# Resonant Activation in Asymmetric Potentials

## Abstract

The resonant activation effect (RA) has been well studied in
different ways during the last two decades. It consists in the
presence of a minimum in the mean time spent by a Brownian
particle to exit from a potential well in the presence of a
fluctuating external force, as a function of the mean frequency
(or the correlation time) of the latter. This work studies the
role played by the asymmetry of a piecewise linear potential in
the RA effect, and, in general, the behavior of the mean first
passage time and the mean velocity of the particle crossing
through the potential barrier. A strong dependence on the
asymmetry of the potential has been found which can be put in
relationship with the current in the ratchet whose the potential
here used is an elementary module. In this case a current reversal
as a function of the frequency of the switching potential occurs.
Comparison of the calculations with the Doering-Gadua model have
been performed, as well as comparison with smooth symmetrical
potentials, by checking for the robustness of the resonant
correlation time. The calculations have been done by solving
numerically the Langevin equation in the presence of an
uncorrelated Gaussian noise. The resonant mean first passage times
show an unexpected behavior as a function of the thermal noise
intensity. The related curves present for the different symmetries
an unexpected inversion of their relative behavior beyond a
certain threshold value of the noise. This means that the current
reversal can only occur for weak noise intensities, lower than
that threshold value.
Pacs: 05.40.-a, 05.45.-a

In the recent past years various theoretical works have been produced around the concept of Resonant Activation (RA) [1, 2, 3, 4, 5, 6, 7], which consists in the presence of a minimum of the mean escape time from a potential well of a Brownian particle when the system is subjected to a randomly switching force, as a function of the mean switching quantity. The RA effect has been also detected experimentally [8, 9, 10] and it is in principle involved in a wide branches of science from physics to biology. The occurrence of the RA together with other stochastic effects such as noise enhanced stability (NES) [11, 10, 12] and stochastic resonance (SR) [13] have been also investigated [9, 14]. The article by Doering & Gadua [1] is considered as one of the most introductory work to the resonant activation phenomenon. They introduced a switching piecewise linear potential in a range with fixed minima in and , and fluctuating amplitude of the maximum. They report the mean escape time for a Brownian particle in the case of fluctuations of the maximum of the potential between and (flat potential), and also between and (well instead of barrier), showing the presence of the RA effect. A slightly different choice was made by Bier & Astumiam [2] who used a potential fluctuating between and with , maintaining so the presence of the barrier in all the dynamics. Both the methods show the resonant activation effect, and have been analytically evaluated in some approximation [6]. Both the choices have in common that the potential is symmetrical in shape and maintains the same value at the two extrema in all the dynamics ().

Aim of this work is to focus on the role played by the asymmetry of the potential on the resonant activation effect using a simple piecewise linear potential .

Many papers with both experimental and analytical investigation concerning the role played by the asymmetry of the potential in stochastic effects have been published during the last years. However, the investigations have been mainly devoted to the effects on the Stochastic resonance phenomenon, that is the presence of a noise induced regular oscillations in a system, which is revealed by means of a maximum in the signal to noise ratio of the output [15, 16, 17, 18, 19], while the relation between the RA and the shape of the potential has been previously performed using a single slope linear potential [20].

Comparison with the Bier-Astumian model have been performed, as well as comparison with smooth symmetrical polynomial potentials. The results obtained have been extended to the most elementary ratchet potential, giving explanation of the current reversal there found as function of the correlation time of the external force.

The fluctuating potential is here given as the sum of a static potential (with, again, ) plus an additional time-dependent giving the two configuration ’up’ and ’down’ between which takes its values. The additional potential has not to be necessarily a stochastic process to give rise to RA [4, 6]. It can be a smooth, continuous potential like a cosine or, instead, a stochastic potential related to a dichotomous force exponentially correlated in time. This last form is widely used in literature and we use it in this work. The related Langevin equation is:

(1) |

where is the Gaussian white noise, with zero mean and correlation function . The intensity is related to thermal bath and damping coefficient (here ) by means of the relation . The random force represents a dichotomous stochastic process, the random telegraph noise (RTN), taking the two values with an exponential correlation function , where the intensity and is the correlation time of the process.

The potential is then defined as:

(2) |

with, explicitly,

(3) |

Here , , , and represents the asymmetry parameter, defined as the distance of the position of the maximum of the potential from the position of the symmetrical maximum . In Fig. 1 (right) we see an example of the static potential with the asymmetry parameter .

