Geodesic
Dynamics of a massive piston in an ideal gas
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 57 (2002) no. 6, pp. 1045-1125 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

This survey is a study of a dynamical system consisting of a massive piston in a cubic container of large size $L$ filled with an ideal gas. The piston has mass $M\sim L^2$ and undergoes elastic collisions with $N\sim L^3$ non-interacting gas particles of mass $m=1$. It is found that under suitable initial conditions there is a scaling regime with time and space scaled by $L$ in which the motion of the piston and the one-particle distribution of the gas satisfy autonomous coupled equations (hydrodynamic equations) such that in the limit $L\to\infty$ the mechanical trajectory of the piston converges in probability to the solution of the hydrodynamic equations for a certain period of time. There is also a heuristic discussion of the dynamics of the system on longer intervals of time. 
@article{RM_2002_57_6_a0,
     author = {N. I. Chernov and J. L. Lebowitz and Ya. G. Sinai},
     title = {Dynamics of a~massive piston in an ideal gas},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {1045--1125},
     year = {2002},
     volume = {57},
     number = {6},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RM_2002_57_6_a0/}
}
TY  - JOUR
AU  - N. I. Chernov
AU  - J. L. Lebowitz
AU  - Ya. G. Sinai
TI  - Dynamics of a massive piston in an ideal gas
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2002
SP  - 1045
EP  - 1125
VL  - 57
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/RM_2002_57_6_a0/
LA  - en
ID  - RM_2002_57_6_a0
ER  - 
%0 Journal Article
%A N. I. Chernov
%A J. L. Lebowitz
%A Ya. G. Sinai
%T Dynamics of a massive piston in an ideal gas
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2002
%P 1045-1125
%V 57
%N 6
%U http://geodesic.mathdoc.fr/item/RM_2002_57_6_a0/
%G en
%F RM_2002_57_6_a0
N. I. Chernov; J. L. Lebowitz; Ya. G. Sinai. Dynamics of a massive piston in an ideal gas. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 57 (2002) no. 6, pp. 1045-1125. http://geodesic.mathdoc.fr/item/RM_2002_57_6_a0/

[1] S. G. Brush, Kinetic Theory, Pergamon, Oxford, 1996

[2] E. Cagliotti, N. Chernov, J. L. Lebowitz, Instability of piston dynamics, www.math.uab.edu/chernov/papers/pubs.html

[3] H. B. Callen, Thermodynamics, Wiley, New York, 1963

[4] N. Chernov, J. L. Lebowitz, “Dynamics of a massive piston in an ideal gas: oscillatory motion and approach to equilibrium”, J. Statist. Phys., 109:3–4 (2002), 507–527 ; www.math.uab.edu/chernov/papers/pubs/ | DOI | MR | Zbl

[5] N. Chernov, “Entropy, Lyapunov exponents, and mean-free path for billiards”, J. Statist. Phys., 88:1–2 (1997), 1–29 | DOI | MR | Zbl

[6] I. P. Kornfeld, Ya. G. Sinai, S. V. Fomin, Ergodicheskaya teoriya, Nauka, M., 1980 | MR | Zbl

[7] B. Crosignani, P. Di Porto, M. Segev, “Approach to thermal equilibrium in a system with adiabatic constraints”, Amer. J. Phys., 64 (1996), 610–613 | DOI

[8] D. Dürr, S. Goldstein, J. L. Lebowitz, “A mechanical model of Brownian motion”, Comm. Math. Phys., 78:4 (1981), 507–530 | DOI | MR | Zbl

[9] R. Feinman, R. Leiton, M. Sends, Feinmanovskie lektsii po fizike, Nauka, M., 1976–1979

[10] Ch. Gruber, “Thermodynamics of systems with internal adiabatic constraints: time evolution of the adiabatic piston”, European J. Phys., 20:4 (1999), 259–266 | DOI | Zbl

[11] Ch. Gruber, L. Frachebourg, “On the adiabatic properties of a stochastic adiabatic wall: Evolution, stationary non-equilibrium, and equilibrium states”, Phys. A, 272 (1999), 392–428 | DOI

[12] Ch. Gruber, J. Piasecki, “Stationary motion of the adiabatic piston”, Phys. A, 268 (1999), 412–423 | DOI

[13] R. Holley, “The motion of a heavy particle in an infinite one dimensional gas of hard spheres”, Z. Wahrscheinlichkeitstheor. verw. Geb., 17 (1971), 181–219 | DOI | MR | Zbl

[14] S. Kerckhoff, H. Masur, J. Smillie, “Ergodicity of billiard flows and quadratic differentials”, Ann. of Math. (2), 124 (1986), 293–311 | DOI | MR | Zbl

[15] E. Kestemont, C. van den Broeck, M. Mansour, “The “adiabatic” piston: and yet it moves”, Europhys. Lett., 49 (2000), 143–149 | DOI | MR

[16] A. N. Kolmogorov, V. M. Tikhomirov, “$\varepsilon$-entropiya i $\varepsilon$-emkost mnozhestv v funktsionalnykh prostranstvakh”, UMN, 14:2 (1959), 3–86 | MR | Zbl

[17] R. Kubo, H, Ichimura, T. Usui, M. Hashizume, Statistical Mechanics, North-Holland, Amsterdam, 1965 | Zbl

[18] L. D. Landau, E. M. Lifshits, Statisticheskaya fizika, Nauka, M., 1976 | MR

[19] J. L. Lebowitz, J. Piasecki, Ya. Sinai, “Scaling dynamics of a massive piston in an ideal gas”, Hard Ball Systems and Lorentz Gas, Encyclopaedia Math. Sci., 101, Springer-Verlag, Berlin, 2000, 217–227 | MR | Zbl

[20] J. L. Lebowitz, “Stationary non-equilibrium Gibbsian ensembles”, Phys. Rev. (2), 114 (1959), 1192–1202 | DOI | MR | Zbl

[21] E. Lieb, “Some problems in statistical mechanics that I would like to see solved”, Phys. A, 263:1–4 (1999), 491–499 | DOI | MR

[22] R. Lupton, Statistics in Theory and Practice, Princeton Univ. Press, Princeton, NJ, 1993 | MR | Zbl

[23] M. Matsumoto, T. Nishimura, “Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator”, ACM Trans. Model. Comput. Simul., 8:1 (1998), 3–30 | DOI | Zbl

[24] E. Nelson, Dynamical Theories of Brownian Motion, Princeton Univ. Press, Princeton, NJ, 1967 | MR | Zbl

[25] J. Piasecki, Ch. Gruber, “From the adiabatic piston to macroscopic motion induced by fluctuation”, Phys. A, 265 (1999), 463–472 | DOI

[26] A. N. Tikhonov, A. B. Vasileva, A. G. Sveshnikov, Differentsialnye uravneniya, Nauka, M., 1985 | MR | Zbl