Relativistic Brownian motion

Abstract

a b s t r a c tOver the past one hundred years, Brownian motion theory has contributed substantially to our understanding of various microscopic phenomena. Originally proposed as a phenomenological paradigm for atomistic matter interactions, the theory has since evolved into a broad and vivid research area, with an ever increasing number of applications in biology, chemistry, finance, and physics. The mathematical description of stochastic processes has led to new approaches in other fields, culminating in the path integral formulation of modern quantum theory. Stimulated by experimental progress in high energy physics and astrophysics, the unification of relativistic and stochastic concepts has re-attracted considerable interest during the past decade. Focusing on the framework of special relativity, we review, here, recent progress in the phenomenological description of relativistic diffusion processes. After a brief historical overview, we will summarize basic concepts from the Langevin theory of nonrelativistic Brownian motions and discuss relevant aspects of relativistic equilibrium thermostatistics. The introductory parts are followed by a detailed discussion of relativistic Langevin equations in phase space. We address the choice of time parameters, discretization rules, relativistic fluctuationdissipation theorems, and Lorentz transformations of stochastic differential equations. The general theory is illustrated through analytical and numerical results for the diffusion of free relativistic Brownian particles. Subsequently, we discuss how Langevin-type equations can be obtained as approximations to microscopic models. The final part of the article is dedicated to relativistic diffusion processes in Minkowski spacetime. Since the velocities of relativistic particles are bounded by the speed of light, nontrivial relativistic Markov processes in spacetime do not exist; i.e., relativistic generalizations of the nonrelativistic diffusion equation and its Gaussian solutions must necessarily be non-Markovian. We compare different proposals that were made in the literature and discuss their respective benefits and drawbacks. The review concludes with a summary of open questions, which may serve as a starting point for future investigations and extensions of the theory.

Quasirelativistic Langevin equation

“…Despite a high motivation toward the construction of a unifying approach, there is currently no consensus on the form of Langevin and master equations describing a relativistic Brownian particle. Several versions were proposed in recent years [1], but their status and validity range are often obscure. The difficulties are many, and some are fundamental to relativistic many-body dynamics [1,2].…”

“…Several versions were proposed in recent years [1], but their status and validity range are often obscure. The difficulties are many, and some are fundamental to relativistic many-body dynamics [1,2]. In the nonrelativistic theory, the standard equations of Brownian motion can be derived microscopically, eliminating (fast) degrees of freedom of the thermal bath with a projection operator or some other technique [3].…”

Time parametrization and stationary distributions in a relativistic gas

“…Among these, are the evolution of dynamical systems towards equilibrium and more specifically their equilibrium probability distribution function (PDF). As reported recently [17,18,23], the description of the motion of relativistic particles with respect to the observer's time, t, and the particle's proper time, τ are completely different. To elucidate, consider the t-averaged PDF of a Brownian particle in terms of the observer's time…”

“…To resolve the uncertainty, semi-relativistic [13] and fully relativistic [14,15] molecular dynamics simulations as well as Monte Carlo studies [16] have been performed by different groups in recent years that unequivocally favored Jüttner distribution. However, some recent investigations on relativistic Brownian motions [17,18] have revealed that stationary distributions can differ depending on the underlying time-parameterizations, a problem which never arises in Newtonian physics due to the existence of a universal time for all inertial observers. On the other hand, the maximum relative entropy principle (MREP) [19,20,21] depicts how symmetry considerations lead to different stationary distributions, each with its own merits [22].…”

A Look at the Cosmic Ray Anisotropy with the Nonlocal Relativistic Transport Approach

“…More formal mathematical generalization of classical Brownian motion to its relativistic counterpart is performed by Dunkel and Hänggi [6]. However, this equation describes propagation of CRs through a system of mutually independent scatterers, whereas real points of scattering are bounded by magnetic field lines along which the particles fly.…”