Diffusion and Kinetics of Reaction Models in Contemporary Statistical Physics

Clarkson University, Potsdam NY

Investigators

Abstract

Reaction-diffusion systems underlie diverse natural and man-made phenomena from self-organization and pattern formation (embryo morphology, stripes and dots in the coats of animals, etc.), to heterogeneous catalysis in the chemical industry, to electron-hole recombination in semiconductors, to the anomalous kinetics of reactions and signal transduction inside living cells, to the spread of opinions, or diseases in populations, to ecological competition between species and attract considerable interest in contemporary statistical physics. The diffusion-limited regime is challenging, as it is dominated by fluctuations at all length scales, and there exists no comprehensive approach to deal with it effectively. Research is proposed into diffusion-controlled processes, such as diffusion-limited coalescence on the line, A+A <-> A, and the method of empty intervals that we introduced for its analysis. Coalescence has long been an example of an analytically tractable model that exhibits many of the hallmarks of nonequilibrium systems, including anomalous kinetics, ordering and self-organization, propagation of Fisher fronts, and kinetic phase transitions. We have further extended the method of intervals to annihilation, branching annihilating walks, and the q-state Potts model in the zero-temperature limit. It has since been embraced by other research groups, for the study of models not amenable to analysis by complementary techniques, such as the method of free Majorana fermions. Research will be conducted into several related problems, including a chance at an exact solution of directed percolation, the main representative class of nonequilibrium kinetic phase transitions; further generalizations of the method of intervals to new reaction schemes; the shielding effect and its use for solving infinite hierarchical PDE systems; the role of fluctuations in reaction fronts, and the applicability of reaction-diffusion mean-field rate equations; extreme statistics of vicious walkers; and a new model of history-dependent random walks that can too be analyzed exactly, with numerous potential applications. Research will also be conducted into a number of practical applications, such as fast computer algorithms for the simulation of reaction-diffusion systems, kinetics of granular gases and evolution under linear mixing, diffusion in nanopore channels and slow drug release, structure and function of large complex networks (the Internet, social nets, etc.), and modeling the architecture and connections between regions of the brain. The proposed work is particularly well suited for educational purposes. The PI has a strong record of working and publishing with graduate and undergraduate students. Nine undergraduate and four graduate students have been involved in related projects in the past four years. The bulk of the requested funding is earmarked to their training. The proposed work involves national and international collaborations with colleagues from Clarkson University, Boston University, Los Alamos National Lab, the University of San Diego, the University of Granada and Universitat Polit`ecnica de Catalunya (Spain), Minas Gerais (Brazil), Bar-Ilan University (Israel), Ecole Normale (France), and their many students and postdocs. The PI and his collaborators and students have presented their results in numerous seminars and meetings. Our recent findings have been featured in IoP Select and Physics Web, Physical Review Focus, Nature Science Update, News in Brief, and in a science radio show.