arXiv:1808.08139v2 [cond-mat.stat-mech] 7 Nov 2018 nmlu iuinadteMsseeti a in aging effect of Moses model the and diffusion Anomalous hlp .Meyer G. Philipp ei .Bassler E. Kevin 1 2 Germany D-01187, Dresden 3 4 USA 77204, Abstract. E-mail: motneo h oe ffc nteaoaosdffso fex of scaling diffusion anomalous a the through discussed. in is effect process Moses the e the calcu of t of Moses are importance distribution are S They the effects effects. and simulations. of numerical three constitutive scaling with Noah the three confirmed of and the combination Joseph, methods a super-diffus of the to or each due sub- quantifying as generally either is have known find can we correlat model is The long-time syste what respectively. have dynamical through propert can chaotic these of diffusion increments Each a non-stationary. The be of mode and distributions iterates The map. are causes. Manneville that constitutive fundamental increments its of into aging of a-lnkIsiu ¨rPyi opee ytm,N¨oth Systeme, komplexer f¨ur Physik Max-Planck-Institut eateto hsc,Uiest fHutn oso,T 7 TX Houston, Houston, of University Physics, of Department ea etrfrSprodciiy nvriyo Houston of University Superconductivity, for Center Texas eateto ahmtc,Uiest fHutn Houston Houston, of University Mathematics, of Department [email protected] edcmoeteaoaosdffsv eairfudi mode a in found behavior diffusive anomalous the decompose We 2 1 , 3 iuh Adlakha Vidushi , , 4 2 , 3 ogrKantz Holger , e a as anomalous cause can ies izrSr 38, Str. nitzer eain ial,the Finally, relation. ae sn analytic using lated oso,TX Houston, , v eair which behavior, ive eietlsystems perimental rcs sasum a is process l e eae othe to related hen X724 USA 77204, TX , ,tePomeau- the m, aigexponents caling 24 USA 7204, os fat-tailed ions, 1 and ffects, l Anomalous diffusion and the Moses effect in a model of aging 2
1. Introduction in the Noah effect. When L > 1/2, the increment distribution has “fat tails” and anomalous scaling can According to the Central Limit Theorem, the result. The Moses exponent M quantifies the effect distribution of a process that is the sum of many of non-stationarity of the increment distributions in random increments will have a variance that grows the Moses effect. When M > 1/2 the increment linearly in time. Such processes are said to diffuse distribution widens with time, and for M < 1/2 it normally. Many experimental systems, however, are shrinks with time. The exponents J, L and M are known to diffuse anomalously. Examples include cold related to H through the scaling relation atoms in dissipative optical lattices [1, 2], motion H = J + L + M − 1 . (1) in a crowded environment such as the cytoplasm of biological cells [3, 4, 5], blinking quantum dots [6, 7, 8], If H 6=1/2 the diffusion is anomalous. If H < 1/2 the and intra-day trades in financial markets [9, 10, 11]. process is sub-diffusive, and if H > 1/2 it is super- Understanding the nature of the dynamics of these diffusive. Robust statistical methods that analyze systems that leads to anomalous diffusion is a topic ensembles of realizations of a stochastic process can of intense interest. be used to independently determine each of the four For stochastic processes that have stationary in- exponents [11]. crements, that is, increments with a time-independent Physical systems with aging behavior can have distribution, Mandelbrot [12] decomposed the nature non-stationary, time-dependent behavior and diffuse of anomalous diffusion into two root causes, or, ef- anomalously. Aging systems can also be non- fects. He recognized that it can be caused either by Markovian, having long-time increment correlations, long-time increment correlations or by increment dis- which also can contribute the anomalous behavior [13, tributions that have sufficiently fat tails, so that their 14, 15]. In this paper, we decompose the anomalous variance is infinite. He called the effect due to incre- diffusion found in a simple model of aging behavior [16, ment correlations the Joseph effect, and the effect due 17] and find that it is due to a rich combination of the to fat-tailed increment distributions the Noah effect. Joseph, Noah and Moses effects. The model process Both effects violate the premises of the Central Limit consists of increments that are generated by a nonlinear Theorem. map. Although the map is a deterministic system, it is A third way that the premises of the Central Limit intermittently chaotic, and the increments it generates Theorem can be violated is if