2 research outputs found
Relativistic Brownian motion: From a microscopic binary collision model to the Langevin equation
The Langevin equation (LE) for the one-dimensional relativistic Brownian
motion is derived from a microscopic collision model. The model assumes that a
heavy point-like Brownian particle interacts with the lighter heat bath
particles via elastic hard-core collisions. First, the commonly known,
non-relativistic LE is deduced from this model, by taking into account the
non-relativistic conservation laws for momentum and kinetic energy.
Subsequently, this procedure is generalized to the relativistic case. There, it
is found that the relativistic stochastic force is still \gd-correlated
(white noise) but does \emph{no} longer correspond to a Gaussian white noise
process. Explicit results for the friction and momentum-space diffusion
coefficients are presented and discussed.Comment: v2: Eqs. (17c) and (28) corrected; v3: discussion extended, Eqs. (33)
added, thereby connection to earlier work clarified; v4: final version,
accepted for publication in Phys. Rev.
Relativistic Brownian Motion
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
fluctuation-dissipation 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. Due to
the finiteness of velocities in relativity, 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.Comment: review article, 159 pages, references updated, misprints corrected,
App. A.4. correcte