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Home » Issues » Archive of Issues » 2014 - 2 issue » pp. 121-145

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Russia Saint Petersburg
Year
2014
Issue
2
Pages
121-145
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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
  1. Kurth R. Introduction to the mechanics of stellar systems, London-New York-Paris: Pergamon Press, 1957, X+174 p.
  2. 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.
  3. Jacobi C.G.J. Vorlesungen uber Dynamik, Zweite Ausgabe, Berlin: Verlag von G. Reiner, 1884. Translated under the title Lektsii po dinamike, Leningrad-Moscow: ONTI, 1936, 272 p.
  4. Wintner A. The analytical foundations of celestial mechanics, Princeton (NJ): Princeton Univ. Press, 1941. Translated under the title Analiticheskie osnovy nebesnoi mekhaniki, Moscow: Nauka, 1967, 524 p.
  5. Hilmy H.F. Problema $n$ tel v nebesnoi mekhanike i kosmogonii ($N$ body problem in celestial mechanics and cosmogony), Moscow: Izd. Akad. Nauk SSSR, 1951, 156 p.
  6. Zhabko A.P., Kirpichnikov S.N. Lektsii po dinamicheskim sistemam. Chast' 4: Ergodicheskaya teoriya (Lectures on dynamical systems. Part 4: Ergodic theory), Saint Petersburg: Saint Petersburg State University, 2004, 128 p.
  7. Pollard H. The behavior of gravitational systems, J. Math. Mech., 1967, vol. 17, no. 6, pp. 601-608.
  8. Saari D.G. Expanding gravitational systems, Trans. Amer. Math. Soc., 1971, vol. 156, pp. 219-240.
  9. Saari D.G. On oscillatory motion in gravitational systems, J. Diff. Equat., 1973, vol. 14, no. 2, pp. 275-292.
  10. Bonnor W.B. Jeans’ formula for gravitational instability, Monthly Notices Roy. Astron. Soc., 1957, vol. 117, no. 1, pp. 104-116.
  11. Poincare H. La Voie Lactee et la theorie de gaz, Bull. Soc. Astron. France, 1906, vol. 20, pp. 153-165.
  12. Eddington A.S. The dynamics of globular stellar system, Monthly Notices Roy. Astron. Soc., 1913. vol. 74, no. 1, pp. 5-16.
  13. Schwarzschild K. Stationare Geschwindigkeitverteilung im Sternsystem, Probleme der Astronomie, Berlin: Verlag von Julius Springer, 1924, pp. 94-105.
  14. Gerasimovich B.P. Statistical ensembles of stellar astronomy, Mirovedeniye, 1931, vol. 20, no. 1, pp. 41-54 (in Russian).
  15. Ogorodnikov K.F. Some modern problems of stellar dynamics, Vestn. Leningrad. Univ., 1947, no. 1, pp. 5-16 (in Russian).
  16. Agekyan T.A. General features of the evolution of rotating systems of gravitating bodies, Sov. Astronomy - AJ, 1960, vol. 4, no. 2, pp. 298-307.
  17. Genkin I.L. Regular and irregular forces in star systems, Trudy Astrofiz. Inst. Akad. Nauk Kazakh. SSR, 1971, vol. 16, pp. 103-110 (in Russian).
  18. Kandrup H.E. The complexion of forces in an anisotropic self-gravitating system, Astrophys. J., 1981, vol. 244, no. 3, pp. 1039-1063.
  19. Ossipkov L.P. Principle problems of galactic dynamics, Matematicheskie metody modelirovaniya galaktik (Mathematical methods of modelling galaxies), Saint Petersburg: SOLO, 2012, pp. 68-112.
  20. Ossipkov L.P. On some fundamental concepts of galactic dynamics, Astron. Nachr., 2013, vol. 334, no. 8, pp. 793-799.
  21. Ahmad A., Cohen L. Integration of the $N$ body gravitational problem by separation of the force into a near and far component, Lecture Notes in Mathematics, 1974, vol. 362, pp. 304-312.
