PHYSICAL REVIEW E 71, 016124 共2005兲
Theory of relativistic Brownian motion: The „1 + 1…-dimensional case
Jörn Dunkel*
Institut für Physik, Humboldt-Universität zu Berlin, Newtonstraße 15, D-12489 Berlin, Germany
Peter Hänggi
Institut für Physik, Universität Augsburg, Theoretische Physik I, Universitätstraße 1, D-86135 Augsburg, Germany
共Received 23 July 2004; published 18 January 2005兲
We construct a theory for the 共1 + 1兲-dimensional Brownian motion in a viscous medium, which is 共i兲
consistent with Einstein’s theory of special relativity and 共ii兲 reduces to the standard Brownian motion in the
Newtonian limit case. In the first part of this work the classical Langevin equations of motion, governing the
nonrelativistic dynamics of a free Brownian particle in the presence of a heat bath 共white noise兲, are generalized in the framework of special relativity. Subsequently, the corresponding relativistic Langevin equations are
discussed in the context of the generalized Ito 共prepoint discretization rule兲 versus the Stratonovich 共midpoint
discretization rule兲 dilemma: It is found that the relativistic Langevin equation in the Hänggi-Klimontovich
interpretation 共with the postpoint discretization rule兲 is the only one that yields agreement with the relativistic
Maxwell distribution. Numerical results for the relativistic Langevin equation of a free Brownian particle are
presented.
DOI: 10.1103/PhysRevE.71.016124
PACS number共s兲: 02.50.Ey, 05.40.Jc, 47.75.⫹f
I. INTRODUCTION
For almost 100 years, Einstein’s theory of special relativity 关1,2兴 is serving as the foundation of our most successful
physical standard models 共apart from gravity兲. The most
prominent and, probably, also the most important feature of
this theory is the absolute character of the speed of light c,
representing an unsurmountable barrier for the velocity of
any 共macroscopic兲 physical process. Due to the great experimental success of the original theory, almost all other physical theories have successfully been adapted to the framework
of special relativity over the past decades. Surprisingly, however, the scientific literature provides relatively few publications on the subject of relativistic Brownian motions 共classical references are 关3–5兴 and more recent contributions
include 关6–13兴兲.
Brownian particles are physical objects 共e.g., dust grains兲
that move randomly through a surrounding medium 共heat
bath兲. Their stochastic motions are caused by permanent collisions with much lighter constituents of the heat bath 共e.g.,
molecules of a liquid兲. The classical theory of Brownian motion or nonrelativistic diffusion theory, respectively, was developed by Einstein 关14兴 and Einstein and von Smoluchowski 关15兴. Since the beginning of the last century, when
their seminal papers were published, the classical theory has
been investigated and generalized by a large number of
physicists 关16–20兴 and mathematicians 关21–23兴. The intense
research led, among others, to different mathematical representations of the Brownian motion dynamics 关Langevin
equations, Fokker-Planck equations 共FPE兲, etc.兴 关18–20兴, to
the notion of the Wiener processes 关21兴, and to new techniques for solving partial differential equations 共FeynmanKac formula, etc. 关22,23兴兲.
*Electronic address: dunkel@physik.hu-berlin.de
1539-3755/2005/71共1兲/016124共12兲/$23.00
With regard to special relativity, standard Brownian motion faces the problem that it permits velocity jumps ⌬v, that
exceed the speed of light c 共see also Schay 关3兴兲. This is due
to the fact that in the nonrelativistic theory the velocity increments ⌬v have a Gaussian distribution, which always assigns a nonvanishing 共though small兲 probability to events
⌬v ⬎ c. This problem is also reflected by the Maxwell distribution, which represents the stationary velocity distribution
for an ensemble of free Brownian particles and permits absolute velocity values v ⬎ c 关20兴.
The first relativistically consistent generalization of Maxwell’s velocity distribution was introduced by Jüttner 关24兴 in
1911. Starting from an extremum principle for the entropy,
he obtained the probability distribution function of the relativistic ideal Boltzmann gas 关see Eq. 共67兲 below兴. In principle, however, Jüttner’s approach made no contact with the
theory of Brownian motion. Fifty years after Jüttner’s work,
Schay 关3兴 performed the first comprehensive mathematical
investigation of relativistic diffusion processes based on
Lorentz-invariant transition probabilities. On the mathematical side, Schay’s analysis was complemented by Hakim 关5兴
and Dudley 关4兴, who studied in detail the properties of
Lorentz-invariant Markov processes in relativistic phase
space. After 40 more years, Franchi and Le Jan 关13兴 have
presented an extension of Dudley’s work to general relativity. In particular, these authors discuss relativistic diffusions
in the presence of a Schwarzschild metric 关25兴. Hence, over
the past 100 years there has been steady 共though relatively
slow兲 progress in the mathematical analysis of relativistic
diffusion processes.
By contrast, one finds in the physical literature only very
few publications that directly address the topic of the relativistic Brownian motion 共despite the fact that relativistic kinetic theory has been fairly well established for more than
30 years 关26–29兴兲. Among the few exceptions are the papers
by Boyer 关8,9兴 and Ben-Ya’acov 关6兴, who have studied the
interaction between two energy-level particles and electro-
016124-1
©2005 The American Physical Society
PHYSICAL REVIEW E 71, 016124 共2005兲
J. DUNKEL AND P. HÄNGGI
magnetic radiation in thermal equilibrium, the latter acting as
a heat bath. In contrast to their specific microscopic model,
we shall adopt a more coarse-grained point of view here by
assuming that the heat bath is sufficiently well described by
macroscopic friction and diffusion coefficients.
Generally, the objective of the present paper can be summarized as follows: We would like to discuss how one can
construct, in a physically straightforward manner, a relativistic theory of Brownian motion for particles moving in a
homogeneous, viscous medium. For this purpose it is sufficient to concentrate on the case of 1 + 1 dimensions 共generalizations to the 1 + 3 dimensions are straightforward and will
be discussed separately in a forthcoming contribution兲. As a
starting point we choose the nonrelativistic Langevin equations of the free Brownian particle. In Sec. II these equations
will be generalized such that they comply with special relativity. As we shall see in Sec. III due to multiplicative noise
for the momentum degree of freedom, the resulting relativistic Langevin equations are not sufficient in order to
uniquely determine the corresponding Fokker-Planck equation 共generalized Ito-Stratonovich dilemma兲. Furthermore, it
it is shown that the stationary solution of a particular form
for the relativistic Fokker-Planck equation coincides with
Jüttner’s relativistic Maxwell distribution 共Sec. III B 3兲. Finally, we also discuss numerical results for the mean-square
displacement in Sec. IV.
It might be worthwhile to emphasize that the systematic
Langevin approach pursued below is methodically different
from those in Refs. 关3–13兴 and also from the kinetic theory
approach 关26–29兴. It is therefore satisfactory that our findings are apparently consistent with rigorous mathematical
results, obtained by Schay 关3兴 and Dudley 关4兴 for the case of
free relativistic diffusion. Moreover, it will become clear in
Sec. IV that numerical simulations of the relativistic Langevin equations constitute a very useful tool for the numerical
investigation of relativistic diffusion processes, provided that
the discretization rule is carefully chosen.
