Nonlinear Brownian motion – mean square displacement
The stochastic dynamics of self-propelled Brownian particles is studied by means of the Langevin and the Fokker-Planck approach. We model the driving by a nonlinear friction function which has a negative part at small velocities, leading to active Brownian motion of the particles. The mean squar...
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irk-123456789-1190402017-06-04T03:04:31Z Nonlinear Brownian motion – mean square displacement Ebeling, W. The stochastic dynamics of self-propelled Brownian particles is studied by means of the Langevin and the Fokker-Planck approach. We model the driving by a nonlinear friction function which has a negative part at small velocities, leading to active Brownian motion of the particles. The mean square displacement is estimated analytically and compared with numerical simulations. Досліджується стохастична динаміка саморухомих броунівських частинок в рамках формалізмів Ланжевена та Фоккер-Планка. Рушійна сила моделюється нелінійною функцією тертя, яка має від’ємні значення при малих швидкостях, що приводить до активного броунівського руху частинок. Середньоквадратичне зміщення частинок оцінюється аналітично та порівнюється з даними чисельних розрахунків. 2004 Article Nonlinear Brownian motion – mean square displacement / W.Ebeling // Condensed Matter Physics. — 2004. — Т. 7, № 3(39). — С. 539–550. — Бібліогр.: 23 назв. — англ. 1607-324X DOI:10.5488/CMP.7.3.539 PACS: 05.40.+j, 05.40.Jc http://dspace.nbuv.gov.ua/handle/123456789/119040 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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English |
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The stochastic dynamics of self-propelled Brownian particles is studied by
means of the Langevin and the Fokker-Planck approach. We model the
driving by a nonlinear friction function which has a negative part at small
velocities, leading to active Brownian motion of the particles. The mean
square displacement is estimated analytically and compared with numerical
simulations. |
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Article |
| author |
Ebeling, W. |
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Ebeling, W. Nonlinear Brownian motion – mean square displacement Condensed Matter Physics |
| author_facet |
Ebeling, W. |
| author_sort |
Ebeling, W. |
| title |
Nonlinear Brownian motion – mean square displacement |
| title_short |
Nonlinear Brownian motion – mean square displacement |
| title_full |
Nonlinear Brownian motion – mean square displacement |
| title_fullStr |
Nonlinear Brownian motion – mean square displacement |
| title_full_unstemmed |
Nonlinear Brownian motion – mean square displacement |
| title_sort |
nonlinear brownian motion – mean square displacement |
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Інститут фізики конденсованих систем НАН України |
| publishDate |
2004 |
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http://dspace.nbuv.gov.ua/handle/123456789/119040 |
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Nonlinear Brownian motion – mean square displacement / W.Ebeling // Condensed Matter Physics. — 2004. — Т. 7, № 3(39). — С. 539–550. — Бібліогр.: 23 назв. — англ. |
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Condensed Matter Physics |
| work_keys_str_mv |
AT ebelingw nonlinearbrownianmotionmeansquaredisplacement |
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2025-07-08T15:08:03Z |
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2025-07-08T15:08:03Z |
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Condensed Matter Physics, 2004, Vol. 7, No. 3(39), pp. 539–550
Nonlinear Brownian motion –
mean square displacement
W.Ebeling∗
Institute of Physics, Humboldt University,
D–12489 Berlin, Germany
Receive April 5, 2004
The stochastic dynamics of self-propelled Brownian particles is studied by
means of the Langevin and the Fokker-Planck approach. We model the
driving by a nonlinear friction function which has a negative part at small
velocities, leading to active Brownian motion of the particles. The mean
square displacement is estimated analytically and compared with numeri-
cal simulations.
