Relativistic Brownian Motion

Physics Reports 471 (2009) 1–73

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Physics Reports

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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).

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 ...... 39 4.4. Moving observers...... 41 4.4.1. Lorentz transformation of the phase space density ...... 41 4.4.2. Lorentz transformation of the Langevin equation...... 41 4.5. Relativistic binary collision model...... 42 5. Non-Markovian diffusion processes in Minkowski spacetime ...... 45 5.1. Reminder: Nonrelativistic diffusion equation ...... 45 5.2. Telegraph equation...... 46 5.3. Relativistic diffusion propagator...... 49 6. Outlook ...... 51 Acknowledgements...... 52 Appendix A. Stochastic integrals and calculus ...... 52 A.1. Ito integral ...... 52 A.1.1. One-dimensional case ...... 52 A.1.2. The n-dimensional case...... 53 A.2. Stratonovich–Fisk integral ...... 53 A.2.1. One-dimensional case ...... 53 A.2.2. The n-dimensional case...... 54 A.3. Backward Ito integral ...... 54 A.3.1. One-dimensional case ...... 54 A.3.2. The n-dimensional case...... 55 A.4. Comparison of stochastic integrals...... 55 A.5. Numerical integration ...... 56 Appendix B. Surface integrals in Minkowski spacetime...... 56 Appendix C. Relativistic thermodynamics...... 59 C.1. Reminder: Nonrelativistic thermodynamics ...... 59 C.2. Relativistic case...... 60 References...... 65

1. Introduction

In his annus mirabilis 1905 Albert Einstein published four manuscripts [1–4] that would forever change the world of physics. Two of those papers [2,3] laid the foundations for the special theory of relativity, while another one [4] solved the longstanding problem of classical (nonrelativistic) Brownian motion.1 Barring gravitational effects [5,6], special relativity has proven to be the correct framework for describing physical processes on all terrestrial scales [7,8]. Accordingly, during the past century extensive efforts have been made to adapt established nonrelativistic theories such as, for example, thermodynamics, quantum mechanics or field theories [9] to the requirements of special relativity. Following this tradition, the present review focuses on recent progress in the theory of special relativistic Brownian motion and diffusion processes [10–34].

1 Einstein’s first paper [1] provided the explanation for the photoelectric effect. Download English Version: https://daneshyari.com/en/article/1875254

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