II. LANGEVIN DYNAMICS
First the main properties of the nonrelativistic Langevin
equations for free Brownian particles are briefly summarized
共Sec. II A兲. Subsequently, we construct generalized Lorentzcovariant Langevin equations 共Sec. II B兲. Finally, the covariant Langevin equations will be rewritten in laboratory coordinates 共Sec. II C兲.
The following notations will be used throughout the paper. Since we confine ourselves to the 共1 + 1兲-dimensional
case, upper and lower Greek indices ␣ , ␤ , . . . can take values
0, 1, where 0 refers to the time component. The
共1 + 1兲-dimensional Minkowski metric tensor with respect to
Cartesian coordinates is taken as
bath 共e.g., small liquid particles兲. In the Langevin approach
the nonrelativistic dynamics of the Brownian particle is described by the stochastic dynamical equations 共see, e.g., 关20兴
Chap. IX兲
dx共t兲
= v共t兲,
dt
m
A. Physical foundations
Consider the nonrelativistic one-dimensional motion of a
Brownian particle with mass m that is surrounded by a heat
dv共t兲
= − ␯mv共t兲 + L共t兲,
dt
共1b兲
where ␯ is the viscous friction coefficient. The Langevin
force L共t兲 is characterized by
具L共t兲典 = 0,
具L共t兲L共s兲典 = 2D␦共t − s兲,
共2兲
with all higher cumulants being zero 共Gaussian white noise兲,
and D being constant. More general models may include
velocity-dependent parameters ␯ and D 共see, e.g.,
关19,30–32兴兲, but we shall restrict ourselves to the simplest
case here. It is worthwhile to summarize the physical assumptions, implicitly underlying Eqs. 共1兲 as follows:
共i兲 The heat bath is homogeneous.
共ii兲 Stochastic impacts between the Brownian particle and
the constituents of the heat bath occur virtually uncorrelated.
共iii兲 On the macroscopic level, the interaction between
Brownian particle and heat bath is sufficiently well described
by the constant viscous friction coefficient ␯ and the white
noise force L.
共iv兲 Equations 共1兲 hold in the rest frame ⌺0 of the heat
bath 共corresponding to the specific inertial system, in which
the average velocity of the heat bath vanishes for all times t兲.
In the following ⌺0 will also be referred to as laboratory
frame.
In the mathematical literature, Eq. 共1b兲 is usually written
as
d关mv共t兲兴 = − ␯mv共t兲dt + dW共t兲,
共3a兲
where W共t兲 is a one-dimensional Wiener process 关19,22,23兴,
i.e., the density of the increments
w共t兲 ⬅ dW共t兲 ⬅ W共t + dt兲 − W共t兲
is given by
P1关w共t兲兴 =
1
冑4␲D dt
冋
exp −
共3b兲
册
w共t兲2
.
4D dt
共3c兲
Here the abbreviation w ⬅ dW has been introduced to simplify the notation in subsequent formulas. From Eq. 共3c兲 one
finds in agreement with 共2兲
具w共t兲典 = 0,
共␩␣␤兲 = 共␩␣␤兲 = diag共− 1,1兲.
Moreover, Einstein’s summation convention is invoked
throughout.
共1a兲
具w共t兲w共s兲典 =
再
0,
t⫽s
2D dt, t = s.
冎
共4兲
Depending on which notation is more convenient for the
current purpose, we shall use below either the physical formulation 共1兲 or the mathematical formulation 共3兲. The two
formulations can be connected by 共formally兲 setting
016124-2
w共t兲 = dW共t兲 = L共t兲dt.
共5兲
THEORY OF RELATIVISTIC BROWNIAN MOTION: …
PHYSICAL REVIEW E 71, 016124 共2005兲
B. Relativistic generalization
It is well known that in inertial coordinate systems, which
are comoving with a particle at a given moment t, the relativistic equations must reduce to the nonrelativistic Newtonian equations 共see, e.g., 关25兴 Chap. 2.3兲. Therefore, our strategy is as follows. Starting from the Langevin equations 共1兲
or 共3a兲, respectively, we construct in the first step the nonrelativistic equations of motion with respect to a coordinate
frame ⌺*, comoving with the Brownian particle at a given
moment t. In the second step, the general form of the covariant relativistic equation motions are found by applying a
Lorentz transformation to the nonrelativistic equations that
have been obtained for ⌺*.
It is useful to begin by considering the deterministic
共noise-free兲 limit case, corresponding to a pure damping of
the particle’s motion. This will be done Sec. II B 1. Subsequently, the stochastic force is separately treated in Sec.
II B 2.
1. Viscous friction
Setting the stochastic force term to zero 共corresponding to
a vanishing temperature of the heat bath兲, the nonrelativistic
Eq. 共1b兲 simplifies to
dv共t兲
m
= − ␯mv共t兲.
dt
trast, in the relativistic theory these equations are exact at
time t only if ⌺* is comoving at time t. In the latter case, we
can use Eqs. 共10兲 to construct relativistically covariant equations of motion. Introducing, as usual, the proper time ␶ by
the definition
d␶ ⬅ dt
共7兲
dp*␣
= f *␣,
d␶
As stated above, in the nonrelativistic theory the last three
equations are assumed to hold in the rest frame ⌺0 of the heat
bath. Now consider another inertial coordinate system ⌺*, in
which the Brownian particle is temporarily at rest at time t or
t* = t*共t兲, respectively, where t* denotes the ⌺*-time coordinate. That is, in ⌺* we have at time t
共9兲
关Conventionally, we use throughout the lax notation g*共t兲
⬅ g*(t*共t兲), where g* is originally a function of t*.兴 With
respect to the comoving frame ⌺*, the heat bath will, in
general, have a nonvanishing 共average兲 velocity V*. Then,
using a Galilean transformation we find that Eq. 共6兲 in ⌺*
coordinates at time t reads as follows:
共10a兲
Similarly, in ⌺* coordinates Eq. 共8兲 is given by
共9兲
dE*
共t兲 = − ␯mv*共t兲„v*共t兲 − V*… = 0.
dt*
1−
v2*
,
c2
共11兲
共f *␣兲 = − m␯共0, v* − V*兲.
共12兲
f ␣* ⫽ − m␯共u␣* − U␣* 兲,
共13兲
because, in general, at time t in ⌺*
u0* − U0* =
c
共9兲
c
c
冑1 − v2*/c2 冑1 − V2*/c2 = c − 冑1 − V2*/c2 ⫽ 0.
−
However, we can write f ␣* in a manifestly covariant form, if
we introduce the friction tensor
共 ␯ *␣␤兲 =
共8兲
共9兲
dv*
共t兲 = − ␯m„v*共t兲 − V*… = ␯mV* .
m
dt*
冑
Let 共u*␣兲 and 共U␣* 兲 denote the 共1 + 1兲-velocity components of
Brownian particle and heat bath, respectively. Now it is important to realize that the covariant force vector f ␣ cannot be
simply proportional to the 共1 + 1兲-velocity difference,
and, by virtue of 共6兲, its time derivative is given by
v*共t兲 ⬅ v*„t*共t兲… = 0.
v2
= dt*
c2
共14兲
The energy of the Brownian particle is purely kinetic,
dv
dE
= mv = − ␯mv2 .
dt
dt
1−
and combining momentum p* = mv* and energy into a
共1 + 1兲-vector 共p␣* 兲 = 共p0 , p*兲 = 共E* / c , p*兲, we can rewrite Eqs.
