Method for Describing Non-Markovian Processes Defined by a System of Linear Integral Equations

We suggest a method for determining characteristic functions of a non-markovian stochastic process when a system of linear integral equations describes it. We show that in this case it is possible to find the solution to this problem using a previously developed method for determining characteristic functions of a process described by a single linear integral equation. We employed the method we developed to describe Brownian motion in equilibrium and non-equilibrium media. We computed spectral density of impulse fluctuations for a Brownian particle in a non-equilibrium medium and determined that in the low-frequency region it is represented by flicker noise. We show that the spectral density of impulse fluctuations for a Brownian particle in a non-equilibrium medium is a linear function of entropy generation

References

[1] Crispin W.G. Handbook of stochastic methods for physics, chemistry, and the natural sciences. Springer-Verlag, 1985. 442 p.

[2] Brauers Y.Y.Kh. Langevin equation of a fluid particle in wall-induced turbulence. Theoretical and Mathematical Physics, 2010, vol. 163, iss. 2, pp. 677–695. DOI:10.1007/s11232-010-0050-2

[3] Marchesoni F., Taloni A. Subdiffusion and long-time anticorrelations in a stochastic single file. Physical Review Letters, 2006, vol. 97, iss. 10, pp. 106101-1--106101-4. DOI: 10.1103/PhysRevLett.97.106101

[4] Pugachev V.S., Sinitsyn I.N. Stokhasticheskie differentsial'nye sistemy [Stochastic differential systems]. Moscow, Nauka Publ., 1990. 632 p.

[5] Bunkin N.F., Morozov A.N. Stokhasticheskie sistemy v fizike i tekhnike [Stochastic systems in physics and technique]. Moscow, Bauman MSTU Publ., 2011. 366 p.

[6] Morozov A.N. Use of the theory of non-Markovian processes in the description of Brownian motion. Journal of Experimental and Theoretical Physics, 1996, vol. 82, no. 4, pp. 703–708.

[7] Morozov A.N., Skripkin A.V. Application of integral transforms to a description of the Brownian motion by a non-Markovian random process. Russian Physics Journal, 2009, vol. 52, no. 2, pp. 184–195. DOI: 10.1007/s11182-009-9217-4

[8] Morozov A.N. Method of describing non-Markovian processes defined by linear integral transformation. Vestn. Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Estestv. Nauki [Herald of the Bauman Moscow State Tech. Univ., Nat. Sci.], 2004, no. 3, pp. 47–56 (in Russ.).

[9] Horsthemke W., Lefever R. Noise-induced transitions. Springer, 1984. 322 р.

[10] Prudnikov A.P., Brychkov Yu.A., Marichev O.I. Integraly i ryady. T. 1. Elementarnye funktsii [Integrals and raws. Vol. 1. Elementary functions]. Moscow, Nauka Publ., 2003. 632 p.

[11] Morozov A.N. Neobratimye protsessy i brounovskoe dvizhenie [Irreversible processes and Brownian motion]. Moscow, Bauman MSTU Publ., 1997. 332 p.

[12] Jahnke E., Emde F., Lösch F. Tafeln höherer funktionen. Stuttgart, Teubner Verlagsgesellschaft. Preis, 1960. 318 p.

[13] Bochkov G.N., Kuzovlev Yu.E. New aspects in 1/f noise studies. Sov. Phys. Usp., 1983, vol. 26, pp. 829–844. DOI: 10.1070/PU1983v026n09ABEH004497

[14] Kuzovlev Yu.E. Why nature needs 1/f noise. Phys. Usp., 2015, vol. 58, no. 7, pp. 719–729. DOI: 10.3367/UFNe.0185.201507d.0773