Brownian rectifiers in the presence of temporally asymmetric unbiased forces

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by  Raishma Krishnan

4 Brownian rectifiers in the presence of temporally asymmetric unbiased forces
0
0 Raishma Krishnan,1,∗ Mangal C. Mahato,2,† and A. M. Jayannavar3,‡
2
1Institute of Physics, Sachivalaya Marg, Bhubaneswar-751005, India
n 2Department of Physics, North-Eastern Hill University, Shillong-793022, India
a 3Institute of Physics, Sachivalaya Marg, Bhubaneswar 751005, India
J

2 Abstract: The efficiency of energy transduction in a temporally asymmetric rocked ratchet
1 is studied. Time asymmetry favours current in one direction and suppresses it in the opposite
direction due to which large efficiency ∼ 50% is readily obtained. The spatial asymmetry in the

] potentialtogetherwithsysteminhomogeneitymayhelpinfurtherenhancingtheefficiency. Finetun-
h
ingofsystemparametersconsideredleadstomultiplecurrentreversalsevenintheadiabaticregime.
c
e
m
PACSnumbers: 05.40.-a,05.60.cd,02.50.Ey.
-
t
a
t I. INTRODUCTION current in one direction but to suppress motion in the
s
. opposite direction. This is in similar spirit as in case
t
a of flashing ratchets proposed by Makhnovskii et al. [10].
Brownianrectifiersorratchetsaredevicesthatconvert
m This is accomplished by applying temporally asymmet-
nonequilibrium fluctuations into useful workin the pres-
- ence of load. Several physical models [1, 2, 3, 4] have ric but unbiased periodic forcings [12, 13, 14]. Interest-
d ingly, such choice of forcings helps in obtaining rectified
been proposed to understand the nature of currents and
n
currents with high efficiency even for spatially symmet-
o their possible reversals with applications in nanoparticle
ric periodic potentials. Still higher efficiency is obtained
c separation devices [4]. The possibility of enhancement
[ ofefficiencywithwhichtheseBrownianrectifiersconvert with asymmetric potentials. The range of parameters of

1 thenonequilibriumfluctuationsintousefulworkhasgen- operation of such ratchets is quite wide sustaining large
loads. In addition, frictional inhomogeneity may further
v erated much interest in this field. This, in turn, has led
enhance the efficiency. We also see multiple current re-
8 to the emergenceof a separatesubfield- stochasticener-
8 getics - on its own right [5, 6]. Using this formalism one versals in the full parameter space of operation even in
1 canreadilyestablishthecompatibilitybetweenLangevin theadiabaticregime. However,multiplecurrentreversals
1 require fine tuning of the parameters.
or Fokker-Planck formalism with the laws of thermody-
0
manicstherebyprovidingatooltostudysystemsfarfrom
4
0 equilibrium. Withthisframeworkonecancalculatevari-
II. THE MODEL
/ ousphysicalquantitiessuchasefficiencyofenergytrans-
t
a duction [7], energy dissipation (hysteresis loss), entropy
m (entropy production) [8], etc. The Brownian motion of a particle in an inhomoge-
neousmediumisdescribedbytheLangevinequation[15]
- Most of the studies yield low efficiencies, in the sub-
d
percentage range, in various types of ratchets. This is
n
duetointrinsicirreversibilityassociatedwithratchetop-
o V′(q)−F(t) k Tγ′(q) k T
eration. Only fine tuning of parameters can lead to a B B
c q˙=− − + ξ(t), (1)
: largeefficiency,theregimeofparameters,however,being γ(q) 2[γ(q)]2 sγ(q)
v
very narrow [9]. Recently Makhnovskii et al. [10] con-
i
X structedaspecialtypeofflashingratchetwithtwoasym- where ξ(t) is a randomly fluctuating Gaussian thermal
r metricdouble-wellperiodic-potential-statesdisplacedby noise with zero mean and correlation, < ξ(t)ξ(t′) >=
a half a period. Such flashing ratchet models were found 2δ(t − t′). The periodic potential V(q) = −sin(q) −
to be highly efficient with efficiency an order of magni- (µ/4)sin(2q). The parameter µ(−1 < µ < 1), charac-
tude higher than in earlier models [5, 6, 11]. The basic terises the degree of asymmetry in the potential. The
ideabehindthisenhancedefficiencyisthatevenfordiffu- friction coefficient γ(q) = γ0(1 − λsin(q + φ)), with
siveBrownianmotionthechoiceofappropriatepotential 0 ≤ λ < 1 where φ is the phase difference. F(t) is
profileensuressuppressionofbackwardmotionandhence the externally applied periodic driving force. The cor-
reduction in the accompanying dissipation. responding Fokker-Planckequation [16] is given by
In the present work, we study the motion of a parti-
∂P(q,t) ∂ 1 ∂P(q,t)
cle in a rocking ratchet rocked purposefully as to favour
= k T (2)
B
∂t ∂qγ(q) ∂q
h
+ [V′(q)−F(t)]P(q,t) .