共10兲 in the covariant form
共6兲
mv共t兲2
,
E共t兲 =
2
冑
0 0
0 ␯
共15兲
,
which allows us to rewrite 共12兲 as
dp*␣
= − m␯*␣␤共u␤* − U*␤兲.
d␶
共16兲
This equation is manifestly Lorentz-invariant, and we drop
the asterisk from now on, while keeping in mind that the
diagonal form of the friction tensor 共15兲 is linked to the rest
frame ⌺* of the Brownian particle. In this respect the friction
tensor is very similar to the pressure tensor, as known from
the relativistic hydrodynamics of perfect fluids 共see, e.g.,
关25兴 Chap. 2.10兲. This analogy yields immediately the following representation:
冉
␯ ␣␤ = ␯ ␩ ␣␤ +
冊
u ␣u ␤
.
c2
共17兲
It is now interesting to consider Eq. 共16兲 in the laboratory
frame ⌺0, defined above as the rest frame of the heat bath.
There we have
共U␤兲 = 共c,0兲,
共10b兲
Note that in the nonrelativistic 共Newtonian兲 theory the left
equalities in Eqs. 共10兲 are valid for arbitrary time t. By con-
冉 冊
共u␤兲 = 共␥c, ␥v兲,
d␶ =
dt
,
␥
␥⬅
1
冑1 − v2/c2 .
共18兲
Combining 共16兲–共18兲 we find that the relativistic equations
of motion in ⌺0 are given by
016124-3
PHYSICAL REVIEW E 71, 016124 共2005兲
J. DUNKEL AND P. HÄNGGI
relativistic velocity curves, given by Eq. 共24兲, exhibit essential deviations from the purely exponential decay, predicted
by the Newtonian theory.
2. Stochastic force
FIG. 1. Velocity curves v共t兲, corresponding to Eq. 共24兲, for the
purely damped motion of a relativistic particle in the rest frame of
the heat bath 共laboratory frame兲. Especially at high velocities 兩v兩
ⱗ c, the relativistic velocity curves deviate from the exponential
decay, predicted by the Newtonian theory.
dp
mv
=−␯
冑1 − v2/c2
dt
共19a兲
dE
mv2
=−␯
冑1 − v2/c2 .
dt
共19b兲
On comparing 共19a兲 with 共6兲 and 共19b兲 with 共8兲, one readily
observes that the relativistic equations 共19兲 do indeed reduce
to the known Newtonian laws in the limit case v2 / c2 Ⰶ 1.
Using the relativistic definitions
E = ␥mc2,
p = ␥mv ,
共20兲
Eqs. 共19兲 can also be rewritten as
dp
= − ␯ p,
dt
共21a兲
dE
v2
= − ␯ pv = − ␯E 2 .
c
dt
共21b兲
dp*共t兲 = − ␯„p*共t兲 − mV*…dt* + w*共t兲,
⇒
E共t兲 =
mc2
冑1 − v2/c2 .
共22兲
P1*关w*共t兲兴 =
1
冑4␲D dt*
共23兲
p共0兲 = p0 ,
and, by using 共20兲, one thus obtains for the velocity of the
particle in the laboratory frame ⌺0 共rest frame of the heat
bath兲
v共t兲 = v0
冋冉 冊
冉
exp −
冊
w*共t兲2
.
4D dt*
v2
v2
1 − 20 e2␯t + 20
c
c
册
共24兲
Figure 1 depicts a semi-logarithmic representation of the velocity v共t兲 for different values of the initial velocity v0. As
one can see in the diagram, at high velocities 兩v兩 ⱗ c the
共27兲
This definition is in agreement with the requirement that in a
comoving inertial system ⌺* the 0-component of the
共1 + 1兲-force vector must vanish 共see, e.g., 关25兴 Chap. 2.3,
and also compare Eqs. 共12兲, 共31兲, and 共32兲 of the present
paper兲. Moreover, if the Lorentz frame ⌺* is comoving with
the Brownian particle at given time t, then the 共equal-time兲
white-noise relations 共4兲 generalize to
具w␣* 共t兲典 = 0,
具w␣* 共t兲w*␤共t兲典 =
再
␣ = 0 and/or ␤ = 0,
兵
otherwise.
2D dt* ,
共28兲
0,
The rhs. of the second equation in 共28兲 makes it plausible to
introduce a correlation tensor by
共D*␣␤兲 =
−1/2
.
共26兲
Note that also in the relativistic theory the momentum increments w共t兲 = dW共t兲 may tend to infinity, as long as the related
velocity increments remain bounded. In other words, in the
relativistic theory one must carefully distinguish between
stochastic momentum and velocity increments 共this is not
necessary in the nonrelativistic theory, because Newtonian
momenta are simply proportional to their velocities兲.
The next step is now to define the increment 共1 + 1兲-vector
by
The solution of 共21a兲 reads
p共t兲 = p0 exp共− ␯t兲,
共25兲
where the momentum increments w* ⬅ dW* represent a
Wiener-process with parameter D, i.e., the increments w*共t兲
have a Gaussian distribution
共w*␣兲 = 共0,w*兲.
In fact, only one of the two Eqs. 共19兲 or 共21兲, respectively,
must be solved due to the fixed relation between relativistic
energy and momentum:
p␣ p␣ = − E2/c2 + p2 = − m2c2
We now construct a relativistic generalization of the stochastic force. To this end, we consider Eq. 共11兲 as an operational definition for the proper time parameter ␶. The generalization procedure will be based on the standard assumption
共postulate兲 that, in temporarily comoving inertial frames ⌺*,
the relativistic equations of motions must reduce to the Newtonian equations of motions. According to this assumption,
for frames ⌺*, comoving with the particle at laboratory time
t, the relativistic stochastic differential equation must reduce
to
冉
0
0
0 2D dt*
冊
,
共29a兲
thus,
具w*␣共t兲w␤* 共t兲典 = D*␣␤ .
共29b兲
Additionally defining an “inverse” correlation tensor by
016124-4
THEORY OF RELATIVISTIC BROWNIAN MOTION: …
冠
共D̂*␣␤兲 =
0
0
0 共2D dt*兲−1
冡
PHYSICAL REVIEW E 71, 016124 共2005兲
共29c兲
,
allows us to generalize the distribution of the increments
from Eq. 共26兲 as follows:
␣
P1+1
* „w* 共t兲… =
冋
1
冑4␲D dt* exp
册
1
− D̂*␣␤w␣* 共t兲w␤* 共t兲 ␦„w0*共t兲….
2
from 共35b兲 and then used that u␣w␣ = 0, see Eq. 共32兲.
By virtue of the above results, we are now in the position
to write down the covariant Langevin equations with respect
to an arbitrary inertial system: If a Brownian particle with
rest mass m, proper time ␶ and 共1 + 1兲-velocity u␤ is surrounded by an isotropic, homogeneous heat bath with constant 1 + 1 velocity U␤, then the relativistic Langevin equations of motions read
共30兲
Here, the Dirac ␦-function on the right-hand side accounts
for the fact that the 0 component of the stochastic force must
vanish in every inertial frame, comoving with the Brownian
particle at time t; compare Eq. 共27兲. This also follows more
generally from the identity
d
d
0 ⬅ 共− mc2兲 = m 共u␣u␣兲 = 2u␣ f ␣ ,
d␶
d␶
0 = u ␣w ␣ .
