Chaos: a bridge from microscopic uncertainty to macroscopic randomness
aa r X i v : . [ n li n . C D ] J a n Chaos: a bridge from microscopic uncertaintyto macroscopic randomness
Shijun LiaoState Key Lab of Ocean Engineering, Dept. of MathematicsSchool of Naval Architecture, Ocean and Civil EngineeringShanghai Jiao Tong University, Shanghai 200240, China( Email address: [email protected])
Abstract
It is traditionally believed that the macroscopic randomness has nothing to do with themicro-level uncertainty. Besides, the sensitive dependence on initial condition (SDIC) of Lorenz chaoshas never been considered together with the so-called continuum-assumption of fluid (on which Lorenzequations are based), from physical and statistic viewpoints. A very fine numerical technique [6] withnegligible truncation and round-off errors, called here the “clean numerical simulation” (CNS), isapplied to investigate the propagation of the micro-level unavoidable uncertain fluctuation (causedby the continuum-assumption of fluid) of initial conditions for Lorenz equation with chaotic solutions.Our statistic analysis based on CNS computation of 10 ,
000 samples shows that, due to the SDIC, theuncertainty of the micro-level statistic fluctuation of initial conditions transfers into the macroscopicrandomness of chaos. This suggests that chaos might be a bridge from micro-level uncertainty tomacroscopic randomness, and thus would be an origin of macroscopic randomness. We reveal in thisarticle that, due to the SDIC of chaos and the inherent uncertainty of initial data, accurate long-term prediction of chaotic solution is not only impossible in mathematics but also has no physicalmeanings. This might provide us a new, different viewpoint to deepen and enrich our understandingsabout the SDIC of chaos. Key Words
Chaos, propagation of uncertainty, fine numerical simulation, multiple scales
Nowadays, it is a common belief [2, 3, 5, 12–14, 16] of scientific society that some “deterministic”dynamic systems have chaotic behaviors: their solutions are exponentially sensitive to initial condi-tions so that accurate long-term prediction of chaotic solution is impossible. Here, the deterministicmeans that the evolution of solutions is fully determined by initial conditions without random or1uncertain elements involved. Such kind of behaviors is called “deterministic chaos” [5], because “thedeterministic nature of these systems does not make them predictable” [16].Such kind of non-periodic solutions was first pointed out by Poincar´e [9] in 1880s for the famousthree-body problem. In 1962 Saltzman [11] found “oscillatory, overstable cellular motions” and “con-sequently an alternating value of the heat transport about a time-mean value” for a free convectionflow with very large Rayleigh number. It is a pity that Saltzman [11] paid main attentions on thestable solutions for Rayleigh number smaller than 10. Fortunately, this “oscillatory, overstable” non-periodic solutions of the free convection flow was further studied in details by Lorenz [7] in 1963 forthe weather prediction, governed by the so-called Lorenz equation˙ x = σ ( y − x ) , (1)˙ y = R x − y − x z, (2)˙ z = x y + b z, (3)where σ, R and b are physical parameters, the dot denotes the differentiation with respect to thetime. Although the Lorenz equation is much simpler than those used by Saltzman [11], its solutionalso becomes “oscillatory, overstable” for large Rayleigh number. Especially, using a digit computerand data in 6-digit precision, Lorenz [7] found that small changes in initial conditions leaded to greatdifference in long-term prediction, called today the “butterfly effect”. Based on the “butterfly effect”,Lorenz [7] made a correct conclusion that long-term weather prediction is impossible, although theLorenz equation is only a very simple approximation model of the exact Navier-Stokes equations.All numerical methods have the so-called truncation and round-off error, more or less. Due to theso-called “butterfly effect”, all traditional numerical simulations of chaos are mixed with such kind of“numerical noise”. Unfortunately, as pointed out by Lorenz [8] in 2006, different traditional numericalschemes may lead to not only the uncertainty in prediction but also fundamentally different regimesof the solution. Thus, the traditional numerical simulations of chaos are not “clean” so that some ofour understandings about chaos based on these impure numerical results might be questionable.In order to gain reliable chaotic solutions in a long enough time interval, Liao [6] developed afine numerical technique with extremely high precision, called here the “clean numerical simulation”(CNS). Using the computer algebra system Mathematica with the 400th-order Taylor expansion forcontinuous functions and data in accuracy of 800-digit precision, Liao [6] gained, for the first time,“clean” numerical results of chaotic solution of Lorenz equation (in a special case σ = 10 , R = 28 , b = − /
