$206.8K Funding

1 People

External

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.