  22. Zonn W., Rudnicki K. Astronomia gwiazdowa, Warszawa: Panstwowe wydawnictwo naukowe, 1957. Translated under the title Zvezdnaya astronomiya, Moscow: Inostr. Literatura, 1959, 448 p.
  23. Braun W., Hepp K. The Vlasov equation and its fluctuations in the $1/N$ limit of interacting classical particles, Comm. Math. Phys., 1977, vol. 56, no. 2, pp. 101-113.
  24. Spohn H. Kinetic equations from Hamiltonian dynamics: Markovian limit, Rev. Math. Phys., 1980, vol. 53, no. 3, pp. 589-615.
  25. Chandrasekhar S. New methods in stellar dynamics, Ann. New York Acad. Sci., 1943, vol. 45, pp. 131-161.
  26. Ogorodnikov K.F. On principles of statistical mechanics of stellar systems, Sov. Astron. - AJ, 1957, vol. 1, no. 6, pp. 787-795.
  27. Agekyan T.A. Stellar statistics. Galaxy structure, Kurs astrofiziki i zvezdnoi astronomii. Tom 2 (Course of astrophysics and stellar astronomy. Vol. 2), Moscow: Fizmatgiz, 1962, pp. 427-480.
  28. Ossipkov L.P. Obshchie printsipy matematicheskogo modelirovaniya zvezdnykh system (General principles of mathematical modelling star systems), Saint Petersburg: SOLO, 2010, 102 p.
  29. Vlasov A.A. Statisticheskie funktsii raspredeleniya (Statistical distribution functions), Moscow: Nauka, 1968, 356 p.
  30. Chavanis P.-H. Hamiltonian and Brownian systems with large-range interactions: III. The BBGKY hierarchy for spatially inhomogeneous systems, Physica A, 2008, vol. 387, pp. 787-805.
  31. Camm G.L. Random gravitational forces in a star field, Monthly Notices Roy. Astron. Soc., 1963, vol. 126, no. 3, pp. 283-293.
  32. Chandrasekhar S., von Neumann J. The statistics of the gravitational field arising from a random distribution of stars. I. The speed of fluctuations, Astrophys. J., 1942, vol. 95, no. 3, pp. 489-531.
  33. Ogorodnikov K.F. Dynamics of stellar systems, London: Pergamon Press, 1965, 359 p.
  34. Dibai E.A., Kaplan S.A. Razmernosti i podobie astrofizicheskikh velichin (Dimensions and similarity of astrophysical quantities), Moscow: Nauka, 1976, 400 p.
  35. Chandrasekhar S., Elbert D. Some elementary application of the virial theorem to stellar dynamics, Monthly Notices Roy. Astron. Soc., 1972, vol. 155, no. 4, pp. 435-447.
  36. Ossipkov L.P. Gross-evolution of star clusters, Zvezdnye skopleniya i problemy zvezdnoi evolyutsii (Star clusters and problems of stellar evolution), Sverdlovsk: Ural State University, 1983, pp. 20-38.
  37. Ossipkov L.P. Small virial oscillations of axisymmetric gravitating systems. I, Astrophysics, 2000, vol. 43, no. 2, pp. 215-221.
  38. Antonov V.A., Nuritdinov S.N. Nonlinear oscillations of some homogeneous models of star systems. I. The case of radial oscillations, Vestnik Leningrad. Univ., 1973, no. 7, pp. 131-138 (in Russian).
  39. Ossipkov L.P. Dimensionless functionals for self-gravitating systems, Dinamika, optimizatsiya, upravlenie. Voprosy mekhaniki i protsessov upravleniya, vyp. 22 (Dynamics, optimization, control. Problems of mechanics and control processes, iss. 22), Saint Petersburg: Saint Petersburg State University, 2004, pp. 127-130.