共32兲
Hence, we can rewrite the probability distribution 共30兲 as
冉
c
冑4␲D dt* exp
1
− D̂*␣␤w␣* 共t兲w␤* 共t兲
2
⫻␦„u*␣w*␣共t兲…,
冊
1=
再
兿
␣=0
冕
⬁
d„w␣* 共t兲…
−⬁
冎
冉
冉
D␣␤ = 2D d␶ ␩␣␤ +
u ␣u ␤
c2
冊
冊
共35b兲
Then, in an arbitrary Lorentz frame, the density 共33兲 can be
written as
冋
1
␣
␤
冑4␲D d␶ exp − 2 D̂␣␤w 共␶兲w 共␶兲
c
␣
⫻␦„u␣w 共␶兲…
冋
具w␣共␶兲w␤共␶⬘兲典 =
册
共35c兲
To obtain the last line from the first, we have inserted D̂␣␤
共36c兲
再
␶ ⫽ ␶ ⬘;
兵
D , ␶ = ␶⬘ ,
0,
␣␤
共36d兲
共36e兲
with D␣␤ given by 共35a兲. Note that in each comoving Lorentz frame, in which, at a given moment t, the particle is at
rest, the marginal distribution of the spatial momentum increments, defined by
P1„w共t兲… =
冕
⬁
−⬁
d„w0共t兲…P1+1„w␣共t兲…,
共37兲
reduces to a Gaussian. In the Newtonian limit case, corresponding to v2 Ⰶ c2, one thus recovers from Eqs. 共35兲 and
共36兲 the usual nonrelativistic Brownian motion.
C. Langevin dynamics in the laboratory frame
A laboratory frame ⌺0 is, by definition, an inertial system,
in which the heat bath is at rest, i.e., in ⌺0 we have 共U␤兲
= 共c , 0兲 for all times t. Hence, with respect to ⌺0 coordinates,
the two stochastic differential Eqs. 共36b兲 assume with 共36c兲
the form
册
w␣共␶兲w␣共␶兲
=
exp −
␦„u␣w␣共␶兲….
冑4␲D d␶
4D d␶
c
冊
u ␣u ␤
,
c2
with ␯ denoting the viscous friction coefficient measured in
the rest frame of the particle. This is a first main result of this
work. The stochastic increments w␣共␶兲 ⬅ dW␣共␶兲 are distributed according to 共35c兲 and, therefore, characterized by
共35a兲
1
u u
D̂␣␤ =
␩␣␤ + ␣ 2 ␤ .
2D d␶
c
P1+1„w␣共␶兲… =
冉
␯ ␣␤ = ␯ ␩ ␣␤ +
共34兲
Furthermore, analogous to 共17兲, we have the following more
general representation of the correlation tensors:
共36b兲
where, according to Eq. 共17兲, the friction tensor is given by
共33兲
␣
P1+1
* „w* 共t兲….
共36a兲
具w␣共␶兲典 = 0,
where 共u*␣兲 = 共−c , 0兲 is the covariant 共1 + 1兲-velocity of the
particle a the comoving rest frame. It should be stressed that,
because of the constraint 共32兲, only one of the two increments w␣ ⬅ dW␣ is to be regarded as “independent,” which is
reflected by the appearance of the ␦ function in 共33兲. Also
note that, due to the prefactor c, the normalization condition
takes the simple form
1
p ␣共 ␶ 兲
d␶
m
dp␣共␶兲 = − ␯␣␤„p␤共␶兲 − mU␤…d␶ + w␣共␶兲,
共31兲
which, in the case of the stochastic force, translates to
␣
P1+1
* „w* 共t兲… =
dx␣共␶兲 =
dp = − ␯ pdt + w共t兲,
共38a兲
dE = − ␯ pvdt + cw0共t兲.
共38b兲
Here it is important to note that the stochastic increments
w␣共t兲, appearing on the right-hand side. of 共38兲, are not of
simple Gaussian type anymore. Instead, their distribution
now also depends on the particle velocity v. This becomes
immediately evident, when we rewrite the increment density
共35c兲 in terms of ⌺0 coordinates. Using
016124-5
PHYSICAL REVIEW E 71, 016124 共2005兲
J. DUNKEL AND P. HÄNGGI
共u␣兲 = 共− ␥c, ␥v兲,
␥−1 =
冑
v2
,
c2
1−
共w␣兲 = 共w0,w兲,
共39兲
we find
P1+1„w␣共t兲… = c
冉
␥
4␲D dt
冊 冉
1/2
exp −
w共t兲2 − w0共t兲2
4D dt/␥
冊
⫻␦„c␥w0共t兲 − ␥vw共t兲….
III. DERIVATION OF CORRESPONDING
FOKKER-PLANCK EQUATIONS
共40兲
As we already pointed out earlier, the ␦ function in 共40兲
reflects the fact that the energy increment w0 is coupled to
the spatial 共momentum兲 increment w via
0 = u ␣w ␣ = − c ␥ w 0 + ␥ v w
⇒
w0 =
III, it will become clear that, for example, choosing p = p共t兲
in Eqs. 共46兲 would be consistent with an Ito-interpretation
关20,33,34兴 of the stochastic differential equation 共38a兲. However, we will also see that alternative interpretations lead to
reasonable results as well.
vw
.
c
The objective in this part is to derive relativistic FokkerPlanck equations 共FPE兲 for the momentum density f共t , p兲 of
a free particle in the laboratory frame ⌺0. Before we deal
with this problem in Sec. III B, it is useful to briefly recall
the nonrelativistic case.
A. Nonrelativistic case
共41兲
Consider the nonrelativistic Langevin equation 共1b兲
Hence, w0 can be eliminated from the Langevin equations
共38b兲, yielding
dE = − ␯ pvdt + vw共t兲 = vdp.
共42兲
Using the identity
v=
cp
cp
冑m2c2 + p
dp
2
Thus, in the laboratory frame ⌺0 the relativistic Brownian
motion is completely described by the Langevin equation
共38a兲 already. If we assume that the Brownian particle has
fixed initial momentum p共0兲 = p0 or initial velocity v共0兲 = v0,
respectively, then the formal solution of 共38a兲 reads 共关20兴
Chap. IX.1兲
p共t兲 = p0e−␯t + e−␯t
冕
e␯sw共s兲.
共45兲
The stochastic process 共45兲 is determined by the marginal
distribution P1(w共t兲), defined in Eq. 共37兲. Performing the
integration over the ␦ function in 共40兲, we find
where
冉
1
4␲D␥ dt
exp −
冉
冊
⳵
⳵
⳵
f=
␯ pf + D f ,
⳵p
⳵t
⳵p
冊 冉
1/2
冋 册 冋
v2
c2
exp −
−1/2
= 1+
共47b兲
共48兲
whose stationary solution is the Maxwell distribution
f共p兲 =
冉 冊 冉 冊
␯
2␲D
1/2
exp −
␯ p2
.
2D
共49兲
冊
w共t兲2
,
4D␥ dt
共46a兲
We next discuss three different relativistic Fokker-Planck
equations for the momentum density f共t , p兲, related to the
stochastic processes defined by 共38a兲 and 共46兲.
Our starting point is the relativistic Langevin equation
共38a兲, which holds in the laboratory frame ⌺0 共i.e., in the rest
frame of the heat bath兲. Next we define a stochastic process
by
y共t兲 =
␥= 1−
冊
dt
L共t兲2 .
