Discontinuity, Nonlinearity, and Complexity
Clustering of a Positive Random Field –What is This？
Discontinuity, Nonlinearity, and Complexity 4(3) (2016) 225242  DOI:10.5890/DNC.2016.09.002
V.I. Klyatskin
A. M. Obukhov Institute of Atmospheric Physics RAS, Moscow, Pyzhevsky per. 3, 119017, Russia
Download Full Text PDF
Abstract
It is shown that, in parametrically excited stochastic dynamic systems described by partial differential equations, spatial structures (clusters) can appear with probability one, i.e., almost in every system realization, due to rare events happened with probability approaching to zero. The problems of such type arise in hydrodynamics, magnetohydrodynamics, physics of plasma, astrophysics, and radiophysics.
Acknowledgments
This work was supported by the RSF 142700134.
References

[1]  Klyatskin, V.I. (2013), Clustering of a random positive field as a law of Nature, Theoretical and Mathematical Physics, 176(3), 12521266. 

[2]  Nicolis, G. and Prigogin, I. (1989), Exploring Complexity, an Introduction, FreemanW.H. and Company, New York. 

[3]  Klyatskin, V.I. (1969), On Statistical theory of twodimensional turbulence, Journal of Applied Mathematics and Mechanics, 33(5), 864866. 

[4]  Klyatskin, V.I. (1995), Equilibrium states for quasigeostrophic flows with random topography, Izvestiya, Atmospheric and Oceanic Physics, 31(6), 717722. 

[5]  Klyatskin, V.I. and Gurarie, D. (1996), Random topography in geophysical models, in: Stochastic Models in Geosystems, eds. Molchanov, S.A. and Woyczynski, W.A. IMA Volumes in Math. and its Appl. 85, 149170. N.Y. Springer Verlag. 

[6]  Klyatskin, V.I. and Gurarie D. (1996), Equilibrium states for quasigeostrophic flows with random topography, Physica D, 98, 466480. 

[7]  Klyatskin, V.I. (2011), Lectures on Dynamics of Stochastic Systems, Elsevier, Boston, MA. 

[8]  Klyatskin, V.I. (2014), Stochastic Equations: Theory and Applications in Acoustics, Hydrodynamics, Magnetohydrodynamics, and Radiophysics, Volume 1 (Basic Concepts, Exact Results, and Asymptotic Approximations), Springer: Complexity, Springer Berlin. 

[9]  Klyatskin, V.I. (2014), Stochastic Equations: Theory and Applications in Acoustics, Hydrodynamics, Magnetohydrodynamics, and Radiophysics, Volume 2 (Coherent Phenomena in Stochastic Dynamic Systems), Springer: Complexity, Springer Berlin. 

[10]  Klyatskin, V.I. (2015), Equilibrium distributions for hydrodynamic flows, Discontinuity, Nonlinearity, and Complexity, 4(3), 243255. 

[11]  Klyatskin, V.I. (2012), Spatial structures can form in stochastic dynamic systems due to nearzeroprobability events: (comment on ＊21st century: what is life from the perspective of physics?), PhysicsUspekhi, 55(11), 11521154. 

[12]  Klyatskin, V.I. (2013), On the criterion of stochastic structure formation in random media, Proceedings of the 4th International Interdisciplinary Chaos Symposium, 6973, SpringerVerlag, Berlin. 

[13]  Klyatskin V.I. (2013), On the Statistical Theory of Spatial Structure Formation in Random Media, Russian Journal of Mathematical Physics, 20(3), 295314. 

[14]  Kharif C., Pelinovskyy E. and Slyunaen A. (2009), Rogue Waves in the Ocean, Springer, Berlin. 

[15]  Klyatskin V.I. (2014), Anomalous waves as an object of statistical topography. Problem statement, Theoretical and Mathematical Physics, 180(1), 850861. 

[16]  Ivanittskii, G.R. (2010), 21st century: what is life from the perspective of physics, PhysicsUspekhi, 53(4), 327356. 