Are there any central results/theorems which concern the implication that a dynamical system which is chaotic (in the sense of a largest positive Lyapunov exponent) will exhibit normal diffusion? By 'normal diffusion' I mean linear growth in the mean-square displacement.

There is a 'theme' of reasoning on this topic which usually suggests that such an implication is true, at least under certain conditions, however I can never track down exactly what they are. The gist I suppose is that sufficient "randomness" along the trajectory of a particle leads to a vanishing probability of ballistic flights.

Perhaps the simpler question is: What is an example of a chaotic dynamical system which exhibits diffusion of the anomalous kind? I would like to avoid those intermittent chaotic systems if possible.


2 Answers 2


No, chaos does not ensure normal diffusion.

The typical example for anomalous diffusion are systems whose behavior alternates between jumps/bursts and relatively well-behaved regimes.

A prototypical case is indeed intermittency (see this introduction), but also prominent are chaotic Hamiltonian systems which are not fully chaotic, i.e., with mixed phase spaces – here the chaotic behavior is interrupted by long periods of almost-regular movement caused by the trajectories getting trapped around regular islands (as mentioned in this question and many other references).

This intermittent dynamics is often described by means of Lévy flights, which are random walks where long steps are more likely than for the Brownian motion. The step length probability distribution for these systems is then said to be fat tailed – which confirms your impression that “sufficient ‘randomness’ along the trajectory of a particle leads to a vanishing probability of ballistic flights”, only that chaos alone (being, after all, deterministic) is not necessarily sufficiently “random” to preclude fat tails. The chaotic pendulum, for instance, can be mimicked by

a stochastic model which [alternates] ballistic flights and random diffusion.

The link of anomalous diffusion with Lévy flights, its connection to other phenomena such as ergodicity breaking and the possibility of it being a transient regime for some systems is well articulated in this paper’s introduction:

Their behavior is dominated by large and rare fluctuations that are described by nonexponential decay laws, commonly referred to as Lévy statistics. Other characteristic features of these systems are ergodicity breaking, that is, non-equivalence between time averages and the corresponding ensemble averages, as well as aging, i.e. a manifest dependence of physical observable on the time span between initialisation of the system and the start of the measurement. Both ergodicity breaking and aging are in essence two sides of the same coin as they are intimately connected to non-stationarity or ultra-slow relaxation. These systems exhibit various forms of diffusion anomalies which were also demonstrated in numerous experiments. This kind of dynamics may not survive until the asymptotic long time regime, nonetheless, lately even its transient nature has been predicted theoretically and observed experimentally.

Thus, considering the examples given, chaos might not ensure normal diffusion, but strong chaos typically probably does, at least in the asymptotic regime.


There are plenty of dynamical systems which do not exhibit normal transport. I will take a physical example from fluid mixing. When mean square displacement does not grow linearly with time, it may indicate the presence of non-hyperbolic points (bad/sticky regions). Hyperbolicity implies good fluid mixing. Track particles in the unsteady fluid flow through an obstacle (this system is chaotic ) at a moderate Reynold number and calculate mean square displacement. You will notice that it will exhibit a diffusive regime after a long time. But it will pass through a non-diffusive or anomalous regime. This happens because of the presence of non-hyperbolic points on obstacles that contaminates the fluid mixing.

A diffusive regime implies perfect mixing. However, in nature, no system either natural or artificial is perfectly mixed. Even mixing cream in the coffee is anomalous (see JEAN LUC THIFFEAULT study on mixing) that's why we always see the deposition of cream around the cup boundary.

  • $\begingroup$ There are plenty of dynamical systems which do not exhibit normal transport. This was not the focus of the question, but thanks for bringing up the example of systems which may pass through anomalous regimes before reaching a normal one. As I understand, transport (in the physical sense) is usually defined in terms of asymptotic growth in the MSD, so despite this scenario being interesting, it is not quite what I am after. But as you infer, we live in a finite world and so asymptotic may be too strong a word. Could you point to a clearer reference for the study you mentioned? $\endgroup$
    – algae
    Oct 12, 2019 at 13:19
  • $\begingroup$ Please see this link journals.aps.org/prl/pdf/10.1103/PhysRevLett.99.114501. $\endgroup$
    – Curiosity
    Oct 12, 2019 at 20:00
  • $\begingroup$ I have given a possible answer to your question 'What is an example of a chaotic dynamical system that exhibits diffusion of the anomalous kind?' I just want to add that one should check other statistical properties such as probability distribution, escape rate before coming to the conclusion of anomalous or normal transport. $\endgroup$
    – Curiosity
    Oct 12, 2019 at 20:13
  • $\begingroup$ Thanks. Could you clarify this statement: "A diffusive regime implies perfect mixing"? It seems too strong. For instance in irrational polygonal billiards we have apparent normal diffusion, but it is not known whether these systems are ergodic, let alone mixing? $\endgroup$
    – algae
    Oct 15, 2019 at 23:32
  • $\begingroup$ Ah, my context was fluid flow mixing. When I say perfect mixing, I mean tracer concentrations are the same everywhere in the fluvial domain. No concentration gradient arises or persists, absence of islands and non-hyperbolic points. Or to put in other words when the tracer has had time to sample the full distribution of velocities in the fluid flow. One says it is perfectly mixed. $\endgroup$
    – Curiosity
    Oct 16, 2019 at 1:20

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