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
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
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.