Resonant Activation in Asymmetric Potentials
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] O c t Resonant Activation in Asymmetric Potentials
Alessandro Fiasconaro ∗ Mark Kac Complex Systems Research Center and Marian Smoluchowski Institute of PhysicsJagellonian University, Reymonta 4, 30-059 Krak´ow, Poland ,and
Dipartimento di Fisica e Tecnologie Relative † and CNISM, Universit`a di PalermoViale delle Scienze, I-90128 Palermo, Italy December 11, 2018
Abstract
The resonant activation effect (RA) has been well stud-ied in different ways during the last two decades. Itconsists in the presence of a minimum in the mean timespent by a Brownian particle to exit from a potentialwell in the presence of a fluctuating external force, as afunction of the mean frequency (or the correlation time)of the latter. This work studies the role played by theasymmetry of a piecewise linear potential in the RA ef-fect, and, in general, the behavior of the mean first pas-sage time and the mean velocity of the particle crossingthrough the potential barrier. A strong dependence onthe asymmetry of the potential has been found whichcan be put in relationship with the current in the ratchetwhose the potential here used is an elementary module.In this case a current reversal as a function of the fre-quency of the switching potential occurs. Comparisonof the calculations with the Doering-Gadua model havebeen performed, as well as comparison with smooth sym-metrical potentials, by checking for the robustness of theresonant correlation time. The calculations have beendone by solving numerically the Langevin equation inthe presence of an uncorrelated Gaussian noise. Theresonant mean first passage times show an unexpectedbehavior as a function of the thermal noise intensity.The related curves present for the different symmetriesan unexpected inversion of their relative behavior be-yond a certain threshold value of the noise. This meansthat the current reversal can only occur for weak noiseintensities, lower than that threshold value.Pacs: 05.40.-a, 05.45.-aIn the recent past years various theoretical works havebeen produced around the concept of Resonant Activa-tion (RA) [1, 2, 3, 4, 5, 6, 7], which consists in thepresence of a minimum of the mean escape time froma potential well of a Brownian particle when the sys-tem is subjected to a randomly switching force, as afunction of the mean switching quantity. The RA effect ∗ E-mail address: afi[email protected] † Group of Interdisciplinary Physics, http://gip.dft.unipa.it has been also detected experimentally [8, 9, 10] and it isin principle involved in a wide branches of science fromphysics to biology. The occurrence of the RA togetherwith other stochastic effects such as noise enhanced sta-bility (NES) [11, 10, 12] and stochastic resonance (SR)[13] have been also investigated [9, 14]. The article byDoering & Gadua [1] is considered as one of the most in-troductory work to the resonant activation phenomenon.They introduced a switching piecewise linear potential ina range [0 , L ] with fixed minima in x = 0 and x = L ,and fluctuating amplitude of the maximum. They re-port the mean escape time for a Brownian particle inthe case of fluctuations of the maximum of the potentialbetween V and 0 (flat potential), and also between V and − V (well instead of barrier), showing the presenceof the RA effect. A slightly different choice was made byBier & Astumiam [2] who used a potential fluctuatingbetween V − a and V + a with a < V , maintaining sothe presence of the barrier in all the dynamics. Both themethods show the resonant activation effect, and havebeen analytically evaluated in some approximation [6].Both the choices have in common that the potential issymmetrical in shape and maintains the same value atthe two extrema in all the dynamics ( V (0) = V ( L )).Aim of this work is to focus on the role played by theasymmetry of the potential on the resonant activationeffect using a simple piecewise linear potential .Many papers with both experimental and analyticalinvestigation concerning the role played by the asym-metry of the potential in stochastic effects have beenpublished