4D
B. Relativistic case
0
P1„w共t兲… =
1/2
As is well known 关20,35兴, the related momentum probability
density f共t , p兲 is governed by the Fokker-Planck equation
共44兲
t
冉 冊 冉
dt
4␲D
P„L共t兲… =
E共t兲 = 冑m2c4 + p共t兲2c2 .
⇒
共47a兲
where p共t兲 = mv共t兲 denotes the nonrelativistic momentum,
and, in agreement with 共3c兲, the Langevin force L共t兲 is distributed according to
共43兲
冑m2c2 + p2 ,
we can further rewrite 共42兲 as
dE =
dp
= − ␯ p + L共t兲,
dt
p2
m 2c 2
册
1/2
.
共46b兲
On the basis of Eqs. 共38a兲 and 共46兲 one can immediately
perform computer simulations, provided one still specifies
the rules of stochastic calculus, i.e., which value of p is to be
taken to determine ␥ in 共46兲. In Sec. IV several numerical
results are presented. Before, it is useful to consider in more
detail the Fokker-Planck equations of the relativistic Brownian motion in the laboratory frame ⌺0. By doing so in Sec.
w共t兲
冑␥
共50兲
,
and using 共46b兲, we can rewrite 共38a兲 as
dp = − ␯ pdt + 冑␥y共t兲,
共51a兲
where y共t兲 is distributed according to the momentumindependent density
P1y 关y共t兲兴 =
冉
1
4␲D dt
冊 冉
1/2
exp −
冊
y共t兲2
.
4D dt
共51b兲
Thus, instead of the increments w共t兲, which implicitly de-
016124-6
THEORY OF RELATIVISTIC BROWNIAN MOTION: …
PHYSICAL REVIEW E 71, 016124 共2005兲
pend on the stochastic process p via Eqs. 共46兲, we consider
ordinary p-independent white noise y共t兲, determined by
共51b兲, from now on. Due to the multiplicative coupling of
y共t兲 in 共51a兲, we must next specify rules for the “multiplication with white noise” 关note that, on viewing Eqs. 共11兲, 共35兲,
and 共36兲 as postulates of the relativistic Brownian motion, all
above considerations remain valid, independent of this specification兴.
In Secs. III B 1, III B 2, and III B 3, we shall discuss three
popular multiplication rules, which go back to proposals
made by Hänggi and Thomas 关19,42兴, Van Kampen 关20兴, Ito
关33,34兴, Stratonovich 关36,37兴, Fisk 关38,39兴, Hänggi 关40,41兴,
and Klimontovich 关31兴. As it is well-known from 关19,20,30兴,
these different interpretations of the stochastic process 共51兲
result in different Fokker-Planck equations, i.e., the Langevin
equation 共51兲 per se does not uniquely determine the corresponding Fokker-Planck equation; it is the stochastic interpretation of the multiplicative noise that matters from a
physical point of view.
Nevertheless, the three approaches discussed below have
in common that, formally, the related Fokker-Planck equation can be written as a continuity equation 关42兴
⳵
⳵
f共t,p兲 + j共t,p兲 = 0,
⳵t
⳵p
共52兲
but with different expressions for the probability current
j共t , p兲. It is worthwhile to anticipate that only for the HänggiKlimontovich approach 共see Sec. III B 3兲 the current j共t , p兲
takes such a form that the stationary distribution of 共52兲 can
be identified with Jüttner’s relativistic Maxwell distribution 关24兴.
冕
␥ = ␥„p共t兲…,
where as before
冉
␥共p兲 = 1 +
p2
m 2c 2
冉
f I共p兲 = CI 1 +
冋
1/2
冊 冉 冑
−1/2
k BT ⬅
exp − ␤
1+
冊
p2
,
c m2
2
共57兲
␯ m 2c 2
.
D
共58兲
mc2 D
=
,
␤
m␯
共59兲
with kB denoting the Boltzmann constant. Put differently, the
parameter ␤ = mc2 / 共kBT兲 measures the ratio between rest
mass and thermal energy of the Brownian particle.
2. Stratonovich approach
According to Stratonovich, the coefficient before y共t兲 in
共51a兲 is to be evaluated with the midpoint discretization rule,
i.e.,
␥=␥
冉
冊
p共t兲 + p共t + dt兲
.
2
共60兲
This choice leads to a different expression for the current
关19,36–38兴, namely,
冋
jS共p,t兲 = − ␯ pf + D冑␥共p兲
册
⳵冑
␥共p兲f .
⳵p
共61兲
This Stratonovich-Fisk current jS vanishes identically for
f S共p兲 =
CS
冋 冕
冑␥共p兲 exp
冉
f S共p兲 = CS 1 +
册
dp
共56兲
2
The dimensionless parameter ␤ can be used to define the
scalar temperature T of the heat bath via the Einstein relation
.
⳵
␥共p兲f .
⳵p
冋 冕
␯
CI
exp −
␥共p兲
D
p2
,
c m2
1+
−
␯
D
dp
册
p
,
␥共p兲
共62兲
and, by virtue of 共56兲, the explicit stationary solution of Stratonovich’s Fokker-Planck equation reads
册
p
,
␥共p兲
p2
m 2c 2
冊 冉 冑
−1/4
exp − ␤
1+
冊
p2
. 共63兲
c 2m 2
3. Hänggi-Klimontovich approach
共54兲
The related relativistic Fokker-Planck equation is obtained
by inserting this current into the conservation law 共52兲. The
current 共54兲 vanishes identically for
f I共p兲 =
p2
m 2c 2
␤=
Ito’s choice leads to the following expression for the current
关19,20,33,34兴:
jI共p,t兲 = − ␯ pf + D
冑
where
共53兲
冊
p
= c 2m 2
␥共p兲
we find the following explicit representation of 共55兲:
1. Ito approach
According to Ito’s interpretation of the Langevin equation
共51a兲, the coefficient before y共t兲 is to be evaluated at the
lower boundary of the interval 关t , t + dt兴, i.e., we use the prepoint discretization rule
dp
共55兲
where CI is the normalization constant. Consequently, f I共p兲
is a stationary solution of the Fokker-Planck equation. In
view of the fact that
Now let us still consider the Hänggi-Klimontovich stochastic integral interpretation, sometimes referred to as the
transport form 关40–42兴 or also as the kinetic form 关31兴. According to this interpretation, the coefficient in front of y共t兲
in 共51a兲 is to be evaluated at the upper boundary value of the
interval 关t , t + dt兴; i.e., within the postpoint discretization we
set
␥ = ␥„p共t + dt兲….
共64兲
This choice leads to the following expression for the current
关31,41,42兴:
016124-7
PHYSICAL REVIEW E 71, 016124 共2005兲
J. DUNKEL AND P. HÄNGGI
冋
jHK共p,t兲 = − ␯ pf + D␥共p兲
册
⳵
f .
