Symmetric Brownian motor

0.25 MB·

by  A. Gomez-Marin

Symmetric Brownian Motor
A. Gomez-Marin1 and J. M. Sancho1
1 Departament d’Estructura i Constituents de la Materia, Facultat de Fisica,
Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain
(Dated: 2nd February2008)
5
0 In thispaperwepresent amodel of asymmetric Brownian motor (SBM) which changesthesign
0 ofitsvelocitywhenthetemperaturegradientisinverted. Thevelocity,externalworkandefficiency
2 arestudiedasafunctionofthetemperaturesofthebathsandotherrelevantparameters. Themotor

n shows a current reversal when another parameter (a phase shift) is varied. Analytical predictions
a and results from numerical simulations are performed and agree very well. Generic properties of
J thistypeof motors are discussed.

3
1
I. INTRODUCTION

]
h
c We all do know that it is possible to extract some
e amount of mechanical work from a thermal bath at a
m
temperatureT providedwehaveanotherbathatalower
2
- temperature T < T . Thermal engines are the devices
t 1 2 T ? T
a that perform this task. All this is well known from el- 1 2
t ementary textbooks on thermodynamics. We also know
s
. from statistical mechanics that any object in a thermal Figure 1: Feynman’soriginal ratchet and pawl machine.
t
a bath exhibits random energy fluctuations of the order
m
K T. These fluctuations are relatively very small for
B
- macroscopic objects but of very important relevance for T
d nanometric objects such as biological motors: kinesins, k 1
n
dyneins, etc. [1]. We are also familiar with windmills,
o
c whichareabletoextractusefulworkfromrandomwinds
h
[ byaproperadaptationtothewinddirection. Wecanask
ourselves if it is possible to rectify thermal fluctuations θ h (θ )
2 R
by some appropriate mechanical devices.
v
4 Theengineswhichaimtogetusefulworkbyrectifying Figure 2: Detail of the stopping mechanism based on the
0 thermal fluctuations are called Brownian motors (BM). interaction between theratchet and thepawl.
6
In fact, the paradigm of such speculations is Feynman’s
1
famous ratchet and pawl machine [2]. During the last
1
years a lot of effort has been invested to study what
4
0 has been called as the ratchet effect. This is a mech- have the efficiency of a Carnot engine when operating
/ anism which consists in breaking the spatial and tem- reversibly. He idealized a gadget (see Fig.1) made up
t
a poral inversion symmetry of the system so that directed of two boxes with some gas at different temperatures.
m transport emerges, often enhanced by the thermal fluc- The hotter box contains an axle with vanes in it. The
- tuations. The ratchet mechanism can be implemented bombardments of gas molecules on the vane make the
d in different ways. Here we will make a model through axle rotate with random symmetric fluctuations. At the
n
an equation for a dynamical classical variable (position other end side of the axle there is a second box with a
o
c or angle) moving in a periodic and asymmetric poten- toothed wheel which in principle can turn only one way.
: tial (a ratchet potential) coupled with another degree of The pawl (the stopping mechanism) is under the influ-
v
freedom which will break thermal equilibrium. ence of another temperature (Fig.2). At first glance one
i
X There is an enormous variety of ratchets in the liter- could think that it seems quite likely that the wheel will
r ature: pulsating ratchets (on-off, fluctuating potential, spin round one way and lift a weight even when both
a
traveling potential), tilting ratchets (fluctuating force, gasesareatthesametemperatureandthusviolatingthe
rocking ratchets), Seebeck ratchets, Feynman ratch- Second Law. However, a closer look at the pawl reveals
ets, temperature ratchets, frictional ratchets, quantum that it bounces and so the wheel will rotate randomly
ratchets, collective ratchets, mechanico-chemical ratch- in any direction, doing a lot of jiggling and with no net
ets, ionic bombs and pumps, etc. [3]. turning. Thus, the machine cannot extract work from
In this work we will focus on a new Brownian mo- two baths at the same temperature.
