We study the phenomena of noise induced transport in frictional ratchet
systems. For this we consider a Brownian particle moving in a space dependent
frictional medium in the presence of external white noise fluctuations. To get
the directed transport, unlike in other ratchet models like flashing or rocking
ratchets etc., we do not require the potential experienced by the particle to
be asymmetric nor do we require the external fluctuations to be correlated. We
have obtained analytical expressions for current and the diffusion coefficient.
We show that the frictional ratchet do not exhibit a pronounced coherence in
the transport in that the diffusion spread overshadows the accompanying
directed transport in system with finite spatial extensions.Comment: Based on the poster presentation (by RK) at the Condensed Matter Days
  - 2003 held at Jadavpur University, Kolkata, India from 27-29 August 2003.
  Minor corrections have been don

Dan, Debasis

Jayannavar, A. M.

Krishnan, Raishma

English

arXiv

Jayannavar, A M

IACS Institutional Repository

Transport Coherence in Frictional Ratchets

We study the phenomena of noise induced transport in frictional ratchet systems. For this we consider a Brownian particle moving in a space dependent frictional medium in the presence of external white noise fluctuations. To get the directed transport, unlike in other ratchet models like flashing or rocking ratchets etc., we do not require the potential experienced by the particle to be asymmetric nor do we require the external fluctuations to be correlated. We have obtained analytical expressions for current and the diffusion coefficient. We show that the frictional ratchets do not exhibit a pronounced coherence in the transport in that the diffusion spread overshadows the accompanying directed transport in system with finite spatial extensions

