arXiv:cond-mat/9803302v1 [cond-mat.stat-mech] 25 Mar 1998
Efficiency of Brownian Motors
J.M.R. Parrondo
Dept. de F´ısica At´omica, Nuclear y Molecular, Universidad Complutense, 28040-Madrid, Spain
J.M. Blanco, F.J. Cao, and R. Brito
Dept. de F´ısica Aplicada I, Universidad Complutense, 28040-Madrid, Spain
February 1, 2008
Pacs: 05.40,82.20M
Abstract
The efficiency of different types of Brownian motors is calculated analytically and nu-
merically. We find that motors based on flashing ratchets present a low efficiency and an
unavoidable entropy production. On the other hand, a certain class of motors based on adi-
abatically changing potentials, named reversible ratchets, exhibit a higher efficiency and the
entropy production can be arbitrarily reduced.
In the last years there has been an increasing interest in the so-called “ratchets” or Brownian
motors [1, 2, 3, 4, 5, 6]. These systems consist o f Brownian particles moving in asymmetric
potentials, such as the one depicted in fig. 1 (left), and subject to a source of non-equilibrium,
like exter nal fluctuations or temperature gra dients. As a consequence of these two ingredients
—asymmetric potentials and non-equilibrium—, a flow of particles can be induced.
Most of the cited papers consider systems where the Brownian pa rticles do not gain energy
in a systematic way. Although these systems are called “Brownian or molecular motors”, they
do not convert heat into work, nor induce any energy conversion. Feynman in his Lectu res [7]
already understood that, in order to have an engine out of a ratchet, it is necessary to use its
systematic motio n to store potential energy. This can be achieved if the ratchet lifts a load. Then
the ratchet becomes a thermal engine and Feynman estimated its efficiency (although following
assumptions which have been revealed to contain some inconsistencies [8]). Recently, Sekimoto
[9] has generalized this procedure, defining efficiency for a wide class of ratchets. J¨ulicher et
al [6 ] have also discussed the efficiency of molecular motors and Sokolov and Blumen [10] have
calculated the e fficiency of a deterministically flashing ratchet in contact with thermal baths at
different temper atures. A general co nclus ion is that these motors are intrinsically irreversible, even
in the quasistatic limit [6, 8, 9, 10].
On the other hand, it has been recently introduced [11] a class of deterministically driven
ratchets where the entropy production vanishes in the quasistatic limit, i.e., reversible ratchets.
The aim of this letter is to explore the differences, regarding efficiency, between randomly flashing
ratchets and both r e versible and irreversible deterministically driven ratchets.
Randomly flashing ratchets
Consider two species of Brownian particles, say A and B , moving in the interval [0, L] with periodic
boundary conditions. Particles of type A feel a potential V
A
(x), whereas pa rticles B feel V
B
(x).
Besides, there is a continuous transfer of particles, A
⇀
↽
B, which accounts fo r non-equilibrium
fluctuations. This picture is equivalent to that of a single Brownian particle in a randomly sw itching
potential [3].
1
L 2L 3La
x
V(x)
10 15 20 25 30
V
-0.10
-0.05
0.00
0.05
0.10
J
η
Figure 1: Left: Asymmetric sawtooth potential of the ratchets presented in refs. [1, 2, 3, 4, 5]. In
this letter we consider two types of r atchets: a) one where the p otential is rando mly switched on
and off; and b) one where the potential is deterministically modulated.
Right: Efficiency and curre nt of the ratchet where the the potential on the left is randomly switched
on and off (case a), as a function of the maximum height V of the potential. The reaction rates
are ω
A
= 1.08 and ω
B
= 81.8, a = 1/11, and the e xternal force is F = 4.145.
In ref. [3], it was proved that a flow towards a given direction, say, to the right, occurs for some
asymmetric potentials V
A
and V
B
. If we add a load or force F oppos ite to the flow, the evolution
equation for the probability density of particles A, ρ
A
(x), and particles B, ρ
B
(x), is:
∂
t
ρ
A
(x, t) = −∂
x
J
A
ρ
A
(x, t) − ω[ρ
A
(x, t) − ρ
B
(x, t)]
∂
t
ρ
B
(x, t) = −∂
x
J
B
ρ
B
(x, t) + ω[ρ
A
(x, t) − ρ
B
(x, t)] (1)
where J
i
= −V
′
i
(x) − F − ∂
x
is the current operator, the prime indicates derivative with respect to
x, and ω is the rate of the reaction A
⇀
↽
B. We have taken units of energy, length and time such
that the temperature is k
B
T = 1, the length of the interval is L = 1, and the diffusion coefficient
is D = 1.