3) in a long time interval 0 ≤ t ≤ ≤ t ≤ T c , the initial conditions must be at least in the accuracy of 10 − T c / . Thus,when T c = 1000 LTU, the initial condition must be in the accuracy of 400-digit precision at least.Currently, Liao’s “clean” chaotic solution [6] of Lorenz equation is confirmed by Wang et al [15],who used parallel computation with the multiple precision (MP) library: they gained reliable chaoticsolution up to 2500 LTU by means of the 1000th-order Taylor expansion and data in 2100-digitprecision, and their result agrees well with Liao’s one [6] in 0 ≤ t ≤ physical and statistic points of view, as shown below.Since Lorenz [7] introduced the concept of SDIC of chaos, its meanings has been discussed andinvestigated in many articles and books, mostly from the viewpoints of mathematics, logic andphilosophy, but hardly from physical viewpoints. This might be mainly because most models ofchaos are too simple to accurately describe the complicated physical phenomena. So, to deepen ourunderstandings about the SDIC of chaos, it is valuable to study it from the physical viewpoints.Lorenz equation [7] was originally derived from the Navier-Stokes (N-S) equation describing phe-nomena of fluid motions. The N-S equations are based on such an assumption that the fluid is a continuum , which is infinitely divisible and not composed of particles such as atoms and molecules.Let us consider the uniform laminar flow of air with the velocity 1 (m/s) at the temperature T = 0 ◦ Cand the standard pressure. In this case, there are about 2 . × molecules in a cube of fluid. Thisis a hugh number so that the continuum-assumption of fluid is mostly satisfied in practice. Assumethat all molecules of a cube of fluid have the same velocity, except one which has a tiny velocity fluc-tuation 10 − m/s. Then, the averaged velocity fluctuation of a cube of fluid reads 3 . × − m/s.Such micro-level velocity fluctuation of fluid should be neglected under the continuum-assumption.In other words, in the frame of the continuum-assumption, it has no physical meanings to considerthe observable influence of such a tiny velocity fluctuation, from physical point of view!However, Liao’s CNS computation [6] in the accuracy of 800-digit precision indicates that, mathe-matically , to gain reliable chaotic solution in 0 ≤ T ≤ − is much smaller than3 . × − that is a minimum of the averaged velocity fluctuation of fluid! Thus, as mentionedabove, from physical point of view, such a tiny velocity fluctuation (in the level of 10 − ) has no physical meanings at all under the continuum-assumption that is a base of Lorenz equation! There-fore, a paradox arises: according to the continuum-assumption, the tiny velocity fluctuation in thelevel of 10 − should have no observable influence on the chaotic solution of Lorenz equation; on theother hand, the SDIC and “butterfly effect” indicate that the influence of a tiny velocity fluctuationeven in the level of 10 − must be considered! This is certainly a paradox in logic!In history, many paradoxes first revealed the restrictions of some well-established theories andthen greatly promoted their developments. What is the essence of this paradox from the viewpointof physics? What can we learn from it? Without loss of generality, let us consider the Lorenz equation with chaotic solution in case of R = 28 , b = − / σ = 10. Assume that the observable values of initial condition x = − / , y = − / , z = 891 / given exactly. However, due to the continuum-assumption of fluid, the initial conditions involvethe uncertainty: the statistic fluctuations of velocity and temperature are inherent and unavoidable in essence, although their absolute values are often much smaller than those of the observable valuesof initial condition. According to the central limit theorem in probability theory, we assume thatthe fluctuations of velocity and temperature are in the normal distribution with zero mean and amicro-level deviation σ , such as σ = 10 − used in this article. Thus, the entire initial conditions x (0) = x + ˜ x , y (0) = y + ˜ y and z (0) = z + ˜ z involve random, where ˜ x , ˜ y , ˜ z are random variablesin the normal distribution with zero mean and deviation σ , i.e. h ˜ x i = h ˜ y i = h ˜ z i = 0 , (cid:10) ˜ x (cid:11) = (cid:10) ˜ y (cid:11) = (cid:10) ˜ z (cid:11) = σ . For each random initial condition, the corresponding “clean” chaotic solution is gained by means ofthe CNS [6] with the 60-order Taylor expansion and data in the accuracy of 120-digit precision. Fordetails, please refer to Liao [6]. According to Liao’s work [6], both of the truncation and round-offerror are negligible in 0 ≤ t <
