Application of the diffusion equation to prove scaling invariance on the transition from limited to unlimited diffusion
Edson D. Leonel, Celia Mayumi Kuwana, Makoto Yoshida, Juliano Antonio de Oliveira
aa r X i v : . [ n li n . C D ] M a r Application of the diffusion equation to prove scaling invariance on the transitionfrom limited to unlimited diffusion Edson D. Leonel, C´elia Mayumi Kuwana, Makoto Yoshida, Juliano Antonio de Oliveira Universidade Estadual Paulista (UNESP) - Departamento de F´ısicaAv.24A, 1515 – Bela Vista – CEP: 13506-900 – Rio Claro – SP – Brazil Universidade Estadual Paulista (UNESP) - Campus de S˜ao Jo˜ao da Boa VistaAv. Prof a . Isette Corrˆea Font˜ao, 505 – CEP: 13876-750 – S˜ao Jo˜ao da Boa Vista – SP – Brazil (Dated: March 17, 2020)The scaling invariance for chaotic orbits near a transition from unlimited to limited diffusion ina dissipative standard mapping is explained via the analytical solution of the diffusion equation.It gives the probability of observing a particle with a specific action at a given time. We showthe diffusion coefficient varies slowly with the time and is responsible to suppress the unlimiteddiffusion. The momenta of the probability are determined and the behavior of the average squaredaction is obtained. The limits of small and large time recover the results known in the literaturefrom the phenomenological approach and, as a bonus, a scaling for intermediate time is obtainedas dependent on the initial action. The formalism presented is robust enough and can be appliedin a variety of other systems including time dependent billiards near a transition from limited tounlimited Fermi acceleration as we show at the end of the letter and in many other systems underthe presence of dissipation as well as near a transition from integrability to non integrability. PACS numbers: 05.45.-a, 05.45.Pq, 05.45.Tp
Since the observation and a phenomenological charac-terization [1] of scaling invariance in the chaotic sea neara transition from integrability to non integrability in aFermi Ulam model [2], the formalism using homogeneousand generalized function leading to a set of critical expo-nents [3] has been widely used in a variety of systems toinvestigate dynamical properties near dynamical phasetransitions including oscillating spring mass system [4],billiards [5, 6], scaling in social media [7], in waveguides[8] and in many other systems. In a majority of the casesthe scaling is closely connected with diffusion yielding ap-plications in different subjects of science being thereforeobserved in systems from pollen diffusing [9], in diseasepropagation [10–12], in pests spreading [13] and manyothers hence making the topic of wide interest.In this letter our aim is to characterize analytically atransition from limited to unlimited diffusion by using thediffusion equation [14] applied in a paradigmatic model innonlinear science the so called dissipative standard map-ping [15] and the success of the formalism allowed us toextend its applicability to time dependent billiards. Themapping is written in terms of two equations I n +1 =(1 − γ ) I n + ǫ cos( θ n ) and θ n +1 = ( θ n + I n +1 ) mod(2 π )where γ ∈ [0 ,
1] is the dissipative parameter and ǫ cor-responds to the intensity of the nonlinearity. This sys-tem has two well known transitions [15] for γ = 0 (con-servative case): (i) A transition from integrability for ǫ = 0 where the phase space is foliated to non integra-bility when ǫ = 0 and mixed structure is presented inthe phase space including periodic islands, chaotic seasand invariant spanning curves limiting the diffusion to aclosed region; (ii) at a critical value of ǫ c = 0 . . . . , thesystem admits a transition from local chaos when ǫ < ǫ c to globally chaotic dynamics for ǫ > ǫ c where invariantspanning curves are no longer present and, depending on the initial conditions, chaos can diffuse unbounded inthe phase space. The determinant of the Jacobian ma-trix is det J = (1 − γ ) and for γ = 0 the Liouville’stheorem is violated leading to the existence of attractorsin the phase space. For large enough ǫ , typically ǫ > γ . At such limit one is facing a transi-tion from limited ( γ = 0) to unlimited ( γ = 0) diffusionfor the variable I which is the transition we consider inthis