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FIG. 2: Mean square displacement (MSD) and waiting time distribution (WTD) for randomized deterministic diffusion. The two deterministic dynamical systems that are randomly sampled in time with probability p by applying the recipe of Fig. 1 are illustrated in the inset of (a). All symbols FIG. 1: Diffusion generated by a random dynamical system. 2 The three time series in space-time plots display the position are generated from computer simulations. (a) MSD hxt i for p = 0.6, 0.63, 0.663, 2/3, 0.669, 0.68, 0.7 (top to bottom) for an xt ∈ R of a point particle at discrete time t ∈ N. The trajec- tory in the upper left is generated by the equation of motion ensemble of particles, where each particle experiences a dif- ferent random sequence. There is a characteristic transition xt+1 = D(xt) using a deterministic dynamical system D that yields normal diffusion. The trajectory in the upper right is between normal diffusion and localization via subdiffusion at a critical pc = 2/3. (b) MSD at pc by starting the compu- from xt+1 = L(xt) for a deterministic dynamical system L 2 3 4 where all particles localize in space. The random dynamical tations after different ageing times ta = 0, 10 , 10 , 10 (top system R mixes these two types of dynamics at time t based to bottom). The MSD displays ageing similar to analytical results from continuous time random walk (CTRW) theory on flipping a biased coin: The position xt+1 of the particle at the next time t+1 is determined by choosing with probability [13] (bold lines). (c) WTD η(t) at pc for particles leaving a 1−p the diffusive system D while L is picked with probability unit interval at the same ageing times ta as in (b). The bold lines are again analytical results from CTRW theory [13]. (d) p. The trajectory generated by xt+1 = R(xt) displays inter- mittency, where long regular phases alternate randomly with MSD at the critical probability pc for different types of aver- irregular-looking, chaotic motion. aging over the random variable. For the straight black line with matching symbols each particle experiences a different random sequence (called uncommon noise), cf. Fig. 2(a). The other four lines depict MSDs obtained from applying the same from D = M4(x), where the dynamics is chaotic accord- sequence of random variables to all particles (called common ing to λ(4) = ln 4 > 0. The trajectory in the upper right noise). In these four cases the MSD becomes a random vari- of Fig. 1 corresponds to L = M1/2(x), where the dynam- able breaking self-averaging. ics is non-chaotic due to λ(1/2) = ln 2 < 0. Here all particles contract onto stable fixed points− at integer posi- tions x Z. For defining the random map R the slope a with λ(p) < 0 should lead to localization for long times. ∈ becomes an independent and identically distributed, mul- In Fig. 2 we test this conjecture by comparing numeri- tiplicative random variable: At any time step t we choose cal with analytical results. For our simulations we used 5 5 for our map R = Ma(x) with probability p [0, 1] the 10 iterations of R with 10 initial points, which ∈ slope a =1/2 while with probability 1 p we pick a = 4. ∼were distributed randomly and∼ uniformly in the unit in- − The sequence of random slopes may or may not depend terval [0, 1). Here each particle experienced a different se- on the individual particle if we consider an ensemble of quence of random slopes. We used an arbitrary precision them [40], as we explore below. Random maps of this algorithm with up to 1010000 decimal digits. Figure 2(a) 2 type are also called iterated function systems [41, 42]. depicts the MSD xt under variation of p by confirm- They have been studied by both mathematicians and ing the diffusion scenarioh i conjectured above. However, physicists in view of their measure-theoretic [42–44] and passing through pc the dynamics displays a subtle tran- statistical physical properties [21, 40, 45, 46]. sition: Right at pc we obtain long-time subdiffusion, 2 1/2 One can show straightforwardly that the Lyapunov ex- x (t) t , while around pc this dynamics survives ponent λ(p) of the random map R is zero at probability hfor longi ∼ transient times. Figures 2(b) and (c) reveal that pc =2/3 [30]. Since λ(p) > 0 for p
