Abstract
The piston problem is investigated in the case where the length of the cylinder is infinite (on both sides) and the ratio m/M is a very small parameter, where m is the mass of one particle of the gaz and M is the mass of the piston. Introducing initial conditions such that the stochastic motion of the piston remains in the average at the origin (no drift), it is shown that the time evolution of the fluids, analytically derived from Liouville equation in a previous work, agrees with the Second Law of thermodynamics. We thus have a non equilibrium microscopical model whose evolution can be explicitly shown to obey the two laws of thermodynamics.
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REFERENCES
Ch. Gruber, Thermodynamics of systems with internal adiabatic constraints: time evolution of the adiabatic piston, Eur.J.Phys. 20:259–266 (1999).
J. Piasecki and Ch. Gruber, From the adiabatic piston to macroscopic motion induced by fluctuations, Physica A 265:463–472 (1999).
Ch. Gruber and J. Piasecki, Stationary motion of the adiabatic piston, Physica A 268:412 (1999).
Ch. Gruber and L. Frachebourg, On the adiabatic properties of a stochastic adiabatic wall: evolution, stationary non-equilibrium and equilibrium states, Physica A 272:392 (1999).
Ch. Gruber, S. Pache and A. Lesne, (5a) Deterministic motion of the controversial piston in the thermodynamic limit, J.Stat.Phys. 108:669–701 (2002). (5b) Two-time-scale relaxation towards thermal equilibrium of the enigmatic piston, J.Stat.Phys 112:1199–1228 (2003).
Ch. Gruber and S. Pache, The controversial piston in the thermodynamic limit, Physica A 314:345 (2002).
S. Pache, Propri´et´es hors-´equilibre d'une paroi adiabatique, Diploma thesis EPFL (2000), unpublished.
G. P. Morris and Ch. Gruber, (8a) Strong and weak damping in the adiabatic motion of the simple piston, J.Stat.Phys. 109:649–568 (2002). (8b) A Boltzmann Equation approach to the dynamics of the simple piston, J.Stat.Phys.113:297–333 (2003).
N. Chernov, On a slow drift of a massive piston in an ideal gas that remains at mechanical equilibrium (preprint, Sept. 2003).
N. Chernov, J. L. Lebowitz and Ya. Sinai, (10a) Dynamics of a massive piston in an ideal gas (preprint, Jan. 2003). (10b) Dynamics of a massive piston in an ideal gas, Russ.Math.Surveys 57:1–84 (2002). (10c) Scaling Dynamics of a massive piston in a cube filled with ideal gas: exact results, J.Stat.Phys. 109:529–548 (2002).
N. Chernov and J. L. Lebowitz, Dynamics of a massive piston in an ideal gas: oscillatory motion and approach to equilibrium, J.Stat.Phys. 109:507–527 (2002).
J. Piasecki, (12a) A model of Brownian motion in an inhomogeneous environment J.Phys.Cond.Matter, 14:1 (2002). (12b) Drift velocity induced by collision J.Stat.Phys. 104:1145 (2001). (12c) Stochastic adiabatic wall J.Stat.Phys.101:711 (2000).
A. Plyukhin and J. Schonfield, (13a) The Langevin Equation for the extented Rayleigh model with asymmetric bath (preprint, May 2003). (13b) On the Langevin Equation for the Rayleigh model with finite range interaction (Preprint, May 2003).
J. Piasecki, and Ya. Sinai, A model of non equilibrium statistical mechanics, Dynamics: Model and Kinetic methods for non equilibrium Many Body systems, J. Karkheck (ed), (2000) 191–199.
J. L. Lebowitz, J. Piasecki, and Ya. Sinai, Scaling dynamics of a massive piston in an ideal gas, Encyclopedia of Mathematical Sciences Series (101, Springer-Verlag, Berlin 2000), pp. 217–227.
J. L. Lebowitz, J. Piasecki, and Ya. Sinai, in Hard ball systems and the Lorentz gas, Encyclopedia of Mathematical Sciences Series (101, Springer-Verlag, Berlin 2000).
C. Crosignani, On the validity of the Second Law of thermodynamics in the mesoscopic real, Eur.Phys.Lett. 53:290 (2001).
E. Kestemont, C. Van den Broeck, and M. Malek Mansour, The “adiabatic” piston: and yet it moves, Europhys.Lett. 49:143 (2000).
C. Van den Broeck, E. Kestemont, and M. Malek Mansour, Heat conductivity shared by a piston Europhys.Letters 56:771 (2001).
T. Munakata and H. Ogawa, Dynamical aspects of an adiabatic piston, Phys.Rev.E 64:036119 (2001).
J. A. White, F. L. Roman, A. Gonzales, and S. Velasco, The “adiabatic” piston at equilibrium: spectral analysis and time-correlation function, Europhys.Lett. 59:459–485 (2002).
M. J. Renne, M. Ruijgrok, and Th. Ruijgrok, the enigmatic piston, Acta Physica Polonica B 32:4183 (2001).
V. Balakrishnan, I. Bena, and C. Van den Broeck, Diffusion and velocity correlation functions, Phys.Rev.E 65:031102 (2002).
C. Garrod and M. Rosina, Three interesting problems in statistical mechanics, Am.J.Phys. 67:1240–1244 (1999).
H. B. Callen, Thermodynamics, John Wiley and Sons, New York (1963), Appendix C. See also H. B. Callen, Thermodynamics and Thermostatics, John Wiley and Sons, 2nd edition, New York (1985), pp. 51, 53.
O. L. de Lange and J. Pierrus, Measurement of bulk moduli and ratio of specific heats of gases using R´uchardt's experiment, Am.J.Phys. 68:265–270 (2000).
L. D. Landau and E. M. Lifshitz, Fluid mechanics, Pergamon Press, New York (1959), in §85: Shock waves in a perfect gas, pp. 329 in Chapter X: One-dimensional gas flow.
P. Gaspard, G. Nicolis and J. R. Dorfman, Diffusive Lorentz gases and multibaker maps are compatible with irreversible thermodynamics, Physics A 323:294–322 (2003).
G. Gallavotti, Nonequilibrium thermodynamics? (preprint IHES/P/03/05, 2003).
E. Lieb and J. Yngvason, Physics and mathematics of the Second Law of thermodynamics, Phys.Rep. 310:1–99 (1999).
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Gruber, C., Pache, S. & Lesne, A. On the Second Law of Thermodynamics and the Piston Problem. Journal of Statistical Physics 117, 739–772 (2004). https://doi.org/10.1007/s10955-004-2271-y
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DOI: https://doi.org/10.1007/s10955-004-2271-y