PHYSICAL REVIEW E 83, 041122 (2011)
Resonant activation in piecewise linear asymmetric potentials
Alessandro Fiasconaro
1,2,*
and Bernardo Spagnolo
3,†
1
Centro Universitario de la Defensa de Zaragoza, Ctra. de Huesca s/n, E-50090 Zaragoza, Spain
2
Departamento de F
´
ısica de la Materia Condensada and ICMA (CSIC-Universidad de Zaragoza), E-50009 Zaragoza, Spain
3
Dipartimento di Fisica, Group of Interdisciplinary Physics, Universit
`
a di Palermo and CNISM, Viale delle Scienze, I-90128 Palermo, Italy
(Received 13 June 2010; published 22 April 2011)
This work analyzes numerically the role played by the asymmetry of a piecewise linear potential, in
the presence of both a Gaussian white noise and a dichotomous noise, on the resonant activation phenomenon. The
features of the asymmetry of the potential barrier arise by investigating the stochastic transitions far behind the
potential maximum, from the initial well to the bottom of the adjacent potential well. Because of the asymmetry
of the potential profile together with the random external force uniform in space, we find, for the different
asymmetries: (1) an inversion of the curves of the mean first passage time in the resonant region of the correlation
time τ of the dichotomous noise, for low thermal noise intensities; (2) a maximum of the mean velocity of the
Brownian particle as a function of τ; and (3) an inversion of the curves of the mean velocity and a very weak
current reversal in the miniratchet system obtained with the asymmetrical potential profiles investigated. An
inversion of the mean first passage time curves is also observed by varying the amplitude of the dichotomous
noise, behavior confirmed by recent experiments.
DOI: 10.1103/PhysRevE.83.041122 PACS number(s): 05.40.−a, 05.45.−a, 02.50.Ey
I. INTRODUCTION
Resonant activation (RA) is one of the well-studied noise-
induced phenomena for thermally activated barrier crossing
problems. For a Brownian particle surmounting a fluctuating
potential barrier from an initial metastable state, the signature
of the RA effect is the presence of a minimum of the mean
escape time as a function of the mean switching frequency
of the external force. The RA phenomenon is important for
investigating the transient dynamics in physical, chemical,
and biological systems [1–22]. Very recently, a strategy
against bacterial persistence based on resonant activation was
proposed in Ref. [23]. Moreover, RA was investigated in
a system with a piecewise linear potential and a multistate
noise [24], in a thermal-inertial ratchet [25], and was experi-
mentally observed in single-electron escape from a metastable
state over an oscillating barrier in a silicon-based ratchet
transfer [26]. Specifically, the role of a constant bias force
on the RA phenomenon was investigated in Ref. [25]; in
Ref. [26], an inversion of the curve of the mean first passage
time (MFPT) in the resonant region was experimentally
observed, with differences in the MFPT between the lower-
and resonance-frequency regimes, by increasing the amplitude
of the modulating radio frequency (rf) signal.
The RA phenomenon was theoretically investigated
in Refs. [1–16,23–25] and experimentally revealed in
Refs. [17–22,26]. The occurrence of the RA together with
other stochastic effects such as noise-enhanced stability
[12,20,27–29] and stochastic resonance [30] have been also
investigated [11–14,19,20]. The RA effect is characterized by
the strong correlation between the mean time of the potential
fluctuations and the crossing time over the barrier, and it
is totally different from the dynamic resonance requiring
the matching between the driving frequency and the natural
*
afiascon@unizar.es
†
http://gip.dft.unipa.it
frequency of the system [31]. In the seminal paper by Doering
and Gadua [1], a switching piecewise linear potential in a range
[0,L] with fixed minima in x = 0 and x = L was considered. A
slightly different choice was made by Bier and Astumiam [2],
who used a piecewise linear potential with the height of the
barrier fluctuating between V
0
− b and V
0
+ b,withV
0
the
height of the average barrier and b<V
0
, thus maintaining the
presence of the barrier in all t he dynamics.
Both choices have in common that the potential can be
considered as symmetrical in shape and maintains the same
value at the two extremes in all dynamics [V (0) = V (L)].
The dependence of the RA effect on the shape of potentials
has been previously investigated in Refs. [4,5,7,10], but t here
is lack of studies on the role played by the asymmetry of
the potential barrier in this phenomenon. In a very recent
paper [16], the thermally activated crossing of an asymmetric
fluctuating barrier was investigated, by using two different
scalings of the dichotomous noise.