Key words: statistical mechanics, Brownian motion, nonlinear friction, self-
propelling, Fokker-Planck equations, mean square displacement
PACS: 05.40.+j, 05.40.Jc
1. Introduction
The standard theory of Brownian motion due to Einstein, Smoluchowski, Lange-
vin, Fokker and Planck is based on the model at which a particle of mass m moves in
a dense medium which generates friction and random collisions [1]. The equation of
motion is the Langevin equation and the corresponding equation for the probability
is the Fokker-Planck equation. In order to explain the basic ideas of this essen-
tially linear theory we start with the classical case of force-free Brownian particles
under equilibrium conditions. The phenomenological method yields a Maxwellian
distribution, which is in fact a “canonical distribution function”
f0(v) = C exp
[
− mv2
2kBT
]
. (1)
The alternative method to derive this distribution starts from the Langevin equation
dv
dt
= −γ0v + (2D0)
1/2ξ(t) (2)
∗E-mail: ebeling@physik.hu-berlin.de
c© W.Ebeling 539
W.Ebeling
(γ0 – friction coefficient, D0 – diffusion constant). The corresponding Fokker-Planck
equation reads
∂P (v, t)
∂t
=
∂
∂v
[
γ0vP (v, t) + D0
∂P (v, t)
∂v
]
. (3)
This equation is solved using the Gaussian distribution
f0(v) = C exp
[
−γ0v
2
2D0
]
. (4)
The Gaussian corresponds to the equilibrium Maxwell distribution just in the case
that the so-called Einstein relation
D0 =
kBTγ0
m
(5)
holds. In this paper we will consider several generalizations of these classical re-
sults to nonlineaar friction which are in part based on the work of Klimontovich
[1,2]. These generalizations refer to non-equilibrium cases, when the friction is not
a constant, but a function of the velocities, and when equations (1) and (5) are not
observed. In this case the system might be driven far away from equilibrium.
Let us replace γ0 by a function γ(v) and the constant D0 also by a function D(v).
Then we speak about “nonlinear Brownian motion” [1,2], while in other connection
we speak about “active Brownian movement” [3–6]. The corresponding Langevin
equation of motion reads:
dv
dt
=
1
m
F (v) +
√
2D(v2)ξ(t). (6)
Here F (v) = −mγ(v2)v is the dissipative force acting on the particle and D(v2) is
a diffusion function. The theoretical problems connected with these rather general
equations have many interesting applications in different fields, such as in the theory
of sound [7], in physical chemistry [8,9] and in biophysics [10].
A rather new application refers to the dynamics of charged grains in dusty plas-
mas. During the last decade the physics of charged grains has attracted much ex-
perimental and theoretical interest [11]. The dynamics of Coulomb clusters has been
studied in various experiments [12–14]. Theoretical reviews discussing some of these
phenomena can, e.g., be found in [15,16]. In several recent papers, the stochastic
dynamics of grains have been analyzed in detail [18–22]. In particular, it could be
shown that the Brownian motion of grains in a plasma exhibits many parallels to
the theory of active Brownian particles [23]. Therefore we believe that the theory
explained here may find applications not only to biological problems but also to the
dynamics of grains in plasmas.
540
Nonlinear Brownian motion – mean square displacement
2. Velocity-dependent friction functions
Negative friction driven by energy input:
Nonlinear friction forces driven by an external energy source was first used in
the theory of sound developed by Helmholtz and Rayleigh. Rayleigh postulated for
the friction function a quadratic expression:
γ(v2) = −γ1 + γ2v
2 = γ1
(
v2
v2
0
− 1
)
(7)
with γ1 < 0. This Rayleigh-Helmholtz-type model is a standard model studied in
many papers on Brownian dynamics [2]. Klimontovich studied the solutions of the
Fokker-Planck and Langevin equations for this model in much detail, including sus-
tained oscillations, fluctuations, correlations etc. [2]. We note that v2
0 = γ1/γ2 defines
a special value of the velocities where the friction changes from negative to posi-
tive values. This is the characteristic velocity of the problem, which defines e.g. the
motion on limit cycles. Beside the Helmholtz-Rayleigh model we will study here a
different model for active friction with energy input which was introduced and trea-
ted in [3,4]. This new friction function is based on a model of Brownian motion with
energy depot. We assume that the Brownian particle itself should be capable of ta-
king up external energy, thus storing some of this additional energy into an internal
energy depot, e(t). This energy depot may be altered by three different processes:
1. Take-up of energy from the environment with the rate q.
2. Internal dissipation, which is assumed to be proportional to the internal ener-
gy; the rate of energy loss is c.
3. Conversion of internal energy into motion, where d is the rate of conversion of
internal to kinetic degrees of freedom. This means that the depot energy may
be used in order to accelerate motion with the acceleration dev.
The corresponding balance equation is
de(t)
dt
= q − ce − dev2 (8)
and the equation of motion reads
dv
dt
= −γ0v + dev + (2D0)
1/2ξ(t). (9)
We assume that the acceleration is not subject to noise. This extension of the model
is motivated by investigations of active biological motion, that relies on the supply
of energy, which is dissipated by metabolic processes, but can be also converted
into kinetic energy. Under adiabatic conditions de/dt = 0, the depot model leads to
e0 = q/(c + dv2) and thus to the effective friction function
γ(v2) = γ0 −
qd
c + dv2
. (10)
541
W.Ebeling
Due to the pumping, slow particles are accelerated and fast particles are damped.
