Флуктуації Хольцмарка нестаціонарних гравітаційних полів

V. A. Litovchenko

doi:10.37863/umzh.v73i1.6113

V. A. Litovchenko Yuriy Fedkovych Chernivtsi National University https://orcid.org/0000-0001-9848-3167

DOI: https://doi.org/10.37863/umzh.v73i1.6113

Abstract

UDC 517.937, 519.21

We construct the Holtsmark distributions of non-stationary fluctuations of local interaction of moving objects in a system where the gravitational influence is governed by a power law.
We find a pseudodifferential equation with a Riesz operator of fractional differentiation corresponding to this process.
We also elucidate the general nature of stable symmetric random Levy processes.

References

S. Chandrasekhar, Stohastic problems in physics and astronomy, Rev. Modern Phys., 15 , № 1, 1 – 89 (1943), https://doi.org/10.1103/RevModPhys.15.1 DOI: https://doi.org/10.1103/RevModPhys.15.1

J. Holtsmark, Über die Verbreiterung von Spektrallinier, Ann. Phys., 58, 577 – 630 (1919), https://doi.org/10.1002/andp.19193630702 DOI: https://doi.org/10.1002/andp.19193630702

V. M. Zolotarev, Odnomerny`e ustojchivy`e raspredeleniya, Nauka, Moskva (1983).

P. Levy, Calcul des probabilities, Gauthier-Villars, Paris (1925).

B. Mandelbrot, The Pareto – Levy law and the distribution of income, Int. Econ. Rev., 1, 79 – 106 (1960). DOI: https://doi.org/10.2307/2525289

I. I. Sobel`man, Vvedenie v teoriyu atomny`kh spektrov, Fizmatgiz, Moskva (1963).

M. Kacz, Veroyatnost` i smezhny`e voprosy` v fizike, Mir, Moskva (1965).

V. Feller, Vvedenie v teoriyu veroyatnostej i ee prilozheniya: v 2 t. T. 2, Mir, Moskva (1984).

A. F. Nikiforov, V. G. Novikov, V. B. Uvarov, Kvantovo-statisticheskie modeli vy`sokotemperaturnoj

plazmy` i metody` rascheta rosselandovy`kh probegov i uravnenij sostoyaniya, Fizmatlit, Moskva (2000).

T. A. Agekyan, Teoriya veroyatnostej dlya astronomov i fizikov, Nauka, Moskva (1974).

M. Riesz, Potentiels de divers ordres et leurs fonctions de Green, C. R. Congr. Intern. Math. Oslo, 2, 62 – 63 (1936).

V. A. Litovchenko, Zadacha Koshi z operatorom Rissa drobovogo diferencziyuvannya>, Ukr. mat. zhurn., 57, № 12, 1653 – 1667 (2005).

V. A. Litovchenko, Zadacha Koshi dlya odnogo klassa parabolicheskikh psevdodifferenczial`ny`kh sistem s negladkimi simvolami, Sib. mat. zhurn., 49, № 2, 375 – 394 (2008).

S. G. Samko, A. A. Kilbas, O. I. Marichev, Integraly` i proizvodny`e drobnogo poryadka i nekotory`e ikh

prilozheniya, Nauka i tekhnika, Minsk (1987).

I. M. Gel`fand, G. E. Shilov, Prostranstva osnovny`kh i obobshhenny`kh funkczij, Fizmatgiz, Moskva (1958).

S. D. E`jdel`man, Ya. M. Drin`, Neobkhodimy`e i dostatochny`e usloviya stabilizaczii resheniya zadachi Koshi dlya parabolicheskikh psevdodifferenczial`ny`kh uravnenij, Priblizhenny`e metody` matematicheskogo analiza, 60 – 69 (1974).

Ya. M. Drin`, Vivchennya odnogo klasu parabolichnikh psevdodiferenczial`nikh operatoriv u prostorakh gel`derovikh funkczij, Dop. AN URSR, ser. А, № 1, 19 – 21 (1974).

M. V. Fedoryuk, Asimptotika funkczii Grina psevdodifferenczial`nogo parabolicheskogo uravneniya, Differencz. uravneniya, 14, № 7, 1296 – 1301 (1978).

W. R. Schneider, Stable distributions: Fox function representation and generalization, Lect. Notes Phys., 262, 497 – 511 (1986), https://doi.org/10.1007/3540171665_92 DOI: https://doi.org/10.1007/3540171665_92

R. M. Blumenthal, R. K. Getoor, Some theorems on stable processes, Trans. Amer. Math. Soc., 95, 263 – 273 (1960), https://doi.org/10.2307/1993291 DOI: https://doi.org/10.2307/1993291

A. N. Kochubej, Parabolicheskie psevdodifferenczial`ny`e uravneniya, gipersingulyarny`e integraly` i markovskie proczessy`, Izv. AN SSSR, ser. mat., 52, № 5, 909 – 934 (1988).

Holtsmark fluctuations of non-stationary gravitational fields

Abstract

References