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Relativistic Brownian Motion This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright Author's personal copy Physics Reports 471 (2009) 1–73 Contents lists available at ScienceDirect Physics Reports journal homepage: www.elsevier.com/locate/physrep Relativistic Brownian motion Jörn Dunkel a,∗, Peter Hänggi b a Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, United Kingdom b Institut für Physik, Universität Augsburg, Universitätsstraße 1, D-86135 Augsburg, Germany article info a b s t r a c t Article history: Over the past one hundred years, Brownian motion theory has contributed substantially Accepted 19 December 2008 to our understanding of various microscopic phenomena. Originally proposed as a Available online 6 January 2009 phenomenological paradigm for atomistic matter interactions, the theory has since evolved editor: J. Eichler 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 PACS: processes has led to new approaches in other fields, culminating in the path integral 02.50.Ey formulation of modern quantum theory. Stimulated by experimental progress in high 05.40.-a 05.40.Jc energy physics and astrophysics, the unification of relativistic and stochastic concepts has 47.75.+f re-attracted considerable interest during the past decade. Focusing on the framework of special relativity, we review, here, recent progress in the phenomenological description Keywords: of relativistic diffusion processes. After a brief historical overview, we will summarize Brownian motion basic concepts from the Langevin theory of nonrelativistic Brownian motions and discuss Special relativity relevant aspects of relativistic equilibrium thermostatistics. The introductory parts are Stochastic processes Relativistic Langevin equations followed by a detailed discussion of relativistic Langevin equations in phase space. Fokker–Planck equations We address the choice of time parameters, discretization rules, relativistic fluctuation- Diffusion processes dissipation theorems, and Lorentz transformations of stochastic differential equations. The Relativistic thermodynamics 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. ' 2008 Elsevier B.V. All rights reserved. Contents 1. Introduction............................................................................................................................................................................................. 2 1.1. Historical background ................................................................................................................................................................ 3 1.2. Relativistic diffusion processes: Problems and general strategies .......................................................................................... 5 1.2.1. Non-Markovian diffusion models in Minkowski spacetime..................................................................................... 5 1.2.2. Relativistic Markov processes in phase space............................................................................................................ 6 1.3. Structure of the review............................................................................................................................................................... 6 ∗ Corresponding author. E-mail addresses: [email protected] (J. Dunkel), [email protected] (P. Hänggi). URLs: http://www-thphys.physics.ox.ac.uk/people/JornDunkel/ (J. Dunkel), http://www.physik.uni-augsburg.de/theo1/hanggi/ (P. Hänggi). 0370-1573/$ – see front matter ' 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physrep.2008.12.001 Author's personal copy 2 J. Dunkel, P. Hänggi / Physics Reports 471 (2009) 1–73 2. Nonrelativistic Brownian motion........................................................................................................................................................... 7 2.1. Langevin and Fokker–Planck equations .................................................................................................................................... 7 2.1.1. Langevin equations and discretization rules ............................................................................................................. 7 2.1.2. Nonrelativistic Ornstein–Uhlenbeck process ............................................................................................................ 10 2.2. Microscopic models .................................................................................................................................................................... 12 2.2.1. Harmonic oscillator model.......................................................................................................................................... 13 2.2.2. Elastic binary collision model ..................................................................................................................................... 14 2.3. Remarks and generalizations ..................................................................................................................................................... 18 3. Relativistic equilibrium thermostatistics .............................................................................................................................................. 18 3.1. Preliminaries ............................................................................................................................................................................... 19 3.1.1. Notation and conventions ........................................................................................................................................... 19 3.1.2. Probability densities in special relativity ................................................................................................................... 21 3.2. Stationary systems with confinement....................................................................................................................................... 23 3.2.1. General remarks .......................................................................................................................................................... 23 3.2.2. Jüttner gas .................................................................................................................................................................... 27 4. Relativistic Brownian motion processes in phase space ...................................................................................................................... 31 4.1. Relativistic Langevin and Fokker–Planck equations................................................................................................................. 32 4.1.1. Relativistic Langevin equations: General construction principles ........................................................................... 32 4.1.2. Fokker–Planck equations ............................................................................................................................................ 33 4.1.3. Free motions in an isotropic bath and Einstein relations.......................................................................................... 34 4.2. One-dimensional examples and mean square displacement .................................................................................................. 34 4.2.1. Discretization rules, energy and velocity equations ................................................................................................. 34 4.2.2. Asymptotic mean square displacement ..................................................................................................................... 35 4.2.3. Examples ...................................................................................................................................................................... 36 4.3. Proper-time reparameterization ..............................................................................................................................................
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