FIG. 3: Subdiffusion for different types of randomness. (a) Sample trajectories at pc corresponding to 30 different initial conditions with uncommon noise. (b) Same as (a) with common noise. The envelopes in (a) and (b) correspond to the subdiffusive spreading for uncommon noise shown in Fig. 2(d). (b) displays jump time synchronization of all particles. The inset in (a) yields in double-logarithmic plot the infinite invariant density within a unit cell for uncommon noise for the map R mod 1. the probability distribution η(t) of the times t it takes a not converge for common noise as seen in Fig. 2(d). The particle to escape from a unit interval of R. In both (b) inset in (a) represents the invariant density of the map and (c) there is good agreement with analytical results R mod 1, i.e., within a unit cell, with uncommon noise. from continuous time random walk (CTRW) theory for We see that it decays on average like ρ(x) x−1. This long times, x2 (t+t )α tα and η(t) tα/[(t+t )tα], result and the stepwise structure of ρ(x) are in∼ agreement h t i∼ a − a ∼ a a where ta is the ageing time [13]. This theory furthermore with analytical calculations [43, 44]. At zero Lyapunov predicts that for long times a WTD of η(t) t−γ implies exponent uncommon noise thus leads to a weak spatial a MSD of x2(t) tγ−1 [11–15]. For R this∼ relation is clustering [22] and path coalescence [23] of particles at fulfilled withh γ =i 3∼/2. An exponent of the WTD of 3/2 integer positions x Z. In contrast, for common noise yields a diverging mean waiting time. This as well as the an invariant density∈ does not exist, and we do not find existence of ageing imply weak ergodicity breaking of the any spatial clustering. dynamics [47–49]. We now explore the origin of this type of anomalous However, our map R generates dynamics that goes be- dynamics in terms of dynamical systems theory. As yond conventional CTRW theory. This becomes appar- exemplified by the trajectory shown at the bottom of ent by looking at different types of averaging over the ran- Fig. 1, around pc the dynamics of R is intermittent [52] dom variables shown in Fig. 2(d): While in Fig. 2(a)-(c) meaning long regular phases alternate randomly with each particle experienced a different sequence of random irregular-looking, chaotic motion. A paradigmatic in- slopes, as reproduced by the straight line with matching termittent dynamical system is the Pomeau-Manneville symbols for the MSD in Fig. 2(d), for all the other MSDs map P (x)= x + bxz , b 1 , x [ 1/2, 1/2). Defining z,b ≥ ∈ − in (d) the corresponding random sequences are the same its equation of motion in the same way as for Ma above for all particles. Accordingly, we call the former type of it generates subdiffusion characterized by a MSD and a randomness uncommon noise, the latter common noise. WTD that in suitable scaling limits match to the predic- Crucially, while in Fig. 2(a), based on uncommon noise, tions of CTRW theory with γ = z/(z 1) [11, 12, 15]. the MSD is well-defined for all p, Fig. 2(d) shows that for As shown above, for the map R CTRW− theory correctly common noise sequences it becomes a random variable at predicts the relation between the long-time MSD and the pc in the long time limit that completely depends on the WTD by using γ = 3/2. Trying to understand the ran- random squence chosen. This bears strong similarity to dom map R in terms of Pz,b thus suggests to choose z = 3. what is called breaking of self-averaging for random walks One should now compare the invariant densities ρ(x) of in quenched disordered environments [50], which also im- the two maps mod 1: For Pz,b mod 1 it is known that plies weak ergodicity breaking [51]. ρ(x) x1−z , x 0, which for z 2 becomes a non- Figure 