180 LTU. Thus, the numerical results are “clean” at least in 0 ≤ t ≤ observable influence by numerical noise. Note that, although the standarddeviation σ = 10 − of the uncertain terms ˜ x , ˜ y , ˜ z of initial condition is much smaller than theobservable values x , y , z , it is hugh compared to 10 − : the truncation and round-off errors of thenumerical simulations gained by the 60th-order Taylor formula and the data in accuracy of 120-digitprecision are much smaller than the deviation 10 − and thus are negligible in 0 ≤ t <
180 LTU. Inthis way, we can accurately investigate, for the first time , the influence of the micro-level statisticfluctuation of initial conditions to chaotic solutions, and especially the propagation of uncertaintyfrom the micro-level statistic fluctuation of initial conditions to macroscopic randomness of chaos.Let h x ( t ) i , h y ( t ) i , h z ( t ) i and σ x ( t ) , σ y ( t ) , σ z ( t ) denote the sample mean and unbiased estimate ofstandard deviation of x ( t ) , y ( t ) , z ( t ), respectively, where N = 10 is the number of samples gainedby the CNS. Define the so-called uncertainty intensity ǫ ( t ) = s [ σ x ( t )] + [ σ y ( t )] + [ σ z ( t )] h x ( t ) i + h y ( t ) i + h z ( t ) i . (4) time (Lorenz time unit) U n cer t a i n t y i n t e n s i t y deterministic stationarytransitionrandom Figure 1: The uncertainty intensity ǫ ( t ) in case of the fluctuation of initial conditions in the normaldistribution with zero mean and micro-level deviation σ = 10 − .It is found that there exists such a time interval t ∈ [0 , T d ] with T d ≈
75, in which ǫ ( t ) is so smallthat one can accurately predict the behavior of the dynamic system, but beyond which the uncertaintyintensity increases greatly, as shown in Fig. 1. Thus, using the result at any a point t ∈ (0 , T d ) asthe initial condition and setting t = − t , we can gain the given observable values x , y , z of initialconditions in a high-level of accuracy, meaning that the dynamic system looks like deterministic in 0 ≤ t ≤ T d and that the influence of the uncertain statistic fluctuation of initial condition isnegligible. But, beyond it, the solutions become rather sensitive to the uncertain statistic fluctuation(in the level of 10 − ) of initial condition and look like random , say, the micro-level uncertain statisticfluctuation in initial condition transfers into the observable macroscopic randomness. So, T d is animportant time scale for Lorenz chaos.As shown in Fig. 1, there exists the time T s with T s ≈
120 LTU, beyond which the cumulativedistribution functions (CDF) of x ( t ) , y ( t ), z ( t ) and so on are approximately stationary, i.e. almostindependent of the time. Besides, these CDFs are independent of the observable values x , y , z ofinitial conditions, meaning that all observable information of initial conditions are lost completely.In other words, when t > T d , the asymmetry of time seems to break down so that the time has a one-way direction, i.e. the arrow of time. It suggests that, statistically, the chaotic Lorenz system mighthave two completely different dynamic behaviors before and after T d : it looks like “deterministic”without time’s arrow when t ≤ T d , but thereafter rapidly becomes random with the arrow of time.This strongly suggests that chaos might be a bridge from the micro-level uncertainty to macroscopicrandomness, and thus might be an origin of macroscopic randomness and the time’s arrow. Thisprovides us a new, different viewpoint to enrich and deepen our understandings about the SDIC ofchaos.When T d < t < T s , the CDFs of x ( t ) , y ( t ) , z ( t ), their sample means and unbiased estimates ofstandard deviation are time-dependent, and evolve to the approximately stationary ones for t ≥ T s .This process is called the transition from the deterministic to randomness of chaos.Write x ′ = x − < x >, y ′ = y − < y > and z ′ = z − < z > . It is found that the CDFs of thefluctuations x ′ , y ′ , z ′ are time-dependent when t < T s and become stationary when t > T s . When t > T d , the CDF of x ′ is different from the normal distribution with the standard deviation < x ′ > ,so are the CDFs of y ′ and z ′ , as shown in Fig. 2. It is also found that T d decreases exponentiallywith respect to σ , the standard deviation of the tiny uncertain variables ˜ x , ˜ y , ˜ z of the initialconditions. Besides, the stationary CDFs of x ′ , y ′ , z ′ are independent of the CDFs of ˜ x , ˜ y , ˜ z . Inaddition, more samples are needed to gain accurate mean of the high correlations of x ′ , y ′ , z ′ , such as < x ′ z ′ >, < y ′ z ′ > and especially < x ′ z ′ z ′ >, < x ′ y ′ y ′ >, < y ′ y ′ z ′ >, < y ′ z ′ z ′ > and < x ′ y ′ z ′ > , sincethe higher correlations have the larger standard derivations: this shows the difficulty to propose anaccurate model for the mean < x >, < y >, < z > by means of these higher correlations. This alsoexplains why it is so difficult