Kinetics of the distribution density and entropy of an isolated system of particles in phase space

Minkevich I.G.
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This work lies in the field of nonequilibrium statistical mechanics. It investigates the spatiotemporal behavior of isolated many-particle systems. A phenomenological model has been formulated that combines the laws of classical dynamics and random transitions between various microscopic states of the system. All processes occurring with such a system take place on a closed hypersurface in a coordinate-momentum phase space. This surface corresponds to a fixed value of energy of the system; it is designated as ${\mathrm Surf}_{\mathrm E}^{}$. The kinetic equation for the probability density distribution of a system in a phase space,~$f(\vec{x},\,t)$, is reduced to a form convenient for finding solutions to this equation ($\vec{x}$ is a set of phase variables). Proof is given that the extremum of the entropy of the system, which is found by the calculus of variations, is a maximum. Near its maximum, entropy is proportional to the integral of the squared variation of $f(\vec{x},\,t)$ over the entire volume of the hypersurface~${\mathrm Surf}_{\mathrm E}^{}$. The dependence of entropy on density variation distribution $f(\vec{x},\,t)$ has been found, when this variation has constant values in two subregions of the hypersurface; these values can be rather large. A kinetic equation has been formulated for the $D(\vec{x},\,t)$, deviation of particle distribution density $f(\vec{x},\,t)$ from its equilibrium value. Analytical solutions of this equation have been found for several variants of the dependence of the rate of random transitions on the distance between the initial and final microscopic states of the transition. If this rate does not depend on the above-mentioned distance, the initial shape of $D(\vec{x},\,t)$ is maintained throughout the entire process of the exponential decay of the quantity $D(\vec{x},\,t)$. The transition process is considered in a more general case, when random transitions are possible only in a limited range of distances between microstates. Then the value $D(\vec{x},\,t)$ is equal to the sum of spatial modes, each of which, as the distribution density tends to the equilibrium, decays at its own rate. The higher the order of the mode, the faster it decays. Therefore, in this case, the transition process leads not only to a general decrease of $D(\vec{x},\,t)$, but also to a smoothing of the $D(\vec{x},\,t)$ original shape. The general dependence of the difference between the equilibrium and nonequilibrium entropy values on $D(\vec{x},\,t)$ has been found.

Keywords: system of point particles, classical dynamics, microscopic state, random transitions, probability distribution density, kinetic equation, entropy, thermodynamic equilibrium, transient processes, nonequilibrium state, solutions of the kinetic equation
Citation in English: Minkevich I.G. Kinetics of the distribution density and entropy of an isolated system of particles in phase space // Computer Research and Modeling, 2026, vol. 18, no. 5, pp. 1161-1187
Citation in English: Minkevich I.G. Kinetics of the distribution density and entropy of an isolated system of particles in phase space // Computer Research and Modeling, 2026, vol. 18, no. 5, pp. 1161-1187
DOI: 10.20537/2076-7633-2026-18-5-1161-1187
URL: http://crm-en.ics.org.ru/journal/article/3775/
Creative Commons License This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License.

Copyright © 2026 Minkevich I.G.

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