a stochastic process has model stochastic, noise driven increments in physical non-stationary increments. Chen, et al., in keeping systems. with the biblically themed names of the other effects, The paper is organized as follows. In the next recently named this root cause of anomalous diffusion section we introduce our model system and describe the Moses effect [11]. With the Moses effect, the nature its anomalous diffusive behavior. Then, in the third of anomalous diffusion in processes with non-stationary section, we quantitatively decompose the anomalous increments can now be decomposed as Mandelbrot did diffusion into its root effects using analytic scaling for processes with stationary increments. Any one of arguments for each of the different constitutive effects. the Joseph, Noah or Moses effects, or a combination of In the forth section, we confirm our analytic results them, can cause anomalous diffusion. with numerical simulations. In the final section, we For self-affine processes, which have a distribution discuss our results and the importance of the Moses that scales with a power-law of time tH , where H is effect for anomalous diffusive behavior observed in what Mandelbrot called the Hurst exponent, scaling experimental systems. exponents can be defined to quantify each of the three effects that can cause anomalous scaling. The Joseph 2. Model and its Diffusive Behavior exponent J quantifies the increment correlations in the Joseph effect. When J 6= 1/2 increment correlations Consider a one-dimensional, discrete-time process Xt, exist and can cause anomalous scaling. If J > 1/2 defined for integer time t ∈ [0, ∞). The process is the the increments are positively correlated, while if J < sum of increments {δt} 1/2 they are anti-correlated. The latent exponent L t−1 quantifies the effect of increment distribution fat-tails Xt = δs ; X0 = 0 (2) Xs=0 Anomalous diffusion and the Moses effect in a model of aging 3
0 10 the unstable fixed point at δ = 0. The dynamics can 3 be approximated by a continuous differential equation -1 t = 10 10 4 t = 10 dδ 5 = δ (2δ)z−1 when δ → 0. (6) -2 t = 10 6 dt 10 t = 10 Once δ reaches the outer part |δ| > 0.5, the dynamics -3 (u)
* 10
P is chaotic until a “reinjection” into the regular region
-4 occurs. Hence, this type of system is intermittently 10 chaotic. Generally the chaotic motion is very short,
-5 thus it is sufficient to consider only the waiting periods. 10 Integrating Eq. (6) and assuming a uniform -6 distribution for reinjected δ, the distribution of waiting 10 -5 0 5 u times τ can be calculated [16, 21] z 1−z Figure 1. Scaling function of the distribution of the process X. P (τ) ∝ τ . (7) H The scaling parameter is u = Xt/t . The function is shown at For z > 2 the mean waiting time diverges, which gives four different times t, spanning 3 decades from 103 to 106. Here the PM map parameter is z = 2.5 and the Hurst exponent is rise to all the effects we discuss in this article. During H = 0.38. Lines connecting the data points are shown as guides each waiting period the process Xt performs a “jump” to the eye. due to many small steps δ in the same direction. We define the displacement χ during jumps that start at renewal (reinjection) times t = t as that are iterates of the modified Pomeau-Manneville r
(PM) map [18] χ = Xtr+τ − Xtr . (8)
−4δt + 3 if 0.5 <δt ≤ 1.0 It can be calculated from Eqs. (6,7) to be [22] −1 z 2−z 2−z δt+1 = δt 1+ |2δt| if |δt|≤ 0.5 . (3) 1 1−z 1 1−z χ ∼ (z − 1) τ , (9) −4δt − 3 if − 1 ≤ δt < −0.5 2 z − 2 thus giving a relation between χ and the waiting times. This map has been studied extensively in the past. The joint probability density function (PDF) for χ and It has been linked to anomalous diffusion [16] aging τ then is [17] and weak ergodicity breaking [19]. The initial z 1 2−z 2−z increment δ0 is chosen randomly from a uniform P (χ, τ) ∝ τ 1−z δ(χ − τ 1−z )+ δ(χ + τ 1−z ) , (10) distribution in the interval [−1, 1]. 