  40. Von der Pahlen E. Uber die Enstehung der spharischen Sternhaufen, Zeitschr. f. Astrophys., 1947, vol. 24, no. 1/2, pp. 68-120.
  41. Kurth R. Uber Sternsystemezeitlich ober raumlich veranderlicher Dichte, Zeitschr. f. Astrophys., 1949, vol. 26, no. 1, pp. 100-136.
  42. Vandervoort P.O. Modes of oscillations of a uniformly rotating homogeneous spheroid of stars, Astrophys. J., 1991, vol. 377, no. 1, pp. 49-71.
  43. Zaslavsky G.M. Physics of chaos in Hamiltonian systems, New York: Imperial College Press, 1998. Translated under the title Fizika khaosa v gamil'tonovykh systemakh, Moscow-Izhevsk: Institute of Computer Science, 2004, 288 p.
  44. Mathur S.D. Irreversibility due to mixing in collisionless systems, Monthly Notices Roy. Astron. Soc., 1988, vol. 231, no. 2, pp. 368-372.
  45. Chernin A.D., Valtonen M.J., Zheng J.-Q., Ossipkov L.P. Dynamical evolution of $N$-body gravitational systems starting from Poincare chaos, Stellar Dynamics: from classic to modern, Saint Petersburg: Sobolev Astron. Inst., 2001, pp. 431-436.
  46. Lynden-Bell D. Statistical mechanics of violent relaxation in stellar systems, Monthly Notices Roy. Astron. Soc., 1967, vol. 136, no. 1, pp. 101-121.
  47. Shu F.H. On the statistical mechanics of violent relaxation, Astrophys. J., 1978, vol. 225, no. 1, pp. 83-94.
  48. Ziegler H.J., Wiechen H. Collisionless relaxation of self-gravitating systems with spherically symmetric dynamics, Astrophys. J., 1990, vol. 362, no. 2, pp. 595-603.
  49. Hjorth J., Madsen J. Violent relaxation and stability of elliptical galaxies, Structure, dynamics and chemical evolution of elliptical galaxies, Garching: ESO, 1993, pp. 263-272.
  50. Arad I., Lynden-Bell D. Inconsistency in theories of violent relaxation, Monthly Notices Roy. Astron. Soc., 2004, vol. 362, no. 2, pp. 385-395.
  51. Kozlov V.V. Teplovoe ravnovesie po Gibbsu i Puankare (Heat equilibrium according to Gibbs and Poincare), Moscow-Izhevsk: Institute of Computer Science, 2002, 320 p.
  52. Lynden-Bell D. The stability and vibrations of a gas of stars, Monthly Notices Roy. Astron. Soc., 1962, vol. 124, no. 4, pp. 279-296.
  53. Genkin I.L. Vlasov equation and irreversibility in plasma physics and stellar dynamics, Sov. Physics-Astronomy, 1970, vol. 13, no. 6, pp. 962-963.
  54. Wehrl A. General properties of entropy, Rev. Modern Physics, 1978, vol. 50, no. 2, pp. 221-260.
  55. Grandy W.T. Principle of maximum entropy and irreversible processes, Physics Reports, 1980, vol. 62, no. 3, pp. 175-266.
  56. Vedenyapin V.V. On the uniqueness of Boltzmann's $H$ function, Preprint Inst. Prikl. Mat. Akad. Nauk SSSR (Preprint Inst. Applied Math. Acad. Sci. of the USSR), 1977, no. 3, pp. 1-77.
  57. Tolman R. The principles of statistical mechanics. 2nd ed., London: Oxford Univ. Press, 1957, v+310 p.
  58. Von Neumann I. Mathematische grundlagen der quantenmechanik, Berlin: Verlag von Julius Springer, 1932. Translated under the title Matematicheskie osnovy kvantovoy mekhaniki, Moscow: Nauka, 1964, 368 p.
