Shocks Generate Crossover Behaviour In Lattice Avalanches

A spatial avalanche model is introduced, in which avalanches increase stability in the regions where they occur. Instability is driven globally by a driving process that contains shocks. The system is typically subcritical, but the shocks occasionally lift it into a near or super critical state from which it rapidly retreats due to large avalanches. These shocks leave behind a signature -- a distinct power--law crossover in the avalanche size distribution. The model is inspired by landslide field data, but the principles may be applied to any system that experiences stabilizing failures, possesses a critical point, and is subject to an ongoing process of destabilization which includes occasional dramatic destabilizing events

Crossover Behaviour In Driven Cascades

We propose a model which explains how power-law crossover behaviour can arise in a system which is capable of experiencing cascading failure. In our model the susceptibility of the system to cascades is described by a single number, the propagation power, which measures the ease with which cascades propagate. Physically, such a number could represent the density of unstable material in a system, its internal connectivity, or the mean susceptibility of its component parts to failure. We assume that the propagation power follows an upward drifting Brownian motion between cascades, and drops discontinuously each time a cascade occurs. Cascades are described by a continuous state branching process with distributional properties determined by the value of the propagation power when they occur. In common with many cascading models, pure power law behaviour is exhibited at a critical level of propagation power, and the mean cascade size diverges. This divergence constrains large systems to the subcritical region. We show that as a result, crossover behaviour appears in the cascade distribution when an average is performed over the distribution of propagation power. We are able to analytically determine the exponents before and after the crossover

Spatial evolution of human dialects

The geographical pattern of human dialects is a result of history. Here, we formulate a simple spatial model of language change which shows that the final result of this historical evolution may, to some extent, be predictable. The model shows that the boundaries of language dialect regions are controlled by a length minimizing effect analogous to surface tension, mediated by variations in population density which can induce curvature, and by the shape of coastline or similar borders. The predictability of dialect regions arises because these effects will drive many complex, randomized early states toward one of a smaller number of stable final configurations. The model is able to reproduce observations and predictions of dialectologists. These include dialect continua, isogloss bundling, fanning, the wave-like spread of dialect features from cities, and the impact of human movement on the number of dialects that an area can support. The model also provides an analytical form for S\'{e}guy's Curve giving the relationship between geographical and linguistic distance, and a generalisation of the curve to account for the presence of a population centre. A simple modification allows us to analytically characterize the variation of language use by age in an area undergoing linguistic change

The shape of a memorised random walk

We view random walks as the paths of foraging animals, perhaps searching for food or avoiding predators while forming a mental map of their surroundings. The formation of such maps requires them to memorise the locations they have visited. We model memory using a kernel, proportional to the number of locations recalled as a function of the time since they were first observed. We give exact analytic expressions relating the elongation of the memorised walk to the structure of the memory kernel, and confirm these by simulation. We find that more slowly decaying memories lead to less elongated mental maps.Comment: In the journal published version, there is a misprint in the statement of Theorem 1 p.6, one should take the limit with $c \to \infty.

The formation and arrangement of pits by a corrosive gas

When corroding or otherwise aggressive particles are incident on a surface, pits can form. For example, under certain circumstances rock surfaces that are exposed to salts can form regular tessellating patterns of pits known as "tafoni". We introduce a simple lattice model in which a gas of corrosive particles, described by a discrete convection diffusion equation, drifts onto a surface. Each gas particle has a fixed probability of being absorbed and causing damage at each contact. The surface is represented by a lattice of strength numbers which reduce after each absorbtion event, with sites being removed when their strength becomes negative. The model generates regular formations of pits, with each pit having a characteristic trapezoidal geometry determined by the particle bias, absorbtion probability and surface strength. The formation of this geometry may be understood in terms of a first order partial differential equation. By viewing pits as particle funnels, we are able to relate the gradient of pit walls to absorbtion probability and particle bias