Relativistic Brownian Motion and Diffusion Processes

Relativistic Brownian Motion and Diffusion Processes Dissertation Institut für Physik Mathematisch-Naturwissenschaftliche Fakultät Universität Augsburg eingereicht von Jörn Dunkel Augsburg, Mai 2008 Erster Gutachter: Prof. Dr. Peter Hänggi Zweiter Gutachter: Prof. Dr. Thilo Kopp Dritter Gutachter: Prof. Dr. Werner Ebeling Tag der mündlichen Prüfung: 22.07.2008 Contents Symbols 3 1 Introduction and historical overview 5 2 Nonrelativistic Brownian motion 13 2.1 Langevin and Fokker-Planck equations . ..... 14 2.1.1 Linear Brownian motion: Ornstein-Uhlenbeck process ........ 14 2.1.2 Nonlinear Langevin equations . 19 2.1.3 Othergeneralizations . 22 2.2 Microscopicmodels ............................... 22 2.2.1 Harmonicoscillatormodel . 23 2.2.2 Elastic binary collision model . .. 26 3 Relativistic equilibrium thermostatistics 35 3.1 Preliminaries .................................. 36 3.1.1 Notationandconventions . 36 3.1.2 Probability densities in special relativity . ........ 37 3.2 Thermostatistics of a relativistic gas . ....... 40 3.2.1 Relative entropy, Haar measures and canonical velocity distributions 40 3.2.2 Relativistic molecular dynamics simulations . ....... 45 4 Relativistic Brownian motion 53 4.1 Langevin and Fokker-Planck equations . ..... 54 4.1.1 Construction principle and conceptual aspects . ....... 55 4.1.2 Examples ................................ 59 4.1.3 Asymptotic mean square displacement . .. 61 4.2 Movingobserver................................. 66 4.2.1 Fokker-Planck and Langevin equations . ... 67 4.2.2 Covariantformulation . 68 4.3 Relativistic binary collision model . ....... 69 1 2 CONTENTS 5 Non-Markovian relativistic diffusion 75 5.1 Reminder: nonrelativistic diffusion equation . ......... 76 5.2 Telegraphequation ............................... 78 5.3 Relativisticdiffusionpropagator . ...... 82 6 Summary and outlook 87 Appendices A Special relativity (basics) 93 A.1 Notationanddefinitions . 93 A.2 Lorentz-Poincarétransformations . ....... 95 B Normalization constants 99 B.1 Jüttnerfunction................................ 99 B.2 Diffusionpropagator .............................. 100 C Stochastic integrals and calculus 101 C.1 Itointegral ................................... 102 C.1.1 One-dimensionalcase. 102 C.1.2 The n-dimensionalcase. 103 C.2 Stratonovich-Fiskintegral . .... 104 C.2.1 One-dimensionalcase. 104 C.2.2 The n-dimensionalcase. 105 C.3 BackwardItointegral. .. .. 106 C.3.1 One-dimensionalcase. 106 C.3.2 The n-dimensionalcase. 107 C.4 Comparisonofstochasticintegrals. ...... 107 C.5 Numericalintegration. 110 D Higher space dimensions 111 D.1 Labframe .................................... 111 D.2 Movingobserver................................. 113 Bibliography 115 Lebenslauf 151 Danksagung 152 Symbols M rest mass of the Brownian particle m rest mass of a heat bath particle Σ inertial laboratory frame := rest frame of the heat bath ′ Σ ; Σ∗ moving frame; comoving rest frame of the Brownian particle O; O′ lab observer; moving observer t time coordinate τ proper time of the Brownian particle c vacuum speed of light (set to unity throughout, i.e., c = 1) d number of space dimensions X, x position coordinate V , v particle velocity w observer velocity P , p momentum coordinates E, ǫ particle energy η =(ηαβ) Minkowski metric tensor Λ Lorentz transformation(matrix) γ Lorentz factor. γ(v)=(1 v2)−1/2 − Xα (contravariant) time-space four-vector (Xα)=(t, X), α =0, 1,...,d P α energy-momentum four-vector, (P α)=(E, P ) U α velocity four-vector, U α = P α/M f one-particle phase space probability density ̺ one-particle position probability density φ one-particle momentum probability density ψ one-particle velocity probability density kB Boltzmann constant (set to unity throughout, i.e., kB = 1) T temperature −1 β inverse thermal energy β := (kBT) S relative entropy α friction coefficient 3 4 CONTENTS D noise amplitude D spatial diffusion constant B(s) d-dimensional standard Wiener process with time parameter s P probability measure of the Wiener process Ito (pre-point) interpretation of the stochastic integral ∗ Stratonovich-Fisk (mid-point) interpretation of the stochastic integral ◦ backward Ito (post point) interpretation of the stochastic integral • N set of natural numbers 1, 2,... Z set of integer numbers R set of real numbers λ Lebesgue measure µ, ρ measures X expected value of a random variable X h i Chapter 1 Introduction and historical overview 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, e.g., thermodynamics, quantum