Schrödinger’s interpolating dynamics and Burgers’ flows. (English) Zbl 0939.35201

Garbaczewski, Piotr; Kondrat, Grzegorz; Olkiewicz, Robert

Chaos Solitons Fractals 9, No. 1-2, 29-41 (1998).

Summary: We discuss a connection (and a proper place in this framework) of the unforced and deterministically forced Burgers equation for local velocity fields of certain flows, with probabilistic solutions of the so-called Schrödinger interpolation problem. The latter allows us to reconstruct the microscopic dynamics of the system from the available probability density data, or the input-output statistics in the phenomenological situations. An issue of deducing the most likely dynamics (and matter transport) scenario from the given initial and terminal probability density data, appropriate e.g., for studying chaos in terms of density, is here exemplified in conjunction with Born’s statistical interpretation postulate in quantum theory, that yields stochastic processes which are compatible with the Schrödinger picture of free quantum evolution.

MSC:

35R60	PDEs with randomness, stochastic partial differential equations
35Q53	KdV equations (Korteweg-de Vries equations)
37L55	Infinite-dimensional random dynamical systems; stochastic equations
81Q50	Quantum chaos

Keywords:

forced Burgers equation; probabilistic solutions; Schrödinger interpolation problem; probability density data; chaos

References:

[1]	Kac, M.; Logan, J., ()
[2]	Mackey, M.C.; Glass, L., From clocks to chaos: rhythms of life, (1988), Princeton University Press Princeton · Zbl 0705.92004
[3]	Lasota, A.; Mackey, M.C., Chaos, fractals, and noise, (1994), Springer-Verlag Berlin · Zbl 0784.58005
[4]	Beck, C., (), 3, LNP
[5]	Mikami, T., Commun. math. phys., 135, 19, (1990)
[6]	Jamison, B.; Wahrsch, Z., Verw. geb., 30, 65, (1974)
[7]	Zambrini, J.C., J. math. phys., 27, 3207, (1986)
[8]	Garbaczewski, P.; Klauder, J.R.; Olkiewicz, R., Phys. rev. E, 51, 4114, (1995)
[9]	Schrödinger, E., Ann. inst. Henri poincare, 2, 269, (1932)
[10]	Garbaczewski, P., Acta phys. polon. B, 27, 617, (1996)
[11]	Blanchard, Ph.; Garbaczewski, P., Phys. rev. E, 49, 3815, (1994)
[12]	Garbaczewski, P.; Olkiewicz, R., J. math. phys., 37, 732, (1996)
[13]	Burgers, J.M., The nonlinear diffusion equation, (1974), Reidel Dordrecht · Zbl 0302.60048
[14]	Hopf, E., Commun. pure appl. math., 3, 201, (1950)
[15]	Garbaczewski, P.; Kondrat, G., Phys. rev. lett., 77, 2608, (1996)
[16]	Shandarin, S.F.; Zeldovich, B.Z., Rev. mod. phys., 61, 185, (1989)
[17]	Albeverio, S.; Molchanov, A.A.; Surgailis, D., Prob. theory relat. fields, 100, 457, (1994)
[18]	McKean, H.P., (), 177
[19]	Calderoni, P.; Pulvirenti, M., Ann. inst. Henri Poincaré, 39, 85, (1983)
[20]	Osada, H.; Kotani, S., J. math. soc. jpn., 37, 275, (1985)
[21]	Krzyzański, M.; Szybiak, A., Lincei-rend. sc. fis. mat. e nat., 28, 26, (1959)
[22]	Friedman, A., Partial differential equations of parabolic type, (1964), Prentice-Hall Englewood Cliffs, NJ · Zbl 0144.34903
[23]	Horsthemke, W.; Lefever, R., Noise-induced transitions, (1984), Springer-Verlag Berlin · Zbl 0529.60085
[24]	Haussmann, U.G.; Pardoux, E., Ann. prob., 14, 1188, (1986)
[25]	Föllmer, H., (), 119, LNP
[26]	Hasegawa, H., Progr. theor. phys., 55, 90, (1976)
[27]	Nelson, E., Quantum fluctuations, (1985), Princeton University Press Princeton, NJ · Zbl 0563.60001
[28]	Nelson, E., Dynamical theories of the Brownian motion, (1967), Princeton University Press Princeton, NJ · Zbl 0165.58502
[29]	Garbaczewski, P.; Vigier, J.P., Phys. rev. A, 46, 4634, (1992)
[30]	Garbaczewski, P.; Olkiewicz, R., Phys. rev. A, 52, 3445, (1995)
[31]	Fleming, W.H.; Soner, H.M., Controlled Markov processes and viscosity solutions, (1993), Springer-Verlag Berlin · Zbl 0773.60070