Discretization of phase space usually nullifies chaos in dynamical systems.
We show that if randomness is associated with discretization dynamical chaos
may survive and be indistinguishable from that of the original chaotic system,
when an entropic, coarse-grained analysis is performed. Relevance of this
phenomenon to the problem of quantum chaos is discussed.Comment: 4 pages, 4 figure

Falcioni, M.

Mantica, G.

Pigolotti, S.

Vulpiani, A.

English

arXiv

Discretization of phase space usually nullifies chaos in dynamical systems. We show that if randomness is associated with discretization dynamical chaos may survive and be indistinguishable from that of the original chaotic system, when an entropic, coarse-grained analysis is performed. Relevance of this phenomenon to the problem of quantum chaos is discussed.Peer ReviewedPostprint (published version

Falcioni, Massimo

Vulpiani, Angelo

Mantica, Giorgio

Pigolotti, Simone

UPCommons. Portal del coneixement obert de la UPC

P H Y S I C A L R E V I E W L E T T E R S week ending25 JULY 2003VOLUME 91, NUMBER 4Coarse-Grained Probabilistic Automata Mimicking Chaotic SystemsMassimo Falcioni and Angelo VulpianiDipartimento di Fisica, Universita` di Roma ’’La Sapienza,’’ INFM, Unita` di Roma1 and SMC Center,Piazzale Aldo Moro 2, 00185 Roma, ItalyGiorgio Mantica*Center for Nonlinear and Complex Systems, Universita` dell’Insubria, Via Valleggio 11, 22100 Como, Italy,and I.N.F.M., Unita` di Como, I.N.F.N. Sezione Milano, ItalySimone PigolottiSISSA, Via Beirut 4, 34014 Trieste, Italy(Received 1 October 2002; published 24 July 2003)044101-1Discretization of phase space usually nullifies chaos in dynamical systems. We show that if random-ness is associated with discretization dynamical chaos may survive and be indistinguishable from that ofthe original chaotic system, when an entropic, coarse-grained analysis is performed. Relevance of thisphenomenon to the problem of quantum chaos is discussed.DOI: 10.1103/PhysRevLett.91.044101 PACS numbers: 05.45.–a, 45.05.+xwhere t is discrete time, x belongs to 0; 1D, and D is thewith faster convergence, of the information rate hn Hn Hn1, as n! 1 [15,16]. The KS entropy hKChaos is a common characteristic of motion in con-tinuous spaces. It is signaled by many indicators. Amongthese, positivity of the Kolmogorov-Sinai (KS) entropy isperhaps the most significant, both in theory and applica-tions [1,2]. On the other hand, trajectories in discretespaces are always asymptotically periodic—hence, ofnull KS entropy. They may arise in the discretization ofcontinuous systems, as in the numerical simulation ofdifferential equations, but arguably their role is mostsignificant in the correspondence of classical and quan-tum dynamics.Consider, for instance, the paradigmatic example ofthe quantum Arnol’d cat. Its dynamics are algorithmi-cally equivalent to classical motion on a regular lattice,whose spacing is inversely proportional to the Planckconstant [3,4]. When the spacing diminishes, the latticebecomes denser in continuous, classical phase space. Yet,it has long been recognized that chaos cannot be naivelyrevived in such a limiting procedure [4–6]. A way out ofthis impasse is obtained by randomly perturbing thedynamics [7]: Is this addition enough to bring back thefull algorithmic content, that is, the distinctive signatureof chaos [8]? We attempt here an answer to this questionin the most general and simple terms.Deterministic-probabilistic systems (such as those oc-curring in cellular automata [9]) have long been inves-tigated, with respect to their invariant measures [10] andalso to the entropic content of their motion [11]. We nowextend this study to the regime where continuous, dis-crete, and random effects are simultaneously