∗Electronicaddress: [email protected] i
†Electronicaddress: [email protected] Since we are interested in the adiabatic limit we first
‡Electronicaddress: [email protected] obtain an expression for the probability current density
2
j in the presence of a constant external force F . The The thermodynamic efficiency of energy transduction is
0
expression is given by [5, 6] η = E /E . In our present discussion all the
out in
physical quantities are taken in dimensionless units. In
1−exp[−2πF0] the following section we discuss the results of our calcu-
j = kBT , (3)
2π lation. In order to evaluate currents we use the method
dyI (y)
0 − of Gaussian quadrature.
R
where I (y) is given by
−
−V(y)+F y III. RESULTS AND DISCUSSIONS
0
I (y) = exp
−
k T
(cid:20) B (cid:21) To begin with we consider a homogeneous system in
V(x)−F x
y+2πdx γ(x)exp 0 . (4) the presence of spatially symmetric potential. In Fig. 1
y k T
(cid:20) B (cid:21)
R 0.3
It may be noted that for µ = 0, j(F ) 6= −j(−F ) for ε = 0.4
0 0 ε = 0.9
φ 6= 0, π. This asymmetry ensures rectification of cur- ε = 0.8
rent for the rocked ratchet even in the presence of spa-
tiallysymmetricpotential. WeassumethatF(t)changes 0.2
slow enough, i.e., its frequency is smaller than any other ncy
frequency related to the relaxation rate in the problem Efficie
such that the system is in a steady state at each instant 0.1
of time.
We consider time asymmetric ratchets with a zero
mean periodic driving force [12] given by
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
1+ǫ 1 Load
F(t) = F , (nτ ≤t<nτ + τ(1−ǫ)), (5)
0
1−ǫ 2
FIG. 1: Efficiency vs load for various values of ǫ with fixed
1
= −F0, (nτ + τ(1−ǫ)<t≤(n+1)τ). F0 =0.1.
2
Here, the parameter ǫ signifies the temporal asymme-
try in the periodic forcing. For this forcing in the adia-
batic limit the time averagedcurrent is given by [7, 12] 0.003
<j >= 1[j(F(t))+j(−F(t))] which is equal to E
2 10in E
out
1
<j >= (j +j ), (6) 0.002
1 2
2 Eout
with E, in
0.001
1+ǫ
j = (1−ǫ)j( F ), (7)
1 0
1−ǫ
j = (1+ǫ)j(−F ).
2 0 0
0 0.2 0.4 0.6
Load
The input energy E per unit time is given by [7]
in
FIG. 2: Input and output energy vs load for ǫ = 0.8 with
1 1+ǫ
Ein = F0[( )j1−j2]. (8) fixed F0 =0.1. The negative values of the output energy are
2 1−ǫ
not shown. The output curve is blown up ten times to scale
with theinput curvevalues.
To calculate efficiency a loadL is appliedagainstthe di-
rectionof currentwith overallpotential givenbyV(q)=
−sin(q)−(µ/4)sin(2q)+qL. The current flows against
we plot efficiency as a function of load in the presence
theloadaslongastheloadislessthanthestoppingforce
of temporal asymmetry ǫ for F = 0.1 and T = 0.1.