At certain conditions the active friction function has a zero
v2
0 =
q
γ0
− c
d
(11)
corresponding to stationary velocities v0, where the friction disappears. For 2-dimen-
sional motion the deterministic trajectory of our system is then attracted by a
cylinder in the 4d-space given by
v2
1 + v2
2 = v2
0 . (12)
Here v0 is the value of the stationary velocity which is for the Rayleigh-model given
by v2
0 = γ1/γ2 and for the depot-model by v2
0 = q0/γ0 − c/d.
Negative friction driven by momentum exchange:
Following [21,23] let us consider a plasma with highly charged grains. Then the
friction is determined at first by passive friction due to collisions of the grains with
other particles and due to momentum exchange between grains and the plasma.
Rather important are charging collisions due to ion absorption by grains. In this
case both the friction coefficient γ(v2) and the related diffusion coefficient D(v2) in
the Fokker-Planck equation (22) can be calculated explicitely [21,23]. Taking into
account both contributions we find
γ(v2) = γ0 + γi(v
2), (13)
D(v2) = D0 + Di(v
2). (14)
The explicit result for the friction function due to absorption effects, first derived in
[21], reads
γi(v
2) = −3A [I1(η) + I2(η, Γ)] , (15)
I1(η) =
(
−1
2
+
1
4η
)
√
π
η
Erf (
√
η) − e−η
2η
, (16)
I2(η) =
Γ
η
[
1
2
√
π
η
Erf (
√
η) − e−η
]
(17)
with η = (v2/2v2
T) (vT-thermal velocity) and Erf(x) denoting the error function.
The nonideality Γ and the constant pre-factor A depend on the grain and plasma
parameters
Γ =
ZiZge
2
kBTa
, A =
1
3
√
2π
(
mi
mg
)
a2nivT, (18)
where mg is the grain mass, a is the radius of the grain, ni is the ion density in the
plasma and mi is the ion mass. The terms I1(η) and I2(η) are related to purely ge-
ometrical and charge-dependent collecting cross-sections, respectively. Correspond-
ingly, it is also possible to calculate the related diffusion coefficient [21] The typical
542
Nonlinear Brownian motion – mean square displacement
Figure 1. Friction function of the depot model for different parameters of driving.
behavior of γi(v) is rather similar to the friction function of the depot model shown
in figure 1. The function Di(v) may be approximated by a constant.
Approximations for the friction function:
In many situations it is sufficient to approximate a more complicated friction
coefficient γ(v2) by more simple models [5,23]. In the case of a rather weak driving
force all our models may be unified, since in this case only the behaviour around
v2 = 0 is relevant. For the Rayleigh model, weak driving means γ1 � γ0, and for
the depot model it means
q0
γ0
>
c
d
,
qd
c
− γ0 � 1.
Near to the bifurcation point where the friction changes to negative values, all models
may be unified and by a Taylor expansion we obtain
γ(v) = −α + βv2. (19)
The value of the constants is
α = −γ(v2 = 0), β = γ′(v2 = 0). (20)
Far beyond the bifurcation point the system is strongly attracted to the two stable
points |v| = ±v0 where v0 is the zero of the friction function. In this case the
dissipative force may be well approximated by a piecewise linear approximation
(Schienbein-Gruler model) which reads
F = −mγ1
[
1 − v0
|v|
]
v. (21)
543
W.Ebeling
The constant is determined by the derivative in the zero point:
γ1 = 2γ′(v2
0)v
2
0 .
This linear approximation to the friction force close to the two stable velocities
v = ±v0 is very useful in many applications.
3. Fokker-Planck equations and stationary solutions
The Fokker-Planck equation (FPE) for the probability distribution function
f(r,v) of the grain reads
∂f
∂t
+ v
∂f
∂r
=
∂
∂v
[
γ(v2)vf +
∂D(v2)f
∂v
]
. (22)
A stationary solution of this equation is given by [1,2,5,21,22]
f(v) =
C
D(v2)
exp
(
−1
2
∫
γ(v2)
D(v2)
dv2
)
. (23)
For a given friction function γ(v2) and a given diffusion function D(v2), the ve-
locity distribution function can be readily obtained by integration. In case of a
non-constant friction and diffusion functions, the stationary distribution will be
non-Gaussian.
