Entropic Stochastic Resonance

En tropi Sto  hasti Resonane P .S. Burada, 1 G. S hmid, 1 D. Reguera, 2 M.H. V ainstein, 2 J.M. Rubi, 2 and P . Hänggi 1 1 Institut für Physik, Universität A ugsbur g, Universitätsstr. 1, D-86135 A ugsbur g, Germany 2 Dep artament de Físi a F onamental, F aultat de Físi a, Universidad de Bar  elona, Diagonal 647, E-08028 Bar  elona, Sp ain (Dated: No v em b er 1, 2018) W e presen t a no v el s heme for the app earane of Sto  hasti Resonane when the dynamis of a Bro wnian partile tak es plae in a onned medium. The presene of unev en b oundaries, giving rise to an en tropi on tribution to the p oten tial, ma y up on appliation of a p erio di driving fore result in an inrease of the sp etral ampliation at an optim um v alue of the am bien t noise lev el. This Entr opi Sto hasti R esonan e (ESR),  harateristi of small-sale systems, ma y onstitute a useful me hanism for the manipulation and on trol of single-moleules and nano-devies. P A CS n um b ers: 02.50.Ey , 05.40.-a, 05.10.Gg Sto hasti R esonan e (SR) desrib es the oun terin tu- itiv e phenomenon where an appropriate dose of noise is not harmful for the detetion or transdution of an in- oming, generally w eak signal, but rather of onstrutiv e use in the sense that a w eak signal b eomes amplied up on harv esting the am bien t noise in metastable, non- linear systems [1℄. Sine its rst diso v ery in the early eigh ties SR has b een observ ed in a great v ariet y of sys- tems p ertaining to dieren t disiplines su h as ph ysis,  hemistry , engineering, biology and biomedial sienes [1, 2, 3 , 4, 5, 6 , 7, 8, 9 , 10 ℄. The list of mo dels and ap- pliations is still gro wing. In partiular, SR has found widespread in terests and appliations within biologial ph ysis. The resear h on SR has primarily b een fo used on sys- tems with purely energeti p oten tials. Ho w ev er, in sit- uations frequen tly found in soft ondensed matter and biologial systems, partiles mo v e in onstrained regions su h as small a vities, p ores or  hannels whose pres- ene and shap e pla y an imp ortan t role for the SR- dynamis [10 ℄, sometimes ev en more imp ortan t than the w ell-studied ase of energeti barriers in su h systems [11 , 12 , 13 , 14 ℄. In this w ork, w e demonstrate that ir- regularities in the form of onning, urv ed b oundaries, b eing mo deled via an en tropi p oten tial, an ause noise- assisted, resonan t-lik e b eha viors in the system under on- sideration. Connemen t, an inheren t prop ert y of small- sale systems, an th us onstitute an imp ortan t soure of noise-indued resonan t eets with in teresting appli- ations in the design and on trol of these systems. The phenomenon of SR is ro oted on a sto  hasti syn-  hronization b et w een noise-indued hopping ev en ts and the rh ythm of the externally applied signal, that tak en alone is not suien t for the system to o v erome a p o- ten tial barrier. In the rst plae, noise enables system transitions and it is in fat resp onsible for the observ ed signal ampliation and the emergene of ertain degree of order. In the earliest and basi manifestation of SR, the syn hronization of the random swit hes of a Bro wn- ian partile with a p erio di driving fore w ere observ ed for a bistable p oten tial. Moreo v er, p oten tials of this t yp e are not only found in systems with energy barriers, as ~ G ~ F ( t ) L x L y b x y FIG. 1: S hemati illustration of the t w o-dimensional stru- ture onning the motion of the Bro wnian partiles. The symmetri struture is dened b y a quarti double w ell fun- tion, f. Eq. (2), in