The aim of this work, which is a more refined analy-
sis of a previous preliminary investigation [15] (see also
Ref. [47] in Ref. [16]), is to focus on the role played by the
asymmetry of a piecewise linear potential in the RA effect.
We solve numerically the Langevin equation associated with
this potential and calculate the MFPT and the mean velocity
of the Brownian particle crossing the potential barrier. The
asymmetry of the potential barrier has been investigated by
means of the stochastic transitions far behind the potential
maximum, from the initial well to the bottom of the adjacent
potential well. Owing to the asymmetry of the fluctuating po-
tential profile together with the randomly fluctuating external
force uniform in space, we find the following: (1) an inversion
of the MFPT curves, related to different asymmetric potentials,
as a function of the correlation time τ of the dichotomous
noise, f or low thermal noise intensities, (2) a maximum of
the mean velocity of the Brownian particle as a function of
τ , and (3) a weak current reversal in the miniratchet system
obtained with the asymmetrical potential profiles investigated.
We also find a reversal behavior for the MFPT with different
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ALESSANDRO FIASCONARO AND BERNARDO SPAGNOLO PHYSICAL REVIEW E 83, 041122 (2011)
amplitude of the dichotomous noise. The same behavior has
been observed experimentally in Ref. [26] [see their Fig. 5(a)].
Some comparisons with smooth-shaped symmetric potentials
have been also performed.
II. MODEL AND RESULTS
The overdamped dynamics of the Brownian particle cross-
ing the fluctuating potential barrier is described by the
following Langevin equation:
˙
x =−V
(x) + η(t) + ξ (t) =−U
(x,t) + ξ (t), (1)
with
U(x,t) = V (x) − xη(t)(2)
and
V (x) =
⎧
⎨
⎩
hx
x
m
x x
m
h(L−x)
L−x
m
x x
m
,
(3)
where ξ (t) is the Gaussian white noise, with zero mean and
correlation function ξ(t)ξ (t
)=2Dδ(t − t
), and D = k
B
T
is the noise intensity, with T the absolute temperature and k
B
the Boltzmann constant. In Eq. (1) the time is measured in
terms of the damping parameter γ with the scaling t = t
R
/γ ,
with t
R
the real time. The random force η(t ) is a dichotomous
noise source, which takes here the two values {−a,a} with an
exponential correlation function η(t)η(t
)=(Q/τ )e
−|t−t
|/τ
,
with Q = a
2
τ and correlation time τ . The potential U (x,t)
fluctuates randomly between two configurations, V
+
(x) (up)
and V
−
(x) (down), remaining in one of the two states for the
average time τ before switching to the other. We have
V
±
(x) = V (x) ± ax. (4)
In Eq. (3), the values of parameters are L = 1, h = 2, and
x
m
= L/2 + k.Herek represents the asymmetry parameter,
defined as the distance between the position of the maximum of
the potential x
m
and the position of the symmetrical maximum
x
s
(here x
s
= 0.5).
In Fig. 1 we report the “up” and “down” configurations
of the switching piecewise linear potential used in the
calculations. Specifically we have in panel (a) a symmetric
piecewise linear potential, the same as that used in Refs [1,2],
with V
±
(0) = V
±
(L); in panel (b) a symmetric piecewise linear
potential with the randomly fluctuating external force uniform
in space and V
±
(0) = V
±
(L), obtained from Eqs. (3) and
(4) with k = 0; and in panel (c) an asymmetric piecewise
linear potential with the randomly fluctuating external force
uniform in space, obtained from Eqs. (3) and (4) with k =
−0.25. In panel (c) we report also the average asymmetric
piecewise linear potential. The amplitude of the random
force i s a = 1.2.
The presence of the asymmetry together with the fluctuating
external force uniform in space as shown in panels (b) and (c)
of Fig. 1 has peculiar consequences for the behavior of the RA
phenomenon.
The potential here defined by Eqs. (2) and (3) can be
considered as a base modulus of the piecewise linear ratchet
subjected to a dichotomous noise source, widely used in the
literature [32–34].