Publications of the IAS Fellows

ar
X
iv
:c
on
d-
m
at
/0
30
96
17
v3
  [
co
nd
-m
at.
sta
t-m
ec
h]
  1
9 N
ov
 20
03 Transport Coherence in Frictional Ratchets
Raishma Krishnan,1 Debasis Dan,2 and A. M. Jayannavar1
1Institute of Physics, Bhubaneswar-751005, India∗
2Department of Physics, Indiana University, Bloomington 47405, U.S.A†
Abstract: We study the phenomena of noise induced transport in frictional ratchet systems. For
this we consider a Brownian particle moving in a space dependent frictional medium in the presence
of external white noise fluctuations. To get the directed transport, unlike in other ratchet models
like flashing or rocking ratchets etc., we do not require the potential experienced by the particle
to be asymmetric nor do we require the external fluctuations to be correlated. We have obtained
analytical expressions for current and the diffusion coefficient. We show that the frictional ratchets
do not exhibit a pronounced coherence in the transport in that the diffusion spread overshadows
the accompanying directed transport in system with finite spatial extensions.
PACS numbers: 05.40.Jc, 05.40.Ca, 02.50.Ey
Keywords: Ratchets, Brownian motors, Noise, Transport coherence
I. INTRODUCTION
Ratchets, also termed as Brownian motors, are sys-
tems that exploit the nonequilibrium fluctuations that
are present in the medium to generate directed flow of
Brownian particles in the absence of any net external
force or bias [1, 2]. Recently, such Brownian motors
have been studied extensively because they are believed
to share common features with biological motors. Bio-
logical protien motors convert the chemical energy into
macroscopic work i.e., they transport cargo efficiently
with high reliablility at room temperature in the pres-
ence of a very noisy environment [3, 4].
In thermal equilibrium, Onsager’s principle of de-
tailed balance, where the forward transition in any path-
way is on average balanced by an identical transition in
the backward direction, holds due to which there cannot
be any particle current in the system. Also, an under-
standing of the second law of thermodynamics [5] tells
us that a system when left to itself in thermal equi-
librium would not cause any preferential motion even
in the presence of spatial anisotropy. Thus external
nonequilibrium fluctuations are ubiquitous to drive a
net directed flow [1, 2].
Much has been studied from the viewpoint of statisti-
cal physics in ratchet models to understand how unidi-
∗Electronic address: raishma@iopb.res.in, jayan@iopb.res.in
†Electronic address: ddan@indiana.edu
rectional motion emerges from nonequilibrium fluctua-
tions [1]. Several physical models like flashing ratchets,
rocking ratchets, time asymmetric ratchets etc., where
potential has been taken to be asymmetric in space,
have been developed. In these models to generate noise
induced directional transport the nonequilibrium fluc-
tuations have to be time correlated. In an earlier work
it was shown that in the case of inhomogeneous sys-
tems, where the friction coefficient and/or temperature
varies in space [6], it is possible to get unidirectional
currents even in a symmetric potential in the absence
of a net bias. Moreover, external fluctuations need not
be correlated in time [7, 8]. Such ratchets are termed
as frictional ratchets. In the presence of external para-
metric noise the particle on an average absorbs energy
from the noise source. The particle spends larger time
in the region of space where the friction is higher and
hence the energy absorption from the noise source is
higher in these regions. Therefore, the particle in the
higher friction regions feel effectively higher tempera-
tures. Thus, the problem of particle motion in an in-
homogeneous medium in presence of an external noise
becomes equivalent to the problem in a space depen-
dent temperature. Such systems are known to gener-
ate unidirectional currents [2, 6, 8, 9]. This follows
as a corollary to Landauer’s blow torch theorem that
the notion of stability changes dramatically in the pres-
ence of temperature inhomogenieties. In such cases the
notion of local stability, valid in equilibrium systems,
does not hold. Frictional inhomogenieties are common
2in superlattice structures, semiconductors or motion in
porous media. It is believed that molecular motor pro-
tiens moving close along the periodic structures of mi-
crotubules experience a space dependent friction. Fric-