The friction and diffusion functions discussed above are rather complicated. The
most essential point is the existing negative friction connected with a root of the fric-
tion function. Neglecting for a moment the diffusion effects the dynamical equation
of a grain reads
d
dt
r = v,
d
dt
v = −γ(v2)v. (24)
Multiplying by the velocity v we find for the kinetic energy of the grains
d
dt
(
1
2
v2
)
= −γ(v2)v2. (25)
Thus, we see that the kinetic energy of grains converges to the value mgv
2
0/2; i.e.,
in the deterministic limit D → 0 the root of the friction function determines the
attractor of motion in the velocity space.
For dominant active friction, the dynamics of grains is quite different from that
of ordinary thermal particles. Then a typical grain moves with constant absolute
velocity but changes from time to time its direction due to the effect of noise. For
the case of constant diffusion, D(v2) ≡ D0, we may easily write a Langevin equation
d
dt
v + γ(v)v =
√
2D0ξ(t), (26)
544
Nonlinear Brownian motion – mean square displacement
Figure 2. Stationary distribution function for the depot model and for the
Schienbein-Gruler model.
where, as usual, ξ(t) denotes a standard white noise source.
The solution of the corresponding Fokker-Planck equation (the velocity distri-
bution function) reads for Rayleigh’s model
f(v) = N exp
[
− γ0
4D0
(
v2 − v2
0
)2
]
, (27)
where N is a normalization constant. As one can see, the probability density f(v)
exhibits two peaks near v0 and decays exponentially at high velocities. For the depot
model the solution of equation (23) defining the stationary velocity distribution reads
f(v) = N exp
[
− γ0
2D0
v2 +
q
2D0
ln
(
1 +
d
c
v2
)]
. (28)
The mean square velocity is given by
〈v2〉 = D0
∂
∂γ0
N (29)
and the most probable velocity ṽ is the positive root of the bi-quadratic equation
D0
ṽ2
+
qd
c + dṽ2
= γ0 . (30)
The shape of the distribution functions is shown in figure 2. The depot model con-
tains for q → 0 the Maxwell distribution as a limiting case and yields for this limit
〈v2〉 = 2
kT
m
, ṽ2 =
kT
m
. (31)
545
W.Ebeling
In the opposite case of large q we find
〈v2〉 = ṽ2 = v2
0 . (32)
For this case and also for weak noise, α/D0 � 1, the stationary distribution functions
become very narrow around ±v0. In the so-called δ-limit, corresponding to D0/α →
0, all distribution functions degenerate to
f0(v) = Ñ δ
(
v2 − v2
0
)
, (33)
that is, the dispersion of the kinetic energy of the particles is completely neglected.
From the point of view of physics, in this limit the particles move with constant
absolute velocity, changing only their directions of motion from time to time.
4. Mean square displacement
For standard Brownian motion Einstein derived (in 1905), for the mean square
displacement, the well-known relation
〈(r(t) − r(0))2〉 = d
2kBT
mγ0
t, (34)
where d is the dimension. For the stochastic motion on the plane considered here we
have d = 2. For the case of non-linear Brownian motion it is much more difficult to
obtain explicit solutions. Numerical experiments with the mean square displacement
of the depot model were first carried out in [6]. Mikhailov and Meinköhn [9] were able
to give analytical estimates for the mean square displacement in the limiting case
of δ-distributions of the velocities (negative friction and small noise) corresponding
to equation (33). We will calculate here for the motion on the plane d = 2 the mean
square displacement for the depot model by generalizing the methods developed in
[1,2] and [9]. A particle starting at t = 0 in r(0) will at time t at the vector
r(t) =
∫ t
0
dt1v(t1). (35)
The general expression for the mean square displacement reads [5]
〈(r(t) − r(0)2〉 =
∫ t
0
dt1
∫ t
0
dt2〈v(t1)v(t2)〉. (36)
The correlation function of the velocities may be calculated exactly if the absolute
velocities are constant and the change of directions is an uncorrelated random pro-
cess [9]. We will make here a weaker assumption that the velocities have a rather
narrow distribution around the most probable value ṽ. Assuming that the velocities
are close to the most probable value ṽ
v(t) = ṽn,
546
Nonlinear Brownian motion – mean square displacement
where n = v/|v| is the unit vector pointing in the direction v. This way we get the
mean square displacement
〈(r(t) − r(0)2〉 = ṽ2K(t),
K(t) =
∫ t
0
dt1
∫ t
0
dt2〈n(t1)n(t2)〉 =
∫ t
0
dt1
∫ t
0
dt2〈cos(φ(t1 − t2)〉, (37)
where φ is the angle between the unit vectors. Assuming that radial variable ρ = |v|
is slowly changing, we find following Klimontovich [1] for the distribution of the
angles
∂f(φ, t)
∂t
=
D0
2ρ2
∂2
∂φ2
. (38)
We replace now ρ2 by the mean value 〈v2〉 and introduce the dimensionless time
t0 =
〈v2〉
2D0
.