during the last years. However, the investi-gations have been mainly devoted to the effects on theStochastic resonance phenomenon, that is the presenceof a noise induced regular oscillations in a system, whichis revealed by means of a maximum in the signal to noiseratio of the output [15, 16, 17, 18, 19], while the relationbetween the RA and the shape of the potential has beenpreviously performed using a single slope linear potential[20].Comparison with the Bier-Astumian model have beenperformed, as well as comparison with smooth symmet-rical polynomial potentials. The results obtained havebeen extended to the most elementary ratchet potential,1iving explanation of the current reversal there found asfunction of the correlation time of the external force.The fluctuating potential V ± ( x, t ) is here given as thesum of a static potential V ( x ) (with, again, V (0) = V ( L )) plus an additional time-dependent U ( x, t ) giv-ing the two configuration ’up’ and ’down’ between which V ± ( x, t ) takes its values. The additional potential U ( x, t )has not to be necessarily a stochastic process to give riseto RA [4, 6]. It can be a smooth, continuous potentiallike a cosine or, instead, a stochastic potential relatedto a dichotomous force exponentially correlated in time.This last form is widely used in literature and we use itin this work. The related Langevin equation is:˙ x = − V ′ ( x ) + η ( t ) + ξ ( t ) (1)where ξ ( t ) is the Gaussian white noise, with zero meanand correlation function h ξ ( t ) ξ ( t ′ ) i = 2 Dδ ( t − t ′ ). The in-tensity D is related to thermal bath and damping coeffi-cient γ (here γ = 1) by means of the relation D = γk B T .The random force η ( t ) represents a dichotomous stochas-tic process, the random telegraph noise (RTN), takingthe two values {− a, a } with an exponential correlationfunction h η ( t ) η ( t ′ ) i = ( Q/τ ) e −| t − t ′ | /τ , where the inten-sity Q = a τ and τ is the correlation time of the process.The potential V ± ( x, t ) is then defined as: -1.5-1-0.5 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 x UpDown -1.5-1-0.5 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 x UpDown -1.5-1-0.5 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 x Mean
Figure 1: Piecewise linear potential used in the calcu-lation. The position x = 0 represents the startingpoint of the simulations and there is also put a reflect-ing boundary. With respect to Bier-Doering choice (left)the right extremum of the potential is not fixed (center)and the whole potential flips randomly between the twoshapes { V + , V − } with a correlation time τ . The rightdraw shows an example of asymmetric piecewise linearstatic potential. The parameter k represents the posi-tion of the maximum x m with respect to the position ofthe maximum in the symmetric case x s (here x s = 0 . x m = 0 . k = − . V ± ( x, t ) = V ( x ) + U ( x, t ) = V ( x ) − xη ( t ) . (2)with, explicitly, V ( x ) = (cid:26) h xx m ≤ x m h L − xL − x m x ≥ x m (3)Here L = 1, h = 2, x m = L/ k , and k representsthe asymmetry parameter, defined as the distance of theposition of the maximum of the potential x m from theposition of the symmetrical maximum x s . In Fig. 1 (right) we see an example of the static potential withthe asymmetry parameter k = − . − xη ( t ) giveonly two values (’up’ and ’down’ in fact) in the flipping,being the force η ( t ) uniform overall the x -range of thepotential. In the Doering & Gadua model (as in theBier-Astumian) two values of the force for each potentialslope have to be considered to hold the minimum on theright at the same level ( V ( L ) = const ).The potential V ± ( x, t ) here defined can be consideredas a base modulus of the piecewise linear ratchet sub-jected to RTN widely used in literature [21, 22, 23].The equation (1) has been solved numerically by using -4 -2 M FP T τ RA - Asymmetric PWL Potentials k = -0.25k = 0.00k = 0.25 1.02.00.0 0.2 0.4 0.6 0.8 1.0 V ( x ) x -1 v- τ Figure 2: MFPT showing the resonant activation effectfor three values of the asymmetry parameter k of thepiecewise linear potential ( k = − . k = 0, such assymmetric potential, k = 0 . D = 0 .