2 h i The distribution of the process P (Xt) scales with where δ here is the Dirac δ-function. Eq. (10) is well time known in stochastic renewal processes [23]. From here the value of the Hurst exponent can be P (X )= tH P ∗ X /tH (4) t t calculated from the scaling of the ensemble averaged ∗ 2 where H is the Hurst exponent and P is the scaling mean squared displacement of the process hXt i ∝ function shown in Fig. 1. In the figure, the scaling t2H . As for continuous time random walks [24], the functions converge for large t, but corrections to scaling calculation can be performed in Fourier-Laplace space. are noticeable at smaller t. To measure H empirically, There the mean squared displacement is equal to the one can simply measure the scaling of the width of second derivative of the spatial distribution of the distribution process. Results are known for processes of with PDFs H of the form of Eq. (10) for Levy flights [24] and Levy w[Xt] ∼ t , (5) walks [25]. The results in both cases are the same which can be defined as, say, the distance between the and therefore independent of the exact path the system 75th quantile and the 25th quantile of P (Xt), or, as takes during each waiting time. They thus apply to our 2 the standard deviation of the distribution hXt i, if it system as well [22, 26, 20] is finite. Here hX i = 0 for all t. p t 0.5 if z < 2 The value of H can be calculated from the theory H = 0.5/(z − 1) if 2
3. Joseph, Noah and Moses effects distinguished it from the H defined in Eq. 5 [12]. Of course, in processes with no Noah or Moses effects, J 3.1. Definitions and H become equivalent. Anomalous diffusion can be decomposed into effects that are root causes for the violation of the premises 3.2. Values in the Pomeau-Manneville map of the CLT. In order for the CLT to hold for a In order to calculate the values of the scaling exponents process Xt that is the sum of random increments for the PM map, we utilize the concept of infinite {δt}, the increments must: (1) be independent, (2) invariant densities in infinite ergodic theory. Such have a distribution with a finite variance, and (3) be densities have well defined shape that scale with time. identically distributed. Violation of these premises The increment density P (δt) is a good example of this. is referred as the Joseph, Noah, and Moses effects, It does not satisfy the central limit theorem. However, respectively. Each of these constitutive effects, or a it does satisfy two other limit theorems. combination of them, can cause anomalous diffusion. First, it can be shown [31], that there exists an For self-affine processes, they can be quantified by 1−z infinite invariant density Pinf (|δ|) ∝ |δ| that is scaling exponents, which are related to each other and related to the actual density by the Hurst exponent by Eq. 1. These exponents are 2−z z−1 defined as follows. P (|δt|) ∼ t Pinf (|δ|) (19) To define the exponents, first define the following for |δt| not close to zero. The density P (|δt|) must be random variables: the sum of the absolute values of normalizable since it is a physical density. Therefore, increments it is truncated for small values of δ. For longer aging t−1 times t, the location of the cutoff of the density moves Yt = |δs|, (12) closer and closer towards zero. Thus, the infinite Xs=0 invariant density in the limit of large t has a well and the sum of increment squares defined power law shape, but is not integrable. t−1 There is a second way to obtain an invariant 2 Zt = δs . (13) expression related to the increment density. It was Xs=0 derived by Dynkin [32] in the context of renewal theory, Then, the Moses exponent M and the Latent exponent while Thaler [33] established the connection to the L, which quantify the Moses effect and Noah effect, underlying transformations. The application to the respectively, are defined by the scaling of ensemble- PM map was shown in [34]. It can be shown that by averaged median of these variables transforming the increment δ according to 1 M+1/2 z−1 m[Yt] ∼ t (14) γ =2|δ| (t(z − 1)) , (20) 2L+2M−1 m[Zt] ∼ t , (15) one obtains an invariant distribution or, similarly, of their means, if they are finite. z − 1 π 1 P (γ)= sin (21) The Joseph exponent J, which quantifies the π z − 1 1+ γz−1 Joseph effect, is defined by the scaling of the ensemble- for γ. averaged rescaled range statistic (R/S) [27] Both of these limit theorems are necessary for J E[Rt/St] ∼ t , (16) understanding the scaling of Zt. It is calculated by t t 1 where Rt is the range of the process 2 2 hZti = hδs i = d|δs| δs P (|δs|). (22) s s Z0 Rt = max Xs − Xt − min Xs − Xt (17) Xs=1 Xs=1 1≤s≤t t 1≤s≤t t h i h i The sum just adds a ’+1’ to the exponent describing and St is the standard deviation of increments up to 2 the scaling behavior of hδs i, which was calculated in time t [22]. For z < 2 the scaling is trivial, because P (δ) is 2 2 1 1 stationary and integrable. For 2 1.0 4 10 H M w[X] 0.8 L m[Y] 3 J 10 m[Z] E[R/S] 0.6 2 10 0.4 1 10 0.2 0 10 0.0 2 3 4 0 1 2 3 4 5 6 z 10 10 10 10 10 10 10 t Figure 3. Values of the scaling exponents