present andintermix in nontrivial ways.Let us therefore consider the deterministic mapxt1  fxt; (1)0031-9007=03=91(4)=044101(4)$20.00 dimension of the space. We embed in 0; 1D a uniformrectangular lattice, of spacing , and we label its states byinteger vectors n in 1; b1cD (b	c is the integer part) [12].We then restrict the map f to act on this lattice, and weadd randomness:nt1 1fntt: (2)We stipulate that the uncorrelated, random ‘‘jumps’’ textend to lattice neighbors with total probability p, so thatt  0 with probability 1 p. Note that when p  0 thesystem is purely deterministic. For definiteness, we studythe generalized tent map in one dimension:xt1 axt for 0  x  1a ;a1 xt=a 1 for 1a  x  1;(3)with a  3, the two-dimensional Arnol’d cat map [1]:xt1  xt  ytmod1; yt1  xt  2ytmod1; (4)and their probabilistic lattice automata, Eq. (2).We are now ready to briefly introduce our analyticaltools. Let fE1; E2; . . .g be a finite partition of phase spaceconsisting of identical hypercubic cells of side . Letw  w1; w2; . . . ; wn be a finite symbolic trajectory, oflength n  jwj: wi  j if and only if xi 2 Ej. Let alsopw be the frequency of w, defined by the physicalergodic measure. The n-block entropies Hn, n 1; 2; . . . are defined by the sumsHn  Xw : jwjnpw logpw: (5)The partition entropy h is the limit of Hn=n, or,2003 The American Physical Society 044101-12468100 5 10 15 20Hn(ε, M)nFIG. 1. n-word entropies Hn;M vs n for the discretizedtent map with   1=18, and M  252 (), 1008 (), and4032 (). Hn;M are compared to the lines with slopeh1=18 ’ 0:20 and hK ’ 0:636.P H Y S I C A L R E V I E W L E T T E R S week ending25 JULY 2003VOLUME 91, NUMBER 4is the supremum of the partition entropies, with respect toall countable partitions; hence, h  hK, and hK can beobtained letting  go to zero. However, the function hfor finite  is interesting in its own right [17]. This isbecause it gauges the rate of information production, forobservations of finite accuracy, as a function of the reso-lution desired. In line with our approach in [5] we alsoascribe importance to the full behavior of Hn versus n,and not only to the limit h.It is well-known that the systems (3) and (4) havepositive KS entropy. In contrast, the dynamics of thepurely discrete systems [i.e., Eq. (2) with p  0] are peri-odic, hence of null entropy. Yet, at scales larger than thelattice spacing,  > m : , they approximate the con-tinuous dynamics for a finite time, roughly of the order ofthe logarithm of the period of the trajectory [13,14,18–20]. As a consequence, the entropies Hn are also closeto those of the continuous system, for n  n. The upperboundary n can be estimated requiring that at n  n thenumber of different  histories of length n of the con-tinuous system, N n  exphn, be of the sameorder of the number of discrete states, M 1=D.Here D is the dimension of the attractor or, in the absenceof this latter, the dimension of the space. This leads ton logMh  D logh : (6)Dependence of n on the average period of trajectories,T, follows equally well. Since T MD2=2, where D2 is thecorrelation dimension of the ergodic measure [20],n 2DD2logTh : (7)In the deterministic discrete systems [Eq. (2) withp  0], n may be large enough to observe the entropicgrowth of the continuous map and sometimes to computethe entropy h of the latter to a fair accuracy. However,when n exceeds n, the n-word entropies, Hn, quicklysaturate at a constant value, of the order of logT orlog1= or logM, which reveals the periodic regime ofthe dynamics. It is clear that while the time n is modeldependent, its logarithmic scaling with the parameters isuniversal: the discrete, ‘‘pseudo’’ chaos seems to be veryshort-lived [4–6]. It is at this point that the random jumpst completely change the