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## Archive of Issues

Russia Saint Petersburg
Year
2014
Issue
2
Pages
121-145
 Section Mechanics Title Irregular and regular forces in stellar systems Author(-s) Ossipkov L.P.a Affiliations Saint Petersburg State Universitya Abstract Various ways of definition of irregular (random) and regular (smoothed) forces in stellar systems are critically discussed. The most satisfactory is Eddington's one according to which the regular force is an attraction force of a continuous fluid resulting from spreading a stellar mass over a system. Also, a definition of the regular force as a mathematical expectation of a random force is of interest. It is emphasized that the crossing time, $\tau_c$, a time scale of regular forces, characterizes the rate of collective processes in the system, including collisionless relaxation, that (as a rule) occurs in gravitating systems. The quasi-entropy, i.e., a result of averaging of an arbitrary convex function of a coarse-grained distribution function over the phase space, is discussed as a measure of collisionless stochastization. For non-rotating systems the maximum of quasi-entropy can be reached only for isotropic velocity distributions. Formulas for the first and second variations of quasi-entropy, found by Antonov, are given. If there exists the density variation so that the second variation of quasi-entropy is positive, then the present state of the system is not the most probable. It follows from this assertion that evolution along a sequence of polytropic spheres is not possible without some energy input to the system. We recall the classification of forms of the phase mixing in collisionless systems. The problem of collisional relaxation in gravitating systems is briefly discussed. We state the approach to its analysis on the basis of studying geodesic flows and the ensemble averaging as the next step, which requires the knowledge of distribution of a random force. To avoid truncation of Holtsmark's distribution at small impact parameters the distribution of random force by Petrovskaya was used. In that case the ratio of the effective stochastization time to the crossing time is proportional to $N^{1/3}/(\ln N)^{1/2}$ where $N\gg 1$ is the number of stars in the system. This evolutionary time scale is close to the one found earlier by Genkin. Keywords dynamics of stellar systems, modelling stellar systems, evolution of galaxies UDC 524.3./4.-32 MSC 70F15, 85A05 DOI 10.20537/vm140209 Received 16 January 2014 Language Russian Citation Ossipkov L.P. Irregular and regular forces in stellar systems, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 2, pp. 121-145. References Kurth R. Introduction to the mechanics of stellar systems, London-New York-Paris: Pergamon Press, 1957, X+174 p. Orlov V.V., Rubinov A.V. Zadacha $N$ tel v zvezdnoi dinamike ($N$ body problem in stellar dynamics), Saint Petersburg: Izd. VVM, 2008, 175 p. Jacobi C.G.J. Vorlesungen uber Dynamik, Zweite Ausgabe, Berlin: Verlag von G. Reiner, 1884. 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