⳵p
The current jHK vanishes identically for
冋 冕
f HK共p兲 = CHK exp −
␯
D
dp
册
p
,
␥共p兲
共65兲
共66兲
and, by virtue of 共56兲, the stationary solution explicitly reads
冉 冑
f HK共p兲 = CHK exp − ␤
1+
冊
p2
.
c m2
2
共67a兲
Using the temperature definition in 共59兲 and the relativistic
kinetic energy formula E = 冑m2c4 + p2c2, one can further rewrite 共67a兲 in a more concise form as
冉 冊
f HK共p兲 = CHK exp −
E
.
k BT
共67b兲
The distribution function 共67兲 is known as the relativistic
Maxwell distribution. It was first obtained by Jüttner 关24兴
back in 1911. Pursuing a completely different line of reasoning, he found that 共67兲 describes the velocity distribution of
the noninteracting relativistic gas 共see also 关43兴兲. In contrast
to our approach, which started out with constructing the relativistic generalization of the Langevin equations, Jüttner’s
derivation started from a maximum-entropy-principle for the
gas.
By comparing 共55兲, 共63兲, and 共67a兲 one readily observes
that the stationary solutions f I/S differ from the Jüttner function f HK through additional p-dependent prefactors. In order
to illustrate the differences between the different stationary
solutions, it useful to consider the related velocity probability
density functions ␾I/S/HK共v兲, which can be obtained by applying the general transformation law
␾共v兲 ⬅ f„p共v兲…
冏 冏
⳵p
⳵v
共68兲
in combination with
p=
mv
冑1 − v2/c2 .
The determinant factor 兩⳵ p / ⳵v兩 in 共68兲 is responsible for the
fact that the velocity density functions ␾I/S/HK共v兲 are, in fact,
zero if v2 ⬎ c2.
In Fig. 2 we have plotted the probability density functions
␾I/S/HK共v兲 for different values of the parameter ␤. The normalization constants were determined by numerically integrating ␾共v兲 over the interval 关−c , c兴. As one can observe in
Fig. 2共a兲, for large values of ␤, corresponding to lowtemperature values kBT Ⰶ mc2, the density functions
␾I/S/HK共v兲 approach a common Gaussian shape. On the other
hand, for high-temperature values kBT 艌 mc2 the deviations
from the Gaussian shape become essential. The reason is
that, for a 共virtual兲 Brownian ensemble in the hightemperature regime, the majority of particles assumes velocities that are close to the speed of light. It is also clear that in
other Lorentz frames ⌺⬘, which are not rest frames of the
heat bath, the stationary distributions will no longer stay
FIG. 2. Stationary solutions ␾I/S/HK共v兲 of the relativistic FokkerPlanck equations, according to Ito 共I:solid line兲, Stratonovich
共S:dotted兲, and Hänggi-Klimontovich 共HK:dashed-dotted兲. For low
temperatures, i.e., for ␤ Ⰷ 1, a Gaussian shape is approached 关see
共a兲兴. On the other hand, for very high-temperature values, corresponding to ␤ 艋 1, the distributions exhibit a bistable shape, and the
quantitative deviations between ␾I/S/HK共v兲 increase significantly as
␤ → 0.
symmetric around v = 0. Instead, they will be centered around
the nonvanishing ⌺⬘ velocity V⬘ of the heat bath.
An obvious question then arises, which of the above approaches 共Ito, Stratonovich, or Hänggi-Klimontovich兲 is the
physically correct one. We believe that, at this level of analysis, it is impossible to provide a definite answer to this question. Most likely, the answer to this problem requires additional information about the microscopic structure of the heat
bath 共see, e.g., the discussion of Ito-Stratonovich dilemma in
the context of “internal and external” noise as given in Chap.
IX.5 of van Kampen’s textbook 关20兴兲. At this point, it might
be worthwhile to mention that the relativistic Maxwell distribution 共67兲 is also obtained via the transfer probability
method used by Schay, see Eq. 共3.63兲 and 共3.64兲 in Ref. 关3兴,
016124-8
THEORY OF RELATIVISTIC BROWNIAN MOTION: …
PHYSICAL REVIEW E 71, 016124 共2005兲
and that this distribution also results in the relativistic kinetic
theory 关29兴. By contrast, the recent work of Franchi and Le
Jan 关13兴 is based on the Stratonovich approach. From physical insight, however, it is the transport form interpretation of
Hänggi and Klimontovich that is expected to provide the
physically correct description.
IV. NUMERICAL INVESTIGATIONS
The numerical results presented in this section were obtained on the basis of the relativistic Langevin equation 共51兲,
which holds in the laboratory frame ⌺0. For simplicity, we
confined ourselves here to considering the Ito-discretization
scheme with fixed time step dt 共see Sec. III B 1兲. In all simulations we have used an ensemble size of N = 10 000 particles. Moreover, a characteristic unit system was fixed by
setting m = c = ␯ = 1. Formally, this corresponds to using rescaled dimensionless quantities, such as p̃ = p / mc, x̃ = x␯ / c, t̃
= t␯, ṽ = v / c, etc. The simulation time-step was always chosen as dt = 0.001␯−1, and the Gaussian random variables y共t兲
were generated by using a standard random number generator.
A. Distribution functions
In our simulations we have numerically measured the cumulative velocity distribution function F共t , v兲 in the laboratory frame ⌺0. Given the probability density ␾共t , v兲, the cumulative velocity distribution function is defined by
F共t, v兲 =
冕
v
du␾共t,u兲.
共69兲
−c
In order to obtain F共t , v兲 from numerical simulations, one
simply measures the relative fraction of particles with velocities in the interval 关−c , v兴. Figure 3 shows the numerically
determined stationary distribution functions 共squares兲, taken
at time t = 100 ␯−1 and also the corresponding analytical
curves FI/S/HK共v兲. The latter were obtained by numerically
integrating Eq. 共69兲 using the three different stationary density functions ␾I/S/HK共v兲 from Sec. III.
As one can see in Fig. 3共a兲, for low-temperature values
corresponding to ␤ Ⰷ 1, the three stationary distribution functions are nearly indistinguishable. For high temperatures corresponding to ␤ 艋 1, the stationary solutions exhibit significant quantitative differences, 关see Figs. 3共b兲 and 3共c兲兴.
Because our simulations are based on an Ito-discretization
scheme the numerical values 共squares兲 are best fitted by the
Ito solution 共solid line兲. Also note that the quality of the fit is
very good for the parameters chosen in the simulations, and
that this property is conserved over several magnitudes of ␤.
This suggests that numerical simulations of the Langevin
equations provide a very useful tool if one wishes to study
relativistic Brownian motions in more complicated settings
共e.g., in higher dimensions or in the presence of additional
external fields and interactions兲. In this context, it should
again be stressed that the appropriate choice of the discretization rule is especially important in applications to realistic
systems.
FIG. 3. These diagrams show a comparison between numerical
and analytical results for the stationary cumulative distribution
function F共v兲 in the laboratory frame ⌺0. 共a兲 In the nonrelativistic
limit ␤ Ⰷ 1 the stationary solutions of the three different FPE are
nearly indistinguishable. 共b兲–共c兲 In the relativistic limit case ␤ 艋 1,
however, the stationary solutions exhibit deviations from each
other. Because our simulations are based on an Ito-discretization
scheme, the numerical values 共squares兲 are best fitted by the Ito
solution 共solid line兲.
B. Mean-square displacement
In this section we consider the spatial mean-square displacement of the free relativistic Brownian motion. Because
this quantity is easily accessible in experiments, it has played
an important role in the verification of the nonrelativistic
theory.
