Fluctuations of the Brownian particle velocity influenced by a random Poisson process

Brownian motion of a particle influenced by a Poisson process is described. The stationary solution for the characteristic velocity fluctuation function of a Brownian particle is found when influence on it is described by the Poisson process with normal jump distribution. The first four moments of the Brownian particles velocity distribution function as well as kurtosis and Kullback’s measure for the distribution have been calculated. The results obtained have been used to find the characteristic function of voltage fluctuations in an electrolysis cell. Kullback’s measure for voltage fluctuations in an electrolysis cell is inversely related to the Poisson process intensity and the number of ions in a small volume of electrolyte. The equation for characteristic function of velocity fluctuation distribution of a Brownian particle taking into account fluctuations of the viscous friction coefficient is solved.

References

[1] Morozov A.N. Description of diffusion and Brownian motion as accidental Poisson processes. Vestn. Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Estestv. Nauki [Herald of the Bauman Moscow State Tech. Univ., Nat. Sci.], 1999, no. 2, pp. 85-90 (in Russ.).

[2] Bunkin N.F., Morozov A.N. Stokhasticheskie sistemy v fizike i tekhnike [Stochastic Systems in Physics and Technology]. Moscow, MGTU im. N.E. Baumana Publ., 2011. 366 p.

[3] Kalinkin A.V. Stationary Distribution for a Stochastic System of Complexes Interacting Particles. Vestn. Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Estestv. Nauki [Herald of the Bauman Moscow State Tech. Univ., Nat. Sci.], 2014, no. 4, pp. 3-17 (in Russ.).

[4] Markov Yu.G., Sinitsyn I.N. One- and Multidimensional Distributions of Fluctuations in the Earth’s Irregular Rotation. Dokl. Akad. Nauk [Proc. Russ. Acad. Sci.], 2009, vol. 428, no. 5, pp. 616-619 (in Russ.).

[5] Pugachev V.S., Sinitsyn I.N. Stokhasticheskie differentsial’nye sistemy [Stochastic Differential Systems]. Moscow, Nauka Publ., 1990. 632 p.

[6] 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.).

[7] Morozov A.N. Stationary Fluctuations Distributions of Brownian Particle Velocity in a Medium with Fluctuating Viscous Friction Coefficient. Vestn. Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Estestv. Nauki [Herald of the Bauman Moscow State Tech. Univ., Nat. Sci.], 2014, no. 3, pp. 26-38 (in Russ.).

[8] Morozov A.N. Neobratimye protsessy i brounovskoe dvizhenie: Fiziko-tekhnicheskie problemy [Irreversible Processes and Brownian Motion: Physical and Technical Problems]. Moscow, MGTU im. N.E. Baumana Publ., 1997. 332 p.

[9] Korotaev S.M., Morozov A.N., Serdyuk V.O., Sorokin M.O. Manifestation of the Macroscopic Nonlocality in Some Natural Dissipation Processes. Russian Physics Journal, 2002, no. 5, pp. 431-444.

[10] Morozov A.N. Preliminary Results of Recording the Kullback Measure of Voltage Fluctuations on Electrolytic Cell. Vestn. Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Estestv. Nauki [Herald of the Bauman Moscow State Tech. Univ., Nat. Sci.], 2011, no. 2, pp. 16-24 (in Russ.).

[11] Kamke E. Differentialgleichungen: Losungsmethoden und Losungen, I, Gewohnliche Differentialgleichungen [Handbook of ordinary differential equations]. Leipzig, B.G. Teubner, 1977.

[12] Nikiforov A.F., Uvarov V.B. Spetsial’nye funktsii matematicheskoy fiziki [Special Functions of Mathematical Physics]. Moscow, Nauka Publ., 1984. 344 p.

[13] Malakhov A.N. Kumulyantnyy analiz sluchaynykh negaussovykh protsessov i ikh preobrazovaniy [Cumulant Analysis of Non-Gaussian Random Processes and Their Transforms]. Moscow, Sov. Radio Publ., 1978. 370 p.