mechanics or field theories [9] to the requirements of special relativity. Following this tradition, the present thesis investigates how stochastic concepts such as Brownian motion may be generalized within the framework of special relativity. The subsequent chapters intend to provide a cohesive summary of results obtained during the past three years [10–17], also taking into account important recent contributions by other authors (see, e.g., [18–24]). Historically, the term ‘Brownian motion’ refers to the irregular dynamics exhibited by a test particle (e.g., dust or pollen) in a liquid environment. This phenomenon, already mentioned by Ingen-Housz [25, 26] in 1784, was first analyzed in detail by the Scottish botanist Robert Brown [27] in 1827. About 80 years later, Einstein [4], Sutherland [28] and von Smoluchowski [29] were able to theoretically explain these observations. They proposed that Brownian motion is caused by quasi-random, microscopic interactions with molecules forming the liquid. In 1909 their theory was confirmed experimentally by Per- rin [30], providing additional evidence for the atomistic structure of matter. During the first half of the 20th century the probabilistic description of Brownian motion processes was further elaborated in seminal papers by Langevin [31, 32], Fokker [33], Planck [34], Klein [35], Uhlenbeck and Ornstein [36] and Kramers [37]. Excellent reviews of these early contributions are given by Chandrasekhar [38] and Wang and Uhlenbeck [39]. 1Einstein’s first paper [1] provided the theoretical explanation for the photoelectric effect. 5 6 CHAPTER 1. INTRODUCTION AND HISTORICAL OVERVIEW In parallel with the studies in the field of physics, outstanding mathematicians like Bache- lier [40], Wiener [41–43], Kolmogoroff [44–46], Feller [47], and Lévy [48, 49] provided a rigorous basis for the theory of Brownian motions and stochastic processes in general. Between 1944 and 1968 their groundbreaking work was complemented by Ito [50,51], Gih- man [52–54], Fisk [55,56] and Stratonovich [57–59], who introduced and characterized different types of stochastic integrals or, equivalently, stochastic differential equations (SDEs). The theoretical analysis of random processes was further developed over the past decades, and the most essential results are discussed in several excellent textbook references [60–66]2. The modern theory of stochastic processes goes far beyond the original problem considered by Einstein and his contemporaries, and the applications cover a wide range of different areas including physics [67–74], biology [75,76], economy and finance [77–79]. The present thesis is dedicated to the question how SDE-based Brownian motion models can be generalized within the framework of special relativity. In the physics literature [65], SDEs are often referred to as Langevin equations [31, 32], and we shall use both terms synonymously here. From a mathematical point of view, SDEs [64] determine well-defined models of stochastic processes; from a physicist’s point of view, their usefulness for the description of a real system is a priori an open issue. Therefore, the derivation of nonrelativistic Langevin equations from microscopic models has attracted considerable interest over the past 60 years [13,80–86]. Efforts in this direction helped to clarify the applicability of SDEs to physical problems and led, among others, to the concept of quantum Brownian motion [82,87–99].3 If one aims at generalizing the classical Brownian motion concepts to special relativity, then several elements from relativistic equilibrium thermodynamics and relativistic statis- tical mechanics play an important role. The first papers on relativistic thermodynamics were written by Einstein [109] and Planck [110, 111] in 1907. A main objective of their studies was to clarify the Lorentz transformation laws of thermodynamic variables (temperature, pressure, etc.).4 In 1963 the results of Einstein and Planck were questioned by Ott [115], whose work initiated an intense debate about the correct relativistic transformation behavior of thermodynamic quantities [116–160].5 However, as clarified by van Kampen [137] and Yuen [161], the controversy surrounding relativistic thermodynamics can be resolved by realizing that thermodynamic quantities can be defined in different, 2The history of the mathematical literature on Brownian motions and stochastic processes is discussed extensively in Section 2.11 of Ref. [64]; see also Chapters 2-4 in Nelson

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