v olving the geometrial parameters L x , L y and b . Bro wnian partiles are driv en b y a sin usoidal fore ~ F ( t ) along the longitudinal diretion and a onstan t fore ~ G in the transv ersal diretion. they ma y also arise due to the inuene of en tropi on- strain ts. P artiles diusing freely in a onned medium su h as the one depited in Fig. 1 ma y giv e rise to an ativ ation regime when a onstan t fore ~ G in the trans- v erse diretion is imp osed. W e will sho w that the om- bination of foring and the presene of en tropi eets deriving from the onnemen t and the irregularit y of the b oundaries giv e rise to an eetiv e bistable p oten tial that exhibits the signatures of Sto  hasti Resonane. The dynamis of a partile in a onstrained geometry sub jeted to a sin usoidal osillating fore F ( t ) along the axis of the struture and to a onstan t fore G in the transv ersal diretion an b e desrib ed b y means of the Langevin equation, written in the o v erdamp ed limit as γ d ~ r d t = − G ~ e y − F ( t ) ~ e x + p γ k B T ~ ξ ( t ) , (1) where ~ r denotes the p osition of the partile, γ is the frition o eien t, ~ e x and ~ e y the unit v etors along the x and y -diretions, resp etiv ely , and ~ ξ ( t ) is a Gaussian white noise with zero mean whi h ob eys the utuation- dissipation relation h ξ i ( t ) ξ j ( t ′ ) i = 2 δ ij δ ( t − t ′ ) for i, j = x, y . The expliit form of the longitudinal fore is giv en b y F ( t ) = F 0 sin(Ω t ) where F 0 is the amplitude and Ω is the frequeny of the sin usoidal driving. 2 In the presene of onning b oundaries, this equation has to b e solv ed b y imp osing reeting b oundary ondi- tions at the w alls of the struture. F or the 2D struture depited in Fig. 1, the w alls are dened b y w l ( x ) = L y  x L x  4 − 2 L y  x L x  2 − b 2 = − w u ( x ) , (2) where w l and w u orresp ond to the lo w er and the up- p er b oundary funtions, resp etiv ely , L x orresp onds to the distane b et w een b ottlene k p osition and p osi- tion of maximal width, and L y refers to the narro w- ing of the b oundary funtions and b to the remain- ing width at the b ottlene k, f. Fig. 1 . Consequen tly , 2 w ( x ) = w u ( x ) − w l ( x ) giv es the lo al width of the stru- ture. This  hoie of the geometry is in tended to resem ble the ar het yp e setup for SR in the on text of a double w ell p oten tial. In fat, in the limit of a suien tly large transv ersal fore, the partile is in pratie restrited to explore the region v ery lose to the lo w er b oundary of the struture, reo v ering the eet of an energeti bistable p oten tial. F or sak e of a dimensionless desription, w e heneforth sale all lengths b y the  harateristi length L x , i.e. ˜ x = x/ L x , ˜ y = y /L x whi h implies ˜ b = b /L x and ˜ w l = w l /L x = − ˜ w u , time b y τ = γ L x 2 /k B T R , the or- resp onding  harateristi diusion time at an arbitrary , but irrelev an t referene temp erature T R , i.e. ˜ t = t/τ and ˜ Ω = Ω τ , and fore b y F R = γ L x /τ , transv ersal fore ˜ G = G/F R and a longitudinally ating, sin usoidal fore ˜ F ( ˜ t ) = F ( t ) /F R . In the follo wing w e shall omit the tilde sym b ols for b etter legibilit y . In dimensionless form the Langevin-equation (1) and the b oundary funtions (2) read: d ~ r d t = − G ~ e y − F ( t ) ~ e x + √ D ~ ξ ( t ) , (3) w l ( x ) = − w u ( x ) = − w ( x ) = ǫx 4 − 2 ǫ x 2 − b/ 2 , (4) where w e dened the asp et ratio ǫ = L y /L x and the dimensionless temp erature D = T /T R . In the absene of a time-dep enden t applied bias, i.e. F ( t ) = 0 , it has b een sho wn b y a