scenario.First of all, the null-entropy, periodic system is turnedinto an aperiodic stochastic process of maximal entropyhp. If t extends to k neighbors with equal probabilityp=k, then hp  p lnp=k  1 p ln1 p. Sincehp totally originates from the random jumps at the latticescale m, it can be fully detected only at this scale: hp ’hm  h, for   m.It is clear that hp has no relation with the KS entropyhK. Then, two cases must be considered. In view of theabove, when hp < hK, addition of randomness, and ob-044101-2servational coarse graining, cannot achieve the full en-tropy content of the continuous system. This no-go ruledoes not apply in the opposite case. We find that whenhp  hK and  m , the partition entropies of thecontinuous system and of the discrete ones tend to coin-cide. Notice that the condition  m requires that thenumber of discrete states per cell, MD, be large.To prove these claims, let us first consider the tent map,Eqs. (2) and (3), with p  0:05, and nearest-neighborrandom jumps. In this case hp ’ 0:233 is much smallerthan hK ’ 0:636. Figure 1 plots Hn;M versus n, for dif-ferent values of M, and   1=18. This partition is gener-ating, so that the partition entropy of the continuous map,hcont1=18, is equal to hK. We observe that for n < n,hn is approximately equal to hK, the entropy of thecontinuous system. Later on, for n > n, the curve Hnbends and—rather than tending to a constant, as it wouldif randomness were not present —redirects its growth to adifferent linear regime: Hn ’ h;Mn n  A logM,with A a suitable constant. Numerically, we also find thath;M ’ 0:20  hp, independently of M, even if  1=18 is much larger than m. This is noteworthy: werethe evolution driven only by the probabilistic diffusion,nt1  nt  t, the  entropy would have been 10 timessmaller, hdiff1=18; 252 ’ 0:02. The effect of randomnessis strongly enhanced by the deterministic evolution [21].By raising the value of the jump probability p [22] theentropy hp increases, and it may exceed the KS entropyhK, or hcont for a given . Figure 2 plots h;M versushp, for different values of M. At fixed, low hp, the Mdependence is rather mild, as was observed in Fig. 1, andh;M is smaller than hcont. Raising hp at fixed M towell exceed this value, and then increasing M, we obtainconvergence of h;M to hcont.This convergence is further illustrated in Fig. 3 byfixing hp  1:5, by choosing   1=7 and   1=10,044101-21.21.251.31.351.4,M)0.20.40.60.81.01.21.41.6h n()00.10.20.30.40.50.60.70.80 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8h(ε, M)hpFIG. 2. Partition entropies of the discretized tent map h;Mvs hp, with k  4,   1=7, and M  63 (), M  168 () andM  4032 (  ). The dotted line is drawn at hcont1=7, thepartition entropy of the continuous system for   1=7.P H Y S I C A L R E V I E W L E T T E R S week ending25 JULY 2003VOLUME 91, NUMBER 4and by plotting h;M versus z  M, the number oflattice points enclosed in a cell of the partition. Thedifferent curves tend to coincide for small z, where theentropies overshoot: when coarse graining is too fine (toofew states in a cell) the direct action of randomness isdominating. This effect fades in the opposite direction:randomness becomes a germ that gets scale amplified bythe dynamics, and h; z= tends to hcont, the partitionentropy of the continuous system.This behavior appears to be generic, as indicated by theresults for the two-dimensional Arnol’d cat map, subjectto a random perturbation with k  4 and hp  1:5, plot-ted in Fig. 4. Since   1=4; 1=16; 1=64, all provide gen-erating partitions, the corresponding values of hcont areequal to hK ’ 0:962. As a consequence the data fall on asingle line, starting from the entropy of the random0.50.550.60.650.70.750.80.850.90.9510 100 200 300 400 500h(ε, M)zFIG. 3.  