As before, we consider an ensemble of N-independent
Brownian particles with coordinates x共i兲共t兲 in ⌺0 and initial
conditions x共i兲共0兲 = 0 , v共i兲共0兲 = 0 for i = 1 , 2 , . . . , N. The position mean value is defined as
016124-9
PHYSICAL REVIEW E 71, 016124 共2005兲
J. DUNKEL AND P. HÄNGGI
N
1
x̄共t兲 ⬅
x共i兲共t兲,
N i=1
兺
共70兲
and the related second moment is given by
N
x2共t兲 ⬅
1
关x共i兲共t兲兴2 .
N i=1
兺
共71兲
The empirical mean-square displacement can then be defined
as follows:
␴2共t兲 ⬅ x2共t兲 − „x̄共t兲…2 .
共72兲
Cornerstone results in the nonrelativistic theory of the onedimensional Brownian motion are
lim x̄共t兲 → 0,
t→+⬁
lim
t→+⬁
␴2共t兲
→ 2Dx ,
t
共73a兲
共73b兲
where the constant
Dx =
k BT
D
=
m ␯ m 2␯ 2
共74兲
is the nonrelativistic coefficient of diffusion in coordinate
space 共not to be confused with noise parameter D兲.
It is therefore interesting to consider the asymptotic behavior of the quantity ␴2共t兲 / t for relativistic Brownian motions, using again the Ito-relativistic Langevin dynamics
from Sec. III B 1. In Fig. 4共a兲 one can see the corresponding
numerical results for different values of ␤. As one can observe in this diagram, for each value of ␤, the quantity
␴2共t兲 / t converges to a constant value. This means that, at
least in the laboratory frame ⌺0, the asymptotic mean-square
displacement of the free relativistic Brownian motions increases linearly with t. For completeness, we mention that
according to our simulations the asymptotic relation 共73a兲
holds in the relativistic case, too.
In spite of these similarities between nonrelativistic and
relativistic theory, an essential difference consists of the explicit temperature dependence of the limit value 2Dx. As ilx
, mealustrated in Fig. 4共b兲, the numerical limit values 2D100
sured at time t = 100␯−1, are well fitted by the empirical
formula
Dx =
c2
,
␯共␤ + 2兲
FIG. 4. 共a兲 Mean-square displacement divided by time t as numerically calculated for different ␤-values in the laboratory frame
⌺0 共rest frame of the heat bath兲. As evident from this figure, for the
relativistic Brownian motion the related asymptotic mean-square
displacement grows linearly with t. 共b兲 The coordinate space diffusion constant Dx100共␤兲 was numerically determined at time t
= 100␯−1. The dashed line corresponds to the empirical fitting formula Dx共␤兲 = c2␯−1共␤ + 2兲−1, which reduces to the classical nonrelativistic result Dx ⯝ c2 / 共␯␤兲 = kT / 共m␯兲 for ␤ Ⰷ 2.
⳵
⳵
⳵
cp
f共t,p,x兲 +
f共t,p,x兲 = − jI/S/HK共t,p,x兲,
2
2
2
冑m c + p ⳵x
⳵t
⳵p
共76兲
which might serve as a suitable starting point for such an
analysis. Compared to the relativistic Fokker-Planck equations from Sec. IV, the second term on the left-hand side of
共76兲 is new. In particular, we recover the relativistic FokkerPlanck equations for the marginal density f共t , p兲, see Sec. III
by integrating Eq. 共76兲 over a spatial volume with appropriate boundary conditions. Finally, we mention once again that
also 共76兲, as well as all the other results that have been presented in this section, exclusively refer to the laboratory
frame ⌺0.
共75兲
which reduces to the nonrelativistic result 共74兲 in the limit
case ␤ Ⰷ 2.
We will leave it as an open problem here, to find an analytical justification for the empirically determined formula
共75兲. Instead we merely mention that, on noting 共43兲, the
relativistic Fokker-Planck equations for the full-phase space
density reads
V. CONCLUSION
Concentrating on the simplest case of 1 + 1 dimensions,
we have put forward the Langevin dynamics for the stochastic motion of free relativistic Brownian particles in a viscous
medium 共heat bath兲. Analogous to the nonrelativistic
Ornstein-Uhlenbeck
theory
of
Brownian
motion
关17,19,20,44兴, it was assumed that the heat bath can, in good
approximation, be regarded as homogenous. Based on this
assumption, a covariant generalization of the Langevin equa-
016124-10
THEORY OF RELATIVISTIC BROWNIAN MOTION: …
PHYSICAL REVIEW E 71, 016124 共2005兲
tions has been constructed in Sec. II. According to these
generalized stochastic differential equations, the viscous friction between Brownian particle and heat bath is modeled by
a friction tensor ␯␣␤. For a homogeneous heat bath this friction tensor has the same structure as the pressure tensor of a
perfect fluid 关25兴. In particular, it is uniquely determined by
the value of the 共scalar兲 viscous friction coefficient ␯, measured in the instantaneous rest frame of the particle 共Sec.
II B 1兲. Similarly, the amplitude of the stochastic force is
also governed by a single parameter D, specifying the Gaussian fluctuations of the heat bath, as seen in the instantaneous
rest frame of the particle 共Sec. II B 2兲.
In Sec. II C the relativistic Langevin equations have been
derived in special laboratory coordinates, corresponding to a
specific class of Lorentz frames, in which the heat bath is
assumed to be at rest 共at all times兲. One finds that the corresponding relativistic distribution of the momentum increments now also depends on the momentum coordinate. This
fact is in contrast with the properties of ordinary Wiener
processes 关21,23兴, underlying nonrelativistic standard
Brownian motions with “additive” Gaussian white noise.
However, as shown in Sec. III it is possible to find an equivalent Langevin equation, containing “multiplicative” Gaussian
white noise.
In order to achieve a more complete picture of the relativistic Brownian motion, the corresponding relativistic
Fokker-Planck equations 共FPE兲 have been discussed in Sec.
III 共again with respect to the laboratory coordinates with the
heat bath at rest兲. Analogous to nonrelativistic processes with
multiplicative noise, one can opt for different interpretations
of the stochastic differential equation, which result in different FPE. In this paper, we concentrated on the three most
popular cases, namely, the Ito, the Stratonovich-Fisk, and the
Hänggi-Klimontovich interpretations. We discussed and
compared the corresponding stationary solutions for a free
Brownian particle. It could be established that only the
Hänggi-Klimontovich interpretation is consistent with the
relativistic Maxwell distribution. This very distribution was
derived by Jüttner 关24兴 as the equilibrium velocity distribution of the relativistic ideal gas. Later on, it was also discussed by Schay in the context of relativistic diffusions 关3兴
and by de Groot et al. in the framework of the relativistic
kinetic theory 关29兴.
In Sec. IV we presented numerical results, obtained on the
basis of an Ito prepoint discretization rule. The simulations
indicate that—analogous to the nonrelativistic case—the
relativistic mean-square displacement grows linearly with
关1兴 A. Einstein, Ann. Phys. 共Leipzig兲 17, 891 共1905兲.
关2兴 A. Einstein, Ann. Phys. 共Leipzig兲 18, 639 共1905兲.
关3兴 G. Schay, Ph.D. thesis, Princeton University 共1961兲, available
through University Microfilms, Ann Arbor, Michigan, https://
wwwlib.umi.com
关4兴 R. M. Dudley, Arkiv Feur Matematik 6, 241 共1965兲.
