We use the system-plus-reservoir approach to study the dynamics of a system composed of two independent Brownian particles. We present an extension of the well-known model of a bath of oscillators which is capable of inducing an effective coupling between the two particles depending on the choice made for the spectral function of the bath oscillators. The coupling is nonlinear in the variables of interest, and an exponential dependence on these variables is imposed in order to guarantee the translational invariance of the model if the two particles are not subject to any external potential. The effective equations of motion for the particles are obtained by the Laplace transform method, and, besides recovering all the local dynamical properties for each particle, we end up with an effective interaction potential between them. We explicitly analyze one of its possible forms.9725060

Caldeira, A O

Duarte, O S

Physical Review Letters

English

Repositorio da Producao Cientifica e Intelectual da Unicamp

Effective Coupling between Two Brownian Particles
O. S. Duarte and A. O. Caldeira
Departamento de Fı́sica da Matéria Condensada, Instituto de Fı́sica Gleb Wataghin, Universidade Estadual de Campinas,
CEP 13083-970, Campinas, SP, Brazil
(Received 30 August 2006; published 18 December 2006)
We use the system-plus-reservoir approach to study the dynamics of a system composed of two
independent Brownian particles. We present an extension of the well-known model of a bath of oscillators
which is capable of inducing an effective coupling between the two particles depending on the choice
made for the spectral function of the bath oscillators. The coupling is nonlinear in the variables of interest,
and an exponential dependence on these variables is imposed in order to guarantee the translational
invariance of the model if the two particles are not subject to any external potential. The effective
equations of motion for the particles are obtained by the Laplace transform method, and, besides
recovering all the local dynamical properties for each particle, we end up with an effective interaction
potential between them. We explicitly analyze one of its possible forms.
DOI: 10.1103/PhysRevLett.97.250601 PACS numbers: 05.40.Jc
Usually, the Brownian motion of a given dynamical
variable is modeled by considering the system it de-
scribes—the system of interest—coupled to a thermal
bath responsible for its energy loss. Assuming that any
degree of freedom of this environment is only weakly
perturbed by the system of interest, we can describe it as
a set of independent harmonic oscillators with a coupling
which is bilinear in the reservoir and system variables and
endowed with a particular spectral function [1]. This model
has been successfully used to describe general properties,
classical or quantum mechanical, of dissipative systems
with only 1 degree of freedom subject to arbitrary poten-
tials [1–5]. Indeed, it has been extensively shown in the
literature that, within the range of interest, other ap-
proaches to dealing with dissipative systems described by
a single dynamical variable always furnish us with the
same results as those obtained by the bath of oscillators
with a properly chosen spectral function. It is the case, for
example, of the application of the collective coordinate
method to describing the damped motion of quantum
solitons [6] or microscopic attempts to describe more
realistic systems such as the electron gas of a metallic
environment [7–9].
However, despite all its success, there are certain dis-
sipative systems for which the usual model of the bath of
oscillators can be shown to be inappropriate to account for
the physics we expect from them. Here it should be
stressed that by the usual model we mean independent
oscillators coupled bilinearly in coordinates to the system
of interest.
Suppose one immerses two independent particles in the
same medium, where each of them would separately be-
have as a Brownian particle. Since for each individual
particle we could mimic the effect of the medium by the
bath of oscillators, it would be very natural to try to