entropies h;M vs z  M, for the discretized tentmap, with   1=7 and   1=10 (). Horizontal lines aredrawn at hcont1=7  0:54, and hcont1=10  0:58.044101-3perturbation, hp  1:5, and converging to the KS entropyhK ’ 0:962. The convergence of hn;M to the asymp-totic values is shown in the inset of Fig. 4, together withthe partition entropies obtained for the purely discretesystem (p  0), and for the purely stochastic motion.This comparison proves that h; z= are truly asymp-totic values and provides further evidence in support ofour explanation of the phenomenon, with which we nowconclude.Imposing a finite lattice to the otherwise continuumset of states of a dynamical system, inevitably boundsthe algorithmic complexity of its trajectories, and thevalue of its partition entropies, to the logarithm of thenumber of states. If the lattice is perceived with somefuzziness — or if random errors are allowed, followingthe approach of this Letter — one expects that on largescales continuum properties and chaos might reemergeand be indistinguishable from that of the original sys-tem [23]. We have determined the conditions for this tohappen. First, observational coarse graining must be in-voked. Second, the action of the external randomnessmust be confined to the ‘‘microscopic,’’ unresolved scales.The instability of deterministic dynamics amplifiesthese microscopic, random errors and carries them overto the large, observation scales. Finally, by a sort ofconservation law, the flow of information supplied by2 5 101 2 5 102 2 5 103 2 5 104 2M0.951.01.051.11.15h( 1 2 3 4 5 6 7 8 9 10n0.0FIG. 4. Entropies h;M vs M in the perturbed Arnol’d catmap. Here, M  162 (squares), 322 (circles), 642 (diamonds),1282 (filled squares), 2562 (filled circles), and 5122 (filleddiamonds);   1=4; 1=16; 1=64. The horizontal line is drawnat hK ’ 0:962. In the inset, hn;M vs n for   1=64 and M 162; . . . ; 2562 (coded as before). At the largest value of M, M 2562, the purely discrete system (hearts) shows rapid conver-gence to zero, in accordance with Eqs. (6) and (7). The purelydiffusive system (clubs) on its part is consistently close tohdiff1=64; 2562 ’ 0:085.044101-3P H Y S I C A L R E V I E W L E T T E R S week ending25 JULY 2003VOLUME 91, NUMBER 4the microscopic zitterbewegung must not be less than hK,the maximum entropy production rate of the continuoussystem.For a long time, research in quantum chaos has lookedfor quantum characteristics related to classical chaoticmotion. The fact that none of these could be properlycalled chaos led to the concept of pseudochaos andcast doubts on the very existence of chaos in nature.The results presented in this Letter suggest that onemight try to reverse this approach and consider classicaldynamics as an effective theory that, via truly chaoticdeterministic dynamical systems, models a randomlyperturbed quantum motion under observational coarsegraining.G. M. was supported by PRIN-2000 Chaos and local-ization in classical and quantum mechanics, and byHPRN-CT-2000-00156 Quantum Transport on an atomicscale. M. F and A.V. were supported by INFM (Centro diRicerca e Sviluppo SMC) and MIUR (Cofin. ‘‘FisicaStatistica di Sistemi Complessi Classici e Quantistici’’).0441*Electronic address: giorgio.mantica@uninsubria.it[1] V. I. Arnold and A. Avez, Ergodic Problems of ClassicalMechanics (Benjamin, New York, 1968).[2] R. Badii and A. Politi, Complexity Hierarchical Struc-tures and Scaling in Physics (Cambridge UniversityPress, Cambridge, 1999).[3] J. H. Hannay and M.V. Berry, Physica (Amsterdam) 1D,267–290 (1980).[4] J. Ford, G. Mantica, and G. Ristow, Physica (Amsterdam)50D, 493–520 (1991); J. Ford and G. Mantica, Am. J.Phys. 60, 1086–1098 (1992).