The project is aimed at investigation of mathematical problems in the theory of disordered and noisy systems. Specifically, the following topics will be studied: (1) Properties of geodesics of random Riemannian metrics. In particular, we will study the mechanism of destabilization of the length-minimizing property of geodesics by randomness, using curvature fluctuations. (2) Ground state energy fluctuations in a one-dimensional Anderson (tight-binding) model with a Bernoulli random potential and a sharp form of Lifschits tail estimates, i.e. decay of integrated density of states at the bottom of the spectrum. (3) Stability and invariant measures of randomly perturbed explosive dynamical systems. (4) Dynamics of Brownian particles with position-dependent diffusion coefficients in the Smoluchowski-Kramers (overdamped) limit. Explanation of experimental results and design of new experiments will be performed, supplemented by numerical studies. (5) Disorder-induced order in quantum and classical systems. (6) Connections between quantum networks and percolation theory.
The project studies various aspects, manifestations and consequences of presence of random elements in physical systems. Randomness models disorder, imperfections, defects (in disordered systems and materials) or noise in evolving systems. In the first case, it does not fluctuate with time (quenched randomness); in the second--it does. Depending on this and also on the specific details in which it enters the physics of the system, effects of randomness can be of very different nature, more or less pronounced and, from the point of view of applications, more or less desirable. The influence of randomness on the behavior of physical systems can be profound. To mention just a few examples: presence of noise may give rise to an unexpected force acting on a diffusing particle; presence of disorder may completely change the symmetry of a quantum system (made of superconductors, for example). A third, famous example is that of localization: random defects can change a conductor to an insulator even when their density is very low (i.e. there are very few of them). All these effects--studied in the present project--have very different nature, their detailed mathematical analysis requires different methods, but they are all united by the use of probability theory to formulate and study the relevant mathematical models.
Research projects supported by this NSF award address behavior of physical systems in the presence of disorder or noise. These are ubiquitous components of the physical reality which, due to their complicated nature and diverse origins, do not yield themselves to exact mathematical description. Their effects are accounted for by including judiciously chosen random parameters in the mathematical models. Disorder refers to randomness, which does not vary on the time scales relevant for the studied problem, thus creating a random environment. Presence of disorder may significantly change physical properties of the system. One of the projects studied energies of quantum particles in the presence of a random potential---a paradigmatic example of a disordered system. The particles---electrons or cold atoms, depending on the application---get localized (trapped) by the disorder at various values of energy. The precise distribution of these energies was studied by a new, simple mathematical method. Both noninteracting and interacting particles were studied. The latter case is important for understanding properties of cold atomic systems---Bose-Einstein condensates, intensely studied in recent years. In contrast to disorder, noise refers to random perturbations of system's time evolution. For example, a Brownian particle randomly collides with particles of the fluid in which it is suspended. In another example presence of electrical noise changes the evolution of the current in a circuit. Evolution of systems in the presence of noise is mathematically described by stochastic differential equations. The projects funded by this NSF award studied such systems mathematically, in close collaboration with experimental physicists. The main results of this research are concerned with the noise-induced drift---an additional force, acting on the system, as a result of the energy exchange with the environment due to the presence of noise. Such forces were calculated for a very general class of systems in physically relevant regimes. An excellent agreement with experiment was demonstrated in two cases mentioned above---motion of Brownian particles in a diffusion gradient and evolution of a noisy electrical circuit with a feedback mechanism. This is a convincing proof that the models used capture the essence of the studied phenomenon and bodes well for their future applications. Mathematically, the project resulted in a new, powerful mathematical technique of studying limits of stochastic differential systems. The essence of the studied phenomena was shown to lie in the relations between various time scales relevant for the evolution of the systems. These include the time over which the noise variables are correlated and the delay inherent in the reaction of the system to the noise. The PI and his collaborators have been able to identify the form in which these time scales combine to define the strength of the effective noise-induced drift. These results were summarized in a recent publication in Nature Communications. The general nature ot the obtained results and of the methods by which they were obtained allows to apply related techniques to a variety of other situations, including life science experiments. The projects under way include a study of chemotactic properties of microrobots, a line of fundamental research, motivated by biological and medical considerations. The noise-induced drift is a ubiquitous phenomenon and may be present in quantum systems as well. Another recent project studies quantum Brownian motion and its detailed properties from this point view. One of the most rewarding aspects of scientific work is collaboration with and instruction of young researchers. The projects supported by this NSF award led to three Ph. D. dissertations and the fourth graduate student is scheduled to receive his doctorate in 2015. The results obtained by the Ph. D. candidates were published in best journals and they themselves are on the way to become independent researchers, having obtained postdoctoral positions at first-rate universities. The research described in this report crosses boundaries between mathematics and physics, between rigor and insight, between theory and experiment. It is and is becoming even more interdisciplinary, extending to other sciences and covering new applied grounds. Applying mathematics in the most creative way he can, collaborating with senior researchers and teaching junior ones, working on theory and experiment, applying the results of fundamental investigations was and remains the main goal of the PI.