tor inspired by the so called Feynman’s ratchet (Fig.1). When the temperature of the vanes is higher than the
Feynman stated in his famous lectures [2] that a very temperature ofthe wheel, Feynmanconcludes that some
particular device (the ratchet and pawl machine) could workis performedwithCarnot’sefficiency whenthe ma-
2
chine is lifting the weight very slowly. This is indeed a
θ θ
veryoptimisticresultwhichhasbeenrevised,manyyears 1 k 2
later, in Refs. [4, 5]. Indeed, there are some overlooked
aspects in Feynman’s argument which can be summa-
rized as follows: since the engine seems to be simultane-
τ
ouslyincontactwithtwobathsatdifferenttemperatures T T
1 2
(through the rigid axle), it cannot work in a reversible
way and Carnot’s efficiency will never be achieved. We
Figure 3: Model for thesymmetric Brownian motor (SBM).
will come back to this point along this paper.
A possible mathematical model of the device of Fig.1
andFig.2intermsofoverdampedLangevinequationsis,
II. A SYMMETRIC BROWNIAN MOTOR
dh (SBM)
λ =−∂ V +ξ (t), (1)
1 h 1
dt
Feynman’s ratchet, and similar models, do not fulfill
the following inversion property: T ↔ T −→ v ↔ −v.
2 1
dθ Thus our initial motivation to propose this new model
λ =−∂ V +ξ (t), (2)
2dt θ 2 is to have a motor such that, when switching the tem-
peratures of the two baths, this is, when reversing the
whereξi(t)mimicsthermalfluctuationsanditisassumed temperature gradient, the mean velocity absolute value
tobeawhitenoise,Gaussiandistributedwithzeromean, does not change while the sign of the velocity does. Ob-
satisfying the fluctuation-dissipation theorem, viously, this would imply some geometrical symmetries
in the engine. One example of a microscopic model for
′ ′
<ξ (t)ξ (t)>=2k T λ δ(t−t)δ . (3) a Brownian motor that suffers this change in the sign of
i j B i i ij
the current can be found in [8]. There is also a model
for a diode rectifier that is analytically solvable [9] and
The total potential energy is modeled by,
that has the same main features as the SBM, namely, it
1 e is symmetric and it shows ”reversibility”.
V(θ,h)=τθ+ kh2+ , (4) Our ”symmetric” motor is shown in Fig.3. According
2 |h−hR(θ)|
e l0 −1 to this scheme, the stochastic differential equations that
describethedynamicevolutionofthesystemintheover-
where,hR(θ)representstheperiodicbutasymmetricpro- damped regime are,
file ofthe ratchet. Anexplicitexpressionwillbe givenin
the next section. dθ
1
λ =−∂ V +ξ (t),
ThepotentialV hasthreeterms. τ isthetorquewhich 1 dt θ1 1
gives useful work. The second term accounts for the po- dθ
2
λ =−∂ V +ξ (t). (5)
tential energy stored in the spring (the pawl) with con- 2 dt θ2 2
stantk,whichpushes downthe ratchet. Finally,the last
term is a very repulsive potential at short distances and The noise terms are again white thermal noises, Gaus-
nearlyzeroatlongones. Itisusedtoavoidthatthepawl sian distributed and with zero mean, satisfying the
crosses through the real physical surface of the ratchet. fluctuation-dissipation theorem (3).
At the same time, it couples both degrees of freedom θ The potential V, contains an external torque τ, two
andh. Otherpotentialscanaccountforthesamephysics ratchet-shape potentials and an harmonic interaction
in this mathematical scheme [5, 6]. that couples both degrees of freedom,
Numerical simulations show a non-zero mean velocity
1
of the ratchet device for very high temperature differ- V(θ ,θ )=τθ +V (θ )+V (−θ )+ k(θ +φ−θ )2,
1 2 1 R 1 R 2 1 2
ences. The main conclusion is that, although the motor 2
(6)
can perform useful work, it has an extremely small effi-
where the phase shift φ is one of the most important
ciency, being very far away from Carnot’s efficiency es-
parameters.
timated by Feynman, as pointed out and shown in Refs.
