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Reversible ratchets as Brownian particles in an adiabatically changing periodic potential

Juan M. R. Parrondo

Departamento de Fı

´sica Ato

´mica, Nuclear y Molecular, Universidad Complutense de Madrid, 28040 Madrid, Spain

~Received 6 May 1997!

The existence of transport of Brownian particles in a one-dimensional periodic potential which changes

adiabatically is proven. The net fraction of particles crossing a given point toward a given direction during an

adiabatic process can be expressed as a contour integral of a nonexact differential in the space of parameters

of the potential. Since the work done to change the potential is an exact differential in the space of parameters,

cycles can be designed where transport of particles is induced without any energy consumption. These cycles

can be called reversible ratchets, and a concrete example is described. The repercussions of these results on

equilibrium thermodynamics are discussed. @S1063-651X~98!01406-8#

PACS number~s!: 05.40.1j, 05.20.2y, 05.60.1w, 05.70.2a

Magnasco has drawn the attention of the scientiﬁc com-

munity to a simple phenomenon, namely, that an asymmetric

potential, perturbed by external ﬂuctuations or periodic ex-

ternal forces, can induce a transport of Brownian particles

@1#. Since this seminal work, there have been a number of

papers proposing modiﬁcations and new models @2–11#, de-

signing experiments where the transport can be effectively

observed @12,13#, and calculating general properties and new

features of these models — such us ﬂows @3–5,7–11#, es-

cape rates @14#, and reversing currents @4,7,11,15#. There are

also sparse but signiﬁcant precedents pointing out that non-

equilibrium ﬂuctuations can induce a ﬂow of Brownian par-

ticles @16,17#. All these models are generically called ratch-

ets, since they are somehow inspired by the discussion in

Ref. @16#of a ratchet working as a thermal engine ~originally

proposed by Smoluchowski @18#!.

Two types of ratchets can be distinguished: changing

force ratchets or rocked ratchets, where the external ﬂuctua-

tions or external periodic forces are additive @1,3–5,7,14#;

and ﬂashing ratchets, where a periodic potential is modu-

lated either by a signal periodic in time or by nonthermal

ﬂuctuations @2,3,10–13,15#. It is worth mentioning that the

latter seem to be more relevant both for biological applica-

tions @3#and for segregation experiments @2,12#. In this pa-

per, I will focus only on ﬂashing ratchets, i.e., Brownian

particles in a periodic potential changing in time.

These systems belong to the realm of nonequilibrium

thermodynamics or statistical mechanics. It is believed that

the two basic ingredients for noise-induced transport are

nonequilibrium and anisotropic potentials. In equilibrium,

detailed balance ensures a null local current all over the sys-

tem @19#; thus the ﬁrst requirement seems to be unavoidable.

The second one stems from simple symmetry considerations.

However, in this paper I show that adiabatically moving a

one-dimensional periodic potential can induce transport of

Brownian particles. Moreover, it is possible to design a po-

tential, periodic in space and time, where transport of

Brownian particles can be induced without any energy con-

sumption. These types of systems can be called reversible

ratchets, and an explicit example is discussed below.

The existence of reversible ratchets is of extreme impor-

tance for designing Brownian motors with high efﬁciency.

Feynman @16#calculated under very simple assumptions the

efﬁciency of a ratchet, ﬁnding that it is equal to Carnot efﬁ-

ciency in the quasistatic limit. However, we have revealed

the inconsistency of this arguments by proving the intrinsic

irreversibility of the system under consideration @20#. Most

of the ratchets proposed in the literature are also intrinsically

irreversible ~see discussion below!, and their efﬁciency turns

out to be very low, whereas reversible ratchets posses a com-

paratively high efﬁciency @21#.

I consider Brownian overdamped particles moving in the

interval xP@0,1#under the action of a periodic potential

V(x,R), which depends on a set of parameters collected in a

vector R. If these parameters change in time as R(t) where

tP@0,T#, the probability density

r

(x,t) obeys the Smolu-

chowski equation

]

t

r

~x,t!5

]

x@V8„x;R~t!…1

]

x#

r

~x,t!52

]

xJR~t!

r

~x,t!,

~1!

where JR52V8(x;R)2

]

xis the current operator, and the

prime indicates derivative with respect to x. I have taken

units of time, length, and energy such that the diffusion co-

efﬁcient and the temperature times the Boltzmann constant

are equal to 1.

