PHYSICAL REVIEW E 74, 051106 共2006兲
Relativistic Brownian motion: From a microscopic binary collision model
to the Langevin equation
Jörn Dunkel* and Peter Hänggi
Institut für Physik, Universität Augsburg, Theoretische Physik I, Universitätstraße 1, D-86135 Augsburg, Germany
共Received 4 July 2006; published 3 November 2006兲
The Langevin equation 共LE兲 for the one-dimensional relativistic Brownian motion is derived from a microscopic collision model. The model assumes that a heavy pointlike Brownian particle interacts with the lighter
heat bath particles via elastic hard-core collisions. First, the commonly known, nonrelativistic LE is deduced
from this model, by taking into account the nonrelativistic conservation laws for momentum and kinetic
energy. Subsequently, this procedure is generalized to the relativistic case. There, it is found that the relativistic
stochastic force is still ␦ correlated 共white noise兲 but no longer corresponds to a Gaussian white noise process.
Explicit results for the friction and momentum-space diffusion coefficients are presented and discussed.
DOI: 10.1103/PhysRevE.74.051106
PACS number共s兲: 05.40.Jc, 02.50.Ey, 47.75.⫹f
I. INTRODUCTION
II. NONRELATIVISTIC BROWNIAN MOTIONS
The theories of nonrelativistic Brownian motion and special relativity were introduced more than 100 years ago
关1–7兴. Since then, they have become cornerstones for our
understanding of a wide range of physical processes 关8–12兴.
This fact notwithstanding, the unification of both concepts
still poses a theoretical challenge nowadays 共classical references are 关13–18兴; recent contributions include 关19–32兴; potential applications in high-energy physics and astrophysics
are considered in 关33–38兴兲. The relatively slow progress in
this field can be attributed to the severe difficulties that arise
when one tries to describe N-body systems in a relativistically consistent manner 关39–43兴. For this reason, the derivation of relativistic Langevin equations 共LEs兲 from an underlying microscopic model has remained an unsolved issue
until now.1 However, in the present paper we aim to provide
a solution to this problem.
More precisely, by considering quasielastic, binary collisions between the Brownian and heat bath particles 关48,49兴
we are able to treat the heat bath in a fully relativistic manner
without having to account for the exact details of the relativistic N-body interactions. As shown in Sec. II, for a nonrelativistic framework this approach yields the well-known
nonrelativistic LE with Gaussian white noise as well as the
correct fluctuation-dissipation theorem 共the EinsteinSutherland relation 关1兴兲. In Sec. III, the method is transferred
to the relativistic case, leading to the main result of this
paper, the relativistic LE 关32兴. Remarkably, the relativistic
stochastic force is also ␦ correlated 共“white”兲 but no longer
of Gaussian 共or Wiener 关50兴兲 type. Compared with the nonrelativistic Brownian motion, this is the most important difference. Furthermore, we obtain explicit representations of
friction and 共momentum兲 diffusion coefficients in terms of
expectation values with respect to the heat bath distribution
共see also the Appendix兲.
The objective of this section is to recover the well-known
nonrelativistic LEs from a simple microscopic collision
model for Brownian motions. As shown by several authors in
the past 关44–47,51兴, nonrelativistic LEs can also be derived
by considering a bath of harmonic oscillators with canonical
phase space distribution. Unfortunately, it is problematic to
transfer this approach to the relativistic case, because any
instantaneous linear 共or nonlinear兲 interactions between
Brownian and heat particles would violate the basic principles of special relativity 关41,42兴. To circumvent this problem, we will pursue a different method here, using only the
共non兲relativistic microscopic conservation laws for energy
and momentum, respectively, known to hold for elastic
pointlike, binary collisions 共contact interactions 关42兴兲. Conceptually, our approach is related to that of Pechukas 关48兴
and Pechukas and Tsonchev 关49兴, who considered a similar
model in the context of nonrelativistic quantum Brownian
motion 关52兴. Analogous approaches are also known from
unimolecular rate theory; see, e.g., Sec. V in 关53兴.
A. Microscopic model
For the sake of simplicity only, we will restrict ourselves
throughout to the one-dimensional 共1D兲 case. Generalizations to higher space dimensions are in principle straightforward, but certain calculations will become much more
cumbersome 共see the comments at the end of the Appendix兲.
To start out, consider the following situation in the inertial
laboratory frame ⌺0: A large one-dimensional box volume
V ⬅ 关−L / 2 , L / 2兴 contains an ideal nonrelativistic gas, consisting of N small pointlike particles with identical masses m.
The gas particles—referred to as the heat bath hereafter—
surround a Brownian particle of mass M Ⰷ m. Because of
frequent elastic collisions with heat bath particles, the
Brownian particle performs stochastic motions.
1. Heat bath
*Electronic address: joern.dunkel@physik.uni-augsburg.de
1
For the nonrelativistic Brownian motion, this problem was solved
by Bogolyubov 关44兴, Magalinskii 关45兴, Ford et al. 关46兴, and Zwanzig 关47兴, who considered a bath of harmonic oscillators.