tional inhomogenity changes the dynamics of the parti-
cle nontrivially as compared to the homogeneous case.
This in turn has been shown to give rise to many counter
intuitive phenomena in driven non-equilibrium systems
[10].
Considerable amount of work has been done on the
nature of current, current reversals as well as thermody-
namic efficiency in these systems. However, the question
of reliability or the coherence of transport has recieved
only little attention [9, 11, 12]. Transport of Brownian
particles are always accompanied by a diffusive spread.
The coherence of transport means a large particle cur-
rent accompanied by a minimal diffusive spread. When
a particle on an average moves a distance L due to its
velocity, there will always be an accompanying diffusive
spread. If this diffusive spread is much smaller than
the distance travelled, then the particle motion is con-
sidered to be coherent or optimal or reliable. This is
in turn quantified by a dimensionless quantity, Pec´let
number Pe, which is the ratio of current to the dif-
fusion constant. Higher the Pec´let number, more co-
herent is the transport. The Pec´let numbers for some
of the models studied show low coherence of transport
(Pe = 0.2) [11]. But experimental studies on molecu-
lar motors showed more reliable transport with Pec´let
number ranging from 2 to 6 [13]. Thus study of trans-
port coherence is of importance in identifying a proper
model for the biological motors and also in nanoscale
particle separation devices based on current reversals.
In this work we study the coherence of transport of
a Brownian particle in a medium with frictional inho-
mogeneity and an external parametric white noise. We
show that this system does not exhibit pronounced co-
herence.
II. THE MODEL:
We consider the overdamped dynamics of a Brownian
particle moving in a medium with spatially varying fric-
tional coefficient η(q) at temperature T . Using a micro-
scopic treatment the Langevin equations for the Brown-
ian particle in a space dependent frictional medium has
been obtained earlier [7, 8]. The corresponding over-
damped Langevin equation of motion is given by
q˙ = −
V ′
η(q)
−
kBTη
′(q)
2[η(q)]
2 +
√
kBT
η(q)
f(t) (1)
with < f(t) >= 0, and < f(t)f(t′) >= 2δ(t− t′) where
< ... > denotes the ensemble average and q the coordi-
nate of the particle.
The system is then subjected to an external paramet-
ric additive white noise fluctuating force ξ(t), so that
the equation of motion becomes
q˙ = −
V ′
η(q)
−
kBTη
′(q)
2[η(q)]
2 +
√
kBT
η(q)
f(t) + ξ(t) (2)
with < ξ(t) >= 0 and < ξ(t)ξ(t′) >= 2Γδ(t− t′), where
Γ is the strength of the external white noise ξ(t). The
corresponding Fokker-Planck equation is given by [14]
∂P
∂t
=
∂
∂q
[{
V ′(q)
η(q)
}
P +
{
kBT
η(q)
+ Γ
}
∂P
∂q
]
(3)
For periodic functions V (q) and η(q) with periodicity
2pi, one can readily obtain analytical expression for cur-
rent and is given by [7, 8]
J =
1− exp [−2 pi δ]∫ 2pi
0 dyexp [−ψ(y)]
∫ y+2pi
y
dx exp [ψ(x)]
A(x)
(4)
with the generalized potential ψ(q) as
ψ(q) =
∫ q
dx
V ′(x)
kBT + Γη(x)
(5)
and A(q) as
A(q) =
kBT + Γη(q)
η(q)
(6)
with
δ = ψ(q)− ψ(q + 2pi)
which inturn determines the effective slope of the gen-
eralized potential ψ(q). Hence the sign of δ gives the
direction of current which follows from Eqn(4).
In our present work we have taken the potential
V (q) = V0[1− cos(q)] and η(q) = η0[1−αcos(q−φ)]
−1,
0 < α < 1. For simplicity we have restricted to the
3case where T = 0. For this case the effective potential
ψ(q), δ and A(q) are given by
ψ(q) =
V0
Γη0
[1− cos(q) +
α
4
[cos(2q − φ)− cos(φ)]
−
α
2
qsin(φ)] (7)
δ =
V0piαsin(φ)
Γη0
(8)
A(q) = Γ (9)
Following references [15], one can obtain exact analyt-
ical expressions for the diffusion coefficient D and the
average current J as
D =
∫ q0+L
q0
dx
L
A(x) [I+(x)]
2
I−(x)[∫ q0+L
q0
dx
L
I+(x)
]3 (10)
J = L
1− exp [−Lδ]∫ q0+L
q0
dx
L
I+(x)
(11)
where I+(x) and I−(x) are as given below
I+(x) =
1
A(x)
exp [ψ(x)]
∫ x
x−L
dy exp [−ψ(y)](12)
I−(x) = exp [−ψ(x)]
∫ x+L
x
dy
1
A(y)
exp [ψ(y)](13)
L here represents the period of the potential (= 2pi in
our case). Now, the time taken for a Brownian particle
to travel a distance L is given as τ = L/v and the spread
of the particle in the same time is given as < (∆q)2 >=
2Dτ . For a reliable transport we require < (∆q)2 >=
2Dτ < L2. This in turn implies that Pe = Lv/D > 2
for coherent transport.
III. RESULTS AND DISCUSSIONS
We study the current (J), diffusion constant (D) and