letter. Our main goal in this letter is to fix up anopen problem in the nonlinear community discussing thescaling invariance present in the transition from limitedfor γ = 0 to unlimited diffusion when γ = 0, so far an-alytically for large values of ǫ . As far as we can tell,this scaling investigation has only been described usinga phenomenological approach [16] assuming a set of scal-ing hypotheses allied with a homogeneous function henceleading to a set of critical exponents leaving a lack on theanalytical solution which to the best knowledge of the au-thors has never been made. At the same time, this letterfix up this gap in the literature and the present approachis proved to be valid and can be used in a wide classof other systems including transition from limited to un-limited Fermi acceleration in time dependent billiards aswe shown in the end of the letter, integrability to nonintegrability in nonlinear mappings and many others.The range of parameters we are interested in to val-idate the transition is γ positive and small, typically γ ∈ [10 − , − ] and ǫ >
10, which drives the systemto high nonlinearities and absence of sinks in the phasespace. At such a window of parameters a transition fromlimited, γ = 0 to unlimited, γ = 0, diffusion is observed.A typical plot of the phase space is shown in Figure 1(a)illustrating a chaotic attractor for the parameter ǫ = 10 FIG. 1: (a) Plot of the phase space for the standard dissipativemapping considering the parameters ǫ = 100 and γ = 10 − .(b) Normalized probability distribution for the chaotic attrac-tor shown in (a). and γ = 10 − together with the probability distributionalong the chaotic attractor shown in Figure 1(b). Wesee from Figure 1(a) the density of points concentratearound I ∼ = 0 and is symmetric with respect to the ver-tical axis. The distribution fades soon as it goes apartfrom the origin. The positive Lyapunov exponent mea-sured [17] for the chaotic attractor shown in Fig. 1(a)was λ = 3 . I ∼ = 0 the particle dif-fuses along the chaotic attractor. The natural observableto characterize the diffusion is the average squared ac-tion I rms ( n ) = q M P Mi =1 I i where M corresponds to anensemble of different initial conditions along the chaoticattractor. To obtain such observable we need to solve thediffusion equation that gives the probability to observea specific action I at a given time n , i.e. P ( I, n ). Thediffusion equation is written as ∂P ( I, n ) ∂n = D ∂ P ( I, n ) ∂I , (1)where the diffusion coefficient D is obtained from thefirst equation of the mapping by using D = I n +1 − I n .A straightforward calculation assuming statistical inde-pendence between I n and θ n at the chaotic domain leadsto D ( γ, ǫ, n ) = γ ( γ − I n + ǫ . (2)The expression of I n is obtained also from the firstequation of the mapping assuming that I n +1 − I n = I n +1 − I n ( n +1) − n ∼ = dI dn = γ ( γ − I + ǫ , whose solution is I ( n ) = ǫ γ (2 − γ ) + (cid:18) I + ǫ γ ( γ − (cid:19) e − γ (2 − γ ) n . (3)To compare with the experimental observable Eq. (3)must be averaged over the orbit, leading to < I ( n ) > = 1 n + 1 n X i =0 I ( i ) = γ ( γ − n + 1) ×× (cid:20) I + ǫ γ ( γ − (cid:18) − e − ( n +1) γ (2 − γ ) − e − γ (2 − γ ) (cid:19)(cid:21) . (4)To obtain an unique solution for Eq. (1) we impose thefollowing boundary conditions lim I →±∞ P ( I ) = 0 withthe initial condition P ( I,
0) = δ ( I − I ) that warrants allparticles leave from the same initial action but with M different initial phases θ ∈ [0 , π ]. Although the diffusioncoefficient D depends on n its variation is slow and littlefrom the instant n to n + 1. This property allows us toconsider it constant to obtain the solution of the diffusionequation. However, soon as the solution is obtained, theexpression of D from Eq. (2) is incorporated to the solu-tion. The technique used to solve Eq. (1) is the Fouriertransform [18]. Because the probability is normalized,i.e. R ∞−∞ P ( I, n ) dI = 1, we can define a function R ( k, n ) = F{ P ( I, n ) } = 1 √ π Z ∞−∞ P ( I, n ) e ikI dI. (5)Differentiating R ( k, n ) with respect to n and from theproperty that F n ∂ P∂I o = − k R ( k, n ) we end up thefollowing equation to be solved dRdn ( k, n ) = − Dk R ( k, n ),which leads to R ( k, n ) = R ( k, e − Dk n . (6)Considering the initial condition we have that R ( k,