InFig.4weseethe formoftheratchetpotentialV (θ)
[4, 5, 6, 7]. This motor presents a high thermal conduc- R
used in the numerical simulations. Its analytical expres-
tivity and as a consequence has a very low efficiency.
sion is,
The structure of this paper is the following. In sec-
tion II we present the model for a symmetric Brownian V
0
motor and the numerical results obtained by computer VR(θ)=− V(θ), (7)
2.23
simulations of the equations of such device. Section III
is devoted to the analytical approach to this model. We
end in Section IV with some comments and conclusions. V(θ)=sin(dθ)+0.275sin(2dθ)+0.0533sin(3dθ). (8)
3
1 0.2
0.15
0.5
0
V 0.1
/ 0
R v
V 0.05
-0.5
0
-1 -0.05
0 0.5 1 θ 1(.r5ad)2 2.5 3
-0.1
0 2 4 6 8 10
T / T
Figure 4: Shapeof theratchet potential VR for d=4. 2 1
Figure 5: Mean velocity versus T2/T1 (τ =0).
V controlstheheightofthepotential,disthenumberof
0
teeth per cycle 2π, and the asymmetry of the potential
is controlled by changing the numerical coefficients that
multiply the sinus functions expansion of the potential.
0.15
Noticethattheratchetpotentialthatonevariable(θ )
1
sees is the specular image of the one that the other vari- 0.12
able (θ ) feels, as in the diode rectifier of Ref. [9]. It is
2
in this sense that we call our model symmetric. 0.09
To study the relevance of each parameter we proceed v
0.06
toexpressoursystemintermsofdimensionlessones. In-
troducing the dimensionless time s as t = λ1s, the set 0.03
V0
ofequations(5)becomesmorecompact. Thedimension-
0
lessparametersarenow: T = kBT1,T = KBT2,k = k ,
λ = λλ21, τ = Vτ0 and, offc1ourse,V0φ, wfh2ich isV0in readiaVn0s. -0.030 0.2 0.4 τ0.6 0.8 1
Notice that V controls the energy scale, which is mea-
0
seuredinKe T units. Wealsosee,forinstance,thatonly
B 1
the fraction of the friction coefficients is relevant. Figure6: Meanvelocityofthemotorattheoptimalregimeas
a function of the external torque τ. The stall torque (where
v=0) is around τ =1.2.
A. Numerical results of SBM
Preliminary numerical results indicate that the motor
invertsitsvelocitywhenthetemperaturegradientisalso has a zero mean velocity. The linear dependence of the
inverted(seeFig.5). Someimportantthermodynamicre- velocity around T /T ∼ 1 is is clear signature that the
2 1
quirements are also fulfilled: there is no net motion in motorhastheinversionproperty. Wewillextendthisre-
the limits T2 →T1 (a unique bath), k → 0 (no coupling sultfurther. InFig.6we see the expectednegativelinear
between baths) and k →∞ (only one effective tempera- dependence of the velocity versus torque until the stall
ture). force where the velocity is zero.
To perform the simulations we take λ=1 for simplic-
Moreoverthemoststrikingresultisthe velocityinver-
ity and K T = 1. As in previous models for motors
B 1 e sion as a function of the phase difference parameter φ.
we have explored the parameter domain to get the most
We show in Fig.7 the mean velocity of the motor as a
effective values (larger velocities). This situation corre-
functionofthephaseshiftφinthecasesofT =2T and
sponds to the parameter values V = 2.5, k = 100 and 2 1
0 when T = 2T . One must notice that both cases are
d=16. In Fig.5 and Fig.6 some numerical results of the 1 2
meanvelocityv =. hθ˙ iarepresentedfor φ=0.4. We see ”symmetric”. This result allows to extract some conclu-
1 sions: theexistenceofacurrentinversionasafunctionof
in Fig.5 that there is a maximum around T /T = 2.5
2 1 φ, the great amplification of the speed for the particular
and a minimum at T /T = 0.5. This implies that the
2 1 value of φ = 0.4 and, finally, that the motor is perfectly
velocity is bounded and larger temperature gradients do
symmetric when reversing the temperature gradient.
not imply larger velocities. For very large T the motor
2
doesnotseetheratchetpotentialandcannottakeadvan- Theseresultsdeservetheoreticalexplanationsthatwill
tage of the broken inversion symmetry. Then the motor be presented in section III.
