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Laplace's Deterministic Paradise Lost ? “We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements Laplace’s deterministic paradise lost of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.” Miguel Angel Fuentes Santa Fe Institute, USA Pierre-Simon Laplace, ~1800 Instituto Balseiro and CONICET, Argentina Santa Fe Institute Santa Fe Institute Comments: Heisenberg uncertainty principle In the framework of quantum mechanics -one of the most successful created theories- is not possible to know -at the Einstein was very unhappy about this apparent randomness same time- the position and velocity of a given object. (~1927) in nature. His views were summed up in his famous phrase, 'God does not play dice.' ∆v ∆x m > h v v v (from Stephen Hawking’s lecture) ? Santa Fe Institute Santa Fe Institute In chaotic (deterministic) systems Same functionality (SAME ORGANISM) x(t) Not chaotic trajectory x(0) = c ∆x ± t ? Some expected initial condition + Mutations Santa Fe Institute Santa Fe Institute Same functionality (SAME ORGANISM) Many process can happen between regular-chaotic behavior Chaotic trajectory Some expected initial condition + Mutations Santa Fe Institute Santa Fe Institute 4.1 Parameter Dependence of the Iterates 35 4.1 Parameter Dependence of the Iterates 35 r values are densely interwr values oarevedenselyn andinterwoneovfindsen and onea sensitifinds a sensitive dependenceve dependence ononparameterparametervalues.values. We also discuss the conceptWe also discussof structuralthe conceptuniof structuralversalityuniversalityand calculateand calculatethethe invvariantariantdensitydensityof the of the logistic map at r = 4. logistic map at r = 4. Finally, in Section 4.5 we present a summary that explains the parallels between the Finally, in SectionFeigenbaum4.5 weroutepresentto chaosaandsummaryordinary equilibriumthat explainssecond-orderthephaseparallelstransitions.betweenThis the Feigenbaum route to chapterchaosendsandwithordinarya discussionequilibrium4.1of the measurableParametersecond-orderpropertiesDependenceof the Feigenbaumphaseof thetransitions.Iterrouteatesand a This 35 review of some experiments in which this route has been observed. chapter ends with a discussion of the measurable properties of the Feigenbaum route and a review of some experiments in which this route has been observed. r values are densely4.1interwParameteroven andDependenceone finds a ofsensitithe Iteratesve dependence on parameter values. We also discuss the conceptTo provide anofovstructuralerview in this section,universalitywe present andseveralcalculateresults for thethelogisticinvmapariantobtaineddensity of the logistic map at r = 4.by computer iteration of eq. (4.1) for different values of the parameter r. Figure 23 shows the 4.1 ParameteraccumulationDependencepoints of the iteratesoff nthe(x ) forIteratesn > 300 as a function of r together with the Finally, in Section 4.5 we present a summary{ r 0 } that explains the parallels between the Some process are neitherLiapuno chaoticv enorxponent regularλ obtained via eq. (3.9). TFeigenbaumo provide an oroutevervietow chaosin this section,and ordinarywe presentequilibriumseveral resultssecond-orderfor the logisticphase maptransitions.obtainedThis x (t) x (t) eλt bychaptercomputerendsiterationwith a ofdiscussioneq. (4.1) forof thedifferentmeasurablevalues ofpropertiesthe parameterof ther.FeigenbaumFigure 23 shoroutews theand a green blue accumulationreview of somepointsexperimentsof the iteratesin whichf n(xthis) routefor n >has300beenas observa functioned. of r together with the | − | ∼ { r 0 } Liapunov exponent λ obtained via eq. (3.9). λ : Lyapunov exponent, positive for chaos 4.1 Parameter Dependence of the Iterates To provide an overview in this section, we present several results for the logistic map obtained by computer iteration of eq. (4.1) for different values of the parameter r. Figure 23 shows the n x(t) accumulation points of the iterates f (x0) for n > 300 as a function of r together with the { r } 4.1 Parameter Dependence of the Iterates 35 Liapunov exponent λ obtained via eq. (3.9). r values are densely interwoven and one finds a sensitive dependence on parameter values. We also discuss the concept of structural universality and calculate the invariant density of the logistic map at r = 4. Finally, in Section 4.5 we present a summary that explains the parallels between the Feigenbaum route to chaos and ordinary equilibrium second-order phase transitions. This chapter ends with a discussion of the measurable propertiest of the Feigenbaum route and a review of some experiments in which this route has been observed. Santa Fe Institute Santa Fe Institute 4.1 Parameter