  59. Khinchine A.Ya. Convex functions and evolutionary theorems of statistical mechanics, Izvestiya Akad. Nauk SSSR, Ser. Mat., 1943, vol. 7, no. 3, pp. 111-122.
  60. Penrose O. Foundations of statistical mechanics. A deductive treatment, Oxford: Pergamon Press, 1970, VII+260 p.
  61. Marimoto T. Markov processes and the $H$-theorem, J. Phys. Soc. Japan., 1963, vol. 18, no. 3, pp. 328-331.
  62. Ramshow J.D. Irreversibility and general entropies, Phys. Lett. A., 1993, vol. 175, no. 1, pp. 170-171.
  63. Antonov V.A. Individual and statistical aspects of star motion, Order and chaos in stellar and planetary systems, ASP Conf. Ser., vol. 316, San Fransisco: ASP, 2004, pp. 10-19.
  64. Antonov V.A. Applying the variational method to stellar dynamics and some other problems, Abstract of Cand. Sci. (Phys.-Math.) Dissertation, Leningrad, 1963, 5 p.
  65. Antonov V.A., Nuritdinov S.N., Ossipkov L.P. Classification of mixing kinds in dynamical systems, Dynamika galaktik i zvezdnykh skoplenii (Dynamics of galaxies and star clusters), Alma-Ata: Nauka, 1973, pp. 55-59.
  66. Antonov V.A., Nuritdinov S.N., Ossipkov L.P. On the classification of phase mixing in collisionless stellar systems, Astron. Astrophys. Transact., 1995, vol. 7, no. 2/3, pp. 177-180.
  67. Tremaine S., H\'enon M., Lynden-Bell D. $H$-functions and mixing in violent relaxation, Monthly Notices Roy. Astron. Soc., 1986, vol. 219, no. 2, pp. 285-297.
  68. Kandrup H.E. An $H$-theorem for violent relaxation? Monthly Notices Roy. Astron. Soc., 1987, vol. 225, no. 4, pp. 995-998.
  69. Sridhar S.D. Does $H$-function always increase during violent relaxation? J. Astrophys. Astron., 1987, vol. 8, no. 1, pp. 257-262.
  70. Soker N. $H$-function evolution in collisionless self-gravitating systems, Publ. Astron. Soc. Pacific., 1990, vol. 102, no. 652, pp. 639-645.
  71. Antonov V.A. Stability of spherical clusters relative to finite amplitude perturbations, Voprosy nebesnoi mekhaniki i zvezdnoi dinamiki (Problems of celestial mechanics and stellar dynamics), Alma-Ata: Nauka, 1990, pp. 66-70.
  72. Chavanis P.-H., Sommeria J., Robert R. Statistical mechanics of two-dimensional vortices and collisionless stellar systems, Astrophys. J., 1996, vol. 385, no. 1, pp. 385-399.
  73. Chavanis P.-H., Bouchet F. On the coarse-grained evolution of collisionless stellar systems, Astron. Astrophys., 2005, vol. 430, no. 1, pp. 271-287.
  74. Eddington A.S. The dynamical evolution of stellar systems, Astron. Nachr., 1921, Jubilaumsnummer, pp. 9-10.
  75. Kuzmin G.G. Effect of stellar encounters and evolution of star clusters, Publ. Tartusk. Astron. Observ., 1957, vol. 33, no. 2, pp. 75-102 (in Russian).
  76. Lebowitz J.L., Penrose O. Modern ergodic theory, Physics Today, 1973, vol. 26, no. 2, pp. 23-29.
  77. Wightman A.S. Statistical mechanics and ergodic theory: an expository lecture, Statistical mechanics at the turn of the decade, New York: M. Dekker Inc., 1971, pp. 1-32.