generalize the model to cope with the presence of those
two particles. This generalization is quite straightforward,
and the only point where one should be a bit cautious is
when introducing the well-known counterterm [1,2] in the
generalized model. In order to do that in an unambiguous
way, all one has to do is employ the equivalent model for
the system-bath Lagrangian where the interaction is de-
scribed by a velocity-coordinate coupling and perform the
usual canonical transformations to achieve the desired
coordinate-coordinate coupling with the appropriate coun-
terterm [1,2] (see below). It is then a simple matter to
obtain the equations of motion for each particle under the
influence of the environment using, for example, the
Laplace transform method. These are coupled Langevin
equations that, when written in terms of the center of mass
and relative coordinates q and u of the two particles,
present a somewhat bizarre result: namely, the latter obeys
a free particle equation of motion. Therefore, if we give an
initial velocity to one of those particles, the other must
instantaneously reply in such a way that the relative coor-
dinate evolves linearly in time ignoring completely the
presence of the environment. This is a very unexpected
result to say the least.
It is our intention in this Letter to propose an extension
of the usual model of a bath of oscillators in order to fix this
deficiency. As we will see, on top of succeeding in so
doing, we will also be able to describe an effective cou-
pling between the two particles mediated by the presence
of the bath, and the resulting interaction potential depends
on the specific form of the spectral function of the environ-
ment oscillators. This effect reminds us of the formation of
Cooper pairs or bipolarons in material systems due to the
electron-phonon interaction. Notice that we will not ad-
dress effective interactions between localized objects like a
spatially separated two-state system as in Ref. [10],
although our model can also be applied to this kind of
problem.
In our generalization of the usual model, the system of
interest with a single degree of freedom will be represented
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by the free particle Lagrangian
 LS 
1
2M _x
2: (1)
The heat bath will be described as a symmetrized collec-
tion of independent harmonic modes,
 LR 
1
2
X
k
mk _Rk _Rk !
2
kRkRk; (2)
and for the coupling term we initially assume the interac-
tion Lagrangian
 LI  
X
k
~Ckx _Rk: (3)
Switching to the Hamiltonian formulation and perform-
ing the canonical transformation Pk ! mk!kRk and Rk !
Pk=mk!k, we can show that the coupling term transforms
into
 LI  
1
2
X
k
CkxRk  CkxRk 
X
k
CkxCkx
2mk!2k
;
(4)
where Ckx  ~Ckx!k. In order to represent the effect of
a local interaction of the particle with a spatially homoge-
neous environment, we choose
 Ckx  keikx: (5)
With this choice, it is easy to show that the entire system
is translationally invariant and, moreover, the coupling to
the environment is homogeneous. If the particle is dis-
placed by a distance d, the coupling term transforms into
Ckx dRk  CkxeikdRk, which suggests the defi-
nition of a new set of canonical variables as Rk  eikdRk
that renders the total Lagrangian invariant. It is important
to notice that with a coupling like (5), the potential renor-
malization in (4) is a constant and, therefore, does not
contribute to the particle dynamics. A coupling like (5)
appears, for example, in the interaction of a particle with a
fermionic bath [8].
Now we write the Euler-Lagrange equations for all of
the variables involved in the problem, take their Laplace
transforms, and eliminate the bath variables, in order to
find the following equation of motion for the system of
interest:
 M x
Z t
0
Kxt  xt0; t t0 _xt0dt0  Ft: (6)
Here the nonlinear dissipation kernel, given by
 Kr;  
X
k
k2kk
mk!
2
k
coskr cos!k; (7)
shows that the interaction with the thermal bath induces a
systematic influence on the system which is nonlocal and
noninstantaneous. The function Ft can be interpreted as a
fluctuating force
 