0) = F{ δ ( I − I ) } = √ π e ikI . Inverting the expression of R ( k, n ) we obtain P ( I, n ) = 1 √ π Z ∞−∞ R ( k, n ) e − ikI dk, = 1 √ πDn e − ( I − I Dn . (7)Equation (7) satisfies both the boundary and initial con-dition as well as the diffusion equation (1). It is alsonormalized by construction. The observable we want tocharacterize is I ( n ) = R ∞−∞ I P ( I, n ) dI , which leads to I ( n ) = p D ( n ) n + I . Using D ( n ) obtained from Eq.(2), we end up with the expression of I rms ( n ) as I rms ( n ) = s I + nγ ( γ − n + 1 (cid:20) I + ǫ γ ( γ − (cid:21) (cid:20) − ( n + 1) γ (2 − γ )1 − e − γ (2 − γ ) (cid:21) . (8)Let us discuss specific limits of n and its consequencesto Eq. (8). The first limit is n = 0, which leads to I rms (0) = I , in well agreement to the initial condition.The second limit is n → ∞ . At such a limit we have I rms = s I + γ ( γ − (cid:20) I + ǫ γ ( γ −
2) 11 − e − γ (2 − γ ) (cid:21) , (9)and that when expanding in Taylor series up to first orderthe term 1 − e − γ (2 − γ ) ∼ = γ (2 − γ ) we obtain I rms = 1 p − γ ) ǫγ − / . (10)Let us discuss this result prior move on. It is known inthe literature [16] that the critical exponents α and α can be obtained from the scaling theory. It was supposedthat for large enough n , the stationary state is given by I rms ∝ ǫ α γ α . An immediate comparison of this scalinghypothesis with Eq. (10) leads to a remarkable resultsof α = 1 and α = − , in very well agreement withthe phenomenological prediction discussed in Ref. [16].Interestingly, such a result can also be obtained from theown equations of the mapping imposing that I n +1 = I n = I sat , yielding I sat = √ − γ ) ǫγ − / .The limit of small n is the third limit we consider. As-suming that the initial action I ∼ = 0, hence negligibleas compared to ǫ and doing a Taylor expansion on theexponential of the numerator from Eq. (9) we obtain I rms ( n ) ∼ = q ǫ n . This result proves that for short n , anensemble of particles diffuses along the chaotic attractoranalogously as a random walk motion, hence with dif-fusion exponent β = 1 /
2, i.e., normal diffusion. FromRef. [16] a scaling hypothesis at the limit of small n is I rms ( n ) ∝ ( nǫ ) β , with β = 1 / n but non negligible I such that0 < I < I sat . At such windows of I and n , an additionalcrossover is observed when n ′ x ∼ = 2 I ǫ . This crossover hadalready been observed in [1] when a phenomenological ap-proach was proposed and confirmed analytically in [19].A fifth limit is in the case of I ∼ = 0, leading to a growthin I rms for short n followed by a crossover and a bendtowards the regime of saturation. Such a characteristiccrossover is given by n x ∼ = − γ γ − . From the scalingapproach as discussed in Ref. [16] it is assumed that n x ∝ ǫ z γ z and that z = 0 and z = −
1, as obtainedabove. The last regime of interest is considered when I ≫ ǫ γ (2 − γ ) . At this limit, Equation (8) is rewritten as I rms ( n ) = s I e − ( n +1) γ (2 − γ ) + ǫ (1 − e − ( n +1) γ (2 − γ ) )2 γ (2 − γ ) . (11)The leading term for small n is I rms ( n ) = I e − ( n +1) γ (2 − γ )2 while the stationary state is obtained at the limit oflim n →∞ I rms = ǫ √ − γ ) γ − / , in well agreement withthe previous results.Figure 2(a) shows a plot of I rms vs. n for differentcontrol parameters and initial conditions, as labeled in FIG. 2: (a) Plot of the phase space for the standard dissipativemapping considering the parameters ǫ = 100 and γ = 10 − .(b) Normalized probability distribution for the chaotic attrac-tor shown in (a). Inset of (b) shows an exponential decay tothe attractor. the figure. Filled symbols correspond to the numericalsimulation obtained direct from the iteration of the dy-namical equations of the mapping considering an ensem-ble of M = 10 different initial particles, all startingwith same action I , as shown in Fig. 2(a) and differ-ent initial phases φ ∈ [0 , π ]. Analytical result fromEq. (8) is plotted as continuous line. The overlap ofthe curves is remarkable good. Figure 2(b) shows theoverlap of the curves plotted in (a) onto a single andhence universal curve. The scaling transformations are:(i) I rms → I rms / ( ǫ α γ α ); (ii) n → n/ ( ǫ z γ z ). The in-set of Fig. 2(b) shows the