3 displays space-time plots of 30 trajectories normalizable,∼ infinite≫ invariant density≥ [53, 54]. But for starting at different initial points for (a) uncommon noise z = 3 this yields ρ(x) x−2 while for R mod 1 we have and (b) common noise. While in (a) the different tra- ρ(x) x−1, see the inset∼ of Fig. 3(b) [56]. Hence, the in- jectories look rather irregular yielding a spreading front termittency∼ displayed by R is not of Pomeau-Manneville that matches to the subdiffusion depicted in Fig. 2(d) type but of a fundamentally different microscopic dy- for uncommon noise, (b) shows ‘temporal clustering’ in namical origin. This might relate to deviations between the form of jump time synchronization, i.e., all particles CTRW theory, which on a coarse-grained level works well eventually jump from unit cell to unit cell at the same for the Pomeau-Manneville map, and our numerical re- time step. This matches to the fact that the MSD does sults for R on short time scales in the MSD and the WTD 4 of Fig. 2. It would be interesting to further explore such physically more realistic continuous noise distributions differences, e.g., by the approach outlined in Ref. [55]. choosing 1. uniform noise on a bounded interval, and 2. However, there is another type of intermittency in a non-uniform unbounded log-normal distribution. Fig- dynamical systems that is profoundly different from ures 1 and 2 in Secs. 2 and 3 of [30], respectively, show Pomeau-Manneville dynamics, called on-off intermit- that our mechanism is very robust under variation of the tency [57–62]. It was first reported for two-dimensional type of noise. We may thus conjecture that our scenario coupled maps of subdiffusion generated by random maps holds for any generic type of noise. We also tested whether the strong x +1 = (1 ǫ)f(x )+ ǫf(y ) t − t t localisation due to contraction onto a stable fixed point y +1 = (1 ǫ)f(y )+ ǫf(x ) , (2) t − t t can be replaced by a weaker chaotic localisation to a sub- region in phase space. However, in this case the transi- where xt+1 = f(xt) is chaotic with positive Lyapunov exponent and ǫ [0, 1] [57, 58]. When ǫ is large, the pos- tion between diffusive and localised dynamics is entirely sibly chaotic dynamics∈ is trapped on the synchronization different without displaying any subdiffusion, cf. Fig. 3 in Sec. 4 of [30]. As a general principle, one must thus manifold xt = yt. By decreasing ǫ to a critical param- eter ǫ = ǫ∗ trajectories start to escape from this mani- mix expansion with contraction to generate anomalous fold into the full two-dimensional space. This is called dynamics. Finally, in Sec. 5 of [30] we study a simple a blowout bifurcation and the associated intermittency nonlinear map that exhibits different types of diffusion on-off intermittency [63]. In subsequent works Eqs. (2) in different parameter regions of a bifurcation scenario were boiled down to more specific two-dimensional maps generating chaotic and periodic windows. Randomizing [21, 44, 57, 59–62, 64]. The simplest ones are piecewise this map according to Fig. 1 yields again subdiffusion 2 1/2 −3/2 linear [44, 61, 64], such as [44] with a MSD of xt t and a WTD of η(t) t , cf. Fig. 4 in [30] whichh i ∼ includes Refs [11–13]. This∼ demon- axt (xt < 1 , 0 yt p) strates that the basic mechanism generating anomalous 1 ≤ ≤ xt+1 = xt (xt < 1 ,p
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1. Calculation of the Lyapunov exponent for random maps
For one-dimensional maps M(x) obeying the deterministic equation of motion xt+1 = M(xt) the local Lyapunov exponent is defined by [1–6]