1539-3755/2006/74共5兲/051106共10兲
The coordinates and momenta of the heat bath particles
are denoted by xr 苸 关−L / 2 , L / 2兴 and pr 苸 共−⬁ ; ⬁兲, respectively, where r = 1 , . . . , N. As usual, we make the following
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©2006 The American Physical Society
PHYSICAL REVIEW E 74, 051106 共2006兲
JÖRN DUNKEL AND PETER HÄNGGI
simplifying assumption concerning the heat bath. The probability density function 共PDF兲 of the heat bath particles is a
spatially homogeneous Maxwell distribution, i.e., at each
time t ⬎ 0, the PDF reads
f Nb 共x1, . . . ,pN兲 =
冉 冊兿 冉
␭
L
N N
exp −
r=1
冊
pr2
,
2mkT
X⬘ = X + V␶,
冦
E + ⑀ = Ê + ⑀ˆ ,
Here and below, capital letters refer to the Brownian particle
and lower-case letters to particles forming the heat bath;
quantities without 共with兲 carets refer to the state before 共after兲 the collision. In the nonrelativistic case, we have, e.g.,
before the collision
P = MV,
p = mv,
E=
P2
,
2M
⑀=
p2
,
2m
共3兲
where v and V denote the velocities. Taking into account
conservation of both momentum and 共kinetic兲 energy, one
finds that the change ⌬P ⬅ P̂ − P of the Brownian particle’s
momentum per single collision is given by
⌬P =
2M
− 2m
P+
p.
M+m
M+m
共4兲
B. Derivation of the Langevin equation
The total momentum change ␦ P of the Brownian particle
within the time interval ␶ can be written as
N
␦ P共t兲 ⬅ P共t + ␶兲 − P共t兲 = 兺 ⌬PrIr共t, ␶兲,
+ ⌰共xr − X兲⌰共X⬘ − xr⬘兲⌰共V − vr兲,
where X = X共t兲 , xr = xr共t兲, and
共6a兲
共7a兲
with the function C共P兲 = C̃(V共P兲) given by the integral
formula
C̃共V兲 ⬅
1
2
冕
⬁
V
dvr共vr − V兲f̃ 1b共vr兲 +
1
2
冕
V
dvr共V − vr兲f̃ 1b共vr兲.
−⬁
共7b兲
Here, f̃ 1b共vr兲 is the one-particle velocity PDF of a heat bath
particle. We anticipate that Eqs.共6兲 and 共7兲 remain valid in
the relativistic case as well, but then one has to insert the
relativistic bath distribution in Eq. 共7b兲.
However, in order to recover from Eqs. 共5兲–共7兲 the wellknown nonrelativistic LE, we still have to make a number of
simplifying assumptions.
共i兲 We assume that the time interval ␶ is sufficiently small,
so that 兩␦ P / P兩 Ⰶ 1. In particular, ␶ is supposed to be so small
that there occurs at most only one collision between the
Brownian particle and a specific heat bath particle r. On the
other hand, the time interval ␶ should still be large enough
that the total number of collisions within ␶ is larger than 1.
These requirements can be satisfied simultaneously only if
m / M Ⰶ 1.
共ii兲 We assume that collisions occurring within 关t , t + ␶兴
can be viewed as independent events.
共iii兲 Finally, we will 共have to兲 assume that
共5兲
Ir共t, ␶兲 = ⌰共X − xr兲⌰共xr⬘ − X⬘兲⌰共vr − V兲
冧
␶
␶
具Ir共t, ␶兲典b = C̃共V兲 = C共P兲 ,
L
L
具关prIr共t, ␶兲兴 j典b = 具prjIr共t, ␶兲典b ⯝ 具prj典b具Ir共t, ␶兲典b = 具prj典bC共P兲
r=1
where Ir共t , ␶兲 苸 兵0 , 1其 is the indicator function for a collision
with the heat bath particle r during the interval 关t , t + ␶兴;
i.e., Ir共t , ␶兲 = 1 if a collision has occurred, and, otherwise,
Ir共t , ␶兲 = 0. In the 1D case, the collision indicator can be
written explicitly as
x ⬍ 0,
The expectation of the collision indicator with respect to the
bath distribution, denoted by 具Ir共t , ␶兲典b, gives the probability
that the bath particle r collides with the Brownian particle
between t and t + ␶. As shown in the Appendix, in the limit
␶ → 0, one finds
共2兲
P + p = P̂ + p̂.
0,
⌰共x兲 = 1/2, x = 0,
1,
x ⬎ 0.
2. Kinematics of single-collision events
The momentum and energy balance per 共elastic兲 collision
reads
共6b兲
are the projected particle positions at time t + ␶. The
Heaviside function is defined by
共1兲
where k is the Boltzmann constant, T the temperature,
and ␭ = 共2␲mkT兲−1/2. Thus, it is implicitly assumed
that the heat bath particles are independently and identically
distributed; and the collisions with the Brownian particle
do not significantly alter the bath distribution. These
assumptions are justified, if the collisions between the gas
particles rapidly reestablish a spatially homogeneous bath
distribution.
xr⬘ = xr + vr␶
␶
L
共8兲
for j = 1 , 2 , . . .. Given the explicit representation of the indicator function 共6a兲, it is in principle straightforward to check
the quality of the approximation 共8兲, if a bath distribution has
been specified.
As we shall see immediately, the assumptions 共i兲–共iii兲 are
necessary and sufficient for deriving the well-known nonrelativistic LE from Eqs. 共5兲–共7兲. Upon inserting Eq. 共4兲 into
共5兲 and dividing by ␶ we find
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PHYSICAL REVIEW E 74, 051106 共2006兲
RELATIVISTIC BROWNIAN MOTION: FROM A…
冋
册
N
共9兲
The first term on the right-hand side 共RHS兲 in Eq. 共9兲 can be
identified as the “friction” term, whereas the second term
represents “noise.” On the RHS of Eq. 共9兲, it was assumed
that for each collision occurring within 关t , t + ␶兴, the “initial”
momentum of the Brownian particle is approximately equal
to some suitably chosen value P共t⬘兲 with t⬘ 苸 关t , t + ␶兴; cf. the
assumption 共i兲 above and the discussion at the end of this
section.
