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When looking into the topic of "self-organized criticality," (SOC) one often comes across descriptions of SOC as a state where "the system has no characteristic length or time scale." (Examples here and here).

So, there are systems which--when represented as a bunch of cellular automata in computer simulations--evolve into a state of SOC, at which point their "characteristic length and time scales disappear." (Did I get that much right?)

What, then, are the characteristic length and time scales of such systems before SOC is reached? In the typical sand pile model, for example, or the size distribution of earthquakes and forest fires over time.

What does it then mean for the characteristic length to vanish? Does it mean that the system becomes scale invariant? But that would imply a sort of symmetry or equilibrium (via Noether), and I see that SOC is also characterized as occurring in complex systems that are far from equilibrium.

Is "characteristic length disappearing" somehow analogous to hitting a resonant frequency and blowing up the amplitude?

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When looking into the topic of "self-organized criticality," (SOC) one often comes across descriptions of SOC as a state where "the system has no characteristic length or time scale.

SOC is the phenomenon where a system that is not in thermodynamic equilibrium tends to and reaches a sort of critical state (as a consequence of its stochastic dynamics) where analogies with systems in thermodynamic equilibrium at a critical point can be stablished.

Critical systems usually display power-laws everywhere. In particular power-law spatial and temporal correlations. At difference with exponentials, for example, power-laws disply the scale invariance property. Namely, $f(x)$ is said to be scale invariant if, for any constant $c>0$, $f(cx) \sim f(x)$.

Why the "scale-free" property is so "popular"? The "scale free" property is so "popular" because, whenever it is present, you cannot use well known and convenient tricks---like, for example, the saddle-point approximation or Gaussian approximation---to estimate different statistical quantities of your interest. This is because, to work well or at least without complications, such kind of tricks require the existence of specific scales that dominate the statistics of the problem. Moreover, in critical systems, fluctuations at different part of the systems tend to have "non-negligible correlations" that decay slowly with distance making calculations even harder.

So, there are systems which--when represented as a bunch of cellular automata in computer simulations--evolve into a state of SOC, at which point their "characteristic length and time scales disappear." (Did I get that much right?)

Typically, "SOC-systems" start their dynamics out of the critical point. As the dynamics proceed the system approaches the SOC state. That is why the name "Self-Organized Criticality".

What, then, are the characteristic length and time scales of such systems before SOC is reached? In the typical sand pile model, for example, or the size distribution of earthquakes and forest fires over time.

At the beginning, the initial state of a SOC-system is usually characterized by characteristic scales---i.e. it is characterized by non-power-law relations---because that is how the initial conditions are devised in the models (usually, taking inspiration from the empirical observations and, because we want to show that SOC is an "attractive" state). As the SOC state is approached, many of these relations slowly evolve into power-laws and the scale-invariance emerges. Typically, exponentials slowly become power-laws with exponential cut-offs and the cut-offs grow toward infinite as time evolves.

What does it then mean for the characteristic length to vanish? Does it mean that the system becomes scale invariant? But that would imply a sort of symmetry or equilibrium (via Noether), and I see that SOC is also characterized as occurring in complex systems that are far from equilibrium.

Yes, the existence of power-laws is usually related to scale-free or scale-invariant relations characterizing the system.

I do not understand what you mean by a "symmetry a la Noether".

SOC systems are not far from equilibrium, in the sense that they can be usually mapped into models that are weakly out of equilibrium.

Is "characteristic length disappearing" somehow analogous to hitting a resonant frequency and blowing up the amplitude?

There are cases in which resonance and criticality are related phenomenon but I do not know to which extent the analogy holds. Typically, stochasticity is a crucial ingredient of critical systems so, I would tend to think into stochastic resonance than just to deterministic resonance if I would search for criticality.

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In a noncritical system, a perturbation propagates in a way that can be described by a characteristic scale. Let us say, you can calculate the average size of the propagation (for instance, the distributions could be a gaussian). In a critical system this is not possible because the propagation follows a power law. Because of the shape of a power law distribution, there is no way to define am average, or any other characteristic length. Same with time.

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What does “no characteristic length or time scale” mean?

"No characteristic lenght scale" means that, if you wait long enough, you'll experience phenomena at arbitrarily large (and small) length scales (e.g., powerful earthquakes, huge avalanches). As for the time scale, I take it to mean arbitrarily long memory effects.

Often that means that the length distribution follows a power law, for which the mean is undefined (for exponents smaller than 2), as well as the variance (for exponents smaller than 3).

What does it then mean for the characteristic length to vanish? Does it mean that the system becomes scale invariant?

It doesn't vanish, it becomes undefined, but, yes, the lack of a characteristic length is what makes it be called scale invariant.

But that would imply a sort of symmetry or equilibrium (via Noether), and I see that SOC is also characterized as occurring in complex systems that are far from equilibrium.

The closest it comes to the Noether theorem is perhaps that fact that systems at criticality often belong to so-called universality classes. The system might be either equilibrium or in a stationary state.

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