The flow of pa rticles in the stationary re gime is J = J
A
ρ
st
A
(x) + J
B
ρ
st
B
(x), where ρ
st
A,B
(x) are
the stationary solutions of eq. (1). This flow J is a decreasing function of the external forc e F and
becomes negative if F is stronger than a balancing force, F
bal
. Therefore, if 0 < F < F
bal
, particles
move against the force and, consequently, gain potential energy in a systematic way. The potential
energy gain or output energy per unit of time is E
out
= JF, which vanishes both for F = 0 and
F = F
bal
.
On the other side, switching on and off the potential requires some energy. In our two-sp e c ie s
picture, the reaction A
⇀
↽
B does not conserve energy since V
A
(x) 6= V
B
(x). Therefore, in each
reaction A → B, oc c urring at a point x, an energy V
B
(x) − V
A
(x) is transferred to the system
(or withdrawn, if the sign is neg ative). Similarly, an energy V
A
(x) − V
B
(x) is transferred to the
system in ea ch reaction B → A occurring at x. In the stationary regime, the average number of
such reactions p e r unit of time is, respectively, ωρ
st
A
(x) and ωρ
st
B
(x). Therefore, the input energy
per unit of time is [6, 9]:
E
in
= ω
Z
1
0
dx [V
B
(x) − V
A
(x)]
ρ
st
A
(x) − ρ
st
B
(x)
. (2)
Finally, the efficiency ca n be defined as [6, 9]:
η =
E
out
E
in
. (3)
This efficiency c an be calculated analytically for the system given by eq. (1) with piecewise
potentials. We have performed an exhaustive study for the particular setting V
B
(x) = 0 and V
A
(x)
2
equal to the potential depicted in fig. 1 (left):
V
A
(x) =
V x/a if x ≤ a
V (1 − x)/(1 − a) if x ≥ a
(4)
with a = 1/11. For this system, the maximum efficiency is η
max
= 3 .29%, which is reached for
V = 22, F = 3, and ω = 63. This efficiency can be improved using different reaction rates: ω
A
for
A → B and ω
B
for B → A. In this case, η
max
= 5.315% with V = 16.7, F = 4.145, ω
A
= 1.08,
and ω
B
= 81.8. Obs e rve that, with these values for ω
A,B
, the particle stays much longer within
the potential V
A
(x) than within V
B
(x).
We have plotted in fig . 1 (right) the efficiency and the flow of particles as a function of V
setting the rest of parameters eq ual to these optimal values. Two are the messages from this
figure. Firstly, the maximization of the efficiency is a new criterion to define optimal Brownian
motors and this criterion is, in some cases, less trivial than that of maximizing the flow. Secondly,
the randomly flashing ratchet under study has a rather low efficiency. As we have mentioned
befo re, the heat dissipation per unit of time is E
in
− E
out
. Consequently, the increase of entropy
of the thermal bath, per unit of time, is E
in
− E
out
, since k
B
T = 1. On the other hand, in the
stationary regime there is no change of entropy in the system nor in the external agent which
provides the non-equilibrium fluctuations
1
. Therefore, the net entropy production per unit of time
is E
in
− E
out
. If this entropy production would vanish, i.e., if the sys tem could work in a reversible
way, it should reach a 100% efficiency. On the contrary, the efficiency is b e low 10% and we can
conclude that the motor based on the rando mly flashing ratchet is very inefficient.
One could think that the efficiency would increase in limiting situations where the system is
close to equilibr ium, such us ω → 0 and/or V
A
− V
B
→ 0. However, a perturbative analysis of
eq. (1) shows that η → 0 in both limits. In the first case, ω → 0 , from eq. (1) one can easily
find that J is of order ω, so is F
bal
. Therefore, E
out
, in the interval 0 < F < F
bal
, is of order ω
2
,
whereas one can prove that E
in
is of order ω, g iv ing a zero efficiency in this limit. In the second
case, ∆V (x) ≡ V
A
(x) − V
B
(x) → 0, the input energy E
in
is of order ∆V
2
. However, surprisingly
enough, J is of order ∆V
2
and so is F
bal
, yielding E
out
of order ∆V
4
and, again, a vanishing
efficiency. We conclude that the flashing motor is intrinsically irreversible, as it has been pointed
out for related mo dels in refs. [6, 8, 9, 10].