The next step en route to the conventional LE consists in
replacing the expression in square brackets in Eq. 共9兲 by the
averaged friction coefficient
N
2m
1
具I 共t, ␶兲典b .
␯0共P兲 ⬅ 兺
␶ r=1 m + M r
共10a兲
Since it was assumed that the heat bath particles are
independently and identically distributed, we can rewrite
this as
␯0共P兲 =
N 2m
具I 共t, ␶兲典b ,
␶ m+M r
共10b兲
for some r 苸 兵1 , . . . , N其. The coefficient ␯0 can be interpreted
as an average collision rate weighted by some mass ratio.
Inserting Eq. 共7a兲 into Eq. 共10b兲 yields
␯0共P兲 = nb
2m
C共P兲,
m+M
冉 冊 冋 冉 冊册
冋冉 冊 册
kT
2␲m
+
1/2
exp −
P
erf
2M
m
2kT
m P
2kT M
1/2
P
.
M
具␰共t兲␰共s兲典b =
共8兲
⯝
=
冉 冊兺
冉 冊兺
冉 冊
␦ts 2M
␶2 M + m
2 N
␦ts 2M
␶2 M + m
2 N
共10a兲
冉 冊
共13兲
corresponding to the last term in Eq. 共9兲. The momentum
dependence of the noise enters through the implicit P dependence of the collision indicator functions Ir共t , ␶兲. To keep
subsequent formulas as compact as possible, we shall use the
abbreviation ␰共t兲 ⬅ ␰共P , t兲 in the remainder. Then, averaging
over the bath distribution f Nb and using Eqs. 共8兲, we find for
the mean value
mkT具Ir共t, ␶兲典b
r=1
共14b兲
共15a兲
where ␰共t兲 ⬅ ␰共P , t兲 is a momentum-dependent Gaussian
white noise force, characterized by
具␰共t兲典 = 0,
共15b兲
具␰共t兲␰共s兲典 = 2D0共P兲␦共t − s兲,
共15c兲
with 共momentum-space兲 diffusion coefficient
D0共P兲 =
共12兲
N
2M
1
p I 共t, ␶兲,
␰共P,t兲 ⬅ 兺
␶ r=1 M + m r r
具pr2Ir2共t, ␶兲典b
Ṗ = − ␯0共P兲P + ␰共t兲,
1/2
corresponds to the commonly used Stokes approximation.
It then remains to analyze the “noise force”
the
r=1
␦ts 2M 2
␯ kT,
␶ M+m 0
In particular, setting 共see the Appendix兲
kT
C共P兲 ⬇ C共0兲 =
2␲m
of
with ␦ts 苸 兵0 , 1其 denoting the Kronecker symbol. To obtain
the second line, we have used that Ir2共t , ␶兲 = Ir共t , ␶兲, and the
simplifying assumption 共8兲 that Ir共t , ␶兲 and pr are 共approximately兲 independent random variables with respect to the
bath distribution.
Similar to Eq. 共14b兲, also the higher correlation functions
are determined by the corresponding moments of the Gaussian marginal bath distribution 共1兲. Thus, under the above
assumptions 共i兲–共iii兲, the nonrelativistic stochastic force ␰共t兲
corresponds to Gaussian white noise 共or a Wiener process
关50兴, respectively兲.
Finally, by substituting ␯0 from Eqs. 共11兲 for the expression in square brackets in Eq. 共9兲 and formally letting ␶
→ 0, we recover from Eq. 共9兲 the well-known nonrelativistic
LE 关51,54,55兴
2
共11b兲
共14a兲
Furthermore, assuming mutual independence
collisions, the correlation function is obtained as
共11a兲
where nb = N / L is the density of the bath particles. In the case
of the Maxwell distribution, we can evaluate the integral
共7b兲, and find
C共P兲 =
具␰共t兲典b = 0.
N
2m
2M
1
1
␦ P共t兲
Ir共t, ␶兲 P + 兺
p I 共t, ␶兲.
⯝−
兺
␶
␶ r=1 m + M
␶ r=1 M + m r r
M2
␯0共P兲kT.
M+m
共15d兲
To obtain Eq. 共15c兲, we used that ␦st / ␶ → ␦共t − s兲 for ␶ → 0,
where ␦共t − s兲 is the Dirac function.
In the limit m / M → 0, Eq. 共15d兲 reduces to the standard
fluctuation-dissipation theorem D0 = M ␯0kT 关1,56兴. However,
␯0 and D0 are constants only if one adopts the Stokes approximation 共12兲 共see the Appendix兲. If one goes beyond the
Stokes approximation, then the noise in Eqs. 共15兲 becomes
multiplicative with respect to P, and, therefore, Eqs. 共15兲
must be complemented by a discretization rule in this case
关54,57–64兴. As discussed in 关54,62–64兴, only for the
post-point-discretization rule, corresponding to the choice
␯0共P兲 = ␯0关P共t + ␶兲兴 and D0共P兲 = D0关P共t + ␶兲兴 on the RHS of
Eq. 共15a兲, one recovers the Maxwellian PDF
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JÖRN DUNKEL AND PETER HÄNGGI
⌽⬁共P兲 =
冉
1
2␲ MkT
冊 冉
1/2
exp −
P2
2MkT
冊
共16兲
as the stationary momentum distribution of the Brownian
particle in the limit t → ⬁ 共assuming that m / M → 0兲.
