Towards large deviations in stochastic systems with memory

Abstract

The theory of large deviations can help to shed light on systems in non-equilibrium statistical mechanics and, more generically, on non-reversible stochastic processes. For this purpose, we target trajectories in space time rather than static con figurations and study time-extensive observables. This suggests that the details of the evolution law such as the presence of time correlations take on a major role. In this thesis, we investigate selected models with stochastic dynamics that incorporate memory by means of diff erent mechanisms, devise a numerical approach for such models, and quantify to what extent the memory aff ects the large deviation functionals. The results are relevant for real-world situations, where simpli ed memoryless (Markovian) models may not always be appropriate. After an original introduction to the mathematics of stochastic processes, we explore, analytically and numerically, an open-boundary zero-range process which incorporates memory by means of hidden variables that a ect particle congestion. We derive the exact solution for the steady state of the one-site system, as well as a mean- eld approximation for larger one-dimensional lattices. Then, we focus on the large deviation properties of the particle current in such a system. This reveals that the time correlations can be apparently absorbed in a memoryless description for the steady state and the small uctuation regime. However, they can dramatically alter the probability of rare currents. Di erent regimes are separated by dynamical phase transitions. Subsequently, we address systems in which the memory cannot be encoded in hidden variables or the waiting-time distributions depend on the whole trajectory. Here, the di culty in obtaining exact analytical results is exacerbated. To tackle these systems, we have proposed a version of the so-called \cloning" algorithm for the evaluation of large deviations that can be applied consistently for both Markovian and non-Markovian dynamics. The e cacy of this approach is con rmed by numerical results for some of the rare non-Markovian models whose large deviation functions can be obtained exactly. We nally adapt this machinery to a technological problem, speci cally the performance evaluation of communication systems, where temporal correlations and large deviations are important.