Geodesic
Statistical mechanics of nonlinear dynamical systems
Russian journal of nonlinear dynamics, Tome 5 (2009) no. 3, pp. 385-402.

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With the help of mathematical modeling, we study the behavior of a gas ($\sim 10^6$ particles) in a one-dimensional tube. For this dynamical system, we consider the following cases: – collisionless gas (with and without gravity) in a tube with both ends closed, the particles of the gas bounce elastically between the ends, – collisionless gas in a tube with its left end vibrating harmonically in a prescribed manner, – collisionless gas in a tube with a moving piston, the piston's mass is comparable to the mass of a particle. The emphasis is on the analysis of the asymptotic ($t\to\infty$) behavior of the system and specifically on the transition to the state of statistical or thermal equilibrium. This analysis allows preliminary conclusions on the nature of relaxation processes. At the end of the paper the numerical and theoretical results obtained are discussed. It should be noted that not all the results fit well the generally accepted theories and conjectures from the standard texts and modern works on the subject.
Keywords: one-dimensional collisionless gas, statistical equilibrium, thermodynamical equilibrium, weak limit. 
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     title = {Statistical mechanics of nonlinear dynamical systems},
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V. V. Vaskin; N. N. Erdakova; I. S. Mamaev. Statistical mechanics of nonlinear dynamical systems. Russian journal of nonlinear dynamics, Tome 5 (2009) no. 3, pp. 385-402. http://geodesic.mathdoc.fr/item/ND_2009_5_3_a5/