P = MV␥共V兲,
2uE − 共1 + u2兲P
,
1 − u2
III. RELATIVISTIC BROWNIAN MOTIONS
A. Microscopic model
The basic constituents of the microscopic model are the
same as those outlined in Sec. II A, but in addition we now
have to consider a relativistic heat bath distribution and must
consistently take into account the relativistic collision
kinematics.
共18b兲
where ␥共v兲 ⬅ 共1 − v2兲−1/2. As before, capital letters refer to
the Brownian particle. Inserting Eqs. 共18兲 into the conservation laws 共2兲, and solving for P̂, one finds 关22兴
P̂ =
We shall now apply an analogous reasoning to obtain a
relativistic LE. For this purpose we consider an inertial
共laboratory兲 frame ⌺0 with time coordinate t, as, e.g.,
measured by an atomic clock resting in ⌺0.
E共P兲 = 共M 2 + P2兲1/2 ,
共19兲
where
u共p, P兲 =
P+p
E+⑀
共20兲
is the center-of-mass velocity. Hence, the momentum change
⌬P = P̂ − P of the Brownian particle in a single collision is
given by
2
2
⑀
E
P+
p.
2
2
1−u E+⑀
1−u E+⑀
⌬P = −
共21兲
In the nonrelativistic limit case, where u2 Ⰶ 1, E ⯝ M, and
⑀ ⯝ m, this reduces to Eq. 共4兲.
1. Relativistic heat bath
In the relativistic case, we postulate, analogously to Eq.
共1兲, that with respect to ⌺0 the heat bath distribution is stationary, spatially homogeneous, and independent, so that the
PDF can be written in the product form
B. Derivation of the Langevin equation
Inserting Eq. 共21兲 into Eq. 共5兲, one obtains the relativistic
analog of Eq. 共9兲 as
f Nb 共x1, . . . ,pN兲 = L−N 兿 f 1b共pr兲.
冋
共17a兲
r=1
N
As marginal one-particle momentum PDFs, we will now
consider the ␩-generalized Jüttner-Maxwell distributions
关22,65兴, reading
f 1b共p兲 =
N␩
⑀共p兲
␩
冉 冊
exp −
⑀共p兲
,
kT
␩ 艌 0,
共17b兲
where p 苸 共−⬁ , + ⬁兲, and ⑀共p兲 denotes the relativistic kinetic
energy of a heat bath particle. The normalization constant N␩
is determined by the condition
1=
冕
⬁
dp f 1b共p兲.
2
1
E
+ 兺
p I 共t, ␶兲,
␶ r=1 1 − ur2 E + ⑀r r r
共17c兲
For ␩ = 0, Eq. 共17b兲 reduces to the standard Jüttner-Maxwell
distribution 关65兴. On the other hand, as discussed recently
关22,38兴, the PDF with ␩ = 1 appears to be conserved in relativistic elastic binary collisions. In general, however, the arguments and results presented below remain valid for arbitrary one-particle momentum PDFs f 1b共p兲, i.e., also for
momentum distributions other than the ␩-generalized Jüttner
PDFs 共17b兲.
N
⑀共p兲 = 共m2 + p2兲1/2 ,
1
兺
␶ r=1
␯共P兲 ⬅
=
冓
冓
1−
ur2 E
⑀r
Ir共t, ␶兲
+ ⑀r
2
⑀r
N
I 共t, ␶兲
␶ 1 − ur2 E + ⑀r r
␯共P兲 ⯝
冓
2
⑀r
N
␶ 1 − ur2 E + ⑀r
共7兲
共18a兲
2
冔
冔
b
= nbC共P兲
冓
2
1−
ur2 E
冔
共23兲
,
b
for some r 苸 兵1 , . . . , N其. Next, applying
approximation similar to 共8兲, we obtain
2. Relativistic collision kinematics
Using natural units such that c = 1, the relativistic kinetic
energy, momentum, and velocity are related by
共22兲
where ur ⬅ u共pr , P兲 and ⑀r ⬅ ⑀共pr兲. Formally, the collision
indicator Ir共t , ␶兲 is still determined by Eqs. 共6兲 and 共7兲,
but differences arise due to the fact that we have to
use V = P / 共M 2 + P2兲1/2 and a relativistic bath distribution
now.
Analogous to the nonrelativistic case, we can identify the
first term on the RHS of Eq. 共22兲 as friction, and introduce
an averaged friction coefficient by
−⬁
p = mv␥共v兲,
册
N
2
1
␦ P共t兲
⑀r
⯝−
I 共t, ␶兲 P
兺
␶
␶ r=1 1 − ur2 E + ⑀r r
N
a
product
具Ir共t, ␶兲典b
b
⑀r
+ ⑀r
冔
,
共24兲
b
where nb = N / L is the density of the heat bath particles, and
C共P兲 is determined by Eq. 共7b兲. Figure 1 shows the P de-
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FIG. 1. The momentum-dependent, relativistic friction coefficient ␯共P兲, divided by the total mean collision rate, ␯共P兲 / 关nbC共P兲兴,
as calculated numerically for two different heat bath distributions
f 1b共p兲 and two different bath temperatures, is depicted versus the
scaled momentum P. The solid lines refer to the standard Jüttner
distribution with ␩ = 0, and the dotted lines to ␩ = 1 in Eq. 共17b兲. 共a兲
Weakly relativistic heat bath. In the limit kT Ⰶ mc2 the bath distributions 共17b兲 approach a Maxwellian, and therefore the results
for different ␩ practically coincide. In particular, for P = 0 the
nonrelativistic result is recovered. 共b兲 Strongly relativistic heat bath.
