Crackling noise

Crackling noise arises when a system is subject to an external force and it responds via events that appear very similar at many different scales, in a classical system there are usually two states i.e. on and off, however, sometimes a state can exist in between. There are three main categories such noise can be sorted into, the first is popping where very similar magnitude events occur continuously and randomly, e.g. popcorn. The second is snapping where there is little change in the system until a critical threshold is reached then the whole system flips from one state to another, i.e. all off to all on. The final is crackling which is a composition of popping and snapping, where there are some small and some large events with a relation law predicting their occurrences, this is referred to as universality, this can be observed in many natural phenomena, e.g. crumpling paper, fire, earthquakes and the magnetisation of magnets.

Much like real life, some of these systems are reversible such as demagnetisation by heating a magnet to its Curie temperature while others are irreversible such as an avalanche where the snow can only move down a mountain, but many systems have a positive bias causing it to eventually move from one state to another such as gravity or another external force.

Theory

Barkhausen noise

Research into the study of small perturbations within a large domains began in the late 1910s when Heinrich Barkhausen investigated how the domains, or dipoles, within a ferromagnetic material changed under the influence of an external electric field. When demagnetised, a magnet’s dipoles are pointing in random directions hence the net magnetic force from all the dipoles will be zero. By applying an external electric field by coiling an iron bar with wire and passing a current through it we can create a force perpendicular to the coil (Fleming’s right hand rule for a coil), this causes the dipoles within the magnet to align to the external field.

Contrary to what was thought at the time that these domains flip continuously one by one, Barkhausen found that clusters of domains flipped in small discrete steps. By coiling a secondary coil around the bar connected to a speaker or detector, when a cluster of domains change alignment a change in flux occurs, this disrupts the current in the secondary coil and hence causes a signal output. When played out loud, this is referred to as Barkhausen noise, the magnetisation of the magnet increases in discrete steps as a function of the flux density.

Gutenberg–Richter law

Further research into crackling noise was done in the late 1940s by Charles Francis Richter and Beno Gutenberg who examined earthquakes analytically. Before the invention of the well-known Richter scale, the Mercalli intensity scale was used; this is a subjective measurement of how damaging an earthquake was to property, i.e. II would be small vibrations and objects moving, while XII would be wide spread destruction of all buildings. The Richter scale is a logarithmic scale which measures the energy and amplitude of vibrations dissipated from the epicentre of the earthquake, i.e. a 7.0 earthquake is 10 times more powerful than a 6.0 earthquake. Together with Gutenberg, they went on to discover the Gutenberg–Richter law which is a probability distribution relationship between the magnitude of an earthquake and its probability of occurrence. It states that small earthquakes happen much more frequently and larger earthquakes occur very rarely.

$N=10^{a-bM}$

Gutenberg–Richter law shows an inverse power relation between the number of earthquakes occurring N and its magnitude M with a proportionality constant b and intercept a.

Simulation

To truly simulate such an environment, one would need a continuous infinite 3D system, however due to computational limitations a 2D cellular automata was used which consisted of a 1000x1000 matrix. Each cell stores two pieces of information, the force applied to the cell and the state it is in, the state is an integer value of either +1 (on) or -1 (off) and the force applied is a continuous quantity. The net force is composed of three components, see Equation 2, which can correspond to physical attributes of any crackling noise system; the first is an external force field that increases with time H(t). The second component is a force that is dependent on the state of its neighbouring cells (ΣS) and the third is a random component (h).