关5兴 R. Hakim, J. Math. Phys. 6, 1482 共1965兲.
关6兴 U. Ben-Ya’acov, Phys. Rev. D 23, 1441 共1981兲.
the laboratory coordinate time; the temperature dependence
of the related spatial diffusion constant, however, becomes
more intricate. In principle, the numerical results suggest that
simulations of the Langevin equations may provide a very
useful tool for studying the dynamics of relativistic Brownian particles. In this context it has to be stressed that an
appropriate choice of the discretization rule is especially important in applications to realistic physical systems. If, for
example, agreement with the kinetic theory 关29兴 is desirable,
then a postpoint discretization rule should be used.
From the methodical point of view, the systematic relativistic Langevin approach of the present paper differs from
Schay’s transition probability approach 关3兴 and also from the
techniques applied by other authors 关4,6,7兴. As we shall discuss in a forthcoming contribution, the above approach can
easily be generalized to settings that are more relevant with
regard to experiments 关such as the 共1 + 3兲-dimensional case,
the presence of additional external force fields, etc.兴.
With regard to future work, several challenges remain to
be solved. For example, one should try to derive an analytic
expression for the temperature dependence of the spatial diffusion constant. A suitable starting point for such studies
might be the FPE for the full-phase space density given in
Eq. 共76兲. Another possible task consists of finding explicit
exact or at least approximate time-dependent solutions of the
relativistic FPE. Furthermore, it seems also interesting to
consider extensions to general relativity, as, to some extent,
recently discussed in the mathematical literature 关13兴. In this
context, the physical consequences of the different interpretations 共Ito versus Stratonovich versus HänggiKlimontovich兲 become particularly interesting.
Note added in proof. Recently, we have been informed by
F. Debbasch about two interesting recent papers 关45,46兴 on a
relativistic generalization of the Ornstein-Uhlenbeck process.
These two items are related in spirit to the present work: The
authors of those references have postulated a relativistic
Langevin equation with additive noise and a drift term that
differs from ours; but which also yields the correct relativistic Jüttner distribution. Thus, our HK-approach and theirs
possess the same stationary solution, but notably do exhibit a
different relaxation dynamics.
ACKNOWLEDGMENTS
J. D. would like to thank S. Hilbert, L. Schimansky-Geier,
and S. A. Trigger for helpful discussions.
关7兴
关8兴
关9兴
关10兴
O. Oron and L. P. Horwitz 共2003兲, e-print math-ph/0312003.
T. H. Boyer, Phys. Rev. D 19, 1112 共1979兲.
T. H. Boyer, Phys. Rev. D 19, 3635 共1979兲.
F. Guerra and P. Ruggiero, Lett. Nuovo Cimento Soc. Ital. Fis.
23, 529 共1978兲.
关11兴 L. M. Morato and L. Viola, J. Math. Phys. 36, 4691 共1995兲;
37, 4769共E兲 共1996兲.
关12兴 A. Posilicano, Lett. Math. Phys. 42, 85 共1997兲.
016124-11
PHYSICAL REVIEW E 71, 016124 共2005兲
J. DUNKEL AND P. HÄNGGI
关13兴 J. Franchi and Y. Le Jan 共2004兲, e-print math.PR/0403499.
关14兴 A. Einstein, Ann. Phys. 共Leipzig兲 17, 549 共1905兲.
关15兴 A. Einstein and M. von Smoluchowski, Untersuchungen über
die Theorie der Brownschen Bewegung/Abhandlungen über
die Brownsche Bewegung und verwandte Erscheinungen, 3rd
ed. 共Harri Deutsch, Frankfurt, 1999兲, Vol. 199.
关16兴 M. C. Wang and G. E. Uhlenbeck, Rev. Mod. Phys. 17, 323
共1945兲.
关17兴 G. E. Uhlenbeck and L. S. Ornstein, Phys. Rev. 36, 823
共1930兲.
关18兴 S. Chandrasekhar, Rev. Mod. Phys. 15, 1 共1943兲.
关19兴 P. Hänggi and H. Thomas, Phys. Rep. 88, 207 共1982兲.
关20兴 N. G. Van Kampen, Stochastic Processes in Physics and
Chemistry 共North-Holland, Amsterdam, 2003兲.
关21兴 N. Wiener, J. Math. Phys. 58, 131 共1923兲.
关22兴 M. Grigoriu, Stochastic Calculus: Applications in Science and
Engineering 共Birkhäuser, Boston, 2002兲.
关23兴 I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic
Calculus, 2nd ed. Graduate Texts in Mathematics Vol. 113
共Springer-Verlag, New York, 1991兲.
关24兴 F. Jüttner, Ann. Phys. 共Leipzig兲 34, 856 共1911兲.
关25兴 S. Weinberg, Gravitation and Cosmology 共Wiley, 1972兲.
关26兴 R. W. Lindquist, Ann. Phys. 共N.Y.兲 37, 487 共1966兲.
关27兴 J. M. Stewart, Non-Equilibrium Relativistic Kinetic Theory,
Lecture Notes in Physics Vol. 10 共Springer-Verlag, Berlin,
1971兲.
关28兴 R. L. Liboff, Kinetic Theory 共Prentice Hall, Englewood Cliffs,
NJ, 1990兲.
关29兴 S. R. de Groot, W. van Leeuwen, and C. G. van Weert, Relativistic Kinetic Theory: Principles and Applications 共NorthHolland, Amsterdam, 1980兲.
关30兴 S. A. Trigger, Phys. Rev. E 67, 046403 共2003兲.
关31兴 Y. L. Klimontovich, Phys. Usp. 37, 737 共1994兲.
关32兴 U. Erdmann, W. Ebeling, L. Schimansky-Geier, and F. Schweitzer, Eur. Phys. J. B 15, 105 共2000兲.
关33兴 K. Ito, Proc. Imp. Acad. Tokyo 20, 519 共1944兲.
关34兴 K. Ito, Mem. Am. Math. Soc. 4, 51 共1951兲.
关35兴 F. Schwabl, Statistische Mechanik 共Springer, Berlin, 2000兲.
关36兴 R. L. Stratonovich, Vestn. Mosk. Univ., Ser. 1: Mat., Mekh. 1,
3 共1964兲.
关37兴 R. L. Stratonovich, SIAM J. Control Optim. 4, 362 共1966兲.
关38兴 D. Fisk, Ph.D. thesis, Michigan State University, Dept. of Statistics 共1963兲.
关39兴 D. Fisk, Trans. Am. Math. Soc. 120, 369 共1965兲.
关40兴 P. Hänggi, Helv. Phys. Acta 51, 183 共1978兲.
关41兴 P. Hänggi, Helv. Phys. Acta 53, 491 共1980兲.
关42兴 P. Hänggi and H. Thomas, Phys. Rep. 88, 207 共1982兲, see pp.
292–294.
关43兴 J. L. Synge, The Relativistic Gas 共North-Holland, Amsterdam,
1957兲.
关44兴 P. Hänggi and P. Jung, Adv. Chem. Phys. 89, 239 共1995兲.
关45兴 F. Debbasch, K. Mallick, and J. P. Rivet, J. Stat. Phys. 88, 945
共1997兲.
关46兴 F. Debbasch, J. Math. Phys. 45, 2744 共2004兲.
016124-12