The friction coefficient increases with the temperature of the heat
bath.
pendence of ␯共P兲 / 关nbC共P兲兴 for the bath distributions from
Eq. 共17b兲. This momentum dependence is induced by the
appearance of ur = u共pr , P兲 and E = E共P兲 in the expectation
value on the RHS of Eq. 共24兲. Furthermore, the shape of the
one-particle collision coefficient C共P兲 = 具Ir共t , ␶兲典bL / ␶ is depicted in Fig. 2. As one would intuitively expect, the friction
coefficient grows with the temperature T of the heat bath 共at
constant P兲 as well as with the absolute momentum of the
Brownian particle 共at constant T兲. In the nonrelativistic limit
case, where u2 Ⰶ 1, E ⯝ M, and ⑀ ⯝ m, the relativistic friction
coefficient ␯共P兲 from Eq. 共24兲 reduces to the nonrelativistic
result ␯0共P兲 from Eq. 共11兲.
At this point, it might be worthwhile to emphasize once
again that product approximations of the form
具G共xr,pr兲Ir共t, ␶兲典b ⯝ 具G共xr,pr兲典b具Ir共t, ␶兲典b ,
共25兲
as employed in Eq. 共8兲 and also in the first line of Eq. 共24兲,
can in principle be omitted by using the explicit representation 共6兲 of the collision indicator and Eq. 共A6兲 of the Appendix; if one opts to avoid such approximations then the accuracy of the Langevin model increases 共note that this
statement applies to the nonrelativistic case, too兲. However,
in the following we shall continue to use Eq. 共25兲 in order to
obtain a relativistic LE that is on an equal footing with the
nonrelativistic LE 共15兲.
For this purpose, we interpret the second term on the RHS
of Eq. 共22兲 as “noise,” defining
FIG. 2. Relativistic one-particle collision coefficient C共P兲
= 具Ir共t , ␶兲典bL / ␶, numerically calculated for the same parameters as
in Fig. 1. The solid lines refer to a standard Jüttner distribution with
␩ = 0, and the dotted lines to ␩ = 1 in Eq. 共17b兲. 共a兲 Weakly relativistic heat bath. At small temperatures, the zero-value C共0兲 is approximately equal to the nonrelativistic Stokes value 冑kT / 共2␲m兲.
共b兲 Strongly relativistic heat bath. For 兩P兩 → ⬁ the coefficient C共P兲
converges to 1 / 2.
N
2
1
E
␹共P,t兲 ⬅ 兺
p I 共t, ␶兲.
␶ r=1 1 − ur2 E + ⑀r r r
共26兲
Averaging over the bath distribution f Nb , one finds for the
mean value
␮共P兲 ⬅ 具␹共P,t兲典b =
共25兲
⯝
冓
冓
2
E
N
p I 共t, ␶兲
␶ 1 − ur2 E + ⑀r r r
2
N
E
p
␶ 1 − ur2 E + ⑀r r
⯝ nbC共P兲
冓
2
1−
ur2 E
冔
冔
b
具Ir共t, ␶兲典b
b
E
pr
+ ⑀r
冔
.
共27兲
b
In contrast to the nonrelativistic case, the mean value ␮ of
the relativistic Langevin force ␹共t兲 ⬅ ␹共P , t兲 depends on the
momentum P of the Brownian particle. This can be attributed to the appearance of ur2 = 共P + pr兲2 / 共E + ⑀r兲2 in Eq. 共27兲.
As shown in Fig. 3, the quantity ␮共P兲 / 关nbC共P兲兴 is positive
for P ⬎ 0 and negative for P ⬍ 0. Thus, on average, the relativistic stochastic force tends to accelerate particles in the
direction of their motion, but this effect is compensated by
the increase of the friction coefficient ␯共P兲 at high values of
P 共cf. Fig. 1兲.
Let us next take a closer look at the covariance function
␴ts ⬅ 具关␹共t兲 − 具␹共t兲典b兴关␹共s兲 − 具␹共s兲典b兴典b .
共28兲
In the nonrelativistic case, the stochastic force possesses
a vanishing mean value 具␰共t兲典b = 0. According to Eq. 共27兲,
this is no longer the case for the relativistic noise
051106-5
PHYSICAL REVIEW E 74, 051106 共2006兲
JÖRN DUNKEL AND PETER HÄNGGI
␦ts N 2
␦ts
具␬r 典b具Ir共t, ␶兲典b − ␮2共P兲
␶ ␶
N
␴ts ⯝
共7a兲
=
␦ts
␦ts
nbC共P兲具␬r2典b − ␮2共P兲.
