Relativistic Brownian motion and diffusion processes [Elektronische Ressource] / eingereicht von Jörn Dunkel

Sujets

• Mathematics
• Physik

Informations

• Publié par: universitat_augsburg
• Publié le: 01 janvier 2008
• Nombre de lectures: 27
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Relativistic Brownian Motion and Diffusion Processes
Dissertation
Institut fu¨r Physik
Mathematisch-Naturwissenschaftliche Fakult¨at
Universit¨at Augsburg
eingereicht von
J¨orn Dunkel
Augsburg, Mai 2008Erster Gutachter: Prof. Dr. Peter H¨anggi
Zweiter Gutachter: Prof. Dr. Thilo Kopp
Dritter Gutachter: Prof. Dr. Werner Ebeling
Tag der mu¨ndlichen Pru¨fung: 22.07.2008Contents
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 Other generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 Microscopic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.1 Harmonic oscillator model . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.2 Elastic binary collision model . . . . . . . . . . . . . . . . . . . . . 26
3 Relativistic equilibrium thermostatistics 35
3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1.1 Notation and conventions . . . . . . . . . . . . . . . . . . . . . . . 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 Moving observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.1 Fokker-Planck and Langevin equations . . . . . . . . . . . . . . . . 67
4.2.2 Covariant formulation . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3 Relativistic binary collision model . . . . . . . . . . . . . . . . . . . . . . . 69
12 CONTENTS
5 Non-Markovian relativistic diffusion 75
5.1 Reminder: nonrelativistic diffusion equation . . . . . . . . . . . . . . . . . 76
5.2 Telegraph equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.3 Relativistic diffusion propagator . . . . . . . . . . . . . . . . . . . . . . . . 82
6 Summary and outlook 87
Appendices
A Special relativity (basics) 93
A.1 Notation and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
A.2 Lorentz-Poincar´e transformations . . . . . . . . . . . . . . . . . . . . . . . 95
B Normalization constants 99
B.1 Ju¨ttner function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
B.2 Diffusion propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
C Stochastic integrals and calculus 101
C.1 Ito integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
C.1.1 One-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . 102
C.1.2 The n-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . 103
C.2 Stratonovich-Fisk integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
C.2.1 One-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . 104
C.2.2 The n-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . 105
C.3 Backward Ito integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
C.3.1 One-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . 106
C.3.2 The n-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . 107
C.4 Comparison of stochastic integrals . . . . . . . . . . . . . . . . . . . . . . . 107
C.5 Numerical integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
D Higher space dimensions 111
D.1 Lab frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
D.2 Moving observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Bibliography 115
Lebenslauf 151
Danksagung 152Symbols
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 = (η ) Minkowski metric tensorαβ
Λ Lorentz transformation (matrix)
2 −1/2γ Lorentz factor. γ(v) = (1−v )
α αX (contravariant) time-space four-vector (X ) = (t,X), α = 0, 1,...,d
α αP energy-momentum four-vector, (P ) = (E,P )
α α αU velocity four-vector, U =P /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
k Boltzmann constant (set to unity throughout, i.e., k = 1)B B
T temperature
−1β inverse thermal energy β := (k T)B
S relative entropy
α friction coefficient
34 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
,ρ measures
hXi expected value of a random variable XChapter 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
1classical (nonrelativistic) Brownian motion. 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.
56 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 L´evy [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 dif-
ferent types of stochastic integrals or, equivalently, stochastic differential equations (SDEs).
The theoretical analysis of random processes was further developed over the past decades,
2and the most essential results are discussed in several excellent textbook references [60–66] .
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 fi