Funct. Mater. 2022; 29 (4): 576-585.


Ideal gas in the round vessel: different behaviour

D.M.Naplekov1, V.V.Yanovsky1,2

1 Institute for Single Crystals, NAS Ukraine, 60 Nauky Ave., Kharkov, 61001, Ukraine
2 V. N. Karazin Kharkiv National University, 4 Svobody Sq., Kharkov, 61022, Ukraine


We consider classical ideal gas of a finite number of colliding particles in a stationary vessel. A special case of the round vessel is considered and, for comparison, the results for the rectangular vessel are provided. It is proved that the distributions of energy and velocity of gas particles differ from similar distributions in a rectangular vessel. The paper investigates the case when a finite number of particles in a round vessel has zero total angular momentum. It is shown that these distributions in a round vessel depend on the particle masses and differ from the known classical distributions. As the number of particles increases, the distributions tend to the Boltzmann distribution. Finiteness of the number of particles or degrees of freedom is important for understanding the properties of nanosystems.

gas of colliding particles, round vessel, finite number particles, additional law of conservation, distributions.
1. Gibbs J W 2014, Elementary principles in statistical mechanics (New-York:Dover), 2014.
2. Boltzmann L 1912, Vorlesungen uber Gastheorie, (Leipzig: Ambrosius Barth).
3. Rowlinson J S 2005 The Maxwell-Boltzmann distribution, Molecular Physics}, 103, 2821-2828.
4. Hincin A Y 1943, Mathematical justification of statistical mechanics, (M.-L.: Gostechizdat).
5. Maxwell J C 2003 Illustrations of the Dynamical Theory of Gases, The Scientific Papers of James Clerk Maxwell, (New-York: Dover).
6. Boltzmann L 1876 Uber die Natur der Gasmolekule, Wiener Berichte, 74, 553-560.
7. Jarzynski C 1997 Nonequilibrium Equality for Free Energy Differences, Phys. Rev. Lett., 78, 2690.
8. Esposito M and Van den Broeck C 2010 Three Detailed Fluctuation Theorems, Phys. rev. lett., 104, 090601. 9.\bibitem{copar-Max1}Maxwell J C 1877 The kinetic theory of gases, Nature, 16, 242.
10. Landau L D and Lifshitz E M 2013, Statistical Physics: Volume 5, (Elsevier Science).
11. Becattinia F and Ferroni L 2007 The microcanonical ensemble of the ideal relativistic quantum gas with angular momentum conservation, Eur. Phys. J. C, 52, 597-615.
12. Nakamura T K 2012 Relativistic Statistical Mechanics with Angular Momentum, Prog. Theor. Phys., 127, 153.
13. Thomas M W and Snider R F 1970 Boltzmann Equation and Angular Momentum Conservation, J. of Stat. Phys., 2, 61.
14. Dubrovskii I M 2008 The role of angular momentum conservation law in statistical mechanics, Cond. Matt. Phys., 11, 585-596.
15. Imara N and Blitz L 2011 Angular momentum in giant molecular clouds. I. The milky way, The Astrophysical Journal}, 732, 78.
16. Chevy F, Madison K W and Dalibard J. 2000 Measurement of the Angular Momentum of a Rotating Bose-Einstein Condensate, Phys. Rev. Lett., 85, 2223.
17. Chapman S, Cowling T G, Burnett D and Cercignani C 1990, The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, (Cambridge:Cambridge University Press).
18. Poincare H 1912, Calcul des probabilites, (Paris:Gauthier-Villars).
19. Naplekov D M, Semynozhenko V P and Yanovsky V V 2014 Equation of state of an ideal gas with nonergodic behavior in two connected vessels, Phys. Rev. E, 89, 012920.
20. Chirikov B V, Izrailev F M and Tayursky V A 1973 Numerical experiments on the statistical behaviour of dynamical systems with a few degrees of freedom, Comp. Phys. Comm., 5, 11-16.
21. Kozlov V V 2002 On justification of Gibbs distribution, Regular And Chaotic Dynamics, 7, 1-10.
22. Brush S G 2003, The Kinetic Theory of Gases, an Anthology of Classic Papers with Historical Commentary, (London:Imperial College Press).
23. Puglisi A, et al, 2017 Temperature in and out of equilibrium: A review of concepts, tools and attempts, Physics Reports, 709, 1-60.

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