␶
N
共30兲
The last term vanishes, if we consider the thermodynamic
limit 共TDL兲 of an infinite heat bath, i.e., N , L → ⬁ such that
nb = N / L = const. Thus, reinserting the explicit expression for
␬r, we obtain in this limit
␴ts →
FIG. 3. Mean value of the relativistic stochastic force
␮共P兲 ⬅ 具␹共t兲典b calculated numerically for two different heat bath
distributions f 1b共p兲 and two different bath temperatures. Solid lines
refer to a standard Jüttner distribution with ␩ = 0, and dotted lines to
␩ = 1 in Eq. 共17b兲.
force 具␹共t兲典b ⫽ 0. In order to explicitly calculate ␴ts for
the relativistic case, it is convenient to introduce the
abbreviation
␬r =
共27兲
共25兲
冓冉 兺
冉兺
E
pr .
2
1 − ur E + ⑀r
N
1
␬ I 共t, ␶兲
␶ r=1 r r
冊
2
− ␮2共P兲
冔
具␬r2典b具Ir共t, ␶兲典b
N
⫻ 兺 兺 具␬r典b具Ir共t, ␶兲典b具␬ j典b具I j共t, ␶兲典b
r=1 j⫽r
N
冉
N
+ 具Ir共t, ␶兲典b具␬r典b 兺 具␬ j典b具I j共t, ␶兲典b
共27兲
=
N
冉
j⫽r
冊冔
2
共31兲
.
b
冓
2
1−
ur2 E
⑀r
+ ⑀r
冔
共32a兲
b
for the term in square brackets in Eq. 共22兲, imposing the
TDL for the bath, and letting ␶ → 0 in Eq. 共22兲, we obtain the
relativistic LE
Ṗ = − ␯共P兲P + ␹共t兲,
共32b兲
␮共P兲 ⬅ 具␹共t兲典b = nbC共P兲
冓
2
E
pr
1 − ur2 E + ⑀r
冔
共32c兲
b
and the covariance
with the 共momentum-space兲 diffusion coefficient given by
D共P兲 =
冊
nb
C共P兲
2
冓冉
2
1−
ur2 E
E
pr
+ ⑀r
冊冔
2
.
共32e兲
b
In Fig. 4 the ratio D共P兲 / 关nbC共P兲兴 is plotted for the same
parameters as in Figs. 1 and 3. As it is evident from the
diagrams, this quantity increases with temperature T and
absolute momentum P of the Brownian particle.
冊
IV. SUMMARY
N
冊
␶
␦ts
␶2 2
␶
2
具
␬
典
具I
共t,
␶
兲典
−
␮ 共P兲 + ␮共P兲 兺 ␮共P兲 .
兺
b
r
b
r
2
␶ r=1
N
N
j⫽r N
共29兲
From this, we obtain
␯共P兲 = nbC共P兲
2
␦ts
␶2
= 2 兺 具␬r2典b具Ir共t, ␶兲典b − ␮2共P兲
␶ r=1
N
E
pr
+ ⑀r
共32d兲
b
r=1
N
1−
ur2 E
In principle, any higher correlation function can be calculated in the same manner. It is also evident that the noise
force is non-Gaussian, because the relativistic bath distribution f b共pr兲 that determines the averages 具·典b—and, thus, the
noise correlations—is non-Gaussian.
Finally, by substituting the averaged friction coefficient
− ␶ ␮ 共P兲
2
2
␴共t,s兲 ⬅ 具关␹共t兲 − 具␹共t兲典b兴关␹共s兲 − 具␹共s兲典b兴典b = 2D共P兲␦共t − s兲,
N
␦ts
⯝ 2
␶
冓冉
where, in view of the approximation 共25兲, the non-Gaussian
momentum-dependent noise force ␹共t兲 ⬅ ␹共P , t兲 is characterized by the mean
2
Assuming, as before, that collisions can be viewed as
independent events, the correlation function 共28兲 vanishes at
nonequal times t ⫽ s, and we thus find
␴ts ⯝ ␦ts
␦ts
n C共P兲
␶ b
We conclude the derivation of the relativistic LE with a
set of general remarks.
共i兲 While deriving the relativistic LE 共32兲, we made use of
the stationarity, independence, and homogeneity of the bath
distribution 共17a兲; we did not, however, rely on the specific
properties of the marginal momentum PDF. Hence, the above
results hold true for arbitrary one-particle momentum
distributions f 1b共p兲.
051106-6
PHYSICAL REVIEW E 74, 051106 共2006兲
RELATIVISTIC BROWNIAN MOTION: FROM A…
FIG. 4. Relativistic diffusion coefficients D共P兲 calculated numerically for different heat bath distributions f 1b共p兲 for 共a兲 weakly
and 共b兲 strongly relativistic heat. The solid lines refer to a standard
Jüttner distribution with ␩ = 0, and the dotted lines to ␩ = 1 in Eq.
共17b兲.
共ii兲 In order to be able to use the LEs 共32兲 derived above,
one still needs to calculate the mean collision rate C共P兲 / L,
which is determined by Eq. 共7b兲; see the Appendix. We also
emphasize once again that the approximation 共25兲, leading to
the appearance of C共P兲, can in principle be omitted 共in the
nonrelativistic as well as in the relativistic case兲. More precise results for friction coefficients and noise correlations can
then be extracted from Eq. 共A6兲 in the Appendix.
共iii兲 The stochastic force ␹共t兲 in Eqs. 共32兲 is ␦ correlated
共memory-free兲, but non-Gaussian; i.e., in order to completely
specify the stochastic process one actually has to determine
all higher order correlation functions. This is practically unfeasible. Therefore, in numerical studies and/or practical applications, one could use a Gaussian approximation 共GA兲 of
Eqs. 共32兲, obtained in the following manner. We rewrite Eq.
