This award supports theoretical research and education on fundamental issues of non-equilibrium statistical physics and their applications to basic natural processes, such as the coarsening of magnetic systems, and to phenomena driven by social forces, such as the spread of opinions, technological innovations, and fads.
The PI aims to deepen understanding of the coarsening of the Ising model. While this paradigmatic system has been studied, the PI's recent research has found, quite surprisingly, that coarsening dynamics in three dimensions does not conform to the conventional picture of power-law domain growth. Instead, the system gets stuck in a topologically complex state that consists of two highly interpenetrating domains, with the relaxation time scaling exponentially with the system size. The long-time properties of this system display a variety of puzzling features that the PI will to investigate by a combination of large-scale simulations, scaling arguments, and techniques from topology and differential geometry.
In the two-dimensional Ising model, the PI will exploit a recent connection with continuum percolation theory developed to elucidate the topology of long-time states. Here single-phase domains can organize into stripes of various winding numbers. The PI aims to calculate the probabilities of stripe state and to broadly categorize spanning domains according to the winding numbers. He will also investigate the evolution of single interfaces between oppositely-aligned spin domains in idealized geometries, such the evolution at a single corner. In two dimensions, this system can be mapped onto an exclusion process, an equivalence that the PI will use to solve the interface shape. This approach will be extended to the unexplored case of Ising systems with longer-range interactions, where surprising features appear to lurk. The connection to exclusion processes also leads the PI to explore open questions about the role of additional repulsive interactions.
The PI will also investigate prototypical social dynamics models in which reinforcement plays a decisive role. Reinforcement means that an agent requires multiple prompts from interaction partners before actually changing its state. This new attribute has a profound effect on the ensuing dynamics. For the socially-reinforced voter model, there is surprisingly fast dynamics in the mean-field limit and anomalously slow dynamics on lattice networks. The origin of the latter appears to be an effective surface tension at the interface between opposite-opinion domains.
For innovations, reinforcement delays its onset. When an innovation is abandoned at some rate, the outcome is a transient fad. The population fraction that adopts the fad undergoes a sudden transition as function of the abandonment rate that the PI will explore analytically and numerically. The PI will study how a fad spreads through a spatially dispersed population. Anomalous features arise at a "smoldering" transition that separates a regime where the fad spreads globally from a regime where a fad quickly disappears. Contact with empirical data will be made where possible.
NONTECHNICAL SUMMARY
This award supports theoretical research and education to understand systems that are far from the steady state of equilibrium. Non-equilibrium statistical physics is concerned with evolving systems that contain many interacting elements or particles. For systems in equilibrium, there exist global principles that determine their detailed physical properties. However, when a system is out of equilibrium, either by exposing it to changing external conditions or by adding or extracting energy or mass, the understanding of the evolution is still not well characterized.
A prototypical example is a magnetic system that is initially at high-temperature where it is not magnetized and shows no spatial organization of magnetism at a microscopic level, and is suddenly cooled to below the transition temperature to magnetism. From the initial disorder, competing regions of magnetic order spontaneously grow to form a coarsening mosaic of domains. The PI seeks to understand the geometry and dynamics of this domain mosaic, as well as the properties of the interface between ordered domains. Although this classic problem has been studied before, the PI's preliminary results show that the three-dimensional system contains many surprises that are not anticipated in current theories.
The PI also aims to study a range of dynamic social processes when reinforcement plays an essential role. Reinforcement means that an individual agent must experience multiple prompts from neighboring agents before actually changing its state. Reinforcement leads to very different evolutionary phenomena compared to the case where an agent changes state after a single encounter with a neighboring agent. The effect of this reinforcement mechanism will be explored for both the dynamics of opinion evolution in a population, as well as the spread of irreversible innovations and transient fads. The effect of reinforcement is crucial for fads, because this feature controls whether a fad can quickly fizzle out and barely be noticed or can become widespread before gradually disappearing.
