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Anomalous transport induced by nonhyperbolicity S. R. Lopes1,∗ J. D. Szezech Jr2, R. F. Pereira3, A. A. Bertolazzo1,4, and R. L. Viana1 1Departamento de F´ısica, Universidade Federal do Parana´, Curitiba, PR, Brazil 2Instituto de F´ısica, Universidade de São Paulo, São Paulo, SP, Brazil 3Programa de Po´s-Gradua¸cão em Ciências/F´ısica, Universidade Estadual de Ponta Grossa, Ponta Grossa, PR, Brazil 4Instituto de F´ısica, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil (Dated: January 10, 2012) In this letter we study how deterministic features presented by a system can be used to perform direct transport in a quasi-symmetric potential and weak dissipative system. We show that the presence of nonhyperbolic regions around acceleration areas of the phase space plays an important role in theacceleration of particles giving rise todirect transport in thesystem. Such effect can be 2 observedforalargeintervaloftheweakasymmetricpotentialparameterallowing thepossibility to 1 obtain useful work from unbiased nonequilibrium fluctuation in real systems even in a presence of 0 a quasi-symmetric potential. 2 n a Anomalous transport is an emerging field in physics tentialasymmetryiscomparablewiththeoriginalpoten- J and, generally speaking, refers to nonequilibrium pro- tialenergyofthesystem. Inthisletterweshowthatitis 6 cessesthat cannotbe describedby using standardmeth- notnecessarytohavesuchstrongasymmetry,inthesense ods of statistical physics. The investigation of anoma- that sizeable ratchet currents can be obtained in weakly ] D lous transport processes requires a combination of con- dissipative systems with slightly asymmetric potentials. cept and methods of diverse disciplines, like stochastic In fact, we claim that the presence of ratchet currents is C theory, dynamical systems theory and disordered sys- influenced not so much by the potential asymmetry, but . n tems [1]. Anomaloustransportoccursin a wide realmof rather by the existence of strongly nonhyperbolic regions i l physical systems ranging from a microscopic level (such in the phase space of weakly dissipative systems. n as conducting electrons) to a macroscopic scale (as in As a representative illustration of this effect we con- [ global atmospheric events). One of the well-known phe- sider a periodically kickedrotorsubjected to a harmonic 1 nomenainthiscategoryisanomalousdiffusion,forwhich potential function, whose dynamics is two-dimensional. v the mean-squared-displacement increases with time as a By a hyperbolic region S we mean a set for which the 0 power-law tµ, where µ6=1 [2]. tangent phase space in each point splits continuously 0 5 There is a growing interest in anomalous transport intostableandanunstablemanifoldswhichareinvariant 1 properties of nonlinear systems presenting nonequilib- under the system dynamics: infinitesimal displacements 1. rium fluctuations, the ratchet systems [3–6]. For these in the stable (unstable) direction decay exponentially as 0 systemsitispossibletosurmountthesecondprincipleof time increases forward (backward) [12]. In addition, it 2 thermodynamics provided we do not have any space or is required that the angles between the stable and un- 1 time symmetriesforbiddingdirecttransport[7]. Ratchet stable directions are uniformly bounded awayfrom zero. : v systems occur in a variety ofphysicalproblems,like uni- Chaotic orbits of dissipative two-dimensional mappings, Xi directional transport in molecular motors [7, 8], micro forexample,areoftennonhyperbolicsincethestableand particles segregationin colloidalsolutions [9], and trans- unstablemanifoldsaretangentininfinitely manypoints. r a port in quantum and nanoscale systems [4, 7, 10]. Thedynamicsofaperiodicallykickedrotorwithsmall dissipation and potential asymmetry can be described Previous works have studied ratchet systems with a in a cylindrical phase space (−∞×∞)×[0,2π), whose mixed phase space and weak dissipation. For such sys- discrete-timevariablesp andx arerespectivelythemo- tems it has been shown that the ratchet effect results n n mentum and the angular position of the rotor just after from a connection betweenweak dissipation, chaotic dif- the nth kick, with the dynamics given by the following fusion,andballistictransportduetopresenceofperiodic