Reversible ratchets as Brownian particles in an adiabatically changing periodic potential

Abstract

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.

Full text

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
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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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7300 57
BRIEF REPORTS