-1
0
1
2
3
0 0.2 0.4 0.6 0.8 1
x
(a)
V
+
-
(x)
Up
Down
-2
-1
0
1
2
3
0 0.2 0.4 0.6 0.8 1
x
(b)
Up
Down
-1
0
1
2
3
0 0.2 0.4 0.6 0.8 1
x
(c)
Up
Down
Mean
FIG. 1. (Color online) “Up” and “down” configurations of the
switching piecewise linear potential V
±
(x) used in our calculations.
The position x
0
= 0 represents the starting point of the simulations. A
reflecting boundary is assumed at x = 0 and an absorbing boundary
at x = L. (a) Symmetric piecewise linear potential the same as that
used in Refs. [1,2] with V
±
(0) = V
±
(L); (b) symmetric piecewise
linear potential with the randomly fluctuating external force uniform
in space and V
±
(0) = V
±
(L), obtained from Eqs. (3)and(4) with
k = 0; (c) asymmetric piecewise linear potential with the randomly
fluctuating external force uniform in space, obtained from Eqs. (3)
and (4) with k =−0.25. In panel (c) we report also the average
asymmetric piecewise linear potential. The values of the other
parameters are L = 1, h = 2andx
m
= L/2 + k, x
s
= 0.5, and
a = 1.2.
Equation (1) has been solved numerically by using a time
step t = 10
−3
, and the averages have been performed over
a sampling of at least N = 50 000 realizations. In the ith
realization the particle is put at the starting position x
0
= 0,
and the time t
i
to cross the position x = L is computed. A
reflecting boundary is assumed at x = 0, in the left extremum
of the potential, and an absorbing boundary at the right
extremum x = L, after the particle surmounts the potential
barrier. The ensemble average of the t
i
gives the MFPT,
which presents the evidence of the RA effect, i.e., a well-
pronounced minimum as a function of the correlation time τ ,
for all the cases here studied. The results of our calculations
for asymmetrical and symmetrical potentials with random
external force uniform in space, that is, with V
±
(0) = V
±
(L),
are shown in Fig. 2. We notice that for the three values of the
asymmetry parameter k, we find the resonant correlation time
around the same interval τ
R
∼ 10, but with different values
of the corresponding resonant MFPTs (here called T
R
). These
decrease by increasing the asymmetry parameter k.
We notice that the resonant region shows an inversion of the
behavior of the MFPT curves for the three potential profiles
investigated, with respect to both the low and high mean
correlation times. Indeed, for τ lower than τ
C
L
≈ 10
−1
the
curves show a higher MFPT for positive asymmetry (k = 0.25)
and lower for negative asymmetry (k =−0.25), and the same
qualitative behavior is present for long mean correlation
times (τ higher than τ
C
R
≈ 10
3
). In the intermediate region
τ ∈ [τ
C
L
,τ
C
R
] (L and R mean left and right, respectively),
where we also find the r esonant values T
R
, the situation is
inverted: the highest T
R
value corresponds to the negative
asymmetry parameter and the lowest T
R
to the positive one.
By increasing the absolute value of the asymmetry parameter
k, the corresponding curves show, more pronounced, the same
behavior here presented.
On the other hand, calculations performed with the same
model of Refs. [1,2], that is, by using fixed extremes in
asymmetric piecewise linear potentials and the same potential
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0
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1
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-4
10
-2
10
0
10
2
10
4
10
6
10
8
MFPT
τ
k = -0.25
k = 0.00
k = 0.25
1.0
2.0
0.0
0.2 0.4
0.6
0.8 1.0
V(x)
x
0.00
0.02
0.04
0.06
10
-1
10
1
10
3
v
-
τ
FIG. 2. (Color online) Log-log plot of MFPT as a function of
the correlation time of the dichotomous noise. The RA phenomenon
is observed for three values of the asymmetry parameter k of the
piecewise linear potential [k =−0.25, k = 0 (i.e., symmetric poten-
tial), k = 0.25], with randomly fluctuating external force uniform in
the space. The value of the white noise intensity is D = 0.18. The
amplitude of the dichotomous force is a = 1.2. The left inset shows,
in a semilog plot, the mean velocity of the Brownian particle with
the same asymmetries as a function of the correlation time τ .The
inset on the right shows the three static (average) potential profiles
investigated. (The legend is the same for all the plots.) All the other
parameter values are as in Fig. 1.
heights (a = 0.6) for all the asymmetries, give very different
curves. In fact, as we can see from Fig. 3, no crossings
are present between the MFPT curves for the asymmetries
considered.