dissipativeasymmetrickickedrotormap(DAKRM)[14]: islands [4, 5]. The latter, although occurring in the con- servativecaseonly,stillaffects the dynamicsinthe weak pn+1 =(1−γ)pn−K[sin(xn)+asin(2xn+π/2)],(1) dissipative case since island centers become attracting fixedpointsinthe weakdissipativecase[11]. Ithasbeen xn+1 =xn+pn+1, (2) shown that in the absent of islands anomaloustransport where K is related to the kick strength, 0 ≤ γ ≤ 1 is cannotbeachievedandthetransportcurrentsduetothe a dissipationcoefficient, and a is the symmetry-breaking ratcheteffectinweaklydissipativesystemsarerelatedto parameter of the system. The conservative (γ = 0) and the existence of isoperiodic stable structures [4, 5]. symmetric (a=0) limits yield the well-known Chirikov- Inthesepreviousinvestigationsithasbeenfocusedthe Taylor map [13]. In the following we will keep the dissi- stronglyasymmetriccase,forwhichthemagnitudeofpo- pation small enough (namely γ = 2×10−4) in order to 2 momentum probability distributions σ(p) (Figs. 1(e-h)). For K = 6.40 there is a quasi-symmetric situation, the neighborhood of the two P1 fixed points (left and right) beingseldomvisited[Fig. 1(a)]. Since the left(right)re- gionisresponsibleforanegative(positive)increaseofthe momentum transport, we observe that for this parameter value the momentum distribution function is nearly symmetric, with a Gaussian shape [Fig. 1(e)], resulting in a null transport. Symmetry-breakingeffects start to be noticeable after K = 6.40 and turn to be maximum in the bifurcation at K ≈ 6.92, where only the vicinity of the left P1 is scarcely visited by orbits of the map [Fig. 1(b)]. This effect is triggered by the bifurcation whereby the right P3 fixed point collides with the right P1 point and turns its vicinity easily accessible (not shielded), what is re- FIG. 1. (color online) 2-D histograms (top) and momentum flected in the asymmetric right tail in the momentum probability distributions(bottom) fortheDAKRMwith γ = distribution function [Fig. 1(f)]. The vicinity of the left 0.0002, a= 0.005, and (a,e) K =6.40; (b,f) 6.92; (c,g) 6.96; and (d,h) 7.00. P1 is not yet affected since the collision process did not occur yet for left P1 and P3. As a result, a net transport current is generated. However, this is more an effect of the bifurcation (due to nonhyperbolicity) than of highlight the effect of the periodic islands of the conser- the symmetry-breaking itself. In other words, if there vative case. Moreover, the asymmetry parameter a will is weak symmetry-breaking but no bifurcation (and no be kept small so as to emphasize the role of the nonhy- shield process), the ratchet effect will not occur, at least perbolicphasespaceregionsontheanomaloustransport. with the magnitude we observed in this example. The conservative and asymmetric (a 6= 0) case has Not too far from the bifurcation, (K = 6.96), the been studied in detail: it has two fixed points (we call P1) given by xR,L = sin−1Θ(a,K) and pR,L = 0, where vicinities of both P1 fixed points become now regularly visited [Fig. 1(c)] since positive and negative currents, Θ = 1− 1+8a2±16πa/K /4a, which are stable (cid:16) p (cid:17) caused by the right and left vicinity is counterbalanced. centersinthefollowingparameterintervals: 0<a<1/4, For this case the momentum distribution function is and 6.40 < K < 7.20. These two P1 points are the cen- again approximately symmetric [Fig. 1(g)] but present- ters of two resonantislands that are actually accelerator ing right and left tails. The situation changes again af- modes. Therearealsotwo (left andright)period-3fixed ter the left P1 and P3 fixed points collide [Fig. 1(d)], points (P3) related to secondary resonances around the through the same bifurcation mechanism described for P1 islands. theirrightcounterparts. Theincreaseintheaccessibility Intheweakdissipativecase,thetwoP1 pointsbecome ofthevicinityoftheleftP1 pointleadstoanasymmetric stable foci, their basins of attraction being roughly the momentum distribution function [Fig. 1(h)], restoring a region occupied by the respective islands (in the conser- net transportcurrent. In this lastcase is noticeable that vative case). Moreover, the chaotic region in the con- the seldom visited region is very small, nevertheless it is servative map becomes a chaotic transient in the weakly enoughto inhibit positive currents. Once againthe non- dissipative situation. At K ≈ 6.92 the right P3 points hyperbolicityofrightregionseemstobemoreimportant collides with the right fixed point (pR,xR) by a bifurca- than the asymmetric situation itself. tion. At the bifurcation point the small attraction basin The variation of the average net transport current p of P1 engulfs the stable manifold of the P3 and turns to with the nonlinearity parameter K is depicted in Fig. be accessible to points in a large phase space region. 