Abstract
We continue the study of the time evolution of a system consisting of a piston in a cubical container of large size L filled with an ideal gas. The piston has mass M∼L 2 and undergoes elastic collisions with N∼L 3 gas particles of mass m. In a previous paper, Lebowitz et al. considered a scaling regime, with time and space scaled by L, in which they argued heuristically that the motion of the piston and the one particle distribution of the gas satisfy autonomous coupled differential equations. Here we state exact results and sketch proofs for this behavior.
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REFERENCES
J. L. Lebowitz, J. Piasecki, and Ya. Sinai, Scaling dynamics of a massive piston in an ideal gas, in Hard Ball Systems and the Lorentz Gas, Encycl. Math. Sci., Vol. 101 (Springer, Berlin, 2000), pp. 217–227.
N. Chernov, J. L. Lebowitz, and Ya. Sinai, Dynamic of a Massive Piston in an Ideal Gas, preprint, available at www.math.uab.edu/chernov/pubs/.
Ch. Gruber, Thermodynamics of systems with internal adiabatic constraints: Time evolution of the adiabatic piston, Eur. J. Phys. 20:259–266 (1999).
Ch. Gruber and L. Frachenberg, On the adiabatic properties of a stochastic adiabatic wall: Evolution, stationary non-equilibrium, and equilibrium states, Phys. A 272:392–428 (1999).
Ch. Gruber, J. Piasecki, Stationary motion of the adiabatic piston, Phys. A 268:412–423 (1999).
E. Kestemont, C. Van den Broeck, and M. Mansour, The ''adiabatic'' piston: And yet it moves, Europhys. Lett. 49:143–149 (2000).
E. Lieb, Some problems in statistical mechanics that I would like to see solved, Physica A 263:491–499 (1999).
N. Chernov and J. L. Lebowitz, Dynamics of a Massive Piston in an Ideal Gas: Oscillatory Motion and Approach to Equilibrium, this volume of the journal.
A. N. Kolmogorov and V. M. Tihomirov, e-entropy and e-capacity of sets in function spaces, Amer. Math. Soc. Transl. 17:277–364 (1961).
R. Holley, The motion of a heavy particle in an infinite one dimensional gas of hard spheres, Z. Wahrschein. verw. Geb. 17:181–219 (1971).
D. Dürr, S. Goldstein, and J. L. Lebowitz, A mechanical model of Brownian motion, Commun. Math. Phys. 78:507–530 (1981).
E. Caglioti, N. Chernov, J. L. Lebowitz, and E. Presutti, in preparation.
J. L. Lebowitz, Stationary nonequilibrium Gibbsian ensembles, Phys. Rev. 114:1192–1202 (1959).
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Chernov, N., Lebowitz, J.L. & Sinai, Y. Scaling Dynamics of a Massive Piston in a Cube Filled with Ideal Gas: Exact Results. Journal of Statistical Physics 109, 529–548 (2002). https://doi.org/10.1023/A:1020402312727
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DOI: https://doi.org/10.1023/A:1020402312727