The inversion of the behavior of the MFPT curves,
and consequently the presence of the two intersections,
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1
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3
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4
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-4
10
-2
10
0
10
2
10
4
10
6
10
8
MFPT
τ
PWL - k = -0.25
PWL - k = 0.00
PWL - k = 0.25
1.0
2.0
0.0
0.2 0.4 0.6
0.8
1.0
V(x)
x
FIG. 3. (Color online) Log-log plot of MFPT as a function of the
correlation time. The RA effect is observed for different asymmetrical
piecewise linear (PWL) potentials with fixed extremes and equal
potential heights in the three cases. Differently from the results in
Fig. 2 we do not find any intersection between the three curves, and the
resonant correlation time τ
R
changes for the different asymmetries.
The amplitude of the dichotomous force is a = 0.6, the same as that
corresponding to the symmetric case in Fig. 2. All the other parameter
values are as in Fig. 1.
approximatively at τ
C
L
and τ
C
R
, is uniquely present in the
MFPTs calculated for asymmetric potentials, by using a
fluctuating force uniform over all the range [0,L]. This
inversion does not appear either by using symmetric potentials
having different shapes (see Fig. 8) or by using the asymmetric
ones with fixed extremes and equal potential barrier heights
(Fig. 3). In other words, the comparison between the results
plotted in Figs. 2, 3, and 8 highlights that the crossing feature of
the MFPT curves occurs not merely because of the asymmetry
of the potential profiles, but, instead, because of the presence of
the asymmetry together with the randomly fluctuating external
force uniform in space η(t).
These features explain why we find higher MFPTs for pos-
itive asymmetry than those obtained with negative asymmetry,
for both low and high values of correlation time τ . In fact,
for very low correlation time, i.e., for very fast fluctuations of
the potential profile, the Brownian particle “sees” the average
barrier [1] (see inset on the right of Fig. 2). Because of
the different slopes of the potential, the particle will spend
more time in the interval x<x
m
for potential profile with
positive asymmetry than in the case of negative asymmetry.
As a consequence the MFPTs to r each the boundary at x = 1
for positive asymmetry will be higher than those for negative
asymmetry (see the references in Ref. [28] on noise-enhanced
stability for the role of the slopes of the potential profile
on the MFPT). For very high correlation time, that is, for
very slow fluctuations of the potential barrier, the MFPT is
equal to the average of the crossing times over upper and
lower configurations of the barrier, and the slowest process
determines the value of the average escape time [1]. In that case
we have the same role of the different slopes, and the MFPTs
for positive asymmetry will be higher than those obtained with
the negative one.
Concerning the resonant region of τ , the inversion of the
MFPT curves for the various asymmetries is due to the different
barrier heights of the fluctuating potential. In fact, the resonant
MFPT values T
R
are determined mainly by the mean escape
time over the lower of the two barriers (V
−
)[8], that is,
T
R
∼ (1/V
2
−
)e
V
−
/D
. (5)
According to that, by increasing the value of k, the MFPT at
resonance becomes lower and lower. That is why we find, in
the resonant range of values of τ and at low noise intensities,
that T
R
values are lower for the positive asymmetry case than
for the negative one.
The model here investigated presents interesting features in
the MFPT: First, it has a value of the resonant correlation time
τ
R
not too strongly dependent on the asymmetry parameter
k; second, it presents two correlation time intervals, close to
τ
C
L
≈ 10
−1
and close to τ
C
R
≈ 10
3
, with approximatively the
same MFPT for all the k parameters. Moreover, the crossing
features of the MFPT behavior as a function of the correlation
times do not occur for any value of the thermal noise. A set of
calculations to check the crossing of the MFPT curves has been
performed, and the related results are shown in Fig. 4, where
the typical behavior of the RA effect is shown for different
values of the noise intensity, namely, D = 0.2, 0.25, 0.4, 0.6.
