Analytical
non-perturbative study of the three-dimensional
nonlinear stochastic partial differential equation with additive thermal noise, analogous to that proposed by V. N. Nikolaevskii [1]-[5] to describe longitudinal seismic waves, is presented. The equation has a
threshold of short-wave instability and symmetry, providing long wave dynamics.
New mechanism of quantum chaos generating in nonlinear dynamical systems with infinite number of degrees of freedom is
proposed. The hypothesis is said, that physical turbulence could be identified
with quantum chaos of considered type.
It is shown that the additive thermal noise destabilizes dramatically the ground
state of the Nikolaevskii system thus causing it to make a direct transition from
a spatially uniform to a turbulent state.
References
[1]
Nikolaevskii, V.N. (1989) Recent Advances in Engineering Science. In: Kohand, S.L. and Speciale, C.G., Eds., Lecture Notes in Engineering, No. 39, Springer-Verlag, Berlin, 210.
[2]
Tribelsky, M.I. and Tsuboi, K. (1996) Newscenario to Transition to Slow Turbulence. Physical Review Letters, 76, 1631. http://dx.doi.org/10.1103/PhysRevLett.76.1631
[3]
Tribel’skii, M.I. (1997) Short-Wavelength Instability and Transition to Chaos in Distributed Systems with Additional Symmetry. Physics-Uspekhi, 40, 159-180.
[4]
Toral, R., Xiong, G.X., Gunton, J.D. and Xi, H.W. (2003) Wavelet Description of the Nikolaevskii Model. Journal of Physics A: Mathematical and General, 36, 1323. http://dx.doi.org/10.1088/0305-4470/36/5/310
[5]
Xi, H.W., Toral, R., Gunton, D. and Tribelsky, M.I. (2003) Extensive Chaos in the Nikolaevskii Model. Physical Review E, 61, R17. http://dx.doi.org/10.1103/PhysRevE.62.R17
[6]
Tanaka, D. (2007) Amplitude Equations of Nikolaevskii Turbulence. RIMS Kokyuroku Bessatsu, B3, 121-129.
[7]
Fujisaka, H. (2003) Amplitude Equation of Higher-Dimensional Nikolaevskii Turbulence. Progress of Theoretical Physics, 109, 911-918. http://dx.doi.org/10.1143/PTP.109.911
[8]
Tanaka, D. (2005) Bifurcation Scenario to Nikolaevskii Turbulence in Small Systems. Journal of the Physical Society of Japan, 74, 2223-2225.
[9]
Anishchenko, V.S., Vadivasova, T.E., Okrokvertskhov, G.A. and Strelkova, G.I. (2005) Statistical Properties of Dynamical Chaos. Physics-Uspekhi, 48, 151. http://dx.doi.org/10.1070/PU2005v048n02ABEH002070
[10]
Tsinober, A. (2014) The Essence of Turbulence as a Physical Phenomenon. 6, 169 p. http://www.springer.com/gp/book/9789400771796
[11]
Ivancevic, V.G. (2007) High-Dimensional Chaotic and Attractor Systems: A Comprehensive Introduction. XV, 697 p. http://www.springer.com/gp/book/9781402054556
[12]
Herbst, B.M. and Ablowitz, M.J. (1989) Numerically Induced Chaos in the Nonlinear Schrodinger Equation. Physical Review Letters, 62, 2065. http://dx.doi.org/10.1103/PhysRevLett.62.2065
[13]
Ablowitz, M.J. and Herbst, B.M. (1990) On Homoclinic Structure and Numerically Induced Chaos for the Nonlinear Schrodinger Equation. SIAM Journal on Applied Mathematics, 50, 339-351. http://dx.doi.org/10.1137/0150021
[14]
Li, Y. and Wiggins, S. (1997) Homoclinic Orbits and Chaos in Discretized Perturbed NLS Systems: Part II. Symbolic Dynamics. Journal of Nonlinear Science, 7, 315-370. http://dx.doi.org/10.1007/BF02678141
[15]
Blank, M.L. (1997) Discreteness and Continuity in Problems of Chaotic Dynamics. Translations of Mathematical Monographs. 161 p. http://www.ams.org/bookstore-getitem/item=MMONO-161
[16]
Fadnavis, S. (1998) Some Numerical Experiments on Round-Off Error Growth in Finite Precision Numerical Computation. http://arxiv.org/abs/physics/9807003v1
Gold, B. and Rader, C.M. (1966) Effects of Quantization Noise in Digital Filters. Proceedings of Joint Computer Conference, 26-28 April 1966, 1-8. http://users.ece.utexas.edu/~adnan/comm/sqnr-early-paper-66.pdf
[19]
Bennett, W.R. (1948) Spectra of Quantized Signals. Bell System Technical Journal, 27, 446-472. http://dx.doi.org/10.1002/j.1538-7305.1948.tb01340.x
[20]
Vladimirov, I.G. and Diamond, P. (2002) A Uniform White-Noise Model for Fixed-Point Roundoff Errors in Digital Systems. Automation and Remote Control, 63, 753-765. http://dx.doi.org/10.1023/A:1015493820232
[21]
Moller, M., Lange, W., Mitschke, F., Abraham, N.B. and Hubner, U. (1989) Errors from Digitizing and Noise in Estimating Attractor Dimensions. Physics Letters A, 138, 176-182.