L beyond which the current is in the same direction as 0
s Foragivenǫtheefficiencyslowlyincreaseswithload,at-
thatof the load. Thusin the operating rangeof the load
tains a maximum and then decreases rapidly. The locus
0< L< L theBrownianparticlesmoveinthedirection
s of peak values corresponding to different ǫ values (with
oppositeto the loadtherebystoringenergy. Theaverage
appropriate load L) is found to have a nonmonotonous
work done over a period is given by
behaviour with the maximum (∼ 0.29) being at around
1 ǫ = 0.8 corresponding to a load of L = 0.258. For
Eout = L[j1+j2]. (9) this value of ǫ there is a large external driving force
2
3
0.5
ε = 0.9, µ = 1
0.4 ε = 0.8, µ = 1
ε = 0.6, µ = 1
0.3 0.4
Efficiency0.2 0.04 Efficiency εε==00..88,, µµ==11,, λλ==00..90
0.3
0.02
0.1
ε=0.0, µ=1
0
0 0.01 0.02
0.2
00 0.2 0.4 0.6 0.8 1 0 1 2 3 φ 4 5 6 7
Load
FIG. 3: Efficiency vs load for various ǫ with fixed F0 = 0.1. FIG. 5: Efficiency vs φ for ǫ = 0.8,µ = 1,L = 0.44 for (i)
Insetshowstheefficiencyvsloadforthecaseǫ=0andµ=1. λ=0.0 and (ii) λ=0.9 with fixedF0 =0.1 and T =0.1
0.0006
field, i.e., (1+ǫ)F (= 9F = 0.9) for a time duration
1−ǫ 0 0 0.0004 ε=0.3, µ=−1, λ=0.9, φ=1.005 π
τ(1−ǫ)/2(=0.1τ), inone directionwhichcausesconsid-
erable reduction in the barrier height in that direction. 0.0002
Tanhdissmlidaekedsotwhne pthaertsiclolepecrbosesfotrheetphoetfieneltdialchbaanrrgieesr.eDasuilry- Current 0 0.006
ing the other part of the period of F(t) (=−F =−0.1) −0.0002
0 0.004 λ = 0.9
ε = 0.34
twiohnichrelmasatisnsfolrar0g.9eτththues pnuotlleinfytiinagl btahrericehrafnocrerseovferrseevmerose- −0.0004 Current 0.002 µ = −1
0
motion. There is an optimum value of ǫ (∼ 0.8, for the −0.0006
given parameters), however, for which considerable cur- −0.0008 −0.0020 1 2 3 4 5
rent is obtained. Beyond this value of ǫ the duration for 0 1 2 3 4 5
Temperature
whichtheforceremainslargeissosmallthateventhough
thebarriersmaybenegligibleorvanishingaltogether,the FIG. 6: Current vs temperature curve showing two current
particle may nothaveenoughopportunity to moveaway reversals for µ = −1, λ = 0.9 and ǫ = 0.34 with φ =
beforetheforcegetschanged. Thisisthereasonwhythe 1.005π, F0 = 0.3 and L = 0. Inset shows the current in the
rectifierworksmostefficientlyforanoptimumvalueofǫ. presence of lone asymmetry parameters (λ,ǫ,µ).
For finite ǫ current against the load is obtained for load
L<L . L represents the range of load for which useful
s s
work is performed and is an increasing function of ǫ.
as a function of load. The input energy decreases to
TheusefulworksoobtainedE andtheinputenergy
out
a minimum value for a load larger than L . Moreover,
E are shown in Fig. 2 for a representative value of ǫ s
in
it remains positive, as expected, over the entire range.
