One of the primary goals of mathematical physics (and more broadly areas of applied mathematics relevant for biology or even the social sciences) is to understand how the fundamental microscopic processes of the underlying physical (or biological or social) systems manifest themselves in the complex macroscopic behaviors of such systems that we see in the laboratory or even in the world around us.

Classical thermodynamics or equilibrium statistical mechanics does a wonderful job of describing how macroscopic stationary states, such as the solid, liquid and vapor states of water, arise from the underlying molecular microstructure, depending on extensive parameters.

However many important physical processes take place outside of equilibrium. Physical examples of this include situations where there is a dynamic flow of particles or heat such as in chemical reactions or in the formation of defects, condensates and singularities. Most biological or social processes by their nature are non-equilibrium. At this time there is no general formalism for non-equilibrium statistical mechanics. Hence thinking about concrete non-equilibrium models, even those that are somewhat simplified, is of value.

The projects for this award were focused in three concrete areas.

The first of these concerned models of aggregate formation that can be applied, for instance, to systems of interacting Bose particles, shaken granular gasses or infinite allele models in biology that violate standard principles of equilibrium statistical mechanics. The PI and his collaborators have given a complete classification of models of this type, known as random permutations with general cycle weights, as well as a description of their asymptotic behavior as the size of the permutations grow. In addition they have extended this classification and description to heavy–tailed distributions. A heavy tailed distribution is one for which the tail decay of the distribution is sub-exponential. Such distributions have attracted a greater popular interest recently due to recent financial upheavals and statistical observations such as, ”the top 1% of a population owns 40 % of its collective wealth”, which seem to lie outside predictions based on the central limit theorem.   

The second project concerned growth models of stochastic networks in random environments. These are models applicable to neural networks as well as models of the internet. They also arise in current models of two-dimensional quantum gravity. The analysis of these models is mediated by the systematic study of enumeration problems. So, for instance, the statistical analysis of the distribution of nodal valences (how many edges emerge from each vertex) in a complex network, such as the internet, is grounded in the systematic enumeration of trees of a given number of vertices with binary, ternary, or more generally n-ary branching. The latter typically comes down to solving recurrence relations (difference equations) whose solutions may be encoded by generating functions which are, prima facie, power series whose coefficients are non-negative integers giving the possible sizes of a combinatorial class (for instance, the number of binary trees with n internal (branching) vertices). Here, too, the PI, his collaborators and his students have been able to work out complete closed form formulas for many of these generating functions and, moreover, show their relation to a deeper algebraic and analytical structure known as a quantum group that facilitates the asymptotic analysis.

The third project concerns models of defect condensation in pattern forming systems driven far from threshold. Examples of such models are the Rayleigh-Benard equations a fundamental model relevant to the study of convection patterns in atmospheres and oceans. The PI and his collaborators have numerically identified a fundamental energetic mechanism for macroscopic defect formation from the microscopic equations related to the transition from chevron convection roll patterns to concave-convex disclination pairs. They are currently working on a rigorous mathematical description of this transition.

Last Modified: 12/11/2016
Modified by: Nicholas M Ercolani