to propose a satisfied turbulence model valid for all kinds of turbulentflows, since Lorenz equation is a simplified model from Navier-Stokes equations. Note that, one candirectly obtain all of these correlations from the Lorenz equation, as long as the number of samplesare large enough. In other words, no additional models for < x >, < y >, < z > are needed.It is found that, given σ = 10 − in case of R = 28 , b = − / σ = 10, we gain exactly thesame figure as shown in Fig. 1, even if we use more accurate numerical results obtained by means ofthe CNS with the 120-order Taylor expansion and data in the accuracy of 240-digit precision! Notethat it has no physical meanings to use a micro-level deviation σ of the initial conditions smallerthan 10 − , as pointed out in the section of introduction. Thus, for chaotic dynamic systems, thetransfer from micro-level uncertainty to macroscopic randomness seems unavoidable. In addition,it is fund that, in case of b = − / , σ = 10 and R ≤ .
54 so that solutions are not chaotic, the z’ CD F -20 -15 -10 -5 0 5 10 15 2000.20.40.60.81 t = 80 t = 150 Figure 2: The CDF (solid line) of z ′ at t = 80 LTU and t = 150 LTU, compared with the corre-sponding normal distribution (dashed line) with the zero mean and the standard deviation < z ′ > .micro-level uncertainty never transfers into the macroscopic level. Therefore, the SDIC of chaos isthe key to such kind of transfer. In this article, the sensitive dependence on initial condition (SDIC) of Lorenz chaos is considered to-gether , for the first time, with the so-called continuum-assumption of fluid (on which Lorenz equationsare based) from physical and statistic viewpoints. The so-called “clean numerical simulation” (CNS)proposed by Liao [6] is used to investigate the propagation of the micro-level unavoidable uncertainfluctuation (caused by the continuum-assumption of fluid) of initial conditions with chaotic solutionsof Lorenz equation. Our statistic analysis based on the CNS computation of 10 samples suggeststhat, due to the SDIC, the uncertainty of the micro-level statistic fluctuation of initial conditionstransfers into the macroscopic randomness of chaos. This may deepen and enrich our understandingsabout the SDIC and chaos, from a different viewpoint of physics.The microscopic phenomena are essentially uncertain, although probability distributions are gov-erned by deterministic equations. However, it is traditionally believed that the micro-level uncertaintyhas no relationships with the macroscopic randomness. But, our statistic analysis strongly suggeststhat the micro-level uncertainty might be an origin of the macroscopic randomness, and chaos mightbe a bridge between them. Although the above conclusion is based on Lorenz equation, it hasgeneral meanings. First, we also investigated some other chaotic dynamic systems, and found thesame transfer from micro-level uncertainty to macroscopic randomness for all of them. Secondly, aspointed out by Saltzman [11], the solutions of a dynamic system consist of seven nonlinear differentialequations for the free convention (which is a more accurate model than Lorenz equation) are “oscilla-tory, overstable” (i.e. chaotic) for large enough Rayleigh number. In fact, Saltzman [11] representedthe solution of the original continuous differential equations as a sum of double-Fourier components,and approximated the original problem by a set of nonlinear ordinary different equations with finitenumber of degree of freedom. Both of the Lorenz equation and the above mentioned system of sevenequations are only special cases of it, corresponding to three and seven degree of freedom. Obviously,the larger the degree of freedom, the more accurate the model. It is found that, for large enoughRayleigh number, these dynamic systems given by Saltzman [11] with degree of freedom not lessthan three are chaotic, so that the micro-level uncertainty transfers into macroscopic randomness forall of them. Theoretically speaking, as the number of degree of freedom tends to infinity, this systembecomes the original continuous differential equations. Thus, our conclusion about the transfer frommicro-level uncertainty to macroscopic randomness has general meanings, although it is based on theLorenz equation. This is similar to the Lorenz’s famous conclusion “long-term accurate predictionof weather is impossible” [7], which is based on the Lorenz equation, a very simple model of the N-Sequation, but is correct and has been widely accepted by the scientific community.The similar transfer has been reported in some other fields. For example, as pointed out byBai et al [1], the disorder of materials plays a fundamental role to the so-called sample-specificbehavior of fracture, i.e. the macroscopic failure may be quite different, sample to sample, underthe same macroscopic condition, because the