the Pec´let number (Pe) as a function of the phase dif-
ference φ between the periodic functions V (q) and η(q),
noise strength, Γ and α, the amplitude of the periodic
modulation of η(q). All the physical quantities are taken
in dimensionless form.
In Fig. (1) we plotted J , D and Pe as a function
of phase φ by numerically integrating Eqn 9 & 10 us-
ing quadrature methods. All other physical parameters
are mentioned in the figure caption. As expected, all
the quantities are periodic functions of phase φ with
the current J being zero at φ = 0&2pi [8, 16]. The
Pec´let number Pe exhibits a maximum with the maxi-
mum value being 1. Thus in the parameters considered
in Fig. 1 the transport is not coherent.
In figures 2 and 3 we have plotted J , D and Pe as a
function of α and Γ respectively for some typical values
of physical parameters given in the figure captions. It
is clear from these figures that in the parameter range
we have considered the transport is not very coherent.
However, the transport coherence in our model is found
to be much larger than those in the earlier nonfrictional
ratchet models where the Pec´let number is always less
than 0.2 [11].
0 2 4 6φ
−1
0
1
J
0 2 4 6φ
0.6
0.9
D
0 2φ
0
0.5
1
Pe
FIG. 1: Plot J, D and Pe for Γ = 1 , α = 0.5
40.1 0.5 0.9
α
0
0.2
J
0.1 0.5 0.9
α
0
0.4
D
0.1 0.5 0.9
α
0
2
Pe
FIG. 2: Plot of J, D, and Pe for φ = pi/2 , Γ = 1
0 2 4 6
Γ
0
1
2
J
0 2 4 6
Γ
0
2
4
6
D
0 2 4 6
Γ
0
2
4
Pe
FIG. 3: Plot of J, D and Pe for φ = pi/2 , α = 0.5
IV. CONCLUSIONS
We have studied the reliability of transport in fric-
tional ratchet systems in the presence of external white
noise fluctuations. We restrict our study to the simple
case where the temperature of the heat bath is zero (ab-
sence of thermal noise). The transport coefficients, both
the current as well as diffusion, arises solely due to the
presence of external noise. Our analysis indicates that
for this special case, the transport is not very coherent.
However, the obtained Pe values are much larger than
those obtained from the nonfrictional ratchet models
that were studied before. In the small parameter range
of Γ ∼ 0.05, φ ∼ 0.2, α ∼ 0.9 we find larger coherence
with Pe ∼ 3.5. However, the transport coefficients are
found to be very small. We have not considered here
the mutual interplay between the thermal noise and the
external noise which may lead to an increase in coher-
ence in appropriate parameter range. Work along this
line as well as on different frictional ratchet models is
under progress.
[1] P. Reimann, Phys. Rep. 361, 57 (2002).
[2] A. M. Jayannavar, cond-mat 0107079.
[3] R. D. Astumian and P. Ha¨nggi, Phys. Today, 33 (2002).
[4] M. C. Mahato and A. M. Jayannavar, Resonance 8,
No:7, 33 (2003), M. C. Mahato and A. M. Jayannavar,
Resonance 8, No:9, 4 (2003).
[5] R. P. Feynman, R. B. Leighton and M. Sands, The
Feynman Lectures on Physics (Addison-Wesley, Read-
ing, MA,1963), Vol. 1.
[6] M. Bu¨ttiker, Z. Phys. B 68,161 (1987).
5[7] A. M. Jayannavar and M. C. Mahato, Pramana-J. Phys.
45, 369 (1995).
[8] M. C. Mahato, T. P. Pareek and A. M. Jayannavar, Int.
J. Mod. Phys. B 10, 3857 (1996).
[9] R. Landauer, J. Stat. Phys. 53, 233 (1988), N. G. Van
Kampen, IBM J. Res. Dev. 32, 107 (1998).
[10] D. Dan, PhD Thesis submitted to Utkal University,
Bhubaneswar in 2003.
[11] J. A. Freund and L. Schimansky-Geier, Phys. Rev. E60,
1304 (1999); T. Harms and R. Lipowsky, Phys. Rev.
Lett. 79, 2895 (1997).
[12] B. Linder and L Schimansky-Geier, Phys. Rev. Lett.
89, 230602 (2002), D. Dan and A. M. Jaynnavar, Phys.
Rev. E66, 41106 (2002).
[13] M. J. Schnitzer and S. M. Block, Nature (London) 388,
386 (1997); K. Visscher, M. J. Schnitzer and S. M.
Block, Nature 400, 184 (1999).
[14] H. Riskin, The Fokker-Planck Equation (Springer Ver-
lag, Berlin, 1984).
[15] P. Reimann, C. Van den Broeck, H. Linke, P. Ha¨nggi, J.
M. Rubi and A. Perez-Madrid, Phys. Rev. E65, 31104
(2002); P. Reimann, C. Van den Broeck, H. Linke, P.
Ha¨nggi, J. M. Rubi and A. Perez-Madrid, Phys. Rev.
Lett. 87,10602 (2001); B. Linder and L. Schimansky-
Geier, Fluct. Noise Lett. 1, R25 (2001), D. Dan and A.
M. Jaynnavar, Phys. Rev. E66, 41106 (2002).
[16] D. Dan, M. C. Mahato and A. M. Jayannavar, Phys.
Lett. 258, 217 (1999); D. Dan, A. M. Jayannavar and
M. C. Mahato, Int. J. Mod. Phys. B 14 1585 (2000).


Transport Coherence in Frictional Ratchets

Abstract

Similar works

Full text

Available Versions

IACS Institutional Repository

Publications of the IAS Fellows