[5] A. Crisanti, M. Falcioni, G. Mantica, and A. Vulpiani,Phys. Rev. E 50, 1959 (1994); G. Mantica, Phys. Rev. E61, 6434 (2000).[6] B.V. Chirikov and F. Vivaldi, Physica (Amsterdam)129D, 223 (1999).[7] Originally, it was noticed that introducing a randomcomponent in the quantum evolution restores the diffu-sive properties of kicked rotors: see I. Guarneri, Lett.Nuovo Cimento 40, 171 (1984); D. L. Shepelyansky,Physica (Amsterdam) 8D, 208 (1983); E. Ott, T. M.Antonsen, and J. D. Hanson, Phys. Rev. Lett. 53, 2187(1984). The decoherent approach to quantum chaos flour-ished soon after: see, e.g., T. Dittrich and R. Graham,Phys. Rev. A 42, 4647 (1990); A. R. Kolovsky, Europhys.Lett. 27, 79 (1994); W. H. Zurek and J. P. Paz, Physica01-4(Amsterdam) 83D, 300 (1995); T. A. Brun, I. Percival,and R. Schack, J. Phys. A 29, 2077 (1996); and the re-cent A. K. Pattanayak, B. Sundaram, and B. J.Greenbaum, Phys. Rev. Lett. 90, 14103 (2003). In manyof these works, a randomly perturbed quantum cat playsa major role.[8] J. Ford, Phys. Today 36, No. 4, 40 (1983).[9] P. Grassberger, Physica (Amsterdam) 10D, 52 (1984);P. Rujan, J. Stat. Phys. 49, 139 (1987); A. Georges andP. Le Doussal, J. Stat. Phys. 54, 1011 (1989).[10] D. Ruelle, Commun. Math. Phys. 82, 137 (1981); Yu. I.Kifer, Math. USSR Izv. 8, 1083 (1974).[11] See, e.g., J. P. Crutchfield and N. H. Packard, Physica(Amsterdam) 7D, 201–223 (1983); and the more recentA. Ostruszka, P. Pakon´ski, W. Slomczyn´ski, andK. Z˙ yczkowski, Phys. Rev. E 62, 2018 (2000);G. Boffetta, M. Cencini, M. Falcioni, and A. Vulpiani,Phys. Rep. 356, 367 (2002).[12] If the volume of the space accessible to the continuoussystem is finite, uniform discretization results in a finitenumber of states. It has been remarked that uniformdiscretization is different from, and theoretically moreappealing than, floating-point discretization [13,14].[13] F. Rannou, Astron. Astrophys. 31, 289 (1974).[14] J. H. Lowenstein and F. Vivaldi, Chaos 10, 747 (2000);Nonlinearity 11, 1321 (1998).[15] C. E. Shannon, Bell Syst. Tech. J. 27, 379 (1948); 27, 623(1948).[16] A. N. Kolmogorov, IEEE Transactions on InformationTheory (IEEE, New York, 1956), Vol. IT-2, p. 102.[17] P. Gaspard and X.-J. Wang, Phys. Rep. 235, 291 (1993),and references therein.[18] Y. E. Levy, Phys. Lett. A 88, 1 (1982).[19] J. Coste and M. He´non, in Disordered Systems andBiological Organization, edited by E. Bienestock et al.(Springer, Berlin, 1986), p. 360.[20] C. Grebogi, E. Ott, and J. A. Yorke, Phys. Rev. A 38, 3688(1988).[21] T. Berger, Rate Distortion Theory (Prentice-Hall,Englewood Cliffs, 1971).[22] Note that if p 1=T, contamination of the randomtrajectory by any individual periodic orbit of the discretesystem is unlikely.[23] In M. Cencini, M. Falcioni, E. Olbrich, H. Kanz, andA. Vulpiani, Phys. Rev. E 62, 427 (2000) a similarsituation was found: a nonchaotic continuous system,which can be regarded as an approximation, on largescales of resolution, of a chaotic system, can develop (onthe same scales) the entropic behavior of the chaotic one,by virtue of an added random noise.044101-4

Physical Review Letters

Coarse-grained probabilistic automata mimicking chaotic systems

FALCIONI, Massimo

VULPIANI, Angelo

G. MANTICA

S. PIGOLOTTI

Archivio della ricerca- Università di Roma La Sapienza

Discretization of phase space usually nullifies chaos in dynamical systems. We show that if randomness is associated with discretization dynamical chaos may survive and be indistinguishable from that of the original chaotic system, when an entropic, coarse-grained analysis is performed. Relevance of this phenomenon to the problem of quantum chaos is discussed.Peer Reviewe

Coarse-Grained Probabilistic Automata Mimicking Chaotic Systems

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