My aim is to calculate the net fraction of particles cross-

ing x50 to the right or integrated ﬂow of particles along the

process, which is deﬁned by

f

5

f

(0), with

f

~x![

E

0

Tdt JR~t!

r

~x,t!.~2!

This quantity can be obtained analytically when the potential

is adiabatically changed. Notice ﬁrst that the solution of Eq.

~1!, in the adiabatic limit, is given by the equilibrium Gibbs

state:

r

~x,t!.

r

„x;R~t!…[e2V„x;R~t!…

Z„R~t!…,~3!

with Z(R)5

*

0

1e2V(x;R). This state has zero current every-

where, i.e., JR(t)

r

„x;R(t)…50. Consequently, the total frac-

tion of particles crossing x50 to the right should be zero in

PHYSICAL REVIEW E JUNE 1998VOLUME 57, NUMBER 6

57

1063-651X/98/57~6!/7297~4!/$15.00 7297 © 1998 The American Physical Society

the adiabatic limit. However, it can be shown that this is not

the case. To start, I will prove the following lemma.

Lemma. Consider a Brownian particle in equilibrium with

respect to a potential V0(x) at time t50. If the potential is

suddenly changed to V1(x), then the net fraction of particles

crossing one of the boundaries of the system to the right,

during the relaxation to the new equilibrium state, is given

by

f

5

E

0

1dx

E

0

xdx8eV1~x!

E

0

1dx9eV1~x9!

@

r

1~x8!2

r

0~x8!#,~4!

where

r

i(x)5e2Vi(x)/Ziis the Gibbs state corresponding to

potential Vi(x).

The proof is as follows. Let us deﬁne the function

w

~x!5

E

0

`dt@

r

~x,t!2

r

1~x!#.~5!

If J1is the current operator corresponding to potential

V1(x), the integrated ﬂow of particles through a point xin

the interval can be written as

f

(x)5J1

w

(x), since

J1

r

1(x)50. Applying the operator 2

]

xJ1to Eq. ~5!, one

has

2

]

xJ1

w

~x!5

E

0

`dt

]r

~x,t!

]

t5

r

1~x!2

r

0~x!.~6!

w

(x) can be determined by solving this second-order differ-

ential equation with periodic boundary conditions,

w

(0)

5

w

(1), and imposing that the integral of

w

(x) along the

interval vanishes. These conditions are easily derived form

the deﬁnition of

w

(x)@Eq. ~5!#. Finally, once

w

(x) is ob-

tained, one ﬁnds Eq. ~4!by setting x50in

f

(x)5J

1

w

(x).

Let us now consider the following setup for an adiabatic

change of the potential V(x;R(t)), occurring from t50to

t5T(T→`). The parameter vector Rchanges by jumps

DR. After each jump, the system is allowed to relax before

the next jump takes place. Therefore, the system should relax

for a time much longer than its relaxation time in any of the

potentials V(x;R). This adiabatic limit is achieved if T→`

and DR→0 with T/Nsteps→`,Nsteps being the number of

steps taken to complete the whole process. Using the above

lemma, it is not hard to prove the following theorem.

Theorem: The total fraction of particles crossing x50to

the right, during the complete process in the adiabatic limit

described above, is given by the contour integral

f

5

E

R~0!

R~T!dR•

E

0

1dx

E

0

xdx8

r

1~x;R!¹R

r

2~x8;R!,~7!

where

r

6~x;R!5e6V~x;R!

Z6~R!,Z6~R!5

E

0

1dxe6V~x;R!.

This is the main result of this paper. It tells us that, even in

the adiabatic limit, the net fraction of particles

f

crossing

one of the boundaries of the system in a given direction can

be different from zero. Moreover, it indicates that this frac-

tion of crossing particles in an inﬁnitesimal process

d

f

5

E

0

1dx

E

0

xdx8

r

1~x;R!

d

r

2~x8;R!~8!

is not an exact differential. Consequently, it is possible to

have transport of particles in a cyclic process, R(0)5R(T),

in the adiabatic limit.