18. The intensity of the dichotomousforce is a = 1 .
2. The bottom/left inset show the meanvelocity of the Brownian particle for the same asymme-tries, as a function of the correlation time τ . dt = 10 − and the averages have been performed over asampling of N = 20 ,
000 realizations. In the i -th realiza-tion the particle is put in the starting position x = 0and the time t i to cross the position x = L is computed.A reflecting boundary is put in the left extremum of thepotential while an absorbing boundary is present at theright extremum. The ensemble average of the t i gives theMean First Passage Time (MFPT), which presents, forall the cases here studied, the evidence of the RA effect,i.e. a well drawn minimum as a function of the correla-tion time τ . In fact, as well as the symmetric case, theMFPT obtained with the asymmetrical potentials showa resonant effect which is drawn in Fig. 2. We noticethat for the three values of the asymmetry parameter k ,we find quite the same value of the resonant correlationtime τ R ∼
10, but different values of the correspondingresonant MFPTs ( T R s ), which decrease by increasing the2symmetry parameter k .We note that the resonant region shows an inversionin the behavior of the MFPT curves for the three po-tentials to both the low and high correlation times withrespect to the intermediate one. In fact for τ lower than τ C L ≈ − the curves show a MFPT higher for positiveasymmetry ( k = 0 .
25) and lower for negative asymme-try ( k = − .
25) and the same qualitative behavior isvisible in the long correlation time region τ higher than τ C R ≈ . In the intermediate region τ ∈ [ τ C L , τ C R ],where we also find the resonant values, the situation isinverted: the highest T R value corresponds to the neg-ative asymmetry parameter and the lowest T R to thepositive one.On the other hand, calculation performed with theBier-Astumian and Doering & Gadua model, that is us-ing fixed extrema of the same asymmetric piecewise lin-ear potentials, which fluctuates between the same highs( a = 0 .
6) in all the asymmetries, give strongly differ-ent curves, as visible in Fig. 3, where we can see evena strong displacement of the resonant correlation timeby changing the asymmetry, but no crosses are presentbetween the MFPT curves. The inversion of the MF- -4 -2 M FP T τ RA - PWL Pot. with fixed extrema
PWL - k = -0.25PWL - k = 0.00PWL - k = 0.25 1.02.00.0 0.2 0.4 0.6 0.8 1.0 V ( x ) x Figure 3: MFPT showing the resonant activation effectfor different asymmetrical piecewise linear potential withfixed extrema and equal potential excursion in the threecases. Differently from the results in Fig. 2 we don’tfind any cross between the three curves and also theresonant correlation time τ R changes for the differentasymmetries. The potential increase is here a = 0 .
6, asthe one of the symmetric case in Fig. 2.PTs curves behavior, and consequently the presence ofthe two crosses at approximatively τ C L and τ C R , is souniquely present in MFPTs calculated for asymmetri-cal potentials using uniform fluctuating force overall therange [0 , L ], and it does not appear neither in the sym-metrical ones with different shapes (See Fig. 7), nor inthe asymmetrical ones with fixed extrema and fluctu-ating barriers (Fig. 3). In other words, the comparisonbetween the results plotted in Figs. 2, 3 and 7 put in evi-dence that the cross features of the MFPT curves occursnot 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 fluctuatingexternal force η ( t ) added to the system.The main relevant feature in adding a uniform forcein the range of the constant potential, lies in the factthat in this case the barrier high of the fluctuating po-tential takes different values for different positions of themaximum, i.e. as a function of the asymmetry param-eter k . In fact the resonant MFPT values T R s dependmainly by the lower value taken by the potential ( V − ),being proportional to (1 /V − ) e V − /D [6], and this valuebecomes lower and lower, by increasing the value of k s.This means that at a first sight we can expect that the T R values take a lower value for the positive asymmetrythan for the negative ones. However, as we can see belowin the text, this expectation holds only up to a certainthreshold value of noise intensity ( D T ) and the inversebehavior occurs for higher values ( D > D T ).The model here investigated presents interesting fea-tures in the MFPT: first of all it has a value of the reso-nant mean period τ R not too strongly dependent on theasymmetry parameter k ; then, it presents two period in-tervals, close to τ C L ≈ − , and close to τ C R ≈ having approximatively the same MFPT for all the k -parameters. -4 -2 M FP T τ D = 0.20 k = -0.25k = 0.00k = 0.25 10 -4 -2 M FP T τ D = 0.25 -4 -2 M FP T τ D = 0.40 -4 -2 M FP T τ D = 0.60
