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Introducing randomness into first-order and second-order deterministic differential equations. (English) Zbl 1209.60040
Summary: We incorporate randomness into deterministic theories and compare analytically and numerically some well-known stochastic theories: the Liouville process, the Ornstein-Uhlenbeck process, and a process that is Gaussian and exponentially time correlated (Ornstein-Uhlenbeck noise). Different methods of achieving the marginal densities for correlated and uncorrelated noise are discussed. Analytical results are presented for a deterministic linear friction force and a stochastic force that is uncorrelated or exponentially correlated.

MSC:
60H99 Stochastic analysis
34F05 Ordinary differential equations and systems with randomness
35R60 PDEs with randomness, stochastic partial differential equations
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[1] A. A. Markov, “Extension de la loi de grands nombres aux événements dependants les uns de autres,” Bulletin de La Société Physico, vol. 15, pp. 135-156, 1906.
[2] K. Ito, “On stochastic differential equations,” Memoirs of the American Mathematical Society, vol. 1951, no. 4, p. 51, 1951. · Zbl 0054.05803
[3] R. L. Stratonovich, “A new representation for stochastic integrals and equations,” SIAM Journal on Control and Optimization, vol. 4, pp. 362-371, 1966. · Zbl 0143.19002 · doi:10.1137/0304028
[4] P. Garbaczewski, “On the statistical origins of the de Broglie-Bohm quantum potential. Brownian motion in a field of force as Bernstein diffusion,” Physics Letters A, vol. 178, no. 1-2, pp. 7-10, 1993. · doi:10.1016/0375-9601(93)90718-F
[5] E. Nelson, “Derivation of the Schrödinger equation from Newtonian mechanics,” Physical Review, vol. 150, no. 4, pp. 1079-1085, 1966. · doi:10.1103/PhysRev.150.1079
[6] L. Arnold, Stochastische Differentialgleichungen. Theorie und Anwendung, R. Oldenbourg, Munich, Germany, 1973. · Zbl 0266.60039
[7] B. Øksendal, Stochastic Differential Equations. An Introduction with Applications, Universitext, Springer, Berlin, Germany, 1985. · Zbl 0567.60055
[8] L. Bachelier, “Théorie de la spéculation,” Annales Scientifiques de l’École Normale Supérieure. Troisième Série, vol. 17, pp. 21-86, 1900. · JFM 31.0241.02
[9] A. Einstein, “Über die von molekularkinetischen Theorie der Warme geforderte Bewegungen von in ruhenden Flüssigkeiten suspendierten Teilchen,” Annalen der Physik, vol. 17, pp. 549-560, 1905. · JFM 36.0975.01
[10] A. Einstein, Investigation on the Theory of Brownian Movement, Methuen, London, UK, 1926, Translated by A. D. Cowper. · JFM 53.0876.09
[11] M. Smoluchowski, “Zur kinetischen theorie der Brownschen molekularbewegung und der suspensionen,” Annalen der Physik, vol. 21, pp. 756-780, 1906. · JFM 37.0814.03
[12] M. Smoluchowski, Abhandlungen uber die Brownsche bewegung und verwandte erscheinungen, Akademische Verlags Gesellschaft, Leipzig, Germany, 1923.
[13] E. Wigner, “Differential space,” Journal of Mathematical Physics, vol. 2, pp. 131-174, 1923.
[14] A. W. C. Lau and T. C. Lubensky, “State-dependent diffusion: thermodynamic consistency and its path integral formulation,” Physical Review E, vol. 76, no. 1, Article ID 011123, 17 pages, 2007. · doi:10.1103/PhysRevE.76.011123
[15] J. Masoliver, “Second-order processes driven by dichotomous noise,” Physical Review A, vol. 45, no. 2, pp. 706-713, 1992. · doi:10.1103/PhysRevA.45.706
[16] J. Heinrichs, “Probability distributions for second-order processes driven by Gaussian noise,” Physical Review E, vol. 47, no. 5, pp. 3007-3012, 1993. · doi:10.1103/PhysRevE.47.3007
[17] B. C. Bag, “Upper bound for the time derivative of entropy for nonequilibrium stochastic processes,” Physical Review E, vol. 65, no. 4, Article ID 046118, pp. 1-6, 2002. · doi:10.1103/PhysRevE.65.046118
[18] B. C. Bag, S. K. Banik, and D. S. Ray, “Noise properties of stochastic processes and entropy production,” Physical Review E, vol. 64, no. 2, Article ID 026110, pp. 1-7, 2001.
[19] C. W. Gardiner and P. Zoller, Quantum Noise. A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics, Springer Series in Synergetics, Springer, Berlin, Germany, 3rd edition, 2004. · Zbl 1072.81002
[20] K. Huang, Statistical Mechanics, John Wiley & Sons, New York, NY, USA, 1963.
[21] R. L. Liboff, Kinetic Theory, Prentice-Hall, Englewood Cliffs, NJ, USA, 1990.