The output energy shows a peak with load in the region
0.5 where the input energy is monotonously decreasing. It
0.013
then becomes negative for L > L as anticipated. The
s
0.4 0.011 ε = 0.8 qualitativebehavioursofefficiencyandenergiesshownin
0.3 Current 00..000079 Fthigesfl.a1shainngdr2atacrheets.imilar to those in reference [10] for
Efficiency 0.005 spaIntiaFlilgy.a3sywmemcoentsriidcetrotgheethpeerriwoditihc paotteemntpiaolraVll(yq)atsoymbe-
0.2 0.003
0 0.2 0.4 0.6 0.8 1 1.2 metric external driving force field. The potential asym-
Temperature
metry enhances the efficiency of energy transduction as
0.1 FFF000 === 000...851,,, LLL === 000...344354 well as widens the range of load. This is due to the fact
that for ǫ 6= 0 the presence of asymmetric parameter
0
0.01 0.11 0.21 0.31 0.41 µ(> 0) further reduces the potential barrier for forward
Temperature
motion and enhances the barrier for backward motion.
FIG. 4: Efficiency vs temperature with µ = 1 for (i) ǫ = Moreover, as can be seen from the inset, one can get fi-
0.2,F0 = 0.8, (ii) ǫ = 0.4,F0 = 0.5 and (iii) ǫ = 0.8,F0 = nitecurrentevenwhenǫ=0withfinitestoppingforceLs
0.1. Inset shows current as a function of temperature for in contrast to the symmetric potential case. From Figs.
ǫ = 0.8,F0 =0.1 in theabsence of load. 1 and 3 it is clearthat the temporally asymmetric forces
not only enhance the efficiency of energy transduction
4
butalsowidenthe operationrangeofloadagainstwhich in the adiabatic regime [17]. The inset shows current as
the ratchet system works. a function of individual parameters (ǫ, µ, λ). The plots
InFig.4weplotefficiencyasafunctionofT forvarious indicate that individual parameters cannot bring about
ǫ values in the presence of potential asymmetry (µ>0). current reversals separately. However, the possibility of
The efficiency decreases with temperature. The rele- currentreversalsarisesdue to the combined effect of the
vant physical parameters chosen for optimal efficiency three asymmetry parameters considered. We have also
are mentioned in the caption. From the inset it is to be observed more number of current reversals than shown
notedthatthecurrentpeaksasafunctionoftemperature in Fig. 6 by fine tuning the parameters.
yet efficiency decreases monotonically. This implies that
thermalfluctuationdonotfavourenergytransductionin
this case. It is worth mentioning that for given temper-
IV. CONCLUSIONS
ature and ǫ the efficiency shows peaking behaviour as a
functionofF ;theefficiencybeingzeroforF =0aswell
0 0
We find large efficiency for rocking ratchets driven
asforlargeF forinthese limitsoutputcurrentvanishes
0
by temporally asymmetric periodic field the origin of
in the absence of load.
which can be traced to the suppression of backward
Next, we presentthe effect offrictional inhomogeneity
motion. The observedefficiency is much higher than the
(γ = γ(q);λ 6= 0). In Fig. 5 we plot the efficiency as a
earlier reported values eventhough the ratchet operates
functionofthephasedifferencebetweenthepotentialand
in an intrinsically irreversible domain. This asymmetry
thefrictioncoefficientγ(q)foratypicalcase. Weobserve
factor has also helped in increasing the range of load
that the inclusion of this parameter λ further increases
of operation of the ratchet. We also observe multiple
theefficiencyinarangeofφdependingonotherparame-
current reversals in the adiabatic limit by proper fine
tervalues. Itisworthmentioningthatforinhomogeneous
tuning of different parameters. These reversals are
systems the efficiency peaks with temperature in a lim-
attributed to inherent complex dynamics of the system.
ited range of parameters. With frictional inhomogeneity
therangeoftemperatureinwhichonecanobtainoutput
current with finite efficiency is extended to a large tem- V. ACKNOWLEDGEMENTS
perature where contribution of λ dominates over other
parameters. In Fig 6 we show that by properly choosing AMJ thanks D.-Y. Yang for providing the reference
the parameters we can obtain multiple current reversals [10] prior to publication. MCM thanks Institute of
as a function of temperature. It should be noted that Physics, Bhubaneswar for hospitality where the present
such reversals are not possible in the homogeneous case work was carried out.
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