Figure 4: Resonant activation evolution for various noiseintensities. We observe the disappearing of the cross-ings in the MFPT curves and the shift of the minimaby increasing the noise intensity D . These behaviors areshown in details Fig. 5, where the resonant frequencyand resonant times are plotted as a function of D .However, the crossing features of the MFPT as a func-tion of the mean driving frequency doesn’t occur for anyvalue of the thermal noise intensity D . A set of cal-culation to check this kind of robustness has been per-formed and the related results are shown in Fig. 4, wherethe RA is plotted for different values of the noise. Wecan see that by increasing the thermal noise intensity D , the crosses between the curves are maintained upto a threshold value that we can call D T . For highernoise intensities no crosses appear in the curves. Fur-ther, a shift of the resonant mean switching time as a3unction of noise is visible and a lowering of the related T R , which demonstrate that when the noise is increased,an higher frequency switching is necessary to reach theresonance, and this resonance occurs at a lower meanescape time. The increase of the noise intensity has inthis sense the effect to speed up all the escape featuresfrom the well. The results shown in Fig. 4, can be betterobserved in Fig. 5, where the mean resonant frequencies γ R = 1 / (2 τ R ) and the mean resonant escape times T R shave been plotted as a function of the noise intensity forthe three symmetry values. We can see there that go-ing beyond the threshold noise value D T , the three T R curves invert their relative position. This noise thresh-old corresponds to the presence (for D < D T ) or theabsence (for D > D T ) of the two crossings of the MFPTcurves visible in Fig. 2 and Fig. 4. However we noticethat D T is not unique for all the asymmetries, being thecrossing values slightly different for each couple of thethree curves. We can see that the resonant frequency(inset of Fig. 5) has a slightly different dependence onthe thermal noise intensity D for the different asymme-tries. In the range of D investigated, the three curvescan be easily approximated by a straight line, even if thereal dependence is in general more complicated (see [6]);the slope of this line is higher for the negative asymmetrythan for the positive one. -1 T R D Resonant frequency and Mean first passage time vs D k = -0.25k = 0.00k = 0.25 γ R D k = -0.25k = 0.00k = 0.25 γ R D Figure 5: Resonant mean passage times T R s as a functionof the noise intensity D for the three asymmetry valuesinvestigated. In the inset the corresponding resonant fre-quencies γ R s. The crossing value D T ≈ .
27 representsthe threshold of thermal noise discriminating if the twocrossings in MFPT of Fig. 2 are present (
D < D T ) orabsent ( D > D T ).The presence of a resonant behavior, as well as thecross value at τ C L , is also found in the plot of the meanvelocity of the Brownian particle. Left inset of Fig.2shows, in fact, this measure as a function of the cor-relation time of the fluctuating dichotomous force, cal-culated as ¯ v = N − P Ni =1 L/t i . For all the asymmetryparameters, we see the presence, before the saturatingbehavior, of a weak maximum which corresponds to theresonant correlation time τ R . We can also see that for low values of the correlation times ( τ < τ C L ) the meanvelocity is higher for negative asymmetry and lower forpositive ones, while for ( τ > τ C L ) is the inverse. Thisfeature gives rise to a reversal current in the ratchet, aspredicted in other works [24, 25, 26] and whose occur-rence has been also demonstrated experimentally [27].In fact the difference between the mean velocities of thepositive asymmetry and the negative one change signat the τ C L value. In an asymmetrical ratchet this