Abstract
A symmetric relation between time-dependent problems described by the linearized Boltzmann equation is obtained for a gas in a fixed bounded domain. General representations of the total mass, momentum, and energy in the domain, as well as their fluxes through the boundary, in terms of an appropriate Green function are derived from that relation. Several application examples are presented. Similarities to the fluctuation–dissipation theorem in the linear response theory and its generalization to gas systems of arbitrary Knudsen numbers are also discussed. The present paper is an extension of the previous work of the author (Takata in J. Stat. Phys. 136: 751–784, 2009) to time-dependent problems.
Similar content being viewed by others
References
Kennard, E.H.: Kinetic Theory of Gases. McGraw-Hill, New York (1938)
Sone, Y.: Thermal creep in rarefied gas. J. Phys. Soc. Jpn. 21, 1836–1837 (1966)
Epstein, P.S.: Zur Theorie des Radiometer. Z. Phys. 54, 537–563 (1929)
Sone, Y.: Flow induced by thermal stress in rarefied gas. Phys. Fluids 15, 1418–1423 (1972)
Bakanov, S.P., Vysotskij, V.V., Deryaguin, B.V., Roldughin, V.I.: Thermal polarization of bodies in the rarefied flow. J. Non-Equilib. Thermodyn. 8, 75–83 (1983)
Loyalka, S.K.: Kinetic theory of thermal transpiration and mechanocaloric effect. I. J. Chem. Phys. 55, 4497–4503 (1971)
Roldughin, V.I.: On the theory of thermal polarization of bodies in a rarefied gas flow. J. Non-Equilib. Thermodyn. 19, 349–367 (1994)
Takata, S.: Note on the relation between thermophoresis and slow uniform flow problems for a rarefied gas. Phys. Fluids 21, 112001 (2009)
Takata, S.: Symmetry of the linearized Boltzmann equation and its application. J. Stat. Phys. 136, 751–784 (2009)
Takata, S.: Symmetry of the linearized Boltzmann equation II. Entropy production and Onsager–Casimir relation. J. Stat. Phys. 136, 945–983 (2009)
McCourt, F.R.W., Beenakker, J.J.M., Köhler, W.E., Kuščer, I.: Nonequilibrium Phenomena in Polyatomic Gases, vol. 2. Clarendon, Oxford (1991)
Sharipov, F.: Onsager–Casimir reciprocity relations for open gaseous systems at arbitrary rarefaction. I. General theory for single gas. Physica A 203, 437–456 (1994)
Zhdanov, V.M., Roldughin, V.I.: Non-equilibrium thermodynamics and kinetic theory of rarefied gases. Phys. Usp. 41, 349–378 (1998)
Roldughin, V.I., Zhdanov, V.M.: Non-equilibrium thermodynamics and kinetic theory of gas mixtures in the presence of interfaces. Adv. Colloid Interface. Sci. 98, 121–215 (2002)
Sharipov, F.: Onsager–Casimir reciprocal relations based on the Boltzmann equation and gas-surface interaction: Single gas. Phys. Rev. E 73, 026110 (2006)
Kubo, R., Toda, M., Hashitsume, N.: Statistical Physics II, Nonequilibrium Statistical Mechanics, 2nd edn. Springer, Berlin (1991), Chap. 4
Pottier, N.: Nonequilibrium Statistical Physics. Oxford University Press, New York (2010)
Sugimoto, H., Sone, Y.: Vacuum pump without a moving part driven by thermal edge flow. In: Capitelli, M., (ed.) Rarefied Gas Dynamics, pp. 168–173. AIP, Melville (2005)
Abe, T.: A verification of the Monte Carlo direct simulation by using the fluctuation dissipation theorem. In: Oguchi, H., (ed.) Rarefied Gas Dynamics, pp. 191–198. University of Tokyo Press, Tokyo (1984)
Alexander, F.J., Garcia, A.L., Alder, B.J.: Cell size dependence of transport coefficients in stochastic particle algorithms. Phys. Fluids 10, 1540–1542 (1998)
Hadjiconstantinou, N.G.: Analysis of discretization in the direct simulation Monte Carlo. Phys. Fluids 12, 2634–2638 (2000)
Sone, Y.: Molecular Gas Dynamics. Birkhäuser, Boston (2007); Supplementary Notes and Errata: Kyoto University Research Information Repository (http://hdl.handle.net/2433/66098)
Cercignani, C.: The Boltzmann Equation and Its Applications. Springer, New York (1988)
Chapman, S., Cowling, T.G.: The Mathematical Theory of Non-Uniform Gases, 3rd edn. Cambridge University Press, Cambridge (1995)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Takata, S. Symmetry of the Unsteady Linearized Boltzmann Equation in a Fixed Bounded Domain. J Stat Phys 140, 985–1005 (2010). https://doi.org/10.1007/s10955-010-0009-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-010-0009-6