Ft  
@
@x
X
k

Ckx ~Rk0  Ckx ~Rk0~~
cos!kt
2
 Ckx _Rk0  Ckx _Rk0~~
sin!kt
2!k

; (8)
where ~Rk  Rk  Ckx0=mk!2k. Ft depends explicitly
on the initial conditions of the bath variables, and its
statistical properties are obtained from the initial state of
the total system. Now we need to choose a suitable distri-
bution of oscillators which, in the continuum limit, leads us
to the Brownian motion. The kernel in (6) can be written as
 
K 
X
k
Z 1
0
d!2k2kk
Im0k !
!
coskxt  xt0
	 cos!t t0; (9)
where Im0k !  =2mk!k!!k is the imagi-
nary part of the dynamical response 0k ! of a harmonic
oscillator with wave number k. If we now assume that the
bath oscillators are, in fact, only approximately noninter-
acting and replace only their response functions by those of
damped oscillators k!, one has
 Imk! 
k!
mk!2 !2k
2 !22k
; (10)
where k is the relaxation frequency of that oscillator.
Our main interest is to study the system for times longer
than the typical time scale of the reservoir. In other words,
we wish to study the low-frequency limit of Eq. (10) in
which Imk! / !, and, therefore, we can assume
 Imk! 
 fk!!: (11)
Here we have, as usual, introduced a high frequency cutoff
 as the characteristic frequency of the bath. A functional
dependence like (11) for the dynamical response of the
bath has been employed in Refs. [8,9] for fermionic envi-
ronments. The particular choice of the dynamical suscep-
tibility of the bath allows us to separate its time and length
scales and to obtain a Markov dynamics when we replace
(11) in (9) and integrate with respect to ! taking the limit
! 1. With these considerations the equation of motion
(6) reads
 M xt   _xt  Ft; (12)
where we have identified  
P
kk
2kkfk. Notice
that, with this modification we obtain a relation between
the damping constant and some microscopic parameters of
the oscillator bath. With the current prescription and sup-
posing that the bath is initially in thermal equilibrium, it is
easy to show that, for high temperatures, the fluctuating
force Ft satisfies the relations hFti  0 and
hFtFt0i  2kBTt t0, which are characteristics
of white noise. Notice that this is valid only if we assume
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a classical distribution of oscillators as the initial state of
the bath.
In conclusion, the system-reservoir model with nonlin-
ear coupling presented here allows us to reproduce the
result one would have obtained by coupling the particle
of interest bilinearly to a bath of noninteracting harmonic
oscillators with the spectral function J!  ! [1,2].
Now we are going to study the dynamics of a system
with 2 degrees of freedom immersed in a dissipative envi-
ronment. In this case, the Lagrangian of the system of
interest is
 LS 
1
2M _x
2
1 
1
2M _x
2
2; (13)
and the coupling term
 LI  
1
2
X
k
Ckx1  Ckx2~~Rk  Ckx1
 Ckx2~~Rk: (14)
Notice that we have not included any counterterm in
(14) since our system is translationally invariant. The
equations of motion for this Lagrangian are then
 
M xi 
Z t
0
Kxit  xit0; t t0 _xit0dt0

Z t
0
Kxit  xjt
0; t t0 _xjt
0dt0

@
@xi
Vxit  xjt  Fit; (15)
where i  j  1; 2, the fluctuating force Fit has the form
given by (8) replacing xt by xit, and
 Vrt  
X
k
Z 1
0
d!2kk
Im0k !
!
coskrt: (16)
In the long-time limit, the second term in (15) can be
written as _xi, providing us with the usual dissipative term.
The third term represents a cross-dissipative term that
depends on the velocity of the second particle and the
relative distance between them. The last term is clearly
an effective interaction induced by the coupling with the
thermal reservoir. To see the explicit form of these terms,
we need to evaluate expressions (9) and (16) with
Im0k ! replaced by (11).
Performing the frequency integrals and assuming a one-
dimensional environment of length L, we can transform the
k summations into integrals with the usual replacementP
k !
L
2
R
dk. In order to proceed with the evaluation of
those integrals, it is convenient to define a function gk,
 gk 
L
2
kkfk; (17)
which satisfies the condition
R
1
0 gkk
2dk  1 derived
from the definition of the damping constant  [see (12)].
A reasonable choice for this function is
 gk  Aek=k0 ; (18)
where A  1=2k30 is the normalization constant and k0
determines the characteristic length scale of the environ-
ment. For example, in the case of fermionic environments,
k0 is of the order of kF [8]. One should bear in mind that the
choice of gk is guided either by the knowledge of the
microscopic details of the environment or by some phe-
nomenological input about the effective interaction be-
tween the two particles.
With the form chosen for gk, the kernel of the third
term in (15) is, again only for high temperatures,
 Kx1t  x2t0; t t0  2t t0ut; (19)
where ut  x1t  x2t,
 ut  

1
k20u
2  12

4u2k20
k20u
2  13

; (20)
and the effective potential reads
 Vut  
2
k20k
2
0u
2t  1
: (21)
The strength of the effective potential depends on the
characteristic length and time scales. Therefore, the con-
tribution of this coupling to the dynamics of the Brownian
particles is important only for times longer than the typical
time scale of the reservoir and distances of the order of (or
shorter than) its characteristic length k10 . The fluctuating
forces still satisfy the white noise properties, but now they
present an additional property associated with the distance
between the particles. The forces F1t and F2t are also
spatially correlated; that is,
 hF1tF2t0i  2ut~~kBTt t0: (22)
In Fig. 1, we see the noise correlation strength as a
function of the distance between the particles. For short
distances, the noise correlation has the standard Brownian
behavior. However, for longer distances, the correlation
function becomes negative, and this anticorrelation indu-
 