oarsening of the de- sription [15 , 16 , 17 ℄ that the Langevin equation (1) an b e redued to an eetiv e 1D-F okk er-Plan k equation, reading in dimensionless form ∂ P ( x, t ) ∂ t = ∂ ∂ x  D ∂ P ∂ x + V ′ ( x, D ) P  , (5) where V ( x, D ) = − D ln  2 D G sinh  Gw ( x ) D   , (6) and the prime refers to the deriv ativ e with resp et to x . This equation desrib es the motion of a Bro wnian partile in a bistable p oten tial of en tropi nature. Re- mark ably , the eetiv e p oten tial do es not only dep end on the energeti on tribution of the fore G , but also on the temp erature and the geometry of the struture in a non-trivial w a y . Notably , for the v anishing width at the t w o opp osite orners of the geometry in Fig. (1) this en tropi p oten tial approa hes innit y , th us in trinsi- ally aoun ting for a natural reeting b oundary . It is imp ortan t to emphasize that this bistable p oten tial w as not presen t in the 2D Langevin dynamis, but arises due to the en tropi restritions asso iated to the onnemen t. In general, after the oarse-graining the diusion o e- ien t will dep end on the o ordinate x as w ell, but sine in our ase | w ′ ( x ) | ≪ 1 , this orretion an b e safely negleted, f. Ref. [ 15 , 17 , 18 , 19 , 20 ℄. It is in teresting to analyze the t w o limiting situations that an b e obtained up on v arying the v alue of the ra- tio b et w een the energy asso iated to the transv ersal fore and the thermal energy . F or the energy-dominated ase, i.e. Gw ( x ) /D ≫ 1 , Eq. ( 6) yields V ( x ) = − Gw ( x ) (ne- gleting irrelev an t onstan ts). In the opp osite limit, i.e. for Gw ( x ) /D ≪ 1 , the orresp onding en tropi p oten tial funtion reads V ( x, D ) = − D ln[2 w ( x )] . Two-state appr oximation.- It is instrutiv e to analyze the o urrene of sto  hasti resonane in the on text of the t w o-state appro ximation. F or a p oten tial V ( x ) with barrier heigh t ∆ V the esap e rate of an o v erdamp ed Bro wnian partile from one w ell to the other in the pres- ene of thermal noise, and in the absene of a fore, is giv en b y the o v erdamp ed Kramers rate [21 , 22 , 23 ℄, read- ing in dimensionless units, r K ( D ) = p V ′′ ( x min ) | V ′′ ( x max ) | 2 π exp  − ∆ V D  , (7) where V ′′ is the seond deriv ativ e of the eetiv e p o- ten tial funtion, and with x max and x min indiating the p osition of the maxim um and minim um of the p oten tial, resp etiv ely . F or the p oten tial giv en b y Eq. (6) and the shap e de- ned b y Eq. (2), the orresp onding Kramers rate for tran- sitions from one basin to the other reads, in dimensionless units, r K ( D ) = G ǫ π p 2 sinh ( Gb/D ) sinh [ G ( b + 2 ǫ ) /D ] sinh 2 [ G ( b + 2 ǫ ) /D ] . (8) Sp e tr al ampli ation.- The o urrene of Sto  hasti Resonane an b e deteted in the sp etral ampliation η [22℄. It is dened b y the ratio of the p o w er stored in the resp onse of the system at frequeny Ω and the p o w er of the driving signal, and whi h for the p erio dially driv en t w o-state mo del, f. Ref. [1 ℄, is giv en in dimensionless units as η = 1 D 2 4 r 2 K ( D ) 4 r 2 K ( D ) + Ω 2 . (9) Next w e demonstrate the o urrene of the resonane in the sp etral ampliation that signals the phenomenon of ESR. W e demonstrate that ESR is neither a p euliarit y of the t w o-state appro ximation nor of the equilibration 3 0 250 500 750 η η 0 0 . 05 0 . 1 0 . 15 0 . 2 D D Ω = 0 . 0001 Ω = 0 . 001 Ω = 0 . 01 0 250 500 750 0 0 . 05 0 . 1 0 . 15 0 . 2 FIG. 2: (olor online) In dimensionless units, the dep endene of the sp etral ampliation η on noise lev el D for dieren t driving frequenies, for the transv ersal fore G = 1 . 