  78. Gibbs J.W. Elementary principles of statistical mechanics developed with a special reference to the rational foundation of thermodynamics, Yale: Yale Bicentennial Publ., 1902, XVII+436 p. Translated under the title Osnovnye printsipy statisticheskoi mekhaniki so spetsial'nym prilozheniem k ratsional'nomu obosnovaniyu termodynamiki, Moscow-Leningrad: Gostekhizdat, 1946, 204 p.
  79. Krylov N.S. Works on foundations of statistical physics, Princeton: Princeton Univ. Press, 1979, VI+160 p.
  80. Goldstein S., Lebowitz J.A., Aizenman M. Ergodic properties of infinite systems, Dynamical systems, theory and applications. Lecture Notes in Physics, vol. 38, Berlin: Springer, 1975, pp. 112-143.
  81. Ossipkov L.P. Phase mixing of the second kind in stellar systems. I, Astrophysics, 1972, vol. 8, no. 1. pp. 80-84.
  82. Ossipkov L.P. On the physical interpretation of some kinds of mixing in stellar systems, Sov. Astron. - AJ, 1976, vol. 19, no. 4, pp. 530-532.
  83. Nuritdinov S.N. On phase mixing in nonlinear nonstationary stellar systems, Astron. Tsirkulyar, 1979, no. 1081, pp. 1-2 (in Russian).
  84. Kirbigekova I.I. Non-linear, non-radial evolution of disk-like models of galaxies, Astron. Astrophys. Transact., 1995. vol. 7, no. 4, pp. 299-301.
  85. Mirtadjieva K.T., Kirbijekova I.I., Nuritdinov S.N. Towards theory of compulsive phase mixing for non-stationary stellar systems, Order and chaos in stellar and planetary systems, ASP Conf. Ser., vol. 316, San Francisco: ASP, 2004, pp. 363-365.
  86. Sellwood J.A., Preto M. Scattering of stars by transient spiral waves, Disks of galaxies: kinematics, dynamics and perturbations, ASP Conf. Ser., vol. 275, San Francisco: ASP, 2002, pp. 94-105.
  87. Nuritdinov S.N. On the role of chaos and instability in the evolution of nonlinear nonstationary stellar systems, Instability, chaos and predictability in Celestial Mechanics and Stellar Dynamics, Commaсk (N.Y.): Nova Sci. Publ., 1993, pp. 39-44.
  88. Marochnik L.S. On relaxation of stellar systems without star-star encounters, Dokl. Akad. Nauk Tadjik. SSR, 1967, vol. 10, no. 8, pp. 15-17 (in Russian).
  89. Marochnik L.S. Relaxation of stars in plane subsystems of the Galaxy in the spiral structure, Astrophysics, 1969, vol. 5, no. 3, pp. 242-246.
  90. Tremaine S., Ostriker J.P. Relaxation in stellar systems and the shape and rotation of the inner disk halo, Monthly Notices Roy. Astron. Soc., 1999, vol. 306, no. 4, pp. 662-668.
  91. Pfenniger D. Relaxation and dynamical friction in non-integrable stellar systems, Astron. Astrophys., 1986, vol. 165, no. 1-2, pp. 74-83.
  92. Udry S., Pfenniger D. Stochasticity in elliptical galaxies, Astron. Astrophys., 1988, vol. 198, no. 1, pp. 135-149.
  93. Oseledets V.I. Multiplicative ergodic theorem. Lyapounov exponents for dynamical systems, Trudy Moskovskogo Matematicheskogo Obshchestva, 1968, vol. 19, pp. 179-210 (in Russian).
  94. Binney J., Lacey C. The diffusion of stars through phase space, Monthly Notices Roy. Astron. Soc., 1988, vol. 230, no. 4, pp. 597-627.
  95. Kandrup H.E., Wilmes D.E. Collisional relaxation in non-integrable potential, Astron. Astrophys., 1994, vol. 283, no. 1, pp. 59-66.