4
0.15 0.004
0.1
0.003
0.05
v η
0 0.002
-0.05
0.001
-0.1
-0.15 0
0 0.1 0.2φ (rad0.3) 0.4 0.5 0 0.25 0.5 τ0.75 1 1.25
Figure 7: Current inversion profile v(φ) when T2 =2T1 (cir- Figure 8: Efficiency η as a function of the torque τ. The
cles) and when T1 = 2T2 (triangles). The angle goes from 0 numerics are doneat theoptimal regime of theSBM.
to 0.52 because we are considering d = 12 and therefore the
periodicity is 2π =0.52. The dashed lines are a guide to the
12
eyes.
Our interesthere is to find the efficiency of the motor,
which is defined, in general, as ratio between the work
extracted versus the energy input,
B. Energetics of the SBM: the efficiency
hWi
η = . (10)
It is not enough to know whether a motor will move, hRi
nor even how fast it will run. One would also like to
The work W is quite easy to calculate since it is just
know how efficiently such machine can operate. There
the torque times the angle displaced. Then the work
are many different approaches in the literature to study
performed during a time interval t −t can be obtained
f i
energetics and, in particular, efficiencies [6, 10, 11]. We
as,
will follow Sekimoto’s characterization of the energetics
of thermal ratchet motors [6]. This scenario is the most hWi=τhθ (t )−θ (t )i=τhvi(t −t ). (11)
1 f 1 i f i
suitableschemebothbecauseitisintuitiveandalsofrom
its potential applicability. The main idea is to find how TheevaluationofhRineedsamuchmorecarefulanal-
much energy, received from an external source, the mo- ysis. Following [6], we can write,
tor will be able to employ to produce useful mechanical
tf ∂V(θ (t),θ (t))
work, i.e. to lift a load. In order to compute this, we R= [ 1 2 ]dθ (t). (12)
2
must understand where the energy comes from, where it Zti ∂θ2
can go and what ingredients account for that. Sekimoto
So byusing (6), we findthat the energythat the bathat
deals with energies and not with power. When one tries
T transfers to the mechanism in the bath at T is,
2 1
to study the problem in units of power, velocities ap-
pear explicitely in the computations and, since they are tf ′
R= [−k(θ (t)+φ−θ (t))+V (−θ (t))]dθ (t). (13)
instantaneously not continuous because of the stochas- 1 2 R 2 2
Z
ticity of the system, one finds that some of the terms ti
are extremely bad behaved and it is impossible to get Wehavetoevaluatethisintegralfordifferentrealizations
a reasonable numerical convergence in the simulations. andperformusualstatisticalaverages. Wehaveobserved
Moreover, we have found it easier to directly compute that this quantity is always large thus implying impor-
the stochasticintegralsandafterwardsperformthe aver- tant looses of heat.
age. To progress further let us first note that the potential
Let’s analyze the energetics of such kind of systems. energy U does not contribute because we have a cyclic
′
We would expect there’s some energy R that is released engine. In fact V2(−θ2(t)) is periodic and bounded and
fromthebathatT (theoneatahighertemperature)and thusitcanbediscardedinthelongtimeintervalstudied.
