Abstract:
Since its introduction in the 70’s, large deviation theory has become a prominent tool in the study of scaling problems in physical systems, and the associated quantities found natural interpretations. Indeed, the two main functions of the theory, the cumulant function an the rate function, respectively generalize free energy and entropy for non-equilibrium systems. Information on these functions provide insight on the system at hand. In particular, in this talk, we will focus on the functions associated to fluctuations in time of ergodic averages, which characterize a form of phase transition.
Such functions are in general very difficult to compute by direct Monte Carlo simulation, since an efficient sampling requires simulating an exponential number of trajectories with respect to time. This led to designing specific algorithms for this problem. A large family of methods rely on cloning, or population dynamics, which introduce a selection mechanism between replicas to favor a desired behavior. Another range of methods resort to importance sampling: an additional drift is introduced in the dynamics so as to « push » it towards relevant regions of the state space. However, this drift is in general difficult to estimate. I will present an importance sampling strategy based on previous works of Borkar and collaborators for estimating this optimal drift, a work in collaboration with H. Touchette. The originality of our work resides in building on the fly estimators of the drift, cumulant function, its derivative, and the rate function at once. We provide relevant applications on equilibrium and non-equilibrium toy examples from statistical physics.