  96. Mahon M.E., Abernathy R.A., Bradley B.O., Kandrup H.E. Transient ensemble dynamics in time-dependent galactic potential, Monthly Notices Roy. Astron. Soc., 1995, vol. 275, no. 2, pp. 443-453.
  97. Habib S., Kandrup H.E., Mahon M.E. Chaos and noise in galactic potential, Astrophys. J., 1997, vol. 480, no. 1, pp. 155-166.
  98. Byl J., Ovenden M.W. Time variations of the velocity distribution due to phase mixing, Monthly Notices Roy. Astron. Soc., 1973, vol. 164, no. 3, pp. 289-294.
  99. Ossipkov L.P. Evolution of moving clusters in the regular field of the Galaxy, Zvezdnye agregaty (Star aggregates), Sverdlovsk: Ural State University, 1980, pp. 114-121.
  100. Ivannikova E.I., Maksumov M.N. On the possible criticality of a marginally stable stellar disk, Stellar dynamics: from classic to modern, Saint Petersburg: Sobolev Astron. Institute, 2006, pp. 367-373.
  101. Serafin R.A. On the rectilinear non-collision motion, Cel. Mech. Dyn. Astron., 2001, vol. 80, pp. 97-108.
  102. Mikisha A.M., Tsitsin F.A. On a formula for relaxation time, Vestn. Moskovsk. Univ., Ser. III, Fizika, Astronomiya, 1965, no. 5, pp. 74-77 (in Russian).
  103. Rastorguev A.S., Sementsov V.N. Estimating the stochastization time in stellar systems, Astronomy Letters, 2006, vol. 32, no. 1, pp. 14-17.
  104. Ostriker J.P., Davidson A.F. Time of relaxation. I. Unbounded medium, Astrophys. J., 1968, vol. 151, no. 1, pp. 679-686.
  105. Sivukhin D.V. Coulomb collisions in fully ionized plasmas, Voprosy Teorii Plazmy, 1964, vol. 4, pp. 81-187 (in Russian).
  106. Karayanidi A.D. On the influence of external background on two-body encounters, Trudy Astrofiz. Inst. Akad. Nauk Kazakh. SSR, 1982, vol. 39, pp. 30-37 (in Russian).
  107. Sagintaev B.S., Chumak O.V. Some kinetic effects in non-homogeneous gravitating systems, Trudy Astrofiz. Inst. Akad. Nauk Kazakh. SSR, 1983, vol. 40, pp. 12-20 (in Russian).
  108. Charlier C.V.L. Statistical mechanics based on the law by Newton, Lunds Univ. Arskraft, 1917, vol. 13, no. 5, pp. 1-88.
  109. Kandrup H.E. Discreteness fluctuations and relaxation in stellar dynamical systems, Monthly Notices Roy. Astron. Soc., 1988, vol. 235, no. 4, pp. 1151-1167.
  110. Weinberg M.D. Nonlocal and collective relaxation in stellar systems, Astrophys. J., 1993, vol. 410, no. 2, pp. 543-551.
  111. Nelson R.W., Tremaine S. Linear response, dynamical friction, and the fluctuation dissipation theorem in stellar dynamics, Monthly Notices Roy. Astron. Soc., 1999, vol. 306, no. 1, pp. 1-21.
  112. Kurth R. Dimensional analysis and group theory in astrophysics, Oxford: Pergamon Press, 1972. Translated under the title Analiz razmernostei v astrofizike, Moscow: Mir, 1972, 232 p.
  113. Genkin I.L. Relaxation in a regular field, Sov. Phys. - Doklady, 1971, vol. 16, pp. 261-262.
  114. Chandrasekhar S. Stochastic problems in physics and astronomy, Rev. Mod. Phys., 1943, vol. 15, no. 1, pp. 1-89. Translated under the title Stokhasticheskie problemy v fizike i astronomii, Moscow: Inostr. Literatura, 1947, 168 p.
