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Uspekhi Mat. Nauk, 2002, Volume 57, Issue 6(348), Pages 3–86 (Mi umn572)  

This article is cited in 20 scientific papers (total in 20 papers)

Dynamics of a massive piston in an ideal gas

N. I. Chernova, J. L. Lebowitzb, Ya. G. Sinaic

a University of Alabama at Birmingham
b Rutgers, The State University of New Jersey
c Princeton University, Department of Mathematics

Abstract: 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.

DOI: https://doi.org/10.4213/rm572

Full text: PDF file (892 kB)
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English version:
Russian Mathematical Surveys, 2002, 57:6, 1045–1125

Bibliographic databases:

UDC: 519.248.23
MSC: Primary 37N10, 35Q35; Secondary 37A99, 76N10, 60K40, 82C40, 80A10
Received: 16.05.2002

Citation: N. I. Chernov, J. L. Lebowitz, Ya. G. Sinai, “Dynamics of a massive piston in an ideal gas”, Uspekhi Mat. Nauk, 57:6(348) (2002), 3–86; Russian Math. Surveys, 57:6 (2002), 1045–1125

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. Gruber C., Pache S., Lesne A., “On the second law of thermodynamics and the piston problem”, J. Statist. Phys., 117:3-4 (2004), 739–772  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    2. Neishtadt A.I., Sinai Y.G., “Adiabatic piston as a dynamical system”, J. Statist. Phys., 116:1-4 (2004), 815–820  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    3. Caglioti E., Chernov N., Lebowitz J.L., “Stability of solutions of hydrodynamic equations describing the scaling limit of a massive piston in an ideal gas”, Nonlinearity, 17:3 (2004), 897–923  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    4. Brito R., Renne M.J., Van den Broeck C., “Dissipative collapse of the adiabatic piston”, Europhys. Lett., 70:1 (2005), 29–35  crossref  adsnasa  isi  scopus  scopus
    5. Balakrishnan V., Van den Broeck C., “Analytic calculation of energy transfer and heat flux in a one-dimensional system”, Phys. Rev. E, 72:4 (2005), 046141, 8 pp.  crossref  zmath  adsnasa  isi  scopus  scopus
    6. Kozlov V.V., “To the piston problem”, Dokl. Math., 72:1 (2005), 634–637  mathnet  mathscinet  zmath  isi  elib  elib
    7. Hurtado P.I., Redner S., “Simplest piston problem. I. Elastic collisions”, Phys. Rev. E, 73:1 (2006), 016136, 8 pp.  crossref  adsnasa  isi  scopus  scopus
    8. Hurtado P.I., Redner S., “Simplest piston problem. II. Inelastic collisions”, Phys. Rev. E, 73:1 (2006), 016137, 8 pp.  crossref  adsnasa  isi  scopus  scopus
    9. Caglioti E., Rousset F., “Quasi-stationary states for particle systems in the mean-field limit”, J. Stat. Phys., 129:2 (2007), 241–263  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    10. Crosignani B., Di Porto P., “Random fluctuations of diathermal and adiabatic pistons”, Found. Phys., 37:12 (2007), 1707–1715  crossref  zmath  adsnasa  isi  scopus  scopus
    11. Wright P., “The periodic oscillation of an adiabatic piston in two or three dimensions”, Comm. Math. Phys., 275:2 (2007), 553–580  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    12. Caglioti E., Rousset F., “Long time behavior of particle systems in the mean field limit”, Commun. Math. Sci., 5, suppl. 1 (2007), 11–19  crossref  mathscinet  isi
    13. Chernov N., Dolgopyat D., Brownian Brownian motion. I, Mem. Amer. Math. Soc., 198, no. 927, 2009, viii+193 pp.  mathscinet  isi
    14. I. S. Mamaev, “Metody parallelnykh vychislenii dlya zadach statisticheskoi mekhaniki”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2009, no. 4, 167–175  mathnet
    15. Igor Gorelyshev, “On the dynamics in the one-dimensional piston problem”, Nonlinearity, 24:8 (2011), 2119  crossref  mathscinet  zmath  isi  scopus
    16. Xuwen Chen, Walter Strauss, “Approach to Equilibrium of a Body Colliding Specularly and Diffusely with a Sea of Particles”, Arch Rational Mech Anal, 2013  crossref  mathscinet  isi  scopus  scopus
    17. M. Ebrahim Foulaadvand, M. Mehdi Shafiee, “One-dimensional Brownian motion in hard rods: The adiabatic piston problem”, EPL, 104:3 (2013), 30002  crossref  adsnasa  isi  scopus  scopus
    18. Masato Itami, Shin-ichi Sasa, “Nonequilibrium Statistical Mechanics for Adiabatic Piston Problem”, J Stat Phys, 2014  crossref  mathscinet  isi  scopus
    19. Carati A., Maiocchi A., Galgani L., “Statistical thermodynamics for metaequilibrium or metastable states”, Meccanica, 52:6 (2017), 1295–1307  crossref  mathscinet  zmath  isi  scopus
    20. Tsuji T., Saita Sh., Kawano S., “Thermophoresis of a Brownian Particle Driven By Inhomogeneous Thermal Fluctuation”, Physica A, 493 (2018), 467–482  crossref  mathscinet  isi  scopus  scopus
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