2(a) for different values of the asymmetry parameter. The vicinity of the fixed points plays a key role in The net current is ensemble-averaged over a large num- the anomalous transport mechanism, in the same way ber(106)ofinitialconditions,eachofthembeingfollowed as the islands do for the conservative case. More pre- by a short time (t=103) to preventthe system to settle cisely, the wide accessibility of this vicinity near the bi- down into any fixed point. On varying K we obtain a furcationis responsiblefor largeratchetcurrents,just as series of positive and negative net transport currents re- the role of the accelerator modes in the non-dissipative sultingfromtheratcheteffect. Curiouslythenetcurrent map. Figs. 1(a-d) depict 2-D histograms for 5000 or- fluctuates less for both very small and large asymmetry, bits (each orbit containing 103 points) of the DAKRM being more sensitive to K for intermediate values of a. from initial conditions chosen in the phase plane region Aswehaveseen,theappearanceofnetcurrentsisdueto 0 < x < 2π, −π < p < π, as well as the corresponding thefactthattheleftandrightP3 fixedpoints(thatbifur- 3 (a) Moreover,thenetcurrentamplitudedoesnotchangeap- 50 preciably along the blue and red lines in Fig. 2(b) when aisvaried,emphasizingtheroleofthenonhyperbolicre- _p 0 a = 0.5 gionsasanessentialrequirementforthe ratcheteffectto a = 0.005 -50 a = 0.0005 arise. Theexistenceofnonhyperbolicregionsinphasespace, (b) however small they may be, constitutes a deterministic mechanism underlying anomalous transport in the pro- duction of net currents through a ratchet effect. In order to quantify the degree of nonhyperbolicity related FIG. 2. (color online) (a) Ensemble averaged net current for to the phenomena we describe in this letter, let us differentvaluesofthenonlinearityandasymmetryparameters consider an initial condition (p0,x0) and a unit vec- coofltohrescDaAle)KRasMawfuitnhcγtio=n0o.f00t0h2e.a(sby)mNmetettrryanasnpdorntocnulrirneenatr(itiny tor v, whose temporal evolution is given by vn+1 = J(p ,x )v /|J(p ,x )v |, where J(p ,x ) is the Ja- parameters. n n n n n n n n cobian matrix of the DAKRM. For n large enough, v is parallelto the Lyapunovvectoru(p,x) associatedto the maximum Lyapunov exponent λ of the map orbit be- u cateinpairsforthe symmetriccase)starttobifurcateat ginning with (p0,x0). Similarly a backward iteration of different values of K. For the interval 6.40 < K < 7.00, the same orbit gives us a new vector v that is parallel n corresponding to Fig. 1, the right P3 fixed point bifur- to the direction s(p,x), the Lyapunov vector associated catesbeforetheleftone,generatingasequenceofpositive to the minimum Lyapunov exponent λ [15, 16]. For re- s and negative net currents with very large peak values. gions where λ < 0 < λ the vectors u(p,x) and s(p,x) s u For instance, the maximum amplitude of the net trans- aretangenttotheunstableandstablemanifolds,respec- port current for a = 0.005 is at least three times larger tively, of a point (p,x). than for similar parameters in the case of high asymme- ThenonhyperbolicdegreeofaregionS canbestudied try (a=0.5) [4]. by computing the local angles between the two mani- Bythewayofcontrast,withahigherasymmetryvalue folds θ(p,x) = cos−1(|u · s|), for (p,x) ∈ S [16]. So, as those in Ref. [4] such large net currents are observed θ(p,x) = 0 denotes a tangency between unstable and only for largernonlinearities (8.5<K <10), hence they stable manifolds at (p,x). Let SR,L = {(p,x) ∈ Ω : ǫ are not primarily related to the bifurcations we present. |(p−x)−(pR,L,xR,L| < ǫ} be a ǫ-radius neighborhood In such case the role of the potential asymmetry over- of the right and left P1 fixed points. Results for θ(p,x) comes the bifurcation mechanism presented here. Nev- and its distribution function ρ(θ) calculated in both re- ertheless appreciable net currents can also be acquired gionsareshowninFig. 