At low noise intensities D, the crossings between the curves
are maintained up to the threshold value D
T
. For higher noise
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ALESSANDRO FIASCONARO AND BERNARDO SPAGNOLO PHYSICAL REVIEW E 83, 041122 (2011)
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-4
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-2
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2
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4
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6
10
8
MFPT
τ
D = 0.20
k = -0.25
k = 0.00
k = 0.25
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1
10
2
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3
10
4
10
-4
10
-2
10
0
10
2
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4
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6
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8
MFPT
τ
D = 0.25
10
0
10
1
10
2
10
-4
10
-2
10
0
10
2
10
4
10
6
10
8
MFPT
τ
D = 0.40
10
0
10
1
10
-4
10
-2
10
0
10
2
10
4
10
6
10
8
MFPT
τ
D = 0.60
FIG. 4. (Color online) Log-log plot of MFPT as a function of
the correlation time for different values of the noise intensity D and
asymmetry parameter k: D = 0.20, 0.25, 0.40, 0.60. The values of k
and all the other parameters are the same as in Fig. 2. We observe the
RA effect for all the noise intensities investigated, the disappearance
of the crossings in the MFPT curves, and the shift of the minima by
increasing the noise intensity D.
intensities, no intersection appear in the curves. A shift of the
resonant mean switching time is observed together with lower
values of the r elated values of T
R
. This is in agreement with
the expression of Eq. (5) of the resonant MFPT and with the
physical picture of the RA phenomenon [1,2,8]; that is, at
an intermediate range of correlation time, the crossing event
is strongly correlated with the potential fluctuations and the
MFPT exhibits a minimum. By increasing the noise intensity,
the T
R
decreases and the resonant correlation time as well.
The increase of the noise intensity has the effect of speeding
up the escape process from the left potential well. These
behaviors are shown in Fig. 5, where the resonant MFPT values
T
R
are plotted as a function of the noise intensity D,forthe
three values of the asymmetry parameter k.
We can see that going beyond the threshold noise value
D
T
,thethreeT
R
curves invert their relative position. This
noise threshold corresponds to the presence (for D<D
T
)or
the absence (for D>D
T
) of the two crossings of the MFPT
curves visible in Figs. 2 and 4.
We notice that D
T
is not unique for all the asymmetries
investigated. We calculated the dependence of τ
C
L
and τ
C
R
on
the noise intensity D and on the fluctuation amplitude a, and
that of the threshold D
T
on the amplitude a. The results are
shown in Fig. 6. We can see that D
T
decreases with decreasing
the amplitude a because the difference of the potential barrier
heights decreases. Of course, when a → 0, the escape process
is essentially to cross the barriers of fixed potential profiles,
so no frequency dependence is expected for the activation
phenomenon. In this case the values of MFPT for the different
asymmetry parameter k are the same as in the Figs. 3 and 4 in
the limit of small τ .
The right bottom panel of Fig. 6 shows the MFPT as
a function of the correlation time τ for different values of
the amplitude a of the fluctuating force. We notice there
another peculiar region of the activation process: a crossing
of the curves at τ
a
≈ 70 with the MFPT having the same
10
-1
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1
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2
10
3
10
4
0.2
0.4
0.6
0.8
1 1.2
T
R
D
k = -0.25
k = 0.00
k = 0.25
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
10
-1
10
0
10
1
10
2
10
3
v
-
τ
D = 0.26
D = 0.24
D = 0.22
D = 0.20
D = 0.18
0
1
2
0 0.5
1
V(x)
x
FIG. 5. (Color online) Semilog plot of the resonant MFPT values
(T
R
) as a function of the noise intensity D for the three asymmetries
investigated. The value D
T
≈ 0.27 represents the threshold of thermal
noise that discriminates if the two crossings in MFPT curves of Fig. 2
are present (D<D
T
)orabsent(D>D
T
). Inset: mean velocity of
the Brownian particle for one value of the asymmetry parameter, k =
0.25, as a function of the correlation time τ for the noise intensities
D = 0.18, 0.20, 0.22, 0.24, 0.26. All the other parameter values are
as in Fig. 1.
value obtained at the high-frequency limit. At that limit, the
Brownian particle “sees” the same average potential profile,
independently of the amplitude a. This peculiar behavior has
been observed experimentally in the resonant escape over an
oscillating barrier in a single-electron ratchet transfer [26][see
their Fig. 5(a)]. Specifically, in that paper, the dichotomously
oscillating barrier is obtained by modulating the constant
voltage applied to the center gate of the Si nanowire with a
10
-2
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-1
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0
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1
10
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10
3
10
4
0.2 0.25 0.3 0.35 0.4
τ
C
L
- - τ
C
R
D
a = 1.2
a = 1.8
a = 2.4
a = 3.0
0.2
0.25
0.3
0.35
0.4
0.45
1
2
3
D
T
a
10
1
10
2
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-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
MFPT
τ
D = 0.30
a = 0.9
a = 1.2
a = 1.8
a = 2.4
a = 3.0
.
