 (1997), 285-298. doi: 10.1007/BF01326764. [4] V. Bagland, P. Degond and M. Lemou, Moment systems derived from relativistic kinetic equations, J. Stat. Phys., 125 (2006), 621-659. doi: 10.1007/s10955-006-9173-0. [5] J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups With Applications, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2006. [6] C. Cercignani, Slow Rarefied Flows. Theory and Application to Micro-Electro-Mechanical Systems, Progress in Mathematical Physics, 41, Birkhäuser Verlag, Basel, 2006. [7] C. Cercignani, The Boltzmann Equation and its Application, Applied Mathematical Sciences, 67, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9. [8] S. K. Diadze and J. G. Méolans, Temperature jump and slip velocity calculations from an anisotropic scattering kernel, Physica A, 358 (2005), 328-346. doi: 10.1016/j.physa.2005.04.013. [9] V. Keicher, Convergence of Positive $C_0$-Semigroups to Rotation Groups, Ph.D thesis, Eberhard Karls Universität Tübingen, 2008. [10] I. Kuscer, Reciprocity in scattering of gas molecules by surfaces, Surface Science, 25 (1971), 225-237. [11] J. Lebowitz and P. G. Bergmann, Irreversible Gibsian ensembles, Ann. Physics, 1 (1957), 1-23. doi: 10.1016/0003-4916(57)90002-7. [12] C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Statist. Phys., 83 (1996), 1021-1065. doi: 10.1007/BF02179552. [13] C. Mouhot, Quantitative linearized study of the Boltzmann collision operator and applications,, Commun. Math. Sci., 2007 (): 73.  doi: 10.4310/CMS.2007.v5.n5.a6. [14] B.-S. Tam, A cone theoretic approach to the spectral theory of positive linear operators: The finite dimensional case, Tawainese J. Math., 5 (2001), 207-277. [15] H. Yamaguchi, K. Anazawa, Y. Matsuda, T. Niimi, A. Polikarpov and I. Graur, Investigation of heat transfer between two coaxial cylinders for measurement of thermal accomodation coefficient, Physics of Fluids, 24 (2012), 062002. doi: 10.1063/1.4726059.
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