dif-ference represents a net velocity flux, provided that theabsence of any reflecting boundary in that case gives riseto changes in the values of the velocity. Both the pres-ences of a maximum for τ ≈ τ R and the cross at τ ≈ τ C L are in total agreement with the behavior of the MFPT.This agreement fails, instead, for values of the correlationtimes higher than τ R . While the MFPT curves increasein a different way and joint together at the second cross,the velocities decrease only a few, reaching a saturationvalue. This is because for high values of the correlationtimes, the particle tends to cross the potential barrierwhen it is in its lower high, so acquiring a relatively highspeed because of the low travelling time. When the po-tential is in the high level, the particle takes a longertime to cross and so the contribution to the mean ve-locity becomes very low and relatively negligible. Thismeans that, for high correlation times, ¯ v maintains arelatively high value which doesn’t change so strongly asthe MFPT. -4 -2 M FP T τ RA - Elementary Ratchet k = 0.25 1.02.0-1.0 -0.5 0.0 0.5 1.0 V ( x ) x -1 v- τ v- τ Figure 6: MFPT showing the resonant activation effectfor the simplest ratchet potential, composed by only twothe single barrier. The MFPT is here the mean timetaken by the Brownian particle starting at x = 0 toreach the position x = 1 or x = −
1. The mean velocityof the particle is plotted in the upper/left inset. Themean velocity has again a maximum at the same reso-nant value τ R . In the very low correlation time region,the system has a weak negative velocity (right/top in-set), as a consequence of the different behavior and thecross of the mean velocities shown in the inset of Fig. 2.The results found above for the single barrier poten-tials, mirrors, of course, to the ratchet potential havingthe same asymmetric profile as elementary module. In4his respect a set of calculations has been performed withthe aim to join together the results of the single barrierdescribed above with the simplest ratchet case, such as aratchet with two barriers only. Fig. 6 shows the resultsin such a case and the bottom/right inset shows the cor-responding elementary ratchet. The system consists oftwo asymmetric barriers without the presence of any re-flecting boundary. The MFPT presents again a resonantcorrelation value τ R which is the same for the single bar-rier case, as we can expect. In this system the MFPT isthe mean time spent by the Brownian particle startingat x = 0 to reach the position x = 1 or x = −
1, in-differently. The particle, of course will follow the easiestpath, and the MFPT represents the minimum time of thetwo single barrier case seen above. This also means thatthe curve is lowered and the RA effect less pronounced.The mean velocity, plotted in the upper-left inset of theFig. 6, shows again a maximum at the same resonantvalue τ R . For very low correlation time the mean ve-locity has a weak negative velocity (right/top inset inFig.6). This means that a current reversal appears at acertain correlation time τ rev . This features follows fromthe different behavior of the mean velocity in the twospecular asymmetric single barrier potentials seen above(inset of Fig.2), where the presence of the cross value τ C L indicates a current reversal as a function of τ . Thedifference in value between τ rev and τ C L , as well as thedifference in the absolute value of the mean velocity ofthe Brownian particle, have to be imputed to the pres-ence of the reflecting boundary in the single barrier casewhich change the traveling times of the particle and, so,the related mean velocities. -4 -2 M FP T τ RA - Various potential shapes
Piecewise linear potQuadratic potentialQuartic potentialPower 6 potential 1.02.00.0 0.2 0.4 0.6 0.8 1.0 V ( x ) x Figure 7: MFPT showing the RA effect for differentshape of symmetrical potential. We note than for τ . τ R the logarithmic distances between two MFPTs is approx-imatively a constant. This evidence suggests that in thatregion an exponential from factor should be inserted inthe analytical expression of the mean first passage timeto take into account the shape of different potentials.The parameters are the same than in Fig.2.As a last remark concerning the relationship betweenthe resonant activation effect and the shape of the poten- tial, some calculations have been performed using sym-metrical smooth potentials. The static potentials usedhave the form: V N ( x ) = h N x N L N (cid:16) − xL (cid:17) N (4)where in our calculation h = 2. The values used are: N = 1 , ,