共32b兲 equivalently as
Ṗ = − ¯␯共P兲P + 冑2D共P兲¯␨共t兲,
共33a兲
where
¯␯共P兲 ⬅ ␯共P兲 −
␹共t兲 − ␮共P兲
␮共P兲 ¯
, ␨共t兲 ⬅
冑2D共P兲 .
P
共33b兲
Reminiscent of standardized Gaussian white noise, the effective noise force ¯␨共t兲 is characterized by
具¯␨共t兲典 = 0,
具¯␨共t兲¯␨共s兲典 = ␦共t − s兲,
共33c兲
but the higher moments are non-Gaussian. Accordingly, the
GA is achieved by replacing ¯␨共t兲 in Eq. 共33a兲 with standardized momentum-independent Gaussian white noise ␨共t兲. The
resulting stochastic differential equation is a standard LE
with multiplicative Gaussian white noise. Hence, after having specified a discretization rule, one can easily write down
the corresponding Fokker-Planck equation as well as the
corresponding stationary distribution 关64兴.
The GA obtained in this way neglects higher-order cumulants of the noise, so it cannot be expected that the “truncated” LE yields exactly the same relaxation behavior and/or
the same stationary solution as the full relativistic LE 共32兲
关63,66兴. Nevertheless, this approximation should provide
useful estimates. In particular, if the stationary momentum
distribution of the Brownian particle can be guessed by other
arguments 关22兴, then the GA can be made self-consistent
with respect to this distribution by fixing a suitably generalized Einstein relation for the friction and noise coefficients.
In this case it suffices to calculate, e.g., ¯␯共P兲, because the
corresponding noise amplitude D̄共P兲 is then uniquely determined by the Einstein relation 共i.e., by the stationary
distribution兲.
Due to the multiplicative noise coupling, the results obtained from the GA will also depend on the choice of the
discretization rule 关54,57–64兴. Loosely speaking, this discretization dilemma is the price that one has to pay for mapping the large number of collisions between t and t + ␶ onto a
single instant of time. Our experience with the nonrelativistic
LE 共cf. remarks at the end of Sec. II B兲 suggests that the
“transport” or “kinetic” interpretation, corresponding to the
post-point-discretization rule 关54,62–64兴, should be preferable in the relativistic case as well.
共iv兲 In principle, it should be straightforward to generalize
the above approach to higher space dimensions, by expressing the momentum vector after the collision, P̂, in terms of
the momenta before the collision, P and p, analogous to Eq.
共21兲. In the 2D or 3D case, complications may arise mostly
due to the fact that one also has take into account the corresponding collision angles and cross sections 共e.g., when determining the collision rates; see the comments at the end of
the Appendix兲.
共v兲 According to our above results, the previously proposed “relativistic” LEs 关21,26,27,30兴 should be viewed as
approximations, which can be fruitful for generating or simulating ensembles of relativistic particles in a simple manner.
It is also evident now why these earlier approaches were
intrinsically limited. Debbasch et al. 关30兴 have postulated
that the relativistic stochastic force in the rest frame of the
bath is ordinary Gaussian white noise with a constant amplitude D, whereas we in our prior work 关21,27兴 assumed
Gaussian noise in the 共comoving兲 rest frame of the Brownian
particle. As follows from the derivation presented here, neither of these assumptions is accurate if one properly takes
into account both the relativistic conservation laws and the
relativistic momentum distributions of the heat bath particles. However, if we consider suitably chosen momentumdependent friction and diffusion coefficients, then the previously proposed relativistic LEs 关21,26,27,30兴 become
equivalent to the Gaussian approximation of the relativistic
LE 共32兲.
ACKNOWLEDGMENTS
The authors would like to thank S. Hilbert and P. Talkner
for helpful discussions.
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JÖRN DUNKEL AND PETER HÄNGGI
⳵Ir
共t,0兲 = 共vr − V兲⌰共X − xr兲␦共xr − X兲⌰共vr − V兲
⳵␶
APPENDIX: CALCULATION OF THE COLLISION RATE
We aim to derive an explicit expression for the expectation value 具Ir共t , ␶兲典b in the limit ␶ → 0, as, e.g., required in
Eqs. 共10兲.
By definition, the function Ir共t , ␶兲 苸 兵0 , 1其 indicates
whether or not the Brownian particle has collided with the
heat bath particle r during the time interval 关t , t + ␶兴. The
positions of the Brownian and heat bath particles at time t are
denoted by X and xr, respectively. Ignoring the possibility of
a collision, for small enough ␶, the new positions at time
t + ␶ would be given by
X⬘ = X + V␶,
xr⬘ = xr + vr␶ ,
Ir共t, ␶兲 = ⌰共X − xr兲⌰共xr⬘ − X⬘兲⌰共vr − V兲
+ ⌰共xr − X兲⌰共X⬘ − xr⬘兲⌰共V − vr兲,
冦
⌰共x兲 = 1/2, x = 0,
1,
x ⬎ 0.
+ ⌰共0兲共V − vr兲␦共X − xr兲⌰共V − vr兲,
and, with ⌰共0兲 = 1 / 2, the useful result
1
⳵Ir
共t,0兲 = 共vr − V兲␦共xr − X兲⌰共vr − V兲
2
⳵␶
1
+ 共V − vr兲␦共X − xr兲⌰共V − vr兲.