t−1 1 ′ λ(x) = lim ln M (xk) , (S1) t→∞ t | | =0 kX where M ′(x) denotes the derivative of the map. By definition here the Lyapunov exponent λ depends on the initial condition x = x0 of the map, hence it is called local. Eq. (S1) represents a time average along the trajectory x0, x1,...,xk,...,xt−1 of the map. If the map is ergodic the dependence on initial conditions will disappear [4–6]. { } ′ For the piecewise linear map Ma(x) defined by Eq. (1) in the main text we have that Ma(x) = a, hence λ(a) = ln a as noted in the main text on p.1. We now apply Eq. (S1) to calculate the Lyapunov exponent for our random maps R = Ma(x) as defined on p.1 and 2 in the main text, where Ma(x) is again the piecewise linear map given by Eq. (1) in the main text. Therein the slope a becomes a random variable drawn from a probability distribution χ(ξ) , a = ξ, at any time step t, a = at. We first choose for χ dichotomic noise, where a is sampled independently and identically distributed with probability p [0, 1] from a = a and with probability 1 p from a = a . Feeding this information into Eq. (S1) we obtain ∈ loc − exp tloc texp λdich = lim ln aloc + ln aexp t→∞ t t = p ln a + (1 p) ln a , (S2) loc − exp where tloc and texp yield respective numbers of events out of a total of t time steps for which the particle experiences the contracting and thus localising, respectively the expanding map [4]. In the long time limit these fractions are identical with the corresponding sampling probabilities p and 1 p leading to our final result. Note that in our piecewise − linear map R = Ma(x) all particles are exposed to the same uniform slope a irrespective of initial conditions, hence λdich = λdich(x). From this equation we now calculate the critical sampling probability pc at which λdich = 0 by choosing aloc =1/2 and aexp = 4 as in the main text, which defines our first model. It follows 0= p ln 1/2+(1 p) ln 4 , (S3) − which is solved to pc =2/3 as stated on p.2 in the main text. By observing that dichotomic noise is defined by the probability distribution χ (ξ)= pδ(ξ a )+(1 p)δ(ξ a ) (S4) dich − loc − − exp we can rewrite Eq. (S2) more generally as
aexp+ǫ λ = dξ χ (ξ) ln ξ (S5) dich dich | | Zaloc−ǫ with 0 <ǫ 1. From the above line of arguments it follows that for an arbitrary noise distribution χ(ξ) , ξ I we ≪ ∈ can calculate the Lyapunov exponent λχ of a random dynamical system xt+1 = R(ξt, xt) with map ξx 0 x< 1 R(ξ, x)= 2 (S6) ξ(x 1)+1 1 ≤ x< 1 − 2 ≤ lifted onto the whole real line according to R(x +1)= R(x)+1 by
λ = dξ χ(ξ) ln ξ . (S7) χ | | ZI 2
Equations (S5),(S7) thus express the Lyapunov exponent of a random map R by ensemble averages over the corre- sponding noise distributions instead of time averages along trajectories, which more generally presupposes ergodicity of the dynamics [4–6]. For maps that are more complicated than our piecewise linear case Ma(x) Eq. (1) in the main text, calculating λχ will require integration over the full invariant density ρ(x, η) of the random map, which can be a very non-trivial object [7].
2. Random map with uniform distribution of slopes
We now consider a second model that is more general than the one defined by dichotomic noise. For this purpose we sample the slopes of the piecewise linear map Eq. (S6) from noise χ(ξ) uniformly distributed over an interval ξ I = [a,b] ,a> 0, that is, we choose ∈ 1 ξ χ (ξ)= . (S8) ∼ unif b a − For this model the Lyapunov exponent can be calculated from Eq. (S7) to b 1 λ = dξ ln ξ = b log b b (a log a a) . (S9) unif b a − − − Za −
103 2 10 t1 102
101
1 1/2 0 〉 10 t 10 2 t (t) x η 〈 10-1 t-3/2 10-2 100 10-3 t0 -4 2 3 4 5 10 10 10 10 10 101 102 103 104 105 t t
Supplementary Figure S1: Mean square displacement (MSD) and waiting time distribution (WTD) for the random map Eq. (S6) where the slopes are sampled from the uniform noise distribution Eq. (S8); see the inset on the left for an illustration. All 2 symbols in the plots are generated from computer simulations. (left) MSD hxt i for b = 2.5, 2.45, 2.38, 2.36384, 2.35, 2.33, 2.3 (top to bottom) for an ensemble of particles, where each particle experiences a different random sequence. In complete analogy to Fig. 2(a) of the main text there is a characteristic transition between normal diffusion and localization via subdiffusion at a critical bc ≃ 2.36384. (right) The WTD η(t) at bc ≃ 2.36384 showing a power law with exponent −3/2, cp. with Fig. 2(c) in the main text.
