Integrable Structure of Interacting Particle Systems

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Statistical mechanics is that area of theoretical physics that predicts macroscopic behavior of physical systems given the microscopic dynamics. The dynamics can be either classical or quantum. Equilibrium statistical mechanics (the foundational work going back to L. Boltzmann and J.W. Gibbs) has a well-formulated, in principle, procedure to go from the microscopic dynamics to the description of the macroscopic system. Carrying out this procedure can be extremely difficult; nevertheless, the broad outlines are well understood. Most physical systems are not in equilibrium and the main interest is their long-time behavior. The Gaussian distribution (the familiar bell-shaped curve) is important because of its universality; that is, it applies to a wide variety of seemingly unrelated problems. The underlying common theme for all these problems is the fact that the objects under study have some degree of "independence"; or stated in more physical terms, are non-interacting (or weakly interacting). These types of problems are well-understood both physically and mathematically. The foundations of nonequilibrium statistical physics are not as well understood as equilibrium statistical physics, in particular when the processes involve interacting particles. This project is directed at understanding the processes which are strongly interacting; and as such, there is no satisfactory general theory as there exists for the non-interacting cases. Simple models occupy a bigger role in understanding systems far from equilibrium. The asymmetric simple exclusion process (ASEP), a model of interacting particles on a lattice, is one such model. ASEP is one of the simplest, nontrivial stochastic models in which to study transport phenomena as it models processes far from equilibrium. This project, building on earlier work of the one-point correlation, will analyze a class of higher-order correlations. The first higher-order correlations are block probabilities for step initial conditions. By a block of length L we mean a contiguous block of L particles (starting, say, at position m on the lattice). Recently the probability of this event was computed. The project will generalize this block probability result to the case of step Bernoulli initial conditions. This same result is also the starting point of the project where similar higher-order correlations will be analyzed. For example, the correlation of the first particle with the particle at position m. The methods employed are both combinatorial and analytical. In particular new combinatorial identities generalizing those found for block probabilities are an essential element of the project. The possible long term impact of this project is to discover new universal limit laws that generalize the Tracy-Widom distributions that appear in the limit laws for the one-point fluctuations.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.