Archimedes’ principle for ideal gas. (English) Zbl 07512345

Summary: We prove Archimedes’ principle for a macroscopic ball in ideal gas consisting of point particles with non-zero mass. The main result is an asymptotic theorem, as the number of point particles goes to infinity and their total mass remains constant. We also show that, asymptotically, the gas has an exponential density as a function of height. We find the asymptotic inverse temperature of the gas. We derive an accurate estimate of the volume of the phase space using the local central limit theorem.


82Cxx Time-dependent statistical mechanics (dynamic and nonequilibrium)
60Fxx Limit theorems in probability theory
60Gxx Stochastic processes
Full Text: DOI


[1] Angel, O.; Burdzy, K.; Sheffield, S., Deterministic approximations of random reflectors, Trans. Am. Math. Soc., 365, 12, 6367-6383 (2013) · Zbl 1408.37063
[2] Burdzy, K.; Chen, Z-Q; Pal, S., Archimedes’ principle for Brownian liquid, Ann. Appl. Probab., 21, 6, 2053-2074 (2011) · Zbl 1241.60041
[3] Burdzy, K., Duarte, M., Gauthier, C.-E., Graham, R., Martin, J.S. Fermi acceleration in rotating drums (2021). arXiv:2112.09821 Math
[4] Chernov, N., Lebowitz, J.L.: Dynamics of a massive piston in an ideal gas: oscillatory motion and approach to equilibrium. J. Stat. Phys., 109(3-4), 507-527 (2002). Special issue dedicated to J. Robert Dorfman on the occasion of his sixty-fifth birthday · Zbl 1101.82319
[5] Chernov, N., Lebowitz, J.L., Sinai, Y.: Scaling dynamics of a massive piston in a cube filled with ideal gas: exact results. J. Stat. Phys., 109(3-4), 529-548 (2002). Special issue dedicated to J. Robert Dorfman on the occasion of his sixty-fifth birthday · Zbl 1211.76093
[6] Collins, G.W.: The Virial Theorem in Stellar Astrophysics. Pachart Pub. House, Tucson (1978). Astronomy and astrophysics series; v. 7
[7] Gorelyshev, I., On the dynamics in the one-dimensional piston problem, Nonlinearity, 24, 8, 2119-2142 (2011) · Zbl 1223.37072
[8] Itami, M.; Sasa, S., Nonequilibrium statistical mechanics for adiabatic piston problem, J. Stat. Phys., 158, 1, 37-56 (2015) · Zbl 1317.82047
[9] Knudsen, M. The Kinetic Theory of Gases: Some Modern Aspects. Methuen & Co., London (1934). (Methuen’s Monographs on Physical Subjects) · JFM 60.1420.10
[10] Lambert, J.H. Photometria sive de mensure de gratibus luminis, colorum umbrae. Eberhard Klett (1760)
[11] Lieb, E.H.: Some problems in statistical mechanics that I would like to see solved. Phys. A, 263(1-4), 491-499 (1999). STATPHYS 20 (Paris, 1998)
[12] Lebowitz, J.L., Piasecki, J., Sinai, Y. Scaling dynamics of a massive piston in an ideal gas. In: Hard Ball Systems and the Lorentz Gas, volume 101 of Encylopaedia of Mathematical Sciences, pp. 217-227. Springer, Berlin (2000) · Zbl 1127.82308
[13] Lebovits, L., Sinaĭ, Y., Chernov, N. Dynamics of a massive piston immersed in an ideal gas. Uspekhi Mat. Nauk, 57(6(348)), 3-86 (2002)
[14] Mayer, JE; Mayer, MG, Statistical Mechanics (1977), New York: Wiley, New York · Zbl 0972.82500
[15] Neishtadt, AI; Sinai, YG, Adiabatic piston as a dynamical system, J. Stat. Phys., 116, 1-4, 815-820 (2004) · Zbl 1142.82351
[16] Petrov, V.V.: Sums of Independent Random Variables. Springer, New York. Translated from the Russian by A, p. 82. A. Brown, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band (1975) · Zbl 0322.60043
[17] Alexander Plakhov. Exterior billiards. Springer, New York, 2012. Systems with impacts outside bounded domains · Zbl 1257.37002
[18] Ruelle, D: Statistical Mechanics. World Scientific Publishing Co., Inc., River Edge, NJ; Imperial College Press, London (1999). Rigorous results, Reprint of the 1989 edition
[19] Sinaĭ, YG, Topics in Ergodic Theory. Princeton Mathematical Series (1994), Princeton, NJ: Princeton University Press, Princeton, NJ
[20] Sinaĭ, YG, Dynamics of a massive particle surrounded by a finite number of light particles, Teoret. Mat. Fiz., 121, 1, 110-116 (1999)
[21] Statuljavičus, VA, Limit theorems for densities and the asymptotic expansions for distributions of sums of independent random variables, Teor. Verojatnost. i Primene, 10, 645-659 (1965)
[22] Siraždinov, SH; Šahaĭdarova, N., On the uniform local theorem for densities, Izv. Akad. Nauk UzSSR Ser. Fiz. Mat. Nauk, 9, 6, 30-36 (1965)
[23] Šahaĭdarova, N., Uniform local and global theorems for densities, Izv. Akad. Nauk UzSSR Ser. Fiz. Mat. Nauk, 5, 10, 90-91 (1966)
[24] Šervašidze, TL, The uniform estimation of the rate of convergence in a multidimensional local limit theorem for densities, Teor. Verojatnost. i Primenen., 16, 765-767 (1971) · Zbl 0273.60012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.