Choosing a =0.1 and keeping it fixed by varying the upper bound b, we obtain the critical parameter bc at which λunif =0to bc 2.36384. As for dichotomic noise we expect the resulting dynamics at bc to be subdiffusive, which is confirmed in Fig.≃ S1. We conclude that our main result of subdiffusion in a random map is robust against generalising the noise distribution from a dichotomic one to uniform noise on a bounded interval.
3. Random map with log-normal distribution of slopes
Our third model is defined by sampling the slopes in the random map Eq. (S6) from a log-normal distribution, 1 (ln ξ µ)2 ξ χlogn(ξ) = Lognormal(µ,1) = exp − . (S10) ∼ ξ√2π − 2 3
The Lyapunov exponent can again be calculated from Eq. (S7) to
∞ 1 (ln ξ µ)2 λlogn = dξ exp − ln ξ = µ. (S11) 0 ξ√2π − 2 Z For λlogn = 0 this yields trivially a critical parameter value of µc = 0 at which we expect the random map to generate subdiffusion. This is indeed confirmed in Fig. S2. We conclude that our main result of subdiffusion in a random map is also robust against generalising the noise distribution from uniform noise on a bounded interval to non-uniform noise on an unbounded interval.
103 103 102 t1 101
102 0
〉 10 2 t (t) x η
〈 -1 t1/2 10 101 10-2 t-3/2 10-3 t0 -4 2 3 4 5 10 10 10 10 10 101 102 103 104 105 t t
Supplementary Figure S2: MSD and WTD for the random map Eq. (S6) where the slopes are sampled from the log-normal distribution Eq. (S10); see the inset on the left for an illustration. All symbols in the plots are generated from computer 2 simulations. (left) MSD hxt i for µ = 0.1, 0.03, 0.01, 0, −0.07, −0.015, −0.03 = (top to bottom) for an ensemble of particles, where each particle experiences a different random sequence. In complete analogy to Fig. 2(a) of the main text there is a characteristic transition between normal diffusion and localization via subdiffusion at a critical µ = 0. (right) The WTD η(t) at µ = 0 showing a power law with exponent −3/2, cp. with Fig. 2(c) in the main text.
4. Random map with chaotic trapping
We now investigate whether the vanishing of the Lyapunov exponent exploited in the examples studied above is really necessary for having subdiffusion in our random maps. For this purpose we introduce a fourth model which is similar to our very first model Eqs. (S4),(S6) but for which we choose two slopes at which the corresponding deterministic maps are both expanding, aloc = 3/2 and aexp = 4. Note that for aloc = 3/2 the deterministic map Eq. (1) in the main text has a positive Lyapunov exponent λ(3/2) = ln 3/2, see our calculation at the beginning of Sec. 1. Nevertheless, this map does not generate any long-time diffusion, since particles cannot escape from any unit interval due to the fact that the map does not exceed the unit interval. Hence these sets are trivially decoupled by generating chaotic trapping of particles on any unit interval. Figure S3 demonstrates that for this purely chaotic model there is no transition between normal diffusion and localisation via subdiffusion. In marked contrast to all our previous models we see that for p > 0 the map always generates normal diffusion in the long time limit. Instead of subdiffusion the MSD displays longer and longer transients for shorter times at which diffusion is more and more slowed down when approaching the strictly localised dynamics at p = 1. This numerical finding is confirmed by a WTD that is exponential even very close to the localisation value at p = 1 as also shown in Fig. S3. It is in fact well known that for chaotic dynamical systems characterised by a positive Lyapunov exponent the WTD is always exponential while for non-chaotic systems exhibiting regular dynamics it decays as a power law [8, 9]. For an exact analytical calculation of the exponential WTD in an open version of the deterministic map Eq. (1) (main text) we refer to Refs. [1, 4, 6]. To explicitly calculate the WTDs for this and our other random maps, possibly along these lines, in order to quantitatively confirm our numerical results from first principles remains an interesting open problem. We thus conclude that the condition of having a zero Lyapunov 4