3for fourvalues ofthe nonlinear for asymmetry parameter as small as a=0.0005or even parameter K. The dark regions in Fig. 3 correspond to smaller, although the peak values decrease considerably. strongly nonhyperbolic region surrounding the accelera- Such decrease in the net transport current for small val- tion region. ues of a is expected since for the case of a=0 no trans- For K =6.40, near the fixed points (pR,L,xR,L) there port can be observed due to the symmetry of the stan- is a strong nonhyperbolic regionwhich shields the accel- dard map. erationarea[Figs. 3(a-b)]. Infact,almostallθvaluesfor The sensitive dependence of the net transport cur- the right area (red curve) and left (blue curve) are con- rent on the nonlinearity parameter in the weak asym- finedintheintervalθ <π/16,configuringastronglynon- metry case reminds us of a similar behavior for the hyperbolic region around both fixed points [Fig. 3(c)]. diffusion coefficient of the conservative and symmetric By way of contrast, when K = 6.92, only the left fixed (Chirikov-Taylor)map, caused by the existence of accel- point neighborhood is shielded, resulting in a large pos- erator modes. A more complete comparison is shown in itive transport since only the right acceleration region Fig. 2(b). If the potential asymmetry is large (a . 0.5) is regularly visited [Figs. 3(d-e)]. The right P1 and P3 sizeable net currents are obtained only for higher values fixedpointssufferabifurcationandalltangenciesofsta- ofK. Thisisduetothefactthatlargeasymmetriesshift bleandunstablemanifoldsdisappearfromtherightarea, the small islands present in the symmetric system phase allowing the trajectories to visit the acceleration region. space to some other intervals of K. On the other hand, Accordingly Fig. 3(f) presents different distributions of if the asymmetry is very weak (a & 0) we obtain large θ values for left and right regions. The red curve (right) net currents (both positive and negative) for K-values presentsadistributionpeakaroundθ =π/8considerably considerably lower. The negative transport observed in greaterthantheblueone,leadingtoabsenceofshielding Ref. [4] corresponds to the pale cyan area in Fig. 2(b) in the right region. for a=0.5 and around K =6.5, hence it is not directed In the case of K =6.96 neither of the areas surround- correlatedtothebifurcationmechanismwedescribehere. ingthefixedpointsareshieldedbythetangenciesofman- 4 for weakly dissipative systems. Let the inner region surround the P3 fixed point. When the stable and unstable manifolds ofP3 arealmosttangent,the inner regionsur- rounding P3 mapped to an outer region tends to zero since the angle between the manifolds goes to zero. This clearlyinhibits transportfrominnertoouterregions. As a result the acceleration region is practically not visited and no net transport currents are observed when there are nearly parallel manifolds. In conclusion we have shown that anomalous trans- portdisplayedbyaquasi-symmetricpotentialandweakly dissipative system is strongly related to the topology of the acceleration regions around fixed points displayed by the system. The presence of nonhyperbolic regions caused by almost parallel unstable and stable manifolds can inhibit a chaotic trajectory to visit the neighborhood of the acceleration region surrounding fixed points of the system. This mechanism is closely related to the scenario described by the Poincaré-Birkhoff theorem in area-preserving two-dimensional maps. This dynamical phenomenon yields large net transport current in some direction even though the potential has an extremely small degree of symmetry-breaking. Hence such net cur- rentscanyieldusefulworkfromunbiasednonequilibrium fluctuation even with quasi-symmetric potentials, which enlarges the realm of dynamical systems displaying the ratchet effect. This work is partially supported by CNPq, CAPES and Funda¸caõ Araucária. FIG.3. (coloronline)θ(p,x)valuesevaluatedfrom105 initial conditionsuniformlydistributedaround(0.2radius)SR,Land ǫ thedistribution function of θ. ∗ lopes@fisica.ufpr.br [1] R.Klages,G.Radons,andI.M.Sokolov(Eds.),Anoma- lous Transport (Wiley-VCH,Weinheim, 2008). 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