2
thatispartiallyconvertedintomechanicalworkW tolift Thus the final expressionfor the efficiency that has to
the load. However, according to the Second Law, some be evaluated numerically is,
of this input energy must be dissipated D into the heat
τhθ (t )−θ (t )i
bwailtlhaalstoTb1e. Iafcthhaensgyestienmthheaspoatneninttiarilnesnicerpgoytednetniaoltetdhebrye η = −kh ttif(θ1(1t)+f φ−1θ2(it))dθ2(t)i. (14)
U. So the law of the total energy conservation is, R
In Fig. 8, the efficiency η as a function of the torque τ
R=D+W +U. (9) is plotted. The specific values we use are around those
5
that were shown to be the optimal ones; V =3, T =2,
0 2
k =100,φ=0.4andd=12. Theplotshowsaparabolic
0.03
curve for the efficiency with a maximum at the middle
value of the stall torque. Notice that the maximum of 0.02
the efficiency is still extremely low. Let’s also underline
that at the stall force (τ ≃ 1.25), the efficiency is zero, 0.01
which means that unavoidable heat transfer occurs [4]. v
0
Therefore,evenwhenhvi→0there’sanon-zeroheatflux
transfered from the hotter bath to the cooler one. This -0.01
is the reason why these motors have so low efficiencies.
-0.02
-0.03
III. ANALYTICAL STUDY OF THE CURRENT 0 0.25 0.5 φ 0(.7r5ad) 1 1.25 1.5
INVERSION
Figure9: Comparisonofvvsφobtainedfromnumericalsim-
As we have seen in the numerical results the φ pa-
ulationsofthemodel(5)(circles)andfromtheanalyticalex-
rameter controls a spectacular phenomena. The SBM pression (23) (solid line). The parameters chosen are d = 4,
exhibitsvelocityinversionwhenthisparameterisvaried. k=100, V0 =3 and T2=2T1.
Weknowthatthisphenomenonexistsandhasbeenstud-
ied in the literature [12]. Our purpose now is to predict
analytically that this happens in our model. Any kind
The explicit expressions for these quantities and further
ofexactcalculationinthis systemseemsimpossible and,
mathematicaldetailsareintheAppendix. Summarizing,
so, some approximations have to be assumed.
we have found that the mean angular velocity v can be
ConsidertheequationsthatdefineourBrownianmotor
expressed in the standard form [12, 14, 15],
(5) written in the form,
2π
θ˙1 =f(θ1)−k(θ1+φ−θ2)+ξ1(t), (15) vφ = d N(1−eβφ), (23)
where N is a kind of normalization constant that does
θ˙2 =−f(θ2)+k(θ1+φ−θ2)+ξ2(t), (16) notdepend verymuchonthe parameterφ. The relevant
quantity β, which is a function of φ, is given by,
where f(θ) is the force exerted by the ratchet potential
and there is no torque. Let us define now the change of L −F(x)− Q(x)G(x)
variables [13], βφ = dx 2k .
Z0 1 1+ T2 +(1− T2)G(x) +O(1/k2)
θ +θ 4h T1 T1 k i
x= 1 2, (17) (24)
2 This expression can be numerically evaluated using the
expressions for the different functions that appear in it.
θ −θ +φ The value of the parameter N has been estimated by
1 2
y = . (18)
2 using only one numerical value of the velocity obtained
by the simulation (the minimum value at φ = 0.25). If
Therelevantvariablexdescribestheevolutionofthecen-
β changes its sign as a function of φ then the inversion
ter of mass and the ”irrelevant” variable y describes the
phenomenon is predicted.
relative motion of the two-particle system. Introducing
In Fig.9 we compare the predicted theoretical depen-
a redefinition of the noises,
dence of the velocity as a function of the phase φ to the
values obtained by numerical simulations of the model.
ξ (t)+ξ (t)
η (t)= 1 2 , (19) The analyticalresult, consideringthe amountofapprox-
1
2
imations involved, traces very accurately the current in-
version phenomenon. The fit is very encouraging and
ξ1(t)−ξ2(t) enlighting. The positions of the maxima are quite well
η (t)= , (20)
2 2 determined, the inversionof current is clearly coincident
with the simulations and even the little shift near φ=0
and using the fact that y is very small, we can make a and φ = 2π is faithfully reproduced. This means that
Taylor expansion up to first order, getting the pair of d
in our approximations we have kept the most dominant
equations,
ingredients. Note that the x-axis goes from 0 to 1.57
because we have imposed d = 4 (2π = 1.57) in order
x˙ =F(x)+yG(x)+η1(t), (21) 4
to safely use the analytical expressions derived in the
Appendix. However, for d = 12 the computation of β
y˙ =Q(x)+y(R(x)−2k)+η (t). (22) predicts qualitatively the current inversiontoo.