[22] S. Chandresekhar, “Stochastic problems in physics and astronomy,” Reviews of Modern Physics, vol. 15, pp. 1-89, 1943. · Zbl 0061.46403 · doi:10.1103/RevModPhys.15.1
[23] R. Dorfman, An Introduction to Chaos in no Equilibrium Statistical Mechanics, Cambridge University Press, Cambridge, UK, 1999. · Zbl 0973.82001
[24] E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Physical Review, vol. 40, no. 5, pp. 749-759, 1932. · Zbl 0004.38201 · doi:10.1103/PhysRev.40.749
[25] M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Physics Reports A, vol. 106, no. 3, pp. 121-167, 1984. · doi:10.1016/0370-1573(84)90160-1
[26] H. W. Lee, “Theory and application of the quantum phase-space distribution functions,” Physics Reports A, vol. 259, no. 3, pp. 147-211, 1995. · doi:10.1016/0370-1573(95)00007-4
[27] L. Glass and M. C. Mackey, From Clocks to Chaos: The Rhythms of Life, Princeton University Press, Princeton, NJ, USA, 1988. · Zbl 0705.92004
[28] H. E. Stanley and P. Meakin, “Multifractal phenomena in physics and chemistry,” Nature, vol. 335, no. 6189, pp. 405-409, 1988. · doi:10.1038/335405a0
[29] H. Risken, The Fokker-Planck Equation. Methods of Solution and Applications, vol. 18 of Springer Series in Synergetics, Springer, Berlin, Germany, 2nd edition, 1989. · Zbl 0665.60084
[30] H.-O. Peitgen, H. Jürgens, and D. Saupe, Chaos and Fractals. New Frontiers of Science, Springer, New York, NY, USA, 1992. · Zbl 0779.58004
[31] M. Rypdal and K. Rypdal, “Modeling temporal fluctuations in avalanching systems,” Physical Review E, vol. 78, no. 5, Article ID 051127, 11 pages, 2008. · doi:10.1103/PhysRevE.78.051127
[32] M. Rypdal and K. Rypdal, “A stochastic theory for temporal fluctuations in self-organized critical systems,” New Journal of Physics, vol. 10, Article ID 123010, 2008. · Zbl 1220.82063 · doi:10.1088/1367-2630/10/12/123010
[33] P. Garbaczewski and J. R. Klauder, “Schrödinger problem, Lévy processes, and noise in relativistic quantum mechanics,” Physical Review E, vol. 51, no. 5, part A, pp. 4114-4131, 1995. · doi:10.1103/PhysRevE.51.4114
[34] R. Czopnik and P. Garbaczewski, “Frictionless random dynamics: hydrodynamical formalism,” Physica A, vol. 317, no. 3-4, pp. 449-471, 2003. · Zbl 1005.70014 · doi:10.1016/S0378-4371(02)01343-2
[35] P. Garbaczewski, “Differential entropy and dynamics of uncertainty,” Journal of Statistical Physics, vol. 123, no. 2, pp. 315-355, 2006. · Zbl 1124.82014 · doi:10.1007/s10955-006-9058-2
[36] S. A. Hojman, “The construction of a Poisson structure out of a symmetry and a conservation law of a dynamical system,” Journal of Physics A, vol. 29, no. 3, pp. 667-674, 1996. · Zbl 0924.58022 · doi:10.1088/0305-4470/29/3/017
[37] A. Gomberoff and S. A. Hojman, “Non-standard construction of Hamiltonian structures,” Journal of Physics A, vol. 30, no. 14, pp. 5077-5084, 1997. · Zbl 0939.70020 · doi:10.1088/0305-4470/30/14/018
[38] D. T. Gillespie, “Why quantum mechanics cannot be formulated as a Markov process,” Physical Review A, vol. 49, no. 3, pp. 1607-1612, 1994. · doi:10.1103/PhysRevA.49.1607
[39] R. H. Koch, D. J. Van Harlingen, and J. Clarke, “Quantum-noise theory for the resistively shunted josephson junction,” Physical Review Letters, vol. 45, no. 26, pp. 2132-2135, 1980. · doi:10.1103/PhysRevLett.45.2132
[40] R. H. Koch, D. J. Van Harlingen, and J. Clarke, “Measurements of quantum noise in resistively shunted Josephson junctions,” Physical Review B, vol. 26, no. 1, pp. 74-87, 1982. · doi:10.1103/PhysRevB.26.74
[41] E. Nelson, Dynamical Theories of Brownian Motion, Princeton University Press, Princeton, NJ, USA, 1967. · Zbl 0165.58502
[42] E. Nelson, Quantum Fluctuations, Princeton Series in Physics, Princeton University Press, Princeton, NJ, USA, 1985. · Zbl 0563.60001
[43] P. Blanchard, M. Cini, and M. Serva, “The measurement problem in the stochastic formulation of quantum mechanics,” in Ideas and Methods in Quantum and Statistical Physics (Oslo, 1988), pp. 149-172, Cambridge University Press, Cambridge, UK, 1992. · Zbl 0784.60103