exponential decay as predictedby Eq. (11). The control parameters used in the insetwere ǫ = 10 and γ = 10 − and with the initial action I = 10 . The slope of the exponential decay obtainednumerically is a = 9 . × − , which is close to γ (2 − γ ) / ∼ = 9 . × − .Let us now show applicability of the formalism devel-oped to a far more complicate system, indeed a time de-pendent billiard [20]. The boundary confining an ensem-ble of non interacting particle is written as R ( θ, η, t ) =1 + ηf ( t ) cos( pθ ) with p integer. The case of η = 0 corre-sponds to the circle billiard, which is integrable and thathas foliated phase space [21]. For η = 0 and f ( t ) = const. the phase space is of mixed kind exhibiting chaos, in-variant spanning curves and periodic islands [22]. Fermiacceleration [2] is observed when f ( t ) = 1 + ǫ cos( ωt ),where scaling properties [23] are also observed. We shallconsider that f ( t ) = 1 + ǫ cos( ωt + Z ) where Z ∈ [0 , π ] isa random number generated at each collision of the par-ticle with the moving boundary. The dynamics of eachparticle is given in terms of a 4 − D nonlinear mappingfor the variable velocity of the particle V n , instant of thecollision t n , polar angle θ n and angle that the trajectoryof the particle makes α n with a tangent line at the in-stant of the collision. The velocity of the boundary atthe instant of the impact is ~V b ( t ) = d ~R b dt ( t + Z ). Thereflection laws are given by ~V ′ n +1 · ~T n +1 = ~V ′ n · ~T n +1 and ~V ′ n +1 · ~N n +1 = − γ ~V ′ n · ~N n +1 where γ ∈ [0 ,
1] correspond- ing to a restitution coefficient. The case of γ = 1 leadsto a non dissipative case while 0 < γ < ~T and ~N are the tangent and nor-mal unit vectors at the instant of the impact and ′ is toconsider the momentum conservation law at the movingreferential frame. The case γ = 1 leads to unlimited dif-fusion for the velocity of the particles, hence producingFermi acceleration while γ < D ( η, ǫ, γ, n ) = V n γ ) η ǫ , (12)where the expression for V n is V ( n ) = V e − n ( γ − + (1 + γ ) η ǫ − γ ) (cid:18) − e − n ( γ − (cid:19) . (13)As discussed in Ref. [24], the behavior of the V rms ( n ) canbe summarized as: (i) for short n , V rms ( n ) ∝ n β ; (ii) forlarge enough n , it is observed that V sat ∝ (1 − γ ) α ( ηǫ ) α ,(iii) finally the crossover iteration number is written as n x ∝ (1 − γ ) z ( ηǫ ) z . Doing the same procedure we madealong on the paper we end up with the following set ofcritical exponents α = − . z = − α = 1, z = 0and β = 0 . [1] E. D. Leonel, J. K. L. Silva, P. V. E. McClintock, Phys.Rev. Lett , 014101 (2004).[2] E. Fermi, Phys. Rev. , 1169 (1949).[3] F. Reif, Fundamentals of statistical and thermal physics,New York: McGraw-Hill (1965).[4] O. F. de Alcantara Bonfim, Phys. Rev. E , 056212(2009).[5] D. F. M. Oliveira, M. Robnik, International Journal ofBifurcation and Chaos , 1250207 (2012).[6] E. D. Leonel, L. A. Bunimovich, Phys. Rev. E , 016202(2010).[7] D. F. M. Oliveira, K. S. Chan, E. D. Leonel, PhysicsLetters A , 47 (2018).[8] E. D. Leonel, Phys. Rev. Lett , 114102 (2007).[9] W. F. Morris, Ecology , 493 (1993).[10] S. A. El-Kafrawy et al, The Lancet Planetary Health , e521 (2019).[11] Z. Xu, Y. Zhang, IMA J. of Applied Mathematics ,1124 (2015).[12] Y. Lou, X. Q. Zhao, J. Math. Biol. , 543 (2011).[13] W. S. Jo, H. Y. Kim, B. J. Kim, J. Korean Phys. Soc. , 108 (2017).[14] V. Balakrishnan, Elements of nonequilibrium statisticalmechanics, Ane Books India, New Delhi (2008).[15] A. J. Lichtenberg, M. A. Lieberman, Regular and chaoticdynamics (Appl. Math. Sci.) 38, Springer Verlag, NewYork (1992).[16] D. F. M. Oliveira, M. Robnik, E. D. Leonel, Phys. Lett.A , 723 (2012).[17] J.-P. Eckmann and D. Ruelle, Rev. Mod. Phys. , 617,(1985).[18] E. Butkov, Mathematical Physics, Addison-Wesley Pub. Co., (1968)[19] M. S. Palmero, G. I. D´ıaz, P. V. E. McClintock, E. D.Leonel, Chaos , 013108 (2020).[20] A. Y. Loskutov, A. B. Ryabov, and L. G. Akinshin, J.Exp. Theor. Phys. , 966 (1999).[21] N. Chernov, R. Markarian, Chaotic Billiards (AmericanMathematical Society, Vol. 127, 2006) [22] M. V. Berry, Eur. J. Phys. , 91, (1981).[23] E. D. Leonel, D. F. M. Oliveira, A. Loskutov, Chaos ,033142 (2009).[24] E. D. Leonel, M. V. C. G´alia, L. A. Barreiro, D. F. M.Oliveira, Phys. Rev. E94