2
6
Now we will give an intuitive explanation of why the
motor runs either forward or backwards, depending on
the value of the parameter φ. The original problem can
be mapped into a multiplicative noise scenario (see the
Appendix for details) well documented in the literature
[12, 16,17, 18]. Thenwe canstraightforwardidentify in eff
g
nouotrcnaeseedthteoebffeecatisvyempmoteetnritciaalnVyefmfo(rxe),(aAn1d4)t,hweheicffhecdtoivees , eff
modulation g (x) (A13), which is responsible for the V
eff
breaking of the symmetries that allow a non-zero mean
velocity in the motor. What is more, by plotting them
we can very intuitively see whether and why there is a
-10 -5 0 5 10
forward or backwardflux [16, 17, 18]. x (rad)
In Fig.10 we plot, for three significative phase shifts,
the noise coupling function g (x) and the effective po-
eff
tential V (x). A careful look at each of them easily
eff
reveals the current sign in a very intuitive way. The key
point is the fact that when g (x) is large, the noise ef-
eff
fects are amplified on one side of the hill of the periodic
potential and, then, the net current will go in that di- geff
rpeocstiitoionn. Tofhegnef,fi(txi)swmiuthchrecslepaercetrttohaVtefdfe(pxe)ndthinegcounrrtehnet , eff
will go forwards,backwardsor,in the symmetric case, it V
will cancel. Thus, this simple scheme explains the quali-
tative behavior of the current inversion.
Twomorecommentsaretobemade. Justforthesake -10 -5 0 5 10
x (rad)
ofconsistency,wehavecheckedtwobasicconditionsthat
must always hold on these kind of motors. The first one
concernstemperatures;ifwesetT =T , the integralβ
1 2 φ
vanishesbecausethedenominatorbecomesconstantand,
therefore, the average velocity is zero. The second test
is also related with the first one. When we make k tend
to infinity (this implies a rigid coupling and then only eff
one effective temperature), the dependence on x of the g
misuplrteipselincta)tiavnedn,otisheend,istahpepemaorstoargacainnn(oontlmyaodvediatinvyewnhoeisree , eff
V
on average.
IV. COMMENTS AND CONCLUSIONS -10 -5 0 5 10
x (rad)
We have presented and studied a symmetric Brown-
Figure 10: Plots of the shape of the effective multiplicative
ian motor. Its efficiency has been numerically obtained
noise g (x) (A13) (dashed line) and the effective potential
eff
and other specific properties have been studied. This
V (x)(A14)(solidline)inarbitraryscales. Thephaseshifts
eff
motorhasarelevantexternalparameter(aphase)which are (from top to bottom) φ=0.25, φ=0.785 and φ=1.375,
induces the phenomenon of current inversion. We have whichcorrespondtothefastestnegative,zeroandfastestpos-
performedananalyticalcalculationwithappropriateap- itive velocities respectively for d = 4. Arrows indicate the
proximations to get an expression for the mean velocity velocity direction.
in terms of the relevant parameters of the model. This
formalpredictionfits verywellthe datafromthe numer-
ical simulations of SBM model.