To stress the singularity of this result and to prove the

existence of reversible ratchets, let us repeat the same argu-

ments for the energy introduced in the system by changing

the potential. If one has a Brownian particle in equilibrium

with V0(x) and suddenly changes the potential to V1(x), the

energy introduced is equal to

Ein5

E

0

1dx

r

0~x!@V1~x!2V0~x!#.~9!

Part of this energy can be dissipated to the thermal bath in

the relaxation from

r

0(x)to

r

1

(x). The input energy along

the whole adiabatic process described above is given by the

contour integral

Ein5

E

R~0!

R~T!dR•

E

0

1dx@¹RV~x;R!#

r

~x;R!

52

E

R~0!

R~T!dR•¹Rln Z~R!.~10!

This expression has a simple interpretation in the context of

equilibrium statistical mechanics: ¹Rln Z(R) is a general-

ized pressure which, when multiplied by 2dR, gives us the

work done on the system. Remarkably, this work is an exact

differential in the Rspace. Hence the total work done on the

system along an isothermal cycle is always zero. Since the

fraction of crossing particles

d

f

is not an exact differential,

we can have transport without any energy consumption, i.e.,

a reversible ratchet.

Still, one could be suspicious about Eq. ~7!. What is

wrong with the adiabatic solution given by Eq. ~3!and the

argument discussed right below this equation? How can a

system present a net transport of particles if, at any time,itis

globally in thermal equilibrium and every local current van-

ishes? An alternative and more general proof of Eq. ~7!helps

to clarify these questions.

Let us ﬁnd the correction of the adiabatic solution ~3!up

to ﬁrst order on R

˙(t):

r

~x,t!.

r

„x;R~t!…1R

˙~t!•

w

W

„x;R~t!….~11!

Inserting Eq. ~11!into the Fokker-Planck equation ~1!and

neglecting

]

t@R

˙(t)•

w

W

#, one ﬁnds

¹R

r

„x;R~t!…52

]

xJR~t!

w

W

„x;R~t!….~12!

Solving this equation with periodic boundary conditions

w

W

(0)5

w

W

(1), and imposing that the integral of each compo-

nent of

w

W

(x) along the interval xP@0,1#vanishes ~exactly as

in the proof of the lemma!, the correction

w

W

can be found.

Finally, the fraction of particles crossing xto the right during

the process is @see Eq. ~2!#

7298 57BRIEF REPORTS

f

~x!5

E

R~0!

R~T!dR•JR

w

W

~x;R!,~13!

which, using the solution of Eq. ~12!and setting x50, re-

produces Eq. ~7!. We see that the correction R

˙(t)•

w

W

, al-

though vanishing in the adiabatic limit, gives a nonzero frac-

tion of particles

f

crossing x50 during the interval @0,T#.

This proof resembles the derivation of the well-known Ber-

ry’s phase @22#in quantum mechanics.

Before going on with a concrete example, I would like to

stress an important property of Eq. ~7!. From this equation, it

follows that no transport of particles occurs if one slowly

modulates a potential or, more generally, if one slowly

switches between two potentials VA(x) and VB(x) in the

following way: V(x,t)5r(t)VA(x)1@12r(t)#VB(x), with

r(t)P@0,1#periodic in time. This particular case is, remark-

ably, the only one which has been signiﬁcantly studied to

date @2,3,10–13,15#, and it turns out that the efﬁciency of

these ﬂashing ratchets, when considered as engines, has been

found to be very low @21#~see, however, Ref. @19#!.

In order to have a reversible ratchet, the cycle must be a

process along a loop. A ﬁrst and rather trivial example con-

sists of a well or a barrier around a point x5awithin the

interval @0,1#. If the parameter ais moved from 0 to 1, due

to the periodic boundary conditions, we have a cycle with

f

different from zero. This example has been studied before by

Landauer and Bu

¨ttiker in the context of reversible computa-

tion @23#. The application of Eq. ~7!reproduces their expres-

sion for the current @Eq. ~6.8!in Ref. @23##. However, in this

model we are actually pushing the particles in a given direc-

tion, and, therefore, it cannot be considered as a genuine

ratchet.