  115. Gurzadyan V.G., Kocharyan A.A. Collective relaxation of stellar systems revisited, Astron. Astrophys., 2009, vol. 505, no. 1, pp. 203-205.
  116. Sagintaev B.S., Chumak O.V. Stochastic diffusion of an ensemble of circular orbits, Trudy Astrofiz. Inst. Akad. Nauk Kazakh. SSR, 1984, vol. 43, pp. 51-58 (in Russian).
  117. Sagintaev B.S. Evolution of an ensemble of orbits in an irregular field with a finite correlation time, Trudy Astrofiz. Inst. Akad. Nauk Kazakh. SSR, 1988, vol. 49, pp. 106-116 (in Russian).
  118. Ossipkov L.P. On the fundamental paradox of stellar dynamics, Astron. Astrophys. Transact., 2006, vol. 25, no. 2-3, pp. 123-128.
  119. Gurzadyan V.G., Savvidi G.K. Collective relaxation in stellar systems, Astron. Astrophys., 1986, vol. 160, no. 1, pp. 203-210.
  120. Kandrup H.E. Divergence of nearby trajectories for the gravitational $N$-body problem, Astrophys. J., 1990, vol. 364, no. 2, pp. 420-425.
  121. Boccaletti D., Pucacco G., Ruffini R. Multiple relaxation time scales in stellar dynamics, Astron. Astrophys., 1991, vol. 214, no. 1, pp. 48-51.
  122. Ossipkov L.P. Stochastization in homogeneous graviplasmas, Vestn. St-Peterbg. Univ., Ser. 10, Prikl. Mat. Inf. Prots. Upr., 2009, no. 2, pp. 93-103 (in Russian).
  123. Cipriani P., Pucacco G. Some critical remarks on relaxation in $N$-body simulation, Nuovo Cimento, 1994, vol. 109 B, no. 3, pp. 325-330.
  124. Anosov D.V. Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklova, 1967, vol. 40, pp. 1-210 (in Russian).
  125. Pfenniger D. Order and chaos in $N$-body systems, $N$-body problem and gravitational dynamics, Paris: Observ. de Paris, 1993, pp. 1-8.
  126. Ekker G. Theory of fully ionized plasmas, New York-London: Academic Press, 1972. Translated under the title Teoriya polnost'yu ionizovannoi plazmy, Moscow: Mir, 1977, 432 p.
  127. Ovod D.V. On relaxation in a regular field of star systems, Protsessy upravleniya i ustoychivost'. Trudy XLI Mezhdunar. konferentsii aspirantov i studentov (Control processes and stability. Proc. XLI Internat. sci. conference of postgraduate students and students), Saint Petersburg: Saint Petersburg State University, 2010, pp. 199-203.
  128. Ovod D.V. On relaxation in stellar systems, Astron. Tsirkulyar, 2012, no. 1571, pp. 1-3.
  129. Petrovskaya I.V. The close-encounter force function, Sov. Astron. Lett., 1986, vol. 12, no. 4, pp. 237-240.
  130. Agekyan T.A. The probability of a stellar approach with a given change of the absolute velocity, Sov. Astron. - AJ, 1959, vol. 3, no. 1, pp. 46-58.
  131. Kandrup H.E. Stochastic properties of the gravitational $N$-body problem, Astron. Astrophys. Transact., 1995, vol. 7, no. 4, pp. 225-228.
  132. Goodman J., Heggie D.C., Hut P. On the exponential instability of the gravitational $N$-body system, Astrophys. J., 1993, vol. 415, no. 2, pp. 715-733.
  133. Ossipkov L.P. Effective stochastization time for stellar systems with large number of stars, Astron. Tsirkulyar., 2012, no. 1578, pp. 1-3.
  134. Ovod D.V., Ossipkov L.P. Stochastization in gravitating systems, Astron. Nachr., 2013, vol. 334, no. 8, pp. 799-804.
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