Brownian Motion in Minkowski Space

Entropy

Paul O’Hara²  Lamberto Rondoni¹ 

[1] Dipartimento di Scienze Matematiche and Graphene@PoliTO Lab, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy;Department of Mathematics, Northeastern Illinois University, 5500 North St. Louis Avenue, Chicago, IL 60625-4699, USA; E-Mail:

关键词: geodesic;    quaternions;    stopping times;    Markov processes;    pseudo-diffusion equation;   

DOI  :  10.3390/e17063581

来源: mdpi

【 摘 要 】

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.

【 授权许可】

CC BY   
© 2015 by the authors; licensee MDPI, Basel, Switzerland