Dependence of the Iterates Figure 23: a) Iterates of the logistic map, b) Liapunov exponent λ (after W. Desnizza, priv. comm.). To provide an overview in this section, we present several results for the logistic map obtained by computer iteration of eq. (4.1) for different values of the parameter r. Figure 23 shows the accumulation points of the iterates f n(x ) for n > 300 as a function of r together with the { r 0 } There are importantLiapuno theorv exponentems for randomλ obtained variablesvia eq. that(3.9). are valid for chaotic systems Until now we were discussing: Chaos and uncertainty Its connection with the idea of random processes Figure 23: a) Iterates of the logistic map, b) Liapunov exponent λ (after W. Desnizza, priv. comm.). Chaos and Lyapunov exponent I mention that some important theorem for random variables are valid for deterministic variables too. Let move to one of them (VERY important one.) Here Figure 23: a) Iterates of the logistic map, b) Liapunov exponent λ (after W. Desnizza, priv. comm.). Santa Fe Institute Santa Fe Institute Figure 23: a) Iterates of the logistic map, b) Liapunov exponent λ (after W. Desnizza, priv. comm.). Abraham de Moivre (1667) a french mathematician famous for de Moivre's formula (friend of Isaac Newton, Edmund Halley, and James Stirling, nice crowd!) Measurement n (cos x + i sin x) = cos nx + i sin nx x1 x 1.0 2 0.8 0.6 RAPID COMMUNICATIONS xn 0.4 TIRNAKLI, BECK, AND TSALLIS PHYSICAL REVIEW E 75, 040106͑R͒ ͑2007͒ 0.2 The central limit theorem states that the sum of a large prove the existence of a CLT. This means the probability 6 6 !3 !2 !1 N=2x10 ; nini=12x10 2 3 number of independent and identically-distributed random 6 distribution of N=100 ; nini=10x10 eq.(9) ; N=100 variables will be approximately normally distributed (i.e., N 100 6 following a Gaussian distribution) if the random variables 1 eq.(9) ; N=2x10 y ª ͚ f͑xi͒ ͑2͒ ͱN i=1 Normal distribution have a finite variance (~1773). a=2 System 10-1 becomes Gaussian for N ϱ, regarding the initial value x1 → d k ) as a random variable. Here f : is a suitable smooth y R →R ( function with vanishing average which projects from the y d-dimensional phase space to a k-dimensional subspace. If ρ 10-2 Santa Fe Institute d=k=1, the variance ␴2 of this Gaussian is given by ͓8͔ Santa Fe Institute ϱ 2 2 10-3 ␴ = ͗f͑x0͒ ͘ + 2͚ ͗f͑x0͒f͑xi͒͘. ͑3͒ i=1 Here denotes an expectation formed with the natural RAPID COMMUNICATIONS ͗¯͘ 10-4 N invariant density of the map T. x -4 -2 0 2 4 Example: z = 1 i TIRNAKLI,As an BECK,exampleANDtoForillustrateTSALLIS the logisticthese generalmap atresults, positivelet Lusyapunov exponent (CHAOS)PHYSICAL REVIEW E 75, 040106͑R͒ ͑2007͒ √ consider the logistic map ! N y prove the existence of a CLT. This means2 the probability N=2x106 ; n =2x106 xi+1 = T͑xi͒ = 1 − axi ͑4͒ FIG. 1. ͑Color online͒ Probabilityini density of rescaled sums of distribution of N=100 ; n =10x106 iterates of the logistic*0 mapini with a=2 as given by Eq. ͑8͒, on the interval X=͓−1,1͔. For a=2 the system is ͑semi͒con- 6 eq.(9) ; N=100 N N=2ϫ10 and100 N=100. *The5'67number7*\:(*-\*+,of-.$initial$!!!!values/ contributing to jugated to a Bernoulli shift1and strongly mixing. The natural eq.(9) ; N6=2x106 7 PN (z) the histogram is nini=2ϫ10 , respectively nini=10 . The solid lines y f͑xi͒ ͑2͒ invariant density is ª ͚ correspond to Eq. ͑9͒. &'(')*+,-.$$!!!!/0 ͱN i=1 1-234)*5'677 a=2896'(:-;)*<=<!*>*?@A?BC79,(?DEAFGGGC3H+?IF 1 10-1 J3:KL(:;K)* becomes Gaussian for ␳N͑x͒ ϱ=, regarding. the initial value͑5x͒ < M-*B3:KL(:;K x 2 $"% 1 13,14 . Our final result** is that for finite but large N the ͓ ͔ ) → ␲ͱd1 − x k NL:OFA&-P =*!"!!!F as a random variable. Here f : is a suitable smooth y probability density ofQOFy is=**!given"RRRS by R →R ( ** function with vanishing average which projects from the y Ergodic averages of arbitrary observablesP (y)A are given by <! !"!!$RT U!"!!F%R Z:(43 -2 d-dimensional phase space to a k-dimensional subspace. If ρ 10 1 −yHV2 I!"!!S!1F U!3"!!!FW 2 1 1 ␳y͑y͒ = e B 1!"+%%XW% y U!"!!−!RWy + O . ͑9͒ 2 @ !"RR#%R U!"!!X!S @H:7* ͩ 2 ͪ ͩ Nͪ d=k=1, the varianceA␴x of= this␳Gaussianx A x dx.is given by ͓8͔ 6 ͱ␲ ͫ ͱN ͬ ͗ ͑ ͒͘ ͑ ͒ ͑ ͒ Y* ͑ ͒ ͵ !"# −1 ϱ This result is in excellent agreement with numerical experi- 2 2 10-3 The average ␴͗x͘=vanishes.͗f͑x0͒ ͘ +For2͚the͗f͑correlationx0͒f͑xi͒͘. function one͑3͒ ments ͑Fig. 1͒. has i=1 We should mention at this point that the finite-N correc- tions are nonuniversal, i.e., different mappings have different Here ͗ ͘ denotes an expectation formed with the natural -4 ¯ 1 !"! finite-N corrections.10 For example, by applying the techniques invariant density of the ͗mapxi xi ͘T=.
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