Abstract
A free-molecular gas contained in a static vessel with a uniform temperature is considered. The approach of the velocity distribution function of the gas molecules from a given initial distribution to the uniform equilibrium state at rest is investigated numerically under the diffuse reflection boundary condition. This relaxation is caused by the interaction of gas molecules with the vessel wall. It is shown that, for a spherical vessel, the velocity distribution function approaches the final uniform equilibrium distribution in such a way that their difference decreases in proportion to an inverse power of time. This is slower than the known result for a rarefied gas with molecular collisions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Desvillettes, L.: Convergence to equilibrium in large time for Boltzmann and B.G.K. equations. Arch. Ration. Mech. Anal. 110, 73–91 (1990)
Lions, P.-L.: Compactness in Boltzmann’s equation via Fourier integral operators and applications. I. J. Math. Kyoto Univ. 34, 391–427 (1994)
Arkeryd, L., Nouri, A.: Boltzmann asymptotics with diffuse reflection boundary conditions. Monatshefte Math. 123, 285–298 (1997)
Villani, C.: A review of mathematical topics in collisional kinetic theory. In: Friedlander, S., Serre, D. (eds.) Handbook of Mathematical Fluid Dynamics, vol. 1, pp. 71–305. Elsevier, Amsterdam (2002)
Desvillettes, L., Villani, C.: On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation. Invent. Math. 159, 245–316 (2005)
Villani, C.: Convergence to equilibrium: Entropy production and hypocoercivity. In: Capitelli, M. (ed.) Rarefied Gas Dynamics, pp. 8–25. AIP, Melville (2005)
Mouhot, C., Neumann, L.: Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus. Nonlinearity 19, 969–998 (2006)
Villani, C.: Hypocoercivity. Mem. Am. Math. Soc. 202, 950 (2009)
Guo, Y.: Decay and continuity of the Boltzmann equation in bounded domains. Arch. Ration. Mech. Anal. (2009). doi:10.1007/s00205-009-0285-y
Ukai, S.: On the existence of global solutions of mixed problem for non-linear Boltzmann equation. Proc. Jpn. Acad. 50, 179–184 (1974)
Caflisch, R.E.: The Boltzmann equation with a soft potential. II. Nonlinear, spatially-periodic. Commun. Math. Phys. 74, 97–109 (1980)
Lebowitz, J.L., Frisch, H.L.: Model of nonequilibrium ensemble: Knudsen gas. Phys. Rev. 107, 917–923 (1957)
Arkeryd, L., Ianiro, N., Triolo, L.: The trend to a stationary state for the Lebowitz stick model. Math. Methods Appl. Sci. 16, 739–757 (1993)
Bose, C., Grzegorczyk, P., Illner, R.: Asymptotic behavior of one-dimensional discrete-velocity models in a slab. Arch. Ration. Mech. Anal. 127, 337–360 (1994)
Yu, S.-H.: Stochastic formulation for the initial-boundary value problems of the Boltzmann equation. Arch. Ration. Mech. Anal. 192, 217–274 (2009)
Desvillettes, L., Salvarani, S.: Asymptotic behavior of degenerate linear transport equations. Bull. Sci. Math. 133, 848–858 (2009)
Callen, H.B.: Thermodynamics. Wiley, New York (1960). Chap. 6, Sect. 6.1
Feller, W.: On the integral equation of renewal theory. Ann. Math. Stat. 12, 243–267 (1941)
Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 2, 2nd edn. Wiley, New York (1971). Chap. XI
Babovsky, H.: On Knudsen flows within thin tubes. J. Stat. Phys. 44, 865–878 (1986)
Babovsky, H., Bardos, C., Platkowski, T.: Diffusion approximation for a Knudsen gas in a thin domain with accommodation on the boundary. Asymptot. Anal. 3, 265–289 (1991)
Golse, F.: Anomalous diffusion limit for the Knudsen gas. Asymptot. Anal. 17, 1–12 (1998)
Caprino, S., Marchioro, C., Pulvirenti, M.: Approach to equilibrium in a microscopic model of friction. Commun. Math. Phys. 264, 167–189 (2006)
Caprino, S., Cavallaro, G., Marchioro, C.: On a microscopic model of viscous friction. Math. Models Methods Appl. Sci. 17, 1369–1403 (2007)
Aoki, K., Cavallaro, G., Marchioro, C., Pulvirenti, M.: On the motion of a body in thermal equilibrium immersed in a perfect gas. Math. Model. Numer. Anal. 42, 263–275 (2008)
Aoki, K., Tsuji, T., Cavallaro, G.: Approach to steady motion of a plate moving in a free-molecular gas under a constant external force. Phys. Rev. E 80, 016309 (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Tsuji, T., Aoki, K. & Golse, F. Relaxation of a Free-Molecular Gas to Equilibrium Caused by Interaction with Vessel Wall. J Stat Phys 140, 518–543 (2010). https://doi.org/10.1007/s10955-010-9997-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-010-9997-5