An important conclusion is that, albeit the symmet- without the intermediate state of burning fuel. Conse-
ric motor has a larger efficiency than other mechanical quently,onecannotthinkofthesemodelsasrealisticones
Brownian motors, the efficiency of such devices is very for biological molecular motors.
small. Regardless of the particular properties of these Moreover, the mechanical coupling mechanism be-
kindof heatengines,they areanyhowunrealistic models tweenbothbathsactsasaverygoodheatconductoreven
for molecular motors since it is known that these bio- in situations of very small mean velocity. Therefore, the
logical systems do transform chemical energy into work, efficiency is only a smallfractionof that ofCarnot. This
7
is in fact a general feature of heat Brownian motors due 1 G(x)
g (x)= 1− . (A9)
2
to the fact that in order to rectify thermal fluctuations 2(cid:18) 2k (cid:19)
these systems must be put in contact and a lot of heat Let’swritenowtheFokker-Planckequationassociated
is interchanged. The diode rectifier of Ref. [9] does not to Eq. (A6),
present this problem and accordingly it can arrive near
the Carnot efficiency in some limits. ∂tP(x,t)=−∂xJ(x,t). (A10)
where,
V. ACKNOWLEDGMENTS
J(x,t)=H(x)P(x,t)−K T [g (x)∂ g (x)P(x,t)]
B 1 1 x 1
Fruitful discussions with ProfessorsI. Sokolov,J.M.R. −K T [g (x)∂ g (x)P(x,t)]. (A11)
B 2 2 x 2
ParrondoandC.VandenBroeckaregratefullyacknowl-
edged. This researchwassupportedbythe Ministeriode After some manipulations with partial derivatives, the
Educaci´on y Ciencia (Spain) under project BFM2003- probability current above can be rewritten as,
07850-C03-01.
J(x,t)=H(x)P(x,t)−[g (x)∂ g (x)P(x,t)],
eff x eff
(A12)
Appendix A: ANALYTICAL APPROACH where,
Ourstartingpointisthepairofcoupledequations(21) g (x)= K T g2(x)+K T g2(x), (A13)
and (22) where the different functions are, eff q B 1 1 B 2 2
and H(x) can be related to an effective potential,
f(x−φ/2)−f(x+φ/2)
F(x)= , (A1)
2
x
′ ′
V (x)=− H(x)dx. (A14)
eff
Z
′ ′
f (x−φ/2)+f (x+φ/2)
G(x)= , (A2)
2 Thenwehavetosolvetheequation(A12)inthesteady
state for a constantflux J with periodic conditions. The
first step is to reduce this equation to a Bernoulli form
f(x−φ/2)+f(x+φ/2)
Q(x)= , (A3) which can be formally integrated. By imposing periodic
2 boundary conditions, P(x)=P(x+L) (where L= 2π),
d
. we get,
′ ′
R(x)= f (x−φ/2)−f (x+φ/2). (A4) L e−β(x)
2 P0(1−eβ(L))=J dx , (A15)
Z0 geff(x)
Since the variabley has a faster dynamics we will first
eliminate adiabatically equation (22), and we will also where P0 is a constant that can be found by using the
L
discard the function R(x) because the parameter k is normalization condition P(x)dx = 1, and β is a rele-
0
very large. Then equation (22) reduces to, vant function whose exprRession is,
1 1 x −H(x′)
y = η2(t)+ Q(x) (A5) β(x)= dx′ , (A16)
2k 2k Z g2(x′)+(T /T )g2(x′)
0 1 2 1 2
Substituting now this expressionin equation(21) we get .
where we take K T = 1. Then the mean velocity v =
a Langevin equation with two multiplicative noises, B 1
hx˙i≃hθ˙ i(since hθ˙ i≃hθ˙ i)is found to be proportional
1 1 2
x˙ =H(x)+g (x)ξ (t)+g (x)ξ (t), (A6) to (1−eβ(L)). What is left to do is to find the β integral
1 1 2 2
for every φ.
where the new functions are, To simplify the calculation of the integral, we now
make an expansion in powers of 1. Since the value of
1 k
k that makes the motor run faster is around k = 100,
H(x)=F(x)+ Q(x)G(x), (A7)
2k one can safely suppose that the terms of the order (1)2
k
and so on will not notably contribute to the integral for
d < 5. Then, the integral (A16) turns out to be (24),
1 G(x)
g (x)= 1+ , (A8) which is evaluated numerically.
1
2(cid:18) 2k (cid:19)
8
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