FIG. 6. (Color online) Left panel: Semilog plot of the values of
MFPT at the two intersections in Fig. 2, τ
C
L
and τ
C
R
as a function of
the noise intensity for different values of the amplitude of the dichoto-
mous noise: a = 1.2, 1.8, 2.4, 3.0. Right top panel: Threshold value
of the thermal noise intensity D
T
as a function of the amplitude a.
Right bottom panel: MFPT as a function of the correlation time for
different values of the amplitude a. All the other parameter values are
as in Fig. 1
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RESONANT ACTIVATION IN PIECEWISE LINEAR ... PHYSICAL REVIEW E 83, 041122 (2011)
square wave of amplitude A
rf
. By increasing A
rf
, the barrier
height in the “down” configuration decreases, and the MFPT
at resonance becomes lower as shown in Fig. 5(a) of Ref. [26]
and in our Fig. 6.
The presence of a resonant behavior, as well as the crossing
value at τ
C
L
, is also found in the behavior of the mean velocity
of the Brownian particle. The left inset of Fig. 2 shows, in fact,
this measure as a function of the correlation time of the fluctuat-
ing dichotomous force, calculated as ¯v = N
−1
N
i=1
L/t
i
.For
all the values of the asymmetry parameter, we see the presence,
before the saturating behavior, of a weak maximum that
corresponds to the resonant correlation time τ
R
. To check more
carefully this maximum, we performed a set of simulations
with different values of noise intensity, by increasing the
samplinguptoN = 120 000 realizations, for the asymmetry
parameter value k = 0.25. The results are shown in the inset
of Fig. 5. We see that the maximum is more pronounced by
increasing the noise intensity, while the curves show also a shift
in the position of the maximum, according to the analogous
shift in the minimum value of τ . We can also see from Fig. 2
that for low values of the correlation times (τ<τ
C
L
)the
mean velocity is higher for negative asymmetry and lower for
positive one, while for (τ>τ
C
L
) it is the inverse. This feature
gives rise to a current reversal in the ratchet, as predicted
in other works [35–37] and whose occurrence has been also
observed experimentally [38]. In fact, the difference between
the mean velocities for positive asymmetry and for the negative
one changes sign at τ = τ
C
L
. If we consider an asymmetrical
ratchet, this difference represents a net flux velocity, provided
that the absence of any reflecting boundary in that case gives
rise to changes in the values of the net velocity and on the
correlation time where the reversal occurs. Both the presence
of a maximum for τ ≈ τ
R
and the crossing at τ ≈ τ
C
L
are in
total agreement with the behavior of the MFPT. This agreement
fails for values of the correlation times higher than τ
R
.
While the MFPT curves increase in a different way and join
at the second cross, the velocities decrease slightly, reaching a
saturation value. This is because for high values of correlation
time, the particle can easily cross the potential barrier when it
is in its “down” configuration, reaching a relatively high speed
because of the low crossing time. Conversely, when the poten-
tial is in the “up” configuration, the particle takes a longer time
to cross the barrier, and so the contribution to the mean velocity
becomes very small and relatively negligible. This means that,
for high correlation times, ¯v maintains a relatively high value
because of the average between these two limit cases.
The above results, obtained for potential profiles with a
single barrier, can be applied to the ratchet potential having
the same asymmetric profile as elementary module. A set
of calculations has been performed with the aim of joining
the results obtained for the single-barrier case with the
simplest ratchet case, i.e., a ratchet with two barriers only.
