Abstract
In physics, chemistry, and mathematics, the process of Brownian motion is often identified with the Wiener process that has infinitesimal increments. Recently, many models of Brownian motion with finite velocity have been intensively studied. We consider one of such models, namely, a generalization of the Goldstein–Kac process to the three-dimensional case with the Erlang-2 and Maxwell–Boltzmann distributions of velocities alternations. Despite the importance of having a three-dimensional isotropic random model for the motion of Brownian particles, numerous research efforts did not lead to an expression for the probability of the distribution of the particle position, the motion of which is described by the threedimensional telegraph process. The case where a particle carries out its movement along the directions determined by the vertices of a regular n + 1-hedron in the n-dimensional space was studied in [13], and closed-form results for the distribution of the particle position were obtained. Here, we obtain expressions for the distribution function of the norm of the vector that defines particle’s position at renewal instants in semi-Markov cases of the Erlang-2 and Maxwell–Boltzmann distributions and study its properties. By knowing this distribution, we can determine the distribution of particle positions, since the motion of a particle is isotropic, i.e., the direction of its movement is uniformly distributed on the unit sphere in ℝ3. Our results may be useful in studying the properties of an ideal gas.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
W. Feller, An Introduction to Probability Theory and Its Applications, Wiley, New York, 1971, Vol. 2.
E. V. Tolubinskii, The Theory of Transfer Processes [in Russian], Naukova Dumka, Kiev, 1969.
W. Stadje, “Exact probability distributions for non-correlated random walk models,” J. Stat. Phys., 56, 415–435 (1989).
E. Orsingher and A. De Gregorio, “Random flights in higher spaces,” J. Theor. Prob., 20, 769–806 (2007).
R. Garra and E. Orsingher, “Random flights govern by Klein–Gordon-type partial differential equations,” Stoch. Process. Their Appl., 122, 676–713 (2014).
A. Pogorui and R. M. Rodríguez-Dagnino, “Goldstein–Kac telegraph equations and random flights in higher dimensions,” Appl. Math. Comput., 361, 617–629 (2019).
U. Kuchler and S. Tappe, “Bilateral gamma distributions and processes in financial mathematics,” Stoch. Process. Their Appl., 118(2), 261–283 (2008).
A. Tashkandy Yusra, A. Omair Maha, and A. Alzaid Abdulhamid, “Bivariate and bilateral distribution,” Int. J. Statist. Probab., 7(2), 66–79 (2018).
T. M. Sellke and S. H. Sellke, “Chebyshev inequalities for unimodal distributions,” Amer. Statistician, 51(1), 34–40 (1997).
G. Upton and I. Cook, Gauss Inequality. A Dictionary of Statistics. Oxford Univ. Press, Oxford, 2008.
F. Pukelsheim, “The three sigma rule,” Amer. Statistician, 48(2), 88–91 (1994).
D. F. Vysochanskij and Y. I. Petunin, “Justification of the 3σ rule for unimodal distributions,” Theor. Probab. Math. Statist., 21, 25–36 (1980).
I. V. Samoilenko, “Distribution function of Markovian random evolution in Rn,” (2009), https://arxiv.org/pdf/0911.0165.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 17, No. 4, pp. 563–573, October–December, 2020.
Rights and permissions
About this article
Cite this article
Pogorui, A., Rodríguez-Dagnino, R.M. Distribution of random motion at renewal instants in three-dimensional space. J Math Sci 254, 416–424 (2021). https://doi.org/10.1007/s10958-021-05313-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-021-05313-6