Introduction

• Combines two interesting phenomena:

• Criticality is associated with phase-transitions:

  Sub-critical	Critical	Super Critical
  Phase I	Boundary	Phase II
  Pertubation affects local region	Pertubation can have global effect	Pertubation affects local region
  e.g. tapping glass of water
  liquid: no change to state	super-cooled: instantaneous phase transition	ice: no change to state

• What is needed to form SOC? --> Separation of time-scales

  External Driver	Internal Relaxation
  Slow	Fast (avalanche)
  e.g. build-up of stress in earth's crust (10-10000+ years)	e.g. earthquake (30s-2min)

Threshold Dynamics

(Sornette) An important sub-class of SOC is the out-of-equilibrium systems driven by constant rate, with many interacting components. They possess the following fundamental properties:

1. A highly nonlinear behaviour: essentially a threshold response;
2. A very slow driving rate;
3. A globally stationary regime, characterised by stationary statistical properties;
4. Power distributions of event sizes and fractal geometrical propoerties (including long-range correlations).

5. The threshold dynamics is crucial according to [2] since they accommodate the important separation of time-scales

6. Thus, the slipping of the earth's-crust in earth-quake events is a key example of such dynamics.

Example System: Piano

Consider a piano being moved on a carpet floor.

• stress builds up in friction between piano and floor
• eventually releases, as piano jumps
• applied force relieved, piano at rest

NB: Requires a threshold for release .. consider in the case of the piano, if it were on a plate of ice, applied force would be relieved automatically, quickly. With a threshold for activity (release) tension can build.

Sources of SOC

• Originally believed that only one kind of system was responsible for generating SOC, however, now accepted that various systems can give rise to self-organization to a state without characteristic scale;
• Sources of SOC mentioned in [2] (p.331):

  1. Systems where elements tend to synchronize, but are frustrated by constraints such as open boundary conditions and quenched disorder, leading to a dynamical regime at the edge of synchronization, the SOC state;

  2. So-called extremal models -- SOC is exhibited due to the competition between local strengthening and weaking due to interactions;

  3. SOC also results in systems where the order parameter is tuned in a system exhibiting a critical point, to a vanishingly small, but positive value, ensuring that the corresponding control parameter lies exactly at its critical value for the underlying depinning transition (?);

  4. Multiplicatively driven systems, such as the spring-block model driven by temporally increasing spring coefficients;
  5. Conservation laws are also conjectured to be essential for a class of SOC systems -- the scale invariance can sometimes be shown to arise from local conservation laws (e.g. in the sand-pile, sand-grains are conserved except at the boundaries of the pile).