2
冧
共A3兲
f̃ 1b共xr, vr兲 =
Irj共t, ␶兲 = Ir共t, ␶兲
Ir共t,0兲 = 0.
共A4b兲
册
⳵Ir
共t,0兲 ␶ .
Ir共t, ␶兲 ⯝
⳵␶
冏
⳵
⌰共xr⬘ − X⬘兲
⳵␶
as required for calculating the mean value of the stochastic
force and its higher correlation functions 关e.g., compare first
line of Eq. 共29兲兴. For small ␶, we may truncate the Taylor
expansion after the linear term, yielding
冓
具G̃共xr, vr兲Ir共t, ␶兲典b ⯝ G̃共xr, vr兲
冏 冏
冓
共A4c兲
+
␶=0
= 兩共vr − V兲␦共xr − X + 共vr − V兲␶兲兩␶=0
= 共vr − V兲␦共xr − X兲,
and, analogously,
冏
⳵
⌰共X⬘ − xr⬘兲
⳵␶
G̃共xr, vr兲
=
冏
=
␶=0
⳵
⌰共X − xr + 共V − vr兲␶兲
⳵␶
冏
1
2L
1
2L
⳵Ir
共t,0兲
⳵␶
␶.
共A6a兲
b
冕
冕
冔
b
⬁
dvr共vr − V兲 ⫻ G̃共X, vr兲f̃ 1b共vr兲
V
V
dvr共V − vr兲 ⫻ G̃共X, vr兲f̃ 1b共vr兲.
共A6b兲
−⬁
具Ir共t, ␶兲典b
=
␶
␶→0
冓 冔
⳵Ir
共t,0兲
⳵␶
1
= C̃共V兲,
L
b
共A7a兲
where
␶=0
C̃共V兲 ⬅
= 兩共V − vr兲␦共X − xr + 共V − vr兲␶兲兩␶=0
1
2
冕
⬁
V
dvr共vr − V兲f̃ 1b共vr兲 +
1
2
冕
V
dvr共V − vr兲f̃ 1b共vr兲.
−⬁
= 共V − vr兲␦共X − xr兲.
Hence, we find
冔
In particular, by choosing G̃共xr , vr兲 ⬅ 1, we find the collision
rate
lim
冏 冏
⳵Ir
共t,0兲
⳵␶
Making use of the result 共A4d兲, the mean value on the RHS
is given by
r
⳵
=
⌰共xr − X + 共vr − V兲␶兲
⳵␶
␶=0
共A5兲
具G̃共xr, vr兲Irj共t, ␶兲典b = 具G̃共xr, vr兲Ir共t, ␶兲典b ,
Accordingly, the Taylor expansion at ␶ = 0 gives
具 ⳵⳵I␶ 共t , 0兲典b, we note that
冎
and some function G̃共xr , vr兲 such that the expectation value
具G̃共xr , vr兲典b exists. We are interested in expectations of the
form
共A4a兲
holds for j = 1 , 2. . .. Furthermore, for ␶ → 0, we have
In order to determine
再
1, xr 苸 关− L/2,L/2兴,
1 1
f̃ 共vr兲
0, xr 苸 关− L/2,L/2兴,
L b
共A4a兲
The first 共second兲 summand in Eq. 共A2兲 refers to the initial
configuration, where the heat bath particle is located at the
left 共right兲 side of the Brownian particle. Let us list some
properties of the collision indicator Ir共t , ␶兲.
First we note that Ir共t , ␶兲 is idempotent, i.e.,
冋
共A4d兲
Now let us consider a spatially homogeneous one-particle
bath distribution of the form
共A2兲
where ⌰共x兲 is the Heaviside function, defined by
x ⬍ 0,
=⌰共0兲共vr − V兲␦共xr − X兲⌰共vr − V兲
共A1兲
where V and vr are the velocities. Then, the indicator
function Ir共t , ␶兲 can be explicitly represented as
0,
+ 共V − vr兲⌰共xr − X兲␦共X − xr兲⌰共V − vr兲,
共A7b兲
The following comments are in order.
051106-8
PHYSICAL REVIEW E 74, 051106 共2006兲
RELATIVISTIC BROWNIAN MOTION: FROM A…
共i兲 The above derivation is valid for both nonrelativistic
and relativistic heat bath distributions f̃ 1b共vr兲. Upon identify-
共iii兲 The Stokes approximation corresponds to setting
V = 0 in Eq. 共A7b兲, yielding
冕
⬁
ing C共P兲 ⬅ C̃(V共P兲), where P is the momentum of the
Brownian particle, we obtain the rigorous justification for
Eq. 共7a兲. However, in the nonrelativistic case we have V
= P / M, whereas in the relativistic case V = P / 共M 2 + P2兲1/2.
Additionally, we note that the support interval of the relativistic velocity distribution f̃ 1b共vr兲 is given by 关−c , c兴, which
determines the effective upper and lower integral boundaries
in Eq. 共A7b兲
共ii兲 Given a certain bath distribution f̃ 1b共vr兲, the exact result 共A6兲 allows for evaluating the quality of the product
approximations 共8兲 and 共25兲, respectively.
This shows that the Stokes approximation is useful for
slow Brownian particles, but inappropriate at high
velocities.
共iv兲 It is in principle possible to apply the same procedure
to higher space dimensions, but then the expression 共A2兲 for
the indicator unction has to be modified accordingly 共e.g.,
by taking into account the geometric shape of the Brownian
particle兲. As a consequence, analytic calculations will
become much more difficult.
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