[22]
Widrow, B. and Kollar, I. (2008) Quantization Noise: Round off Error in Digital Computation, Signal Processing, Control, and Communications. Cambridge University Press, Cambridge.
[23]
Foukzon, J. (2005) Advanced Numerical-Analytical Methods for Path Integral Calculation and Its Application to Some Famous Problems of 3-D Turbulence Theory. New Scenario for Transition to Slow Turbulence. Preliminary Report. Meeting: 1011, Lincoln, Nebraska, AMS CP1, Session for Contributed Papers. http://www.ams.org/meetings/sectional/1011-76-5.pdf
[24]
Foukzon, J. (2004) New Scenario for Transition to Slow Turbulence. Turbulence Like Quantum Chaos in Three Dimensional Model of Euclidian Quantum Field Theory. Preliminary Report. Meeting: 1000, Albuquerque, New Mexico, SS 9A, Special Session on Mathematical Methods in Turbulence. http://www.ams.org/meetings/sectional/1000-76-7.pdf
[25]
Foukzon, J. (2008) New Scenario for Transition to Slow Turbulence. Turbulence like Quantum Chaos in Three Dimensional Model of Euclidian Quantum Field Theory. http://arxiv.org/abs/0802.3493
[26]
Foukzon, J. (2014) Large Deviations Principles of Non-Freidlin-Wentzell Type. Communications in Applied Sciences, 2, 230-363.
[27]
Foukzon, J. (2014) Large Deviations Principles of Non-Freidlin-Wentzell Type. 227 p. http://arxiv.org/abs/0803.2072
[28]
Colombeau, J. (1985) Elementary Introduction to New Generalized Functions Math. Studies 113, North Holland.
[29]
Catuogno, P. and Olivera, C. (2013) Strong Solution of the Stochastic Burgers Equation. http://arxiv.org/abs/1211.6622v2
[30]
Oberguggenberger, M. and Russo, F. (1998) Nonlinear SPDEs: Colombeau Solutions and Pathwise Limits. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.40.8866
[31]
Walsh, J.B. Finite Element Methods for Parabolic Stochastic PDE’s. Potential Analysis, 23, 1-43. http://link.springer.com/article/10.1007/s11118-004-2950-y
[32]
Suli, E. (2000) Lecture Notes on Finite Element Methods for Partial Differential Equations. University of Oxford, Oxford. http://people.maths.ox.ac.uk/suli/fem.pdf
[33]
Knabner, P. and Angermann, L. (2003) Numerical Methods for Elliptic and Parabolic Partial Differential Equations. http://link.springer.com/book/10.1007/b97419
[34]
Dijkgraaf, R., Orlando, D. and Reffert, S. (2010) Relating Field Theories via Stochastic Quantization. Nuclear Physics B, 824, 365-386. http://dx.doi.org/10.1016/j.nuclphysb.2009.07.018
[35]
Masujima, M. (2009) Path Integral Quantization and Stochastic Quantization. 2nd Edition. http://www.gettextbooks.com/isbn/9783540665427