The difference between the choice of the fluctuating potential here used (Eqs. 2 and 3) and that one by Bier-Astumian is visible in Fig. 1, where the potentials are drawn in the two cases. Here, with respect to the ’mean’ static potential, the additional fluctuations give only two values (’up’ and ’down’ in fact) in the flipping, being the force uniform overall the -range of the potential. In the Doering & Gadua model (as in the Bier-Astumian) two values of the force for each potential slope have to be considered to hold the minimum on the right at the same level ().

The potential here defined can be considered as a base modulus of the piecewise linear ratchet subjected to RTN widely used in literature [21, 22, 23].

The equation (1) has been solved numerically by using and the averages have been performed over a sampling of realizations. In the -th realization the particle is put in the starting position and the time to cross the position is computed. A reflecting boundary is put in the left extremum of the potential while an absorbing boundary is present at the right extremum. The ensemble average of the gives the Mean First Passage Time (MFPT), which presents, for all the cases here studied, the evidence of the RA effect, i.e. a well drawn minimum as a function of the correlation time . In fact, as well as the symmetric case, the MFPT obtained with the asymmetrical potentials show a resonant effect which is drawn in Fig. 2. We notice that for the three values of the asymmetry parameter , we find quite the same value of the resonant correlation time , but different values of the corresponding resonant MFPTs (), which decrease by increasing the asymmetry parameter .

We note that the resonant region shows an inversion in the behavior of the MFPT curves for the three potentials to both the low and high correlation times with respect to the intermediate one. In fact for lower than the curves show a MFPT higher for positive asymmetry () and lower for negative asymmetry () and the same qualitative behavior is visible in the long correlation time region higher than . In the intermediate region , where we also find the resonant values, the situation is inverted: the highest value corresponds to the negative asymmetry parameter and the lowest to the positive one.

On the other hand, calculation performed with the Bier-Astumian and Doering & Gadua model, that is using fixed extrema of the same asymmetric piecewise linear potentials, which fluctuates between the same highs () in all the asymmetries, give strongly different curves, as visible in Fig. 3, where we can see even a strong displacement of the resonant correlation time by changing the asymmetry, but no crosses are present between the MFPT curves.

The inversion of the MFPTs curves behavior, and consequently the presence of the two crosses at approximatively and , is so uniquely present in MFPTs calculated for asymmetrical potentials using uniform fluctuating force overall the range , and it does not appear neither in the symmetrical ones with different shapes (See Fig. 7), nor in the asymmetrical ones with fixed extrema and fluctuating barriers (Fig. 3). In other words, the comparison between the results plotted in Figs. 2, 3 and 7 put in evidence that the cross features of the MFPT curves occurs not merely because of the asymmetry of the potentials, but, instead, because of the presence of the asymmetry together with the uniformity in space of the fluctuating external force added to the system.

The main relevant feature in adding a uniform force in the range of the constant potential, lies in the fact that in this case the barrier high of the fluctuating potential takes different values for different positions of the maximum, i.e. as a function of the asymmetry parameter . In fact the resonant MFPT values s depend mainly by the lower value taken by the potential (), being proportional to [6], and this value becomes lower and lower, by increasing the value of s. This means that at a first sight we can expect that the values take a lower value for the positive asymmetry than for the negative ones. However, as we can see below in the text, this expectation holds only up to a certain threshold value of noise intensity () and the inverse behavior occurs for higher values ().

The model here investigated presents interesting features in the MFPT: first of all it has a value of the resonant mean period not too strongly dependent on the asymmetry parameter ; then, it presents two period intervals, close to , and close to having approximatively the same MFPT for all the -parameters.

However, the crossing features of the MFPT as a function of the mean driving frequency doesn’t occur for any value of the thermal noise intensity . A set of calculation to check this kind of robustness has been performed and the related results are shown in Fig. 4, where the RA is plotted for different values of the noise. We can see that by increasing the thermal noise intensity , the crosses between the curves are maintained up to a threshold value that we can call . For higher noise intensities no crosses appear in the curves. Further, a shift of the resonant mean switching time as a function of noise is visible and a lowering of the related , which demonstrate that when the noise is increased, an higher frequency switching is necessary to reach the resonance, and this resonance occurs at a lower mean escape time. The increase of the noise intensity has in this sense the effect to speed up all the escape features from the well. The results shown in Fig. 4, can be better observed in Fig. 5, where the mean resonant frequencies and the mean resonant escape times s have been plotted as a function of the noise intensity for the three symmetry values. We can see there that going beyond the threshold noise value , the three curves invert their relative position. This noise threshold corresponds to the presence (for ) or the absence (for ) of the two crossings of the MFPT curves visible in Fig. 2 and Fig. 4. However we notice that is not unique for all the asymmetries, being the crossing values slightly different for each couple of the three curves. We can see that the resonant frequency (inset of Fig. 5) has a slightly different dependence on the thermal noise intensity for the different asymmetries. In the range of investigated, the three curves can be easily approximated by a straight line, even if the real dependence is in general more complicated (see [6]); the slope of this line is higher for the negative asymmetry than for the positive one.

