Theory of stochastic systems with singular multiplicative noise

Noisy, interacting, stochastic systems are analyzed for the case in which their noise intensity varies with the hydrodynamic mode amplitude x according to the power law x^2a, x \in [0, 1]. It is shown that the phase space domain of definition of the stochastic variable x forms a self-affine set of fractal dimensionality D = 2(1-a). Using the gauge procedure, a system of calculus is chosen which is not reducible either to the Ito case or the Stratonovich case. By generalizing the microscopic picture of phase transitions it is demonstrated that the system may reduce its symmetry (for 1 < D \leqslant 2) or lose ergodicity (for 0 < D \leqslant 1). Over the entire interval D \in [0, 2], a noise-induced transition is shown to be possible.

PACS: 05.40.+j, 05.70.Fh, 64.60.−i, 82.20.Fd (all)
DOI: 10.1070/PU1998v041n03ABEH000377
URL: https://ufn.ru/en/articles/1998/3/c/
Web of Science

000073306100003
Citation: Olemskoi A I "Theory of stochastic systems with singular multiplicative noise" Phys. Usp. 41 269–301 (1998)

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Оригинал: Олемской А И «Теория стохастических систем с сингулярным мультипликативным шумом» УФН 168 287–321 (1998); DOI: 10.3367/UFNr.0168.199803c.0287

References (60) Cited by (43) Similar articles (20) ↓

A.I. Olemskoi, I.V. Koplyk “The theory of spatiotemporal evolution of nonequilibrium thermodynamic systems” Phys. Usp. 38 1061–1097 (1995)
A.I. Olemskoi, A.Ya. Flat “Application of fractals in condensed-matter physics” Phys. Usp. 36 (12) 1087–1128 (1993)
Sh.M. Kogan “Low-frequency current noise with a 1/f spectrum in solids” Sov. Phys. Usp. 28 170–195 (1985)
Yu.L. Klimontovich “Nonlinear Brownian motion” Phys. Usp. 37 737–766 (1994)
A.V. Khomenko, I.A. Lyashenko “Statistical theory of the boundary friction of atomically flat solid surfaces in the presence of a lubricant layer” Phys. Usp. 55 1008–1034 (2012)
L.I. Klushin, A.M. Skvortsov, A.A. Gorbunov “An exactly solvable model for first- and second-order transitions” Phys. Usp. 41 639–649 (1998)
L.E. Gendenshtein, I.V. Krive “Supersymmetry in quantum mechanics” Sov. Phys. Usp. 28 645–666 (1985)
A.M. Perelomov “Solutions of the instanton type in chiral models” Sov. Phys. Usp. 24 645–661 (1981)
A.I. Olemskoi, I.A. Sklyar “Evolution of the defect structure of a solid during plastic deformation” Sov. Phys. Usp. 35 (6) 455–480 (1992)
V.S. Anishchenko, A.B. Neiman et al “Stochastic resonance: noise-enhanced order” Phys. Usp. 42 7–36 (1999)
G.R. Ivanitskii, A.B. Medvinskii et al “From Maxwell’s demon to the self-organization of mass transfer processes in living systems” Phys. Usp. 41 1115–1126 (1998)
B.M. Smirnov “Scaling method in atomic and molecular physics” Phys. Usp. 44 1229–1253 (2001)
V.S. Dotsenko “Critical phenomena and quenched disorder” Phys. Usp. 38 457–496 (1995)
V.I. Klyatskin, V.I. Tatarskii “Diffusive random process approximation in certain nonstationary statistical problems of physics” Sov. Phys. Usp. 16 494–511 (1974)
S.M. Stishov “Quantum phase transitions” Phys. Usp. 47 789–795 (2004)
A.I. Olemskoi “Self-consistent theory of localization within site and wave approaches” Phys. Usp. 39 651–668 (1996)
A.I. Olemskoi “Supersymmetric field theory of a nonequilibrium stochastic system as applied to disordered heteropolymers” Phys. Usp. 44 479–513 (2001)
I.S. Lyubutin, A.G. Gavriliuk “Research on phase transformations in 3d-metal oxides at high and ultrahigh pressure: state of the art” Phys. Usp. 52 989–1017 (2009)
V.P. Skripov, A.V. Skripov “Spinodal decomposition (phase transitions via unstable states)” Sov. Phys. Usp. 22 389–410 (1979)
D.A. Kirzhnits, Yu.E. Lozovik, G.V. Shpatakovskaya “Statistical model of matter” Sov. Phys. Usp. 18 649–672 (1975)

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