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Entropy 2015, 17(6), 3581-3594; doi:10.3390/e17063581

Brownian Motion in Minkowski Space

1
Department of Mathematics, Northeastern Illinois University, 5500 North St. Louis Avenue, Chicago, IL 60625-4699, USA
2
Dipartimento di Scienze Matematiche and Graphene@PoliTO Lab, Politecnico di Torino, Corso Ducadegli Abruzzi 24, 10129 Torino, Italy
3
National Institute of Nuclear Physics (INFN), Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy
*
Author to whom correspondence should be addressed.
Academic Editor: Carlo Cafaro
Received: 15 April 2015 / Revised: 20 May 2015 / Accepted: 25 May 2015 / Published: 1 June 2015
(This article belongs to the Special Issue Dynamical Equations and Causal Structures from Observations)
View Full-Text   |   Download PDF [229 KB, uploaded 1 June 2015]

Abstract

We construct a model of Brownian motion in Minkowski space. There are two aspects of the problem. The first is to define a sequence of stopping times associated with the Brownian “kicks” or impulses. The second is to define the dynamics of the particle along geodesics in between the Brownian kicks. When these two aspects are taken together, the Central Limit Theorem (CLT) leads to temperature dependent four dimensional distributions defined on Minkowski space, for distances and 4-velocities. In particular, our processes are characterized by two independent time variables defined with respect to the laboratory frame: a discrete one corresponding to the stopping times when the impulses take place and a continuous one corresponding to the geodesic motion in-between impulses. The subsequent distributions are solutions of a (covariant) pseudo-diffusion equation which involves derivatives with respect to both time variables, rather than solutions of the telegraph equation which has a single time variable. This approach simplifies some of the known problems in this context. View Full-Text
Keywords: geodesic; quaternions; stopping times; Markov processes; pseudo-diffusion equation geodesic; quaternions; stopping times; Markov processes; pseudo-diffusion equation
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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O'Hara, P.; Rondoni, L. Brownian Motion in Minkowski Space. Entropy 2015, 17, 3581-3594.

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