-4 -2 2 4
u k0
-0.2
0.2
0.4
0.6
0.8
1
u k0
η
η
FIG. 1. Spatial correlation of F1 and F2.
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ces an anomalous diffusive process in the system which
will ultimately tend to normal diffusion once the particles
are infinitely far apart.
In terms of the relative and center-of-mass coordinates,
the equations of motion read
 M ut   _ut  ut~~ _u V 0ut  Fut (23)
and
 M qt   _qt  ut _q  Fqt; (24)
where Fut  F1t  F2t and Fqt  F1t 
F2t~~=2. From the form of Vut~~ and ut and the
statistical properties of the fluctuating forces hFuti 
hFqti  0, hFutFut0i  4kT ut t0, and
hFqtFqt
0i  kT ut t0, it is evident that at
large distances the equations of motion for the relative and
center-of-mass coordinates represent the motion of
Brownian particles with reduced mass M=2 and total
mass 2M, respectively. In general, the dissipation depends
on the relative distance between the particles, and, for
distances such that uk0  1, we have up to first order in
ut, V 0u / ut. In this approximation, both dissipa-
tion and fluctuations are negligible, and then we have an
undamped oscillatory motion for ut.
In conclusion, we presented here a system-plus-
reservoir model with a nonlinear coupling in the system
coordinates which, in the adequate limit, allows us to
reproduce the phenomenological results known for the
dynamics of a Brownian particle. As the model is mani-
festly translation invariant, it turns out to become the
natural candidate to describe the dissipative motion of a
particle immersed in a homogeneous material medium.
Actually, it can be shown that the exponential form
employed here is the most general form of the interaction
of a moving particle with the harmonic modes of the
environment (see, for example, [8]). Moreover, in this
generalization, the dissipation coefficient is still directly
expressible in terms of a few parameters of the thermal
bath that can be measured experimentally.
It should also be emphasized that the traditional bilinear
model is the linearized version of the one we have just
presented here, and, therefore, it is more appropriate to
describe only local properties of the motion of a single
particle in the environment.
Besides improving and generalizing a microscopic de-
scription of the Brownian motion, our model is still capable
of inducing an effective coupling between two Brownian
particles which arises from the nonlocal effects generated
by the bath and depends on the choice of its dynamical
susceptibility. Although the coupling between particles
mediated by the medium where they are placed is quite
common in many condensed matter systems (formation of
Cooper pairs and bipolarons or RKKY interaction, just to
name a few cases), there is no reason one should expect it
to take place between two Brownian particles. So we have
shown that the choice made for the behavior of the bath
response function in order to reproduce the Brownian
motion naturally generates this effective coupling.
Actually, it is the separable k-dependent part of the sus-
ceptibility that gives the specific form of the static potential
we have obtained here. Retarded effects will show up as
one can no longer separate the k and ! dependence in that
function. These nonlocal effects are also relevant for the
single particle case but only in a non-Markovian
approximation.
We should finally stress that we have reported only our
findings for the classical or high temperature regime of the
model. The quantum mechanical analysis of this system
requires the evaluation of the reduced density operator for
one or two particles of interest down to very low tempera-
tures and will be presented elsewhere [11]. The extension
of this study to deal with many Brownian particles might
also shed some light on the behavior of complex many
particle systems.
We kindly acknowledge Conselho Nacional de
Desenvolvimento Cientı́fico e Tecnológico (CNPq) and
the Millenium Institute for Quantum Information for the
financial support.
[1] A. O. Caldeira and A. J. Leggett, Ann. Phys. (N.Y.) 149,
374 (1983); 153, 445(E) (1983).
[2] A. O. Caldeira and A. J. Leggett, Physica (Amsterdam)
121A, 587 (1983).
[3] A. J. Leggett et al., Rev. Mod. Phys. 59, 1 (1987).
[4] M. P. A. Fisher and W. Zwerger, Phys. Rev. B 32, 6190
(1985).
[5] A. Garg, J. N. Onuchic, and V. Ambegaokar, J. Chem.
Phys. 83, 4491 (1985).
[6] A. H. Castro Neto and A. O. Caldeira Phys. Rev. E 48,
4037 (1993).
[7] U. Weiss, Quantum Dissipative Systems (World Scientific,
Singapore, 1993).
[8] P. Hedegard and A. O. Caldeira, Phys. Scr. 35, 609 (1987).
[9] F. Guinea, Phys. Rev. Lett. 53, 1268 (1984).
[10] M. Dubé and P. C. E. Stamp, Int. J. Mod. Phys. B 12, 1191
(1998).
[11] O. S. Duarte and A. O. Caldeira (to be published).
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Effective Coupling Between Two Brownian Particles.

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Effective Coupling Between Two Brownian Particles.

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