Complex systems with many interacting elements can exhibit behavior that is far different from the behavior of any single element in the system. Examples are numerous: cognitive function in the brain arises from the integrated activity of billions of neurons; group behaviors such as human traffic, swarming locusts, and flocking birds can exhibit cohesive collective phenomena that arise from local interactions between member of the group; even geological processes, such as earthquake rupture or avalanche nucleation, can arise from local interactions between individual particles.
One of our broad goals is to understand how such macroscopic phenomena can arise from properties of individual elements and local interactions between elements. This requires bridging phenomena across both spatial and temporal scales. Rather than resolving all possible complexities in the system, we isolate the mechanisms most important for capturing behavior at a given scale, and we use these to predict and constrain behavior at subsequently larger and smaller scales.
Because collective systems often exhibit behavior that is ordered over long scales, many models are designed to capture this order at a local level. Yet local disorder can play an important role. In particular, local disorder can lead to instabilities that, when propagated across members in in a population, give rise to macroscopic changes in system behavior.
Such instabilities are known to arise in sheared granular materials. Granular zones form at the interface of earthquake faults, where shearing events produce a layer of finely-ground rock (fault gouge). The individual grains within this material are tiny in comparison to a typical earthquake fault, yet their microscopic behavior shapes the initiation, velocity, and energetics of large-scale earthquake rupture.
At the granular level, individual grains move and rearrange in response to external shear forces. These deformations can be elastic, or inelastic (such as nearest-neighbor switching). In the latter case, inelastic deformation produces local regions of the material with higher configurational disorder. As the material is sheared, these disordered regions can propagate through the material, trigger additional disorder, or diffuse out of the material. The density of disordered regions can be described by an effective temperature that quantifies the entropy in configurational, rather than kinetic, degree of freedom. Just like a thermal temperature, this effective temperature can heat up, cool down, and diffuse through a material. This description lends itself to a host of tools from statistical mechanics and thermodynamics for assessing the microscopic and macroscopic properties of a disordered system in terms of this effective temperature.
In previous work, we applied these tools to the study of energy dissipation during earthquake rupture. In this context, significant energy is dissipated to local grain rearrangements, which ultimately lead to the formation of a shear band (a thin layer of material in which the shear stress is significantly reduced). Shear-band formation in turn leads to an abrupt increase in shear velocity, a characteristic of earthquake rupture.
Our previous work showed that local disorder can lead to instabilities that change the macroscopic behavior of an (inactive) system. What role does disorder play in active collective systems, such as flocks or swarms? Variations in individual behavior can lead to local spatial disorder among individuals. If this disorder can propagate from the local to the system level, it could give rise to macroscopic behaviors such as collective directional changes. We are currently investigating these ideas using theories from disordered materials in combination with multi-agent simulations.