We can obtain a less trivial system if the potential de-

pends on two parameters and these parameters change adia-

batically along a loop. As an example of such a reversible

ratchet, I consider the potential of Fig. 1, which depends on

two parameters V1and V2. If these parameters are changed

following the path described in the same ﬁgure, then a trans-

port of particles is induced towards the positive xdirection.

In Fig. 2, I plot the shape of the potential at the four points of

Fig. 1. The way this ratchet works is apparent from this

ﬁgure, and one can see that a transport of particles to the

right is always induced. In Fig. 3, the net fraction of particles

f

crossing the boundaries of the interval to the right in a

period, calculated with Eq. ~7!, has been plotted as a function

of the width aof the barriers or wells of the potential. For

inﬁnite large barrier and wells (V→`), the fraction

f

is

equal to one for any value of abetween 0 and 1

2, as is evident

from Fig. 2: at step 2, the particle is within the well with

probability one, as it is at steps 3 and 4; then it must cross

x50 with probability one when moving from 3 to 4 and it

can never jump back.

In summary, the existence of reversible ratchets has been

proven. Moreover, I have presented a thermodynamic differ-

ential given by Eq. ~8!, which is not exact in the space of

parameters Rof the potential. This is a nontrivial result in

the ﬁeld of equilibrium thermodynamics, and it opens the

possibility of developing a complete thermodynamics of pe-

riodic potentials, including adiabatic changes of temperature,

chemical potential, and other thermodynamic functions.

There is a corollary of the theorem, which is important for

Brownian motors or noise-induced transport. From the above

results, it is clear that, in order to have transport, the change

of the potential must be driven, not only slowly, but also in a

given direction. Therefore, if this change is driven by a

noise, i.e., if Rﬂuctuates along a given path in the parameter

space, we cannot have adiabatic transport unless the noise

were biased toward a given direction. If, for instance, Ris

FIG. 1. Graphical representation of the reversible ratchet de-

scribed in the text: the potential depends on two parameters, V1and

V2, which are the height of two barriers or wells ~left!, and they

adiabatically change along the path depicted on the right (Vbeing

the half side of the square!.

FIG. 2. Shape of the potential at the numbered steps of the

adiabatic process plotted in Fig. 1.

FIG. 3. Net fraction of particles

f

crossing x50 to the right,

calculated using Eq. ~7!, as a function of the width aof barriers or

wells, for different values of the maximum height V.

57 7299BRIEF REPORTS

driven by a chemical coordinate, this bias can be supplied by

reactants with concentrations far from equilibrium ~fuel!,as

pointed out by Magnasco @6#in a different but related con-

text.

The above considerations are only valid in the adiabatic

limit, and Eq. ~7!cannot be applied to irreversible processes.

The discovery of ﬂashing ratchets by Prost et al. @2#and

Astumian and Bier @3#can now be interpreted in a different

way: they found a path in the Rspace which induces a cur-

rent in a given direction no matter how it moves along the

path, if it does so irreversibly. However, this is not strange in

thermodynamics. For instance, the change of entropy is al-

ways positive for an irreversible process, no matter what the

direction of the process. Nevertheless, as mentioned above,

these irreversible ratchets have a low efﬁciency, i.e., the in-

duced transport is very energy consuming @21#.

I appreciate discussions with J.M. Blanco, F. Cao, and R.

Brito on the efﬁciency of Brownian motors. I am also in-

debted to J. Cuesta, M. Man

˜

as, B. Jime

´nez de Cisneros, and

P. Ha

¨nggi, for suggestions which have improved the com-

pleteness and clarity of the paper. This work was ﬁnancially

supported by the DGCYT ~Spain!under Project Nos. PB94-

0265 and PB94-0388.

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@19#This statement is valid only for ﬂashing ratchets. In a rocked

ratchet, there is a net force acting on the particle at any time,

although the time average of this force vanishes. Therefore, it

is not surprising that, even in the adiabatic limit, there is a ﬂow

of particles, as has been shown in Ref. @7#. Essentially, this

ﬂow comes from the fact the mobility of the Brownian par-

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in Ref. @7##. Also see Ref. @11#for a discussion on the adia-

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BRIEF REPORTS