Deterministically driven ratchets
A different strategy to re duce the production of entropy consists of consider ing Brownian particles
in a potential which changes deterministically in time. If the potential is changed very slowly, the
system evolves close to equilibrium and the entropy pro duction is low. From now on, we focus
our attention on Brownian particles in a spatially periodic potential V (x; R(t)) depending on a
set of para meters collected in a vector R which changes in time [11]. The parameters are changed
periodically in time with period T , i.e., R(0) = R(T ).
As in ref. [9], we have to modify our definition of efficiency. Firstly, we deal with energy transfer
per cycle [0, T ] instead per unit of time. Seco ndly, the input energy or work done to the system in
a cycle, as a consequence of the change of the parameters R(t), is:
E
in
=
Z
T
0
dt
∂V (x; R(t))
∂t
ρ(x, t). (5)
The probability density ρ(x, t) verifies the Smoluchowski equation:
∂
t
ρ(x, t) = −∂
x
J
R(t)
ρ(x, t) (6)
1
A physical realization of this external agent is a third species of particles, say C, feeling a potential V
C
(x) =
V
B
(x) − V
A
(x) and participating in the reaction as A + C
⇀
↽
B. If the temperature of C particles is the same as B
and A particles, then detailed balance holds and there is no flow of particles. However, if the temperature of the C
particles is infinity, we recover the flashing ratchet discussed in the text. Therefore, this randomly flashing ratchet
can be considered as a thermal engine in contact with two thermal baths, one at T = 1/k
B
and the other one at
infinite temperature (see also [10] for an interpretation of the deterministically flashing ratchet as a thermal engine).
3
where J
R
= −V
′
(x; R) − F − ∂
x
is the current operator corresponding to the potential V (x; R).
As before, the output energy is the current times the force F , but now the current is not stationary
and we have to integrate along the process:
E
out
=
Z
T
0
dt F J
R(t)
ρ(x, t) = F φ. (7)
where φ is the integrated flow.
With the above expr essions, the efficiency of the system, η = E
out
/E
in
, can be found analyti-
cally for T la rge and weak external forces, where it is expe cted to be high. For the integrated flow
one finds φ = φ
0
− ¯µF T , where ¯µ is the average mobility of the system:
¯µ =
1
T
Z
T
0
dt
Z
+
(R(t))Z
−
(R(t))
(8)
and φ
0
is the integrated flow for F = 0 [11]:
φ
0
=
I
dR ·
Z
1
0
dx
Z
x
0
dx
′
ρ
+
(x; R)∇
R
ρ
−
(x
′
; R), (9)
with
ρ
±
(x; R) =
e
±V (x;R)
Z
±
(R)
; Z
±
(R) =
Z
1
0
dx e
±V (x;R)
.
In eq. (9) the contour integral runs over the closed path {R(t) : t ∈ [0, T ]} in the space of
parameters of the potential. The term proportional to T in the integrated flow, φ = φ
0
− ¯µF T ,
arises because the force F induces a non-zero current which is present along the whole process.
As a consequence , the balancing force is F
bal
= φ
0
/¯µT , and, in order to design a high efficiency
motor, it is necessary to take simultaneously the adiabatic limit T → ∞ and the limit F → 0 with
F T finite. Notice also that the above expres sions are us e le ss if φ
0
= 0. In a previous paper [11], we
have discus sed the conditions fo r φ
0
to be different from zero and called reversible ratchets those
systems where φ
0
6= 0. From now on, we restrict our analytical calculations to reversible ratchets,
although we present below numerical results for an irreversible ratchet.
The input energy for weak force F a nd large T is E
in
= φ
0
F + b/T , with
b = −
Z
T
0
dt Z
−
(R(t))Z
+
(R(t))
(
Z
1
0
dx
Z
x
0
dx
′
ρ
+
(x; R(t)) [∂
t
ρ
−
(x
′
; R(t))]
2
+
Z
1
0
dx
Z
x
0
dx
′
Z
x
′
0
dx
′′
[∂
t
ρ
−
(x; R(t))] ρ
+
(x
′
; R(t)) [∂
t
ρ
−
(x
′′
; R(t))]
)
(10)
which is a positive quantity. Combining the above expressions, one finds for the efficiency:
η =
F (φ
0
− ¯µF T )
φ
0
F + b/T
=
φ
0
α − ¯µα
2
φ
0
α + b
(11)
where α = F T . This expr e ssion is exact in the limit T → ∞, F → 0. No tice that, even for large
T , the irreversible contribution, b/T , to E
in
is of the same order as φ
0
F .