[44] S. Albeverio and R. Høegh-Krohn, “A remark on the connection between stochastic mechanics and the heat equation,” Journal of Mathematical Physics, vol. 15, pp. 1745-1747, 1974. · Zbl 0291.60028 · doi:10.1063/1.1666536
[45] H. Ezawa, J. R. Klauder, and L. A. Shepp, “A path space picture for Feynman-Kac averages,” Annals of Physics, vol. 88, pp. 588-620, 1974. · Zbl 0296.60074 · doi:10.1016/0003-4916(74)90182-1
[46] S. Albeverio, R. Høegh-Krohn, and L. Streit, “Energy forms, Hamiltonians, and distorted Brownian paths,” Journal of Mathematical Physics, vol. 18, no. 5, pp. 907-917, 1977. · Zbl 0368.60091 · doi:10.1063/1.523359
[47] F. Guerra, “Structural aspects of stochastic mechanics and stochastic field theory,” Physics Reports A, vol. 77, no. 3, pp. 263-312, 1981. · doi:10.1016/0370-1573(81)90078-8
[48] E. A. Carlen, “Conservative diffusions,” Communications in Mathematical Physics, vol. 94, no. 3, pp. 293-315, 1984. · Zbl 0558.60059 · doi:10.1007/BF01224827
[49] W. A. Zheng, “Tightness results for laws of diffusion processes application to stochastic mechanics,” Annales de l’Institut Henri Poincaré. Probabilités et Statistique, vol. 21, no. 2, pp. 103-124, 1985. · Zbl 0579.60050 · numdam:AIHPB_1985__21_2_103_0 · eudml:77251
[50] J.-C. Zambrini, “Variational processes and stochastic versions of mechanics,” Journal of Mathematical Physics, vol. 27, no. 9, pp. 2307-2330, 1986. · Zbl 0623.60102 · doi:10.1063/1.527002
[51] A. Blaquière, “Girsanov transformation and two stochastic optimal control problems. The Schrödinger system and related controllability results,” in Modeling and Control of Systems in Engineering, Quantum Mechanics, Economics and Biosciences (Sophia-Antipolis, 1988), A. Blaquière, Ed., vol. 121 of Lecture Notes in Control and Information Sciences, pp. 217-243, Springer, Berlin, Germany, 1989. · Zbl 0719.93089 · doi:10.1007/BFb0041196
[52] P. Blanchard and P. Garbaczewski, “Natural boundaries for the Smoluchowski equation and affiliated diffusion processes,” Physical Review E, vol. 49, no. 5, part A, pp. 3815-3824, 1994. · doi:10.1103/PhysRevE.49.3815
[53] P. Garbaczewski and R. Olkiewicz, “Why quantum dynamics can be formulated as a Markov process,” Physical Review A, vol. 51, no. 5, pp. 3445-3453, 1995. · doi:10.1103/PhysRevA.51.3445
[54] O. Bratteli and D. W. Robinson, Operator Algebras and Quantum-Statistical Mechanics. II, Springer, New York, NY, USA, 1981. · Zbl 0463.46052
[55] L. E. Ballentine, “The statistical interpretation of quantum mechanics,” Reviews of Modern Physics, vol. 42, no. 4, pp. 358-381, 1970. · Zbl 0203.27801 · doi:10.1103/RevModPhys.42.358
[56] H. M. Taylor and S. Karlin, An Introduction to Stochastic Modeling, Academic Press, San Diego, Calif, USA, 3rd edition, 1998. · Zbl 0946.60002
[57] W. Feller, Markov Processes and Semi-Groups. Vol II: In Introduction to Probability Theory and Its Applications, chapter 10, John Wiley & Sons, New York, NY, USA, 2nd edition, 1971. · Zbl 0219.60003
[58] E. M. Lifshitz and L. P. Pitaevskiĭ, Statistical Physics by Landau L.D., Lifshitz E.M., Part 1, vol. 5, Pergamon Press, Oxford, UK, 3rd edition, 1980.
[59] P. S. Marquis de Laplace, A Philosophical Essay on Probabilities, John Wiley & Sons, New York, NY, USA, 1917. · Zbl 0047.37209
[60] H. Poincaré, Science and Method, Dover, New York, NY, USA, 1952. · Zbl 0049.29107
[61] A. Kolmogoroff, “Zufällige Bewegungen (zur Theorie der Brownschen Bewegung),” Annals of Mathematics, vol. 35, no. 1, pp. 116-117, 1934. · Zbl 0008.39906 · doi:10.2307/1968123
[62] A. Kolmogoroff, “Anfangsgrunde der Theorie der Markoffschen ketten mit uendlich vielen moglischen Zustanden,” Matematicheskii Sbornik, vol. 1, pp. 607-610, 1936. · Zbl 0015.21903
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