0 , the driving amplitude F 0 = 10 − 4 , and for the width funtion w ( x ) = − ǫx 4 + 2 ǫx 2 + 0 . 01 with the asp et ratio ǫ = 1 / 4 . The solid lines orresp ond to the 1D mo delling, f. Eq.(10 ) and Eq.(12 ), whereas the dashed lines orresp ond to the t w o-state appro ximation, f. Eq.(9 ). assumption used to deriv e the eetiv e 1D-F okk er-Plan k equation. In order to study the app earane of Sto  hasti Res- onane w e analyzed the resp onse of the system to the applied sin usoidal signal in terms of the sp etral ampli- ation η . In the presene of an osillating fore F ( t ) in the x − diretion there is an additional on tribution to the eetiv e p oten tial in Eq. (5) and the 1D kineti equation in dimensionless units reads ∂ P ( x, t ) ∂ t = ∂ ∂ x  D ∂ P ∂ x + ( V ′ ( x, D ) − F ( t ) ) P  . (10) The n umerial in tegration of the 1D kineti equation (10 ) w as done b y spatial disretization, using a Cheb y- shev ollo ation metho d, and emplo ying the metho d of lines to redue the kineti equation to a system of or- dinary dieren tial equations, whi h w as solv ed using a ba kw ard dieren tiation form ula metho d. This results in the time-dep enden t probabilit y distribution P ( x, t ) and the time-dep enden t mean v alue, dened as h x ( t ) i = Z x P ( x, t )d x . (11) In the long-time limit this mean v alue approa hes the p e- rio diit y of driving [22 ℄ with angular frequeny Ω . After a F ourier-expansion of h x ( t ) i one nds the amplitude M 1 of the rst harmoni of the output signal. Hene, the sp etral ampliation η for the fundamen tal osillation reads: η =  M 1 F 0  2 . (12) The omparison of the 1D mo deling and the t w o-state appro ximation in terms of the sp etral ampliation η , f. Eqs. (9) and (12 ) , demonstrates the apabilit y of the t w o-state appro ximation for small driving frequenies and amplitudes, f. Fig. 2 . 1D mo del ling vs. pr e ise numeris (2D).- In order to  he k the auray of the desription w e ompared the results obtained b y the 1D mo delling with the results of Bro wnian dynami sim ulations, p erformed b y in tegration of the o v erdamp ed Langevin equation (1), for a 2D stru- ture (see Fig. 1 ) whose shap e is desrib ed b y Eq. (2). In our ase w e ha v e used the asp et ratio ǫ = 1 / 4 and the b ottlene k width b = 0 . 02 . The sim ulations w ere arried out b y the use of the standard sto  hasti Euler-algorithm. Fig. 3 depits the dep endene of the sp etral ampli- ation η on the noise strength for dieren t v alues of the driving frequeny , the driving amplitude and the v alue of G . It is imp ortan t to p oin t out that the re- sults obtained from the 1D mo delling using the 1D- appro ximation (lines) are in exellen t agreemen t with the n umerial sim ulations of the full system (2D) (sym- b ols) within a small relativ e error. This agreemen t is due to the fat that the onsidered p oten tial funtion is a smo oth funtion ( | w ′ ( x ) | ≪ 1 ), and in this situation our emplo y ed 1D-appro ximation is exp eted to b eome v ery aurate [ 15 , 17 , 18 ℄. Fig. 3a sho ws the dep endene of the sp etral ampli- ation η on the noise strength D for v arious driving frequenies at a xed transv ersal fore and foring am- plitude F 0 . Here, w e observ e an inrease in the sp etral ampliation whi h giv es rise to the nding of the eet of Sto  hasti Resonane in the presene of en tropi barri- ers. As observ ed for the usual "energy-dominated" SR [1 ℄ the resonane p eak is more pronouned as the applied an- gular frequeny Ω of the input signal dereases. Similarly , Fig. 3 b depits ho w ESR dep ends on the strength of the transv ersal fore G . In terestingly , the maximal ampli- ation inreases up on dereasing the strength G of the transv ersal fore. Ho w ev er, the presene of the transv er- sal fore G is ruial for observing a non-monotoni b e- ha vior of the sp etral ampliation with inreasing noise lev el D . In the limit of G → 0 , the sp etral amplia- tion inreases monotonially with dereasing noise lev el and tends asymptotially to 1 / Ω 2 at small driving am- plitudes F 0 . Finally , Fig. 3 displa ys the dep endene of the sp etral ampliation η on the noise strength D for v arious amplitudes of the driving fore F 0 at a xed v alue of the transv ersal fore and driving frequeny . Both, the ampliation of the signal and the optimal v alue of the noise at whi h it o urs seemingly inrease as the driving amplitude dereases. This no v el ESR eet is  haraterized b y the app ear- ane of a maxim um in the sp etral ampliation as a funtion of the noise strength D , just as in on v en tional energy-dominated SR [1℄. Ho w ev er, in biologial and pratial systems the temp erature whi h on trols the strength of the thermal noise is not a readily v ariable parameter. Our results suggest that, for a giv en tem- p erature, a prop er  hoie of the externally on trolled pa- 4 0 0 . 5 1 η 0 0 . 25 0 . 5 0 . 75 1 D Ω = 0 . 01 Ω = 0 . 1 Ω = 0 . 2 0 0 . 5 1 0 0 . 25 0 . 5 0 . 75 1 G = 5 . 0 F 0 = 1 . 0 (a) 0 0 . 5 1 1 . 5 η 0 0 . 25 0 . 5 0 . 75 1 D G = 2 . 0 G = 3 . 0 G = 5 . 0 0 0 . 5 1 1 . 5 0 0 . 25 0 . 5 0 . 75 1 Ω = 0 . 1 F 0 = 0 . 5 (b) 0 0 . 25 0 . 5 0 . 75 η 0 0 . 25 0 . 5 0 . 75 1 D F 0 = 0 . 5 F 0 = 1 . 0 F 0 = 1 . 5 0 0 . 25 0 . 5 0 . 75 0 0 . 25 0 . 5 0 . 75 1 Ω = 0 . 1 G = 5 . 0 (c) FIG. 3: (olor online) In dimensionless units, the dep endene of the sp etral ampliation η on noise lev el D for v arious v alues of the quan tities, the driving amplitude F 0 , driving frequeny Ω , and the transv ersal fore G . The sym b ols or- resp ond to the results of the Langevin sim ulations for the t w o-dimensional struture with the shap e dened b y the di- mensionless funtion w ( x ) = − ǫx 4 + 2 ǫx 2 + 0 . 01 with the asp et ratio ǫ = 1 / 4 , whereas the lines are the results of the n umerial in tegration of the 1D kineti equation (10 ) . rameters (i.e. the nature of the driving fore, i.e., its amplitude and driving frequeny and the strength of the transv ersal fore) migh t bring the system in to an opti- mal regime where onnemen t and noise m utually in ter- pla y to b o ost noise-assisted transp ort inside a orrugated struture. In onlusion, w e ha v e eluidated a new me hanism leading to the app earane of noise-indued resonan t ef- fets when a Bro wnian partile mo v es in a onned medium in the presene of p erio di driving. The on- strained motion imp edes the aess of the partile to er- tain regions of spae and an b e desrib ed in terms of a bistable en tropi p oten tial. The ativ ated dynamis of the partile in this eetiv e p oten tial then results in a o- op erativ e eet b et w een noise and external mo dulation, yielding an En tropi Sto  hasti Resonane. The eet deteted is gen uine for small-sale systems in whi h shap e and utuations are una v oidable fators ruling their ev o- lution. The adv an tageous p ossibilities of ESR on what onerns optimization and on trol ma y pro vide new p er- sp etiv es in the understanding of systems at the sales of mirometers and nanometers and op en new a v en ues in their manipulation and on trol. 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