106
105
4 1 10 t 104 3 103 10 〉 102 2 t 2
x 1
〈 10 10 0 exp(-0.011t) 1 10 10 (t) η 10-1 -2 100 10 0 10-3 -1 t 10 10-4 2 3 4 5 10-5 10 10 10 10 102 102 102 102 102 102 102 102 102 103 t t
Supplementary Figure S3: MSD and WTD for the random map Eq. (S6) where the slopes are sampled from the dichotomic distribution Eq. (S4) with slopes aloc = 3/2 and aexp = 4; see the inset on the left for an illustration. Thus, in contrast to the map used for the corresponding results presented in Fig. 2 of the main text here the localisation is not due to contraction onto a stable fixed point but the dynamics remains chaotic on a bounded interval; see the text for further explanations. (left) 2 MSD hxt i for p = 0.0, 0.5, 0.9, 0.99, 0.995, 0.9995, 0.9999, 1.0 (top to bottom) for an ensemble of particles, where each particle experiences a different random sequence. In marked contrast to Fig. 2 of the main text, Fig. S1 and Fig. S2 above there is no characteristic transition between normal diffusion and localization via subdiffusion. (right) The WTD η(t) at p = 0.99 showing an exponential distribution, which is again in sharp contrast to the corresponding power law WTDs in the three figures mentioned above. exponent is indispensable for observing subdiffusion in these random maps. This is deeply rooted in a mechanism yielding anomalous diffusion that requires an intricate interplay between contracting and expanding dynamics mixed in time.
5. Random nonlinear map
We finally test whether the observed subdiffusion in random maps also persists in nonlinear maps. As a fifth model we thus replace the map Ma(x) Eq. (1) in the main text by the climbing sine map,
C (x)= x + c sin(2πx) , x R , (S12) c ∈ sketched in the inset of Fig. S4(b) [10]. This map can be derived from discretizing a driven nonlinear pendulum equation in time [11] thus representing a much more general class of nonlinear dynamics than piecewise linear maps. It was found that Cc displays three different regimes of diffusion under parameter variation with an exponent of the MSD of α = 0, 1 or 2 [12, 13]. These regimes correspond to different parameter regions in the map’s bifurcation diagram reproduced in Fig. S4(a): α = 1 occurs in the chaotic regions while α = 0 and 2 match to two different classes of periodic windows, where all particles either converge onto attracting localized periodic orbits, or onto ballistic ones. Figures S4(b) and (c) show numerical results for the MSD and the WTD of the climbing sine map randomized by using the scheme of Fig. 1 in the main text. That is, with probability 1 p we choose the parameter c1 =0.8 sampling − the chaotic region while with probability p we draw from the attractive periodic orbit case at c2 =0.2. Both figures display results for the critical probability pc 0.505702, where the map’s Lyapunov exponent is approximately zero. One can clearly see excellent matching of the≃ exponents t1/2 for the MSD and t−3/2 for the WTD as predicted by conventional CTRW theory, cf. Fig. 2 in the main text.∼ Hence, the basic mechanism∼ that we propose to generate anomalous diffusion due to randomization of deterministic dynamics is also robust for a generic nonlinear map. 5
(a)
4 -1 (b)10 (c) 10 10-2 3 10 10-3 10-4 2 10 -5 〉 10 2 t (t) x
η -6 〈 1 10 -3/2 10 1/2 t t 10-7 -8 100 10 10-9 10-1 10-10 101 102 103 104 105 101 102 103 104 105 t t
Supplementary Figure S4: Subdiffusion in the random climbing sine map. (a) The bifurcation diagram of the climbing sine map depicted in the inset of (b) [12, 13]. The two arrows denote the two specific parameter values c1 = 0.8 and c2 = 0.2 chosen with probability 1 − p, respectively p, to generate a random dynamical system from this map as explained in the text. (b) MSD at the critical probability pc ≃ 0.505702 where the random map’s Lyapunov exponent is approximately zero. The symbols are from simulations, the line is a fit based on CTRW theory. The MSD clearly exhibits subdiffusion with exponent 1/2. (c) The WTD for the same pc showing a power law with exponent −3/2 as predicted by CTRW theory; symbols and lines are as in (b).
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