The presence of a resonant behavior, as well as the cross value at , is also found in the plot of the mean velocity of the Brownian particle. Left inset of Fig.2 shows, in fact, this measure as a function of the correlation time of the fluctuating dichotomous force, calculated as . For all the asymmetry parameters, we see the presence, before the saturating behavior, of a weak maximum which corresponds to the resonant correlation time . We can also see that for low values of the correlation times () the mean velocity is higher for negative asymmetry and lower for positive ones, while for () is the inverse. This feature gives rise to a reversal current in the ratchet, as predicted in other works [24, 25, 26] and whose occurrence has been also demonstrated experimentally [27]. In fact the difference between the mean velocities of the positive asymmetry and the negative one change sign at the value. In an asymmetrical ratchet this difference represents a net velocity flux, provided that the absence of any reflecting boundary in that case gives rise to changes in the values of the velocity. Both the presences of a maximum for and the cross at are in total agreement with the behavior of the MFPT. This agreement fails, instead, for values of the correlation times higher than . While the MFPT curves increase in a different way and joint together at the second cross, the velocities decrease only a few, reaching a saturation value. This is because for high values of the correlation times, the particle tends to cross the potential barrier when it is in its lower high, so acquiring a relatively high speed because of the low travelling time. When the potential is in the high level, the particle takes a longer time to cross and so the contribution to the mean velocity becomes very low and relatively negligible. This means that, for high correlation times, maintains a relatively high value which doesn’t change so strongly as the MFPT.

The results found above for the single barrier potentials, mirrors, of course, to the ratchet potential having the same asymmetric profile as elementary module. In this respect a set of calculations has been performed with the aim to join together the results of the single barrier described above with the simplest ratchet case, such as a ratchet with two barriers only. Fig. 6 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 which is the same for the single barrier case, as we can expect. In this system the MFPT is the mean time spent by the Brownian particle starting at to reach the position or , indifferently. The particle, of course will follow the easiest path, and the MFPT represents the minimum time of the two single barrier case seen above. This also means that the curve is lowered and the RA effect less pronounced. The mean velocity, plotted in the upper-left inset of the Fig. 6, shows again a maximum at the same resonant value . For very low correlation time the mean velocity has a weak negative velocity (right/top inset in Fig.6). This means that a current reversal appears at a certain correlation time . This features follows from the different behavior of the mean velocity in the two specular asymmetric single barrier potentials seen above (inset of Fig.2), where the presence of the cross value indicates a current reversal as a function of . The difference in value between and , as well as the difference in the absolute value of the mean velocity of the Brownian particle, have to be imputed to the presence of the reflecting boundary in the single barrier case which change the traveling times of the particle and, so, the related mean velocities.

As a last remark concerning the relationship between the resonant activation effect and the shape of the potential, some calculations have been performed using symmetrical smooth potentials. The static potentials used have the form:

(4) |

where in our calculation . The values used are: , such as parabolic, quartic and power potentials. As we can see in Fig. 7, the resonant mean time is quite the same for all the cases, again confirming that is a robust value in the model investigated. Another remarkable and well visible feature is that the four curves of the MFPT differ each other of a constant quantity, at least for . This means that, in that region, their logarithmic distance is constant and so an exponential form factor has to be taken into account in order to estimate the MFPT for each potential shape.

Summarizing the results, the shapes of the potential (both symmetrical and asymmetrical ones) play a very important role in the evaluation of the RA effect and MFPT behaviors. With a spatially uniform random telegraph force, the resonant correlation time appears to be a robust value independently on that shape, while this latter acts always in a strong way by modifying the resonant values of the mean first passage times . In the context of uniform forces, the asymmetry of the potential is then responsible for the crosses of the MFPT curves in a certain range of low thermal noise intensities, giving an explanation for the appearance of the current reversal as a function of the correlation time of the fluctuating force in ratchet potentials. These crosses, and, consequently the current reversal in ratchet, are only present at weak noise intensity, as indicated by the presence of an upper noise intensity threshold .

This work has been supported by the Marie Curie TOK grants under the COCOS project (6th EU Framework Programme, contract No: 52/MTKD-CT-2004-517186).

### Footnotes

- E-mail address: afiasconaro@gip.dft.unipa.it
- Group of Interdisciplinary Physics, http://gip.dft.unipa.it

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