Figure 7 shows the results in such a case, and the bottom right
inset shows the corresponding elementary ratchet. The system
consists of two asymmetric barriers without the presence of
any reflecting boundary. The MFPT presents again a resonant
correlation value τ
R
that lies in the same region found in
the single-barrier case. Now the MFPT is computed as the
mean time spent by the Brownian particle to reach the position
x = 1 or x =−1, indifferently, starting at x = 0. The particle
follows, on average, the easiest path, and the MFPT represents
10
0
10
1
10
2
10
3
10
4
10
-4
10
-2
10
0
10
2
10
4
10
6
10
8
MFPT
τ
k = 0.25
1.0
2.0
-1.0 -0.5
0.0 0.5
1.0
V(x)
x
0.00
0.02
0.04
10
-1
10
1
10
3
v
-
τ
-3 10
-4
-2 10
-4
-1 10
-4
0 10
0
1 10
-4
10
-3
10
-2
v
-
τ
FIG. 7. (Color online) Log-log plot of MFPT as a function of the
correlation time for the elementary ratchet composed by two single
barriers only. The RA effect is observed. The MFPT is here the mean
time taken by the Brownian particle starting at x = 0 to reach the
position x = 1orx =−1. The mean velocity of the particle is plotted
in the upper inset. The mean velocity presents again a maximum at the
same resonant value τ
R
. In the region of very low mean correlation
time, the particle shows a very weak negative velocity (left inset),
as a consequence of the different behavior of the mean velocities
for the various asymmetries and the crossing of the curves shown
in the inset of Fig. 2. The value of the noise intensity is D = 0.18,
and the amplitude of the dichotomous force is a = 1.2. All the other
parameters are as in Fig. 1.
here the minimum between the times spent to cross each single
barrier. This means that the curve is lowered, and the RA
effect is less pronounced. The mean velocity, plotted in the left
inset of Fig. 7, shows again a maximum at the same resonant
value τ
R
. For very low mean correlation times the mean
velocity shows a very weak negative value (left bottom inset in
Fig. 7). This means that a current reversal appears at a certain
correlation time τ
rev
. This feature follows from the different
behavior of the mean velocity in the two single asymmetric
potential profiles (with k =−0.25 and k = 0.25) reported in
the left inset of Fig. 2, where the presence of the crossing value
τ
C
L
indicates a flux reversal as a function of τ . The difference
in value between τ
rev
and τ
C
L
, as well as the difference in the
value of the mean velocity of the Brownian particle, have
to be ascribed to the presence of the reflecting boundary
in the single-barrier case, which changes the traveling times
of the particle and, so, the related mean velocities. The region
of the lowest mean correlation time (τ → 0) shows a zero
current, in agreement with the detailed balance principle. In
fact, that region represents the high-frequency fluctuation of
the potential. In this condition the potential “felt” by the
Brownian particle can be considered as a static one, and there
is no net flux at the steady state.
As a last remark concerning the role of the potential shape
on the the resonant activation effect, we report in Fig. 8 the
results of calculations of the MFPTs by using symmetric
smooth potentials. The corresponding static potentials used
have the simple polynomial form
V
2N
(x) = h2
2N
x
N
L
N
1 −
x
L
N
(6)
with h = 2 and N = 1, 2, 3.
041122-5
![FIG. 1. (Color online) “Up” and “down” configurations of the switching piecewise linear potential V±(x) used in our calculations. The position x0 = 0 represents the starting point of the simulations. A reflecting boundary is assumed at x = 0 and an absorbing boundary at x = L. (a) Symmetric piecewise linear potential the same as that used in Refs. [1,2] with V±(0) = V±(L); (b) symmetric piecewise linear potential with the randomly fluctuating external force uniform in space and V±(0) = V±(L), obtained from Eqs. (3) and (4) with k = 0; (c) asymmetric piecewise linear potential with the randomly fluctuating external force uniform in space, obtained from Eqs. (3) and (4) with k = −0.25. In panel (c) we report also the average asymmetric piecewise linear potential. The values of the other parameters are L = 1, h = 2 and xm = L/2 + k, xs = 0.5, and a = 1.2.](/figures/fig-1-color-online-up-and-down-configurations-of-the-12sqoxu2.webp)
![FIG. 2. (Color online) Log-log plot of MFPT as a function of the correlation time of the dichotomous noise. The RA phenomenon is observed for three values of the asymmetry parameter k of the piecewise linear potential [k = −0.25, k = 0 (i.e., symmetric potential), k = 0.25], with randomly fluctuating external force uniform in the space. The value of the white noise intensity is D = 0.18. The amplitude of the dichotomous force is a = 1.2. The left inset shows, in a semilog plot, the mean velocity of the Brownian particle with the same asymmetries as a function of the correlation time τ . The inset on the right shows the three static (average) potential profiles investigated. (The legend is the same for all the plots.) All the other parameter values are as in Fig. 1.](/figures/fig-2-color-online-log-log-plot-of-mfpt-as-a-function-of-the-3tkj8klk.webp)