By using the known time-dependent distribution for the angles we get the correlation
function
K(t) = t20k(t/t0), k(τ) = [exp(−τ) − 1 + τ ] . (39)
This leads to the final expression
〈
(r(t) − r(0))2
〉
= 4ṽ2t20k(t/t0). (40)
In the limit of passive Brownian motion q = 0, this leads to
〈
(r(t) − r(0))2
〉
=
4kBT
mγ2
0
[
t + γ−1
0 (exp(−γ0t) − 1)
]
. (41)
We see that this limit is in full correspondence with the known results for passive
Brownian motion, in particular for large time with equation (34) In the limit of
strong driving (δ−limit we come back to the expression first derived by Mikhailov
and Meinköhn [9]
〈
(r(t) − r(0))2
〉
=
2v4
0
D0
t +
v6
0
D2
0
[
exp
(
−2D0t
v2
0
)
− 1
]
. (42)
This way we have shown that our approximate expression equation (40) is correct
in the limits q → 0 and q → ∞. For intermediate values of the driving parameter
q our result equation (40) is at least a useful approximation. In order to check this
we have carried out several simulations for the depot model. As we see in figures 3
and 4, the points found from numerical experiments are in reasonable agreement
with the theoretical estimates. We underline that the spatial diffusion coefficient
given here refers to the case of individual particles (self-diffusion). The expressions
do not take into account interactions of the particles. For the case of cell movements
the analytical expressions for the stationary velocity distribution and for the mean
square displacement are in good agreement with measurements on granulocytes [10].
We also mention good agreement with earlier computer simulations [6].
547
W.Ebeling
Figure 3. Mean square displacement as a function of time for the parameters
γ0 = 0.5, c = d = 1 and several small values of the driving parameter q. The
given curves give the mean square displacement for q = 0.0; 0.4; 0.6; 0.8 as a
function of time.
Figure 4. Mean square displacement as a function of time for the parameters
γ0 = 0.5, c = d = 1 and several larger values of the driving parameter q. The
given curves give the mean square displacement for q = 1.0; 2.0; 3.0 as a function
of time.
548
Nonlinear Brownian motion – mean square displacement
5. Discussion and conclusions
The aim of this paper is to contribute new results to the theory of nonlinear
Brownian motion. We investigated both the passive and the active case. While for
passive Brownian motion the velocities are Gaussian-ditributed and centered around
zero velocity, in the case of active motion the particles are driven to some charac-
teristic modulus of the velocity v0. Correspondingly all particles have nearly the
same kinetic energy and differ only in directions. The mean square displacements
are rather high. Following the methods developed in [1,2,9] analytical estimates for
the mean square displacement were derived, which show reasonable agreement with
old and new simulations [6].
Acknowledgements
The author is grateful to J.Dunkel, L.Schimansky-Geier, S.Trigger and F.Schweitzer
for helpful discussions.
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Нелінійний броунівський рух: середньоквадратичне
зміщення
В.Ебелінг
Інститут фізики університету Гумбольдта,
D–12489 Берлін, Німеччина
Отримано 5 квітня 2004 р.
Досліджується стохастична динаміка саморухомих броунівських час-
тинок в рамках формалізмів Ланжевена та Фоккер-Планка. Рушій-
на сила моделюється нелінійною функцією тертя, яка має від’ємні
значення при малих швидкостях, що приводить до активного броу-
нівського руху частинок. Середньоквадратичне зміщення частинок
оцінюється аналітично та порівнюється з даними чисельних розра-
хунків.
Ключові слова: статистична механіка, броунівський рух, нелінійне
тертя, саморухомі частинки, середньоквадратичне зміщення
PACS: 05.40.+j, 05.40.Jc
550
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