differentiation due to meso-scopic disorder may begreatly amplified and lead to largely different macroscopic effects. Xia et al [17] studied the failure ofdisordered materials by means of a stochastic slice sampling method with a nonlinear chain model,and found that “there is a sensitive zone in the vicinity of the boundary between the globally stable(GS) and evolution-induced catastrophic (EIC) regions in phase space, where a slight stochasticincrement in damage can trigger a radical transition from GS to EIC”. In other words, the meso-scopic uncertainty of disordered materials transfers into the macroscopic randomness of failure. Asmentioned by He et al [4], the nonlinearity and multi-scale might play a fundamental role in it. So,“(stochastic) fluctuations are important and must not be neglected” for the failure of disorderedmaterials, as pointed out by Sahimi and Arbabi [10]. Another example is the evolution of theuniverse: the micro-level uncertainty at Big Bang, the inherent uncertainty of position and velocityof stars, and the nonlinear property of gravity might be the origin of the macroscopic randomnessof the universe. All of these support our conclusion: the transfer from micro-level uncertainty tomacroscopic randomness might have meanings in general.Traditionally, it is believed that the SDIC of chaos is the origin of the so-called “butter-fly effect”:long-term prediction is impossible due to the SDIC of chaos and the impossibility of getting exact initial data with precision of arbitrary degree. This traditional idea implies that the initial data itself are exact inherently but our human-being can not obtain the exact value. However, as pointed out inthis article, this traditional thought might be wrong: due to the continuum-assumption of fluid, thereexists the statistic fluctuation of the initial data of Lorenz equation, no matter whether we couldprecisely measure the initial data or not. It should be emphasized that such kind of uncertainty is inherent : it has nothing to do with our ability. In this article, it is revealed that, due to the SDIC andthe inherent uncertainty of initial data, accurate long-term prediction of chaotic solution is not only impossible in mathematics but also has no physical meanings. This provides us a new explanationof the SDIC of chaos, from the physical and statistic points of view.The micro-level uncertainty and the physical variables x, y, z of Lorenz equation are at differentscales: the absolute value of the former (at the level of 10 − ) is much smaller than that of thelatter (at the level of 1). Unfortunately, the truncation and round-off errors (often at the level of10 − ) of most traditional numerical techniques for chaos are much larger than such kind of micro-level uncertainty, so that the propagation of the micro-level uncertainty is completely lost in thenumerical noises. The CNS [6] provides us a way to accurately investigate such kind of problemswith multiple scales, since the numerical noises of the CNS are much smaller than the micro-leveluncertainty.Lorenz equation is a simplified model based on the N-S equations describing flows of fluid. Notethat nearly all models of turbulence are deterministic in essence: the micro-level uncertain statisticfluctuation of velocity caused by the continuum-assumption of fluid has been neglected completely.Note also that the uncertainty intensity (4) is rather similar to the definition of turbulence intensity.Since turbulence has a close relationship with chaos, it might be possible that the influence of themicro-level statistic fluctuation of velocity and temperature should be considered: we even shouldcarefully check the theoretical foundation of turbulence and the direct numerical simulation (DNS),such as the continuum-assumption of fluid. Besides, our very fine numerical simulations and relatedanalysis reported in this article suggest that the randomness of turbulence might come essentiallyfrom the micro-level uncertain statistic fluctuation of velocity and temperature: turbulence is such akind of flow of fluid that it is so unstable that the micro-level uncertainty transfers into macroscopicrandomness .Hopefully, this work stimulated by a paradox could provide us some new physical insights andmathematical ways to deepen and enrich our understanding about chaos and turbulence. Acknowledgement
Thanks to the reviewers for their valuable comments and discussions. The author would like toexpress his sincere thanks to Prof. Y.L. Bai and Prof. M.F. Xia (Chinese Academy of Sciences),Prof. Z. Li (Peking University), Prof. H.R. Ma (Shanghai Jiao Tong University) for their valuablediscussions. This work is partly supported by State Key Lab of Ocean Engineering (Approval No.GKZD010053) and Natural Science Foundation of China (Approval No. 10872129).
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