How do large fluctuations occur?
M.I. Dykman and V.N. Smelyanskiy
Department of Physics 
Astronomy, Michigan State University
Large fluctuations are in the root of many physical and biological phenomena, from nucleation near phase transitions to mutations in the DNA sequencies. We will discuss large fluctuations in classical systems away from thermal equilibrium where Boltzmann distribution does not apply. We will provide a general formulation based on the concept of optimal path. This is the path along which a fluctuating system moves, with overwhelming probability, from the attractor to a given state. Optimal paths are physically real, by now they have been observed in experiment. Their role in the theory of fluctuations is similar to that of dynamical trajectories in mechanics. We will analyze the pattern of optimal paths and its singularities. The most common singularities of the pattern of extreme paths familiar from optics and quantum mechanics are caustics. We will show that caustics are not encountered by optimal paths, and instead there occur singularities of a different type. In some important cases both caustics and observable singular lines have self-similar structure. We will discuss the problem of escape from a metastable state, and consider application of the results to particular physical systems. We will also consider application of the results to the problem of optimal control.