3, such as parabolic, quartic and 6 th powerpotentials. As we can see in Fig. 7, the resonant meantime is quite the same for all the cases, again confirm-ing that τ R is a robust value in the model investigated.Another remarkable and well visible feature is that thefour curves of the MFPT differ each other of a constantquantity, at least for τ . τ R . This means that, in thatregion, their logarithmic distance is constant and so anexponential form factor has to be taken into account inorder to estimate the MFPT for each potential shape.Summarizing the results, the shapes of the potential(both symmetrical and asymmetrical ones) play a veryimportant role in the evaluation of the RA effect andMFPT behaviors. With a spatially uniform random tele-graph force, the resonant correlation time τ R appears tobe a robust value independently on that shape, while thislatter acts always in a strong way by modifying the res-onant values of the mean first passage times T R s . In thecontext of uniform forces, the asymmetry of the potentialis then responsible for the crosses of the MFPT curves ina certain range of low thermal noise intensities, giving anexplanation for the appearance of the current reversal asa function of the correlation time of the fluctuating forcein ratchet potentials. These crosses, and, consequentlythe current reversal in ratchet, are only present at weaknoise intensity, as indicated by the presence of an uppernoise intensity threshold D T .This work has been supported by the Marie CurieTOK grants under the COCOS project (6th EU Frame-work Programme, contract No: 52/MTKD-CT-2004-517186). References [1] C.R. Doering and J. C. Gadoua, Phys. Rev. Lett. , 2318 (1992).[2] M. Bier and R.D. Astumian, Phys. Rev. Lett. ,1649 (1993).[3] P. H¨anggi, Chem. Phys. , 157 (1994);[4] P. Reimann, Phys. Rev. Lett. , 4576 (1995).[5] J. Iwaniszewski, Phys. Rev. E , 3173 (1996).[6] M. Bogu˜n´a, J. M. Porra, J. Masoliver, and K. Lin-denberg, Phys. Rev. E , 3990 (1998).[7] M. Bier, I. Derenyi, M. Kostur, R.D. Astumian,Phys. Rev. E , 6422 (1999).58] R.N. Mantegna and B. Spagnolo, Phys. Rev. Lett. , 3025 (2000); J. Phys. IV (France) , 247(1998).[9] C. Schmitt, B. Dybiec, P. H¨anggi, C. Bechinger,Europhys. Lett. (6), 937-943 (2006).[10] Guozhu Sun et al. Phys. Rev. E , 021107 (2007)[11] P. H¨anggi, P. Talkner, and M. Borkovec, Rev.Mod. Phys. , 251 (1990); I. Dayan, M. Git-terman, and G. H. Weiss, Phys. Rev. A , 757(1992).[12] A. Fiasconaro, B. Spagnolo, S. Boccaletti, Phys.Rev. E , 061110 (2005); A. Fiasconaro, D.Valenti, B. Spagnolo, Physica A , 136 (2003);A. A. Dubkov, N. V. Agudov, and B. Spagnolo,Phys. Rev. E , 061103 (2004).[13] L. Gammaitoni, P. H¨anggi, P. Jung F. Marchesoni,Rev. Mod. Phys. (1), 223 (1998).[14] A. Fiasconaro, B. Spagnolo, A. Ochab–Marcinek,E. Gudowska–Nowak, Phys. Rev. E , 041904(2006); A. Ochab–Marcinek, E. Gudowska–Nowak,A. Fiasconaro, B. Spagnolo, Acta Phys. Pol B (5), 1651 (2006);[15] M.I. Dykman, R. Mannella, P.V.E. McClintock,S.M.Soskin, and N.G.Stocks, Phys. Rev. A (4),1701(1991) M.I. Dykman, D.G. Luchinsky, P.V.E.McClintock, N.D.Stein, and N.G.Stocks, Phys.Rev. A (4), R1713 (1992).[16] H. S. Wio and S. Bouzat, Braz. Journ. of Phys. (1), 136 (1999).[17] Jing-hui Li Phys Rev. E , 031104 (2002).[18] A. Nikitin, N.G.Stocks, A.R. Bulsara Phys. Rev.E , 016103 (2003); A. Nikitin, N.G.Stocks, A.R.Bulsara Phys. Rev. E , 041138 (2007).[19] Ning Li-Huan, Xu Wei, and Yao Ming-Li, Chin.Phys. B, (20), 0486 (2008).[20] B. Dybiec, E. Gudowska–Nowak, Phys. Rev. E ,026123 (2002).[21] M.O. Magnasco, Phys. Rev. Lett. , 1477 (1993)[22] T. Czernik, J. Kula, J. Luczka, and P. H¨anggi,Phys. Rev. E , 4057 (1997).[23] J. Kula, T. Czernik, and J. Luczka, Phys. Rev.Lett. , 1377 (1998).[24] P. Reimann, T. C. Elston, Phys. Rev. Lett. ,5328 (1996)[25] M.I. Dykman, H. Rabitz, V.N.Smelyanskiy, andB.E.Vugmeister, Phys. Rev. Lett. , 1178 (1997)[26] D. G. Luchinsky, M. J. Greenall, and P.V. E. Mc-Clintock, Phys. Lett. A , 316 (2000) [27] R. Gommers, P. Douglas, S.Bergamini, M.Goonasekera, P.H. Jones, and F. Renzoni, Phys.Rev. Lett.94