In a given system, i.e., for a set o f parameters φ
0
, ¯µ and b, the maximum efficiency is reached
for α = (b/φ
0
)[
p
1 + φ
2
0
/(¯µb) − 1] and its value is given by
η
max
= 1 − 2
h
p
z(1 + z) − z
i
(12)
with z = b¯µ/φ
2
0
. Eq. (12) clearly shows how the term b in the denominator of eq. (11) prevents
the system to reach an efficiency equal to one. Fortunately, as we will see below in a concrete
example, using strong potentials one can get arbitrarily close to 100% efficiency.
4
V
2
V
1
a/2 a a/2
1
2
3
4
V
{
1
2
3
4
Figure 2: Graphical representation of the r eversible ratchet described in the text: the potential
depends on two parameters, V
1
and V
2
, which are the height o f two barriers/wells (left) and
they change along the path depicted on the center (V being the maximum height/depth of the
barriers /wells). On the right, the shap e o f the potential at the four labelled points.
To check the validity of the above theory and stress the differences between reversible and
irreversible ratchets, we have studied in detail one example of each class.
As an example of irreversible ratchet, consider the modulation of the potential in fig. 1 (left),
i.e., V (x; t) = cos
2
(πt/T )V (x) with V (x) given by eq. (4). In this case, φ
0
is zero and the above
theory cannot be applied. We have numerically integrated the Smoluchowski equation, eq. (6),
using a n implicit scheme with ∆t = 10
−5
, ∆x = 0.002, 0.005, and the Richardson extrapolation
method to correct inaccuracies coming from the finite ∆x. The efficiency has been obtained using
eqs. (3), (5), (7) and the results, as a function of F and for different values of T , are plotted in
fig. 3 (left). The efficiency is maximum for T around 0.5 and it goes to zero as T increases. The
maximum efficiency found by numerical integration is of the same order of magnitude as the one
found for the randomly flashing ratchet. Notice, however, that we cannot explore with numerical
exp eriments the whole space of parameters.
0 2 4 6 8
F
0.00
0.02
0.04
0.06
η
0 2 4 6 8
α
0.0
0.1
0.2
0.3
η
Figure 3: Left (irreversible ratchet): numerical results for the efficiency of the ratchet consisting
of the potential in fig. 1 (left) modulated by z(t) = cos
2
(πt/T ) as a function of the external forc e
F and for different values of the period T : T = 0.00125 (), 0.025 (2), 0.0 5 (3), 0.25 (×), and 0.5
(△).
Right (reversible ratchet): numerical and analytica l res ults for the efficiency o f the ratchet desc ribed
in fig. 2 for V = 5 a = 0.2 as a function of F and for different values of the period T : T = 1 (×),
2 (3), 10 (2), 4 0 (). The thick solid line is the analytical result given by eq. (11) in the limit
T → ∞ and F → 0. Note that η is an increa sing function of T in the reversible ratchet (right) as
opposite to the irreversible case (left).
On the other hand, let us consider the reversible ratchet represented in fig. 2. Here the
potential depends on two parameters, V
1
and V
2
, which are the heights/depths of two triangular
barriers /wells of width a. The ratchet consists of modifying at constant veloc ity the parameters
V
1
and V
2
along the path depicted in the same figure (center). This example is a modification of
the one presented in ref. [11]. Now φ
0
does not vanish and the above theory gives us the efficiency
in the limit T → ∞ and F → 0. For instance, for V = 5 and a = 0.2, we obtain φ
0
= 0.825,
5
![Figure 1: Left: Asymmetric sawtooth potential of the ratchets presented in refs. [1, 2, 3, 4, 5]. In this letter we consider two types of ratchets: a) one where the potential is randomly switched on and off; and b) one where the potential is deterministically modulated. Right: Efficiency and current of the ratchet where the the potential on the left is randomly switched on and off (case a), as a function of the maximum height V of the potential. The reaction rates are ωA = 1.08 and ωB = 81.8, a = 1/11, and the external force is F = 4.145.](/figures/figure-1-left-asymmetric-sawtooth-potential-of-the-ratchets-xxhf2ygj.webp)