quantifying diffusion and the constitutive effects that cause anomalous diffusion in Figure 2. Log-log plot of the width of X, median of Y and the PM map as a function of parameter z. Circles show the of Z, and mean of R/S as a function of time t. The scaling values resulting from numerical simulation and the solid lines of these functions, quantified by the exponents corresponding to 1 the analytic predictions. As z →∞, H → 1 and M → . the slope of the function at large t, determine the exponents 2 quantifying diffusion and the constitutive effects that cause anomalous diffusion. Here the PM map parameter is z = 2.5. Lines connecting the data points are shown as guides to the eye. At the end of this section we want to add a short remark about the parameter J. Throughout this section followed a similar path of reasoning as in Similar methods can be used to find the scaling [22], where the ensemble averaged time averaged mean behavior of Yt. Again, we use the ensemble average squared displacement (EATAMSD) of the PM map was t t 1 shown to be hYti = h|δs|i = d|δs| |δs|P (|δs|). (24) t−∆ 2 Z0 (X +∆ − X ) Xs=1 Xs=1 X2 ≡ s s ∝ tβ∆2H−β , (29) t − ∆ Here the expression is integrable with respect to the D E Xs=0 infinite invariant density for z < 3 and with respect to for z > 2.5. Here t is the total measurement time. the Thaler-Dynkin limit theorem for z > 3. Therefore, This is the result of the scale-invariant Green-Kubo hYti scales as relation for time averaged diffusivity. The scaling of the t if z < 2 EATAMSD is the difference between the scaling of the 1 z−1 ensemble averaged mean squared displacement (= 2H) hYti∝ t if 2 5. Discussion 0.7 H (theory) H (fit) J+L+M-1 (fit) The root causes of anomalous diffusion can be 0.6 decomposed into the Joseph, Noah and Moses effects. Nonstationary increments, such as what occurs in aging processes, can cause anomalous diffusion through 0.5 the Moses effect. Previous studies have found that the anomalous diffusion in intra-day financial market 0.4 prices is solely due to the Moses effect [37, 9, 38, 39, 10, 11]. Here, however, we have found that the anomalous 0.3 diffusion in a simple model of aging is due to a complex 2 3 4 z combination of the Joseph, Noah and Moses effects. It is perhaps surprising that a process consisting Figure 4. Verification of the scaling relation (Eq. 1) for the of increments that are iterates of the PM map can exponents quantifying diffusion in the PM map as a function of parameter z. have such a rich set of behaviors. The PM map is, after all, a deterministic map from the interval [−1, 1] onto itself. The value of increments, thus, is bounded. the symbol size. The scaling exponents describe the The dynamics of the map, however, consists of an asymptotic, large t scaling behavior of these functions. intermittent mixture of regular and chaotic motion, We fit each of the four functions over the two- which can produce anomalous diffusion. For z ≤ decade range from t = 104 to 106 to the form 2, nothing surprising happens. The diffusion of the f(t)= atΩ (30) system is normal, the increments are stationary, and there is no Joseph, Noah or Moses effect. But for z > 2, where a and Ω are fitting parameters, and Ω is the the system ages with time, because the mean waiting asymptotic scaling exponent. Finding Ω for each time in the regular region diverges. This causes the function, we then determined the values of H, M, L increment distribution to become time-dependent and and J using Eqs. 5, 14, 15 and 16. The result of a sub-diffusive Moses effect to occur. the fitting for each for each exponent as a function Remarkably, a Noah effect occurs despite the of z is shown by the filled circles in Fig. 3. The fact that the tails of the increment distribution are analytic predictions are shown by the solid lines in the bounded. However, for z > 2, where the increment figure. Fig. 4 confirms the scaling relation between distribution is time-dependent, rescaling of δ using the exponents, Eq. 1. It shows the theoretically Eq. (20) results in a stationary increment distribution predicted value of H, versus the fitted value of H from that has fat tails! The second moment of this simulations and the result of the fitting for J+L+M −1 stationary distribution Eq. (21) diverges for 2 invariant green-kubo relation for time-averaged diffusiv- ity. Phys. Rev. E, 96:062122, Dec 2017. [23] Ralf Metzler and Joseph Klafter. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. 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