Continuous-time Markov chains are a useful tool to model non-equilibrium phenomena, like the diffusion of particles from higher density regions to lower density regions. The system evolves in time jumping randomly on a discrete set of allowed configurations, with specified transition rates, and interesting dynamical observables are functionals over the trajectories of the system (like the activity, that counts the total number of jumps). Interestingly, the probability distribution of these random observables may satisfy some constraints, largely independent from the specific details of the dynamics (based only on some hidden symmetries), called fluctuation relations. For instance, a fluctuation relation could state that the probability of observing a heat current from a cold region to a hot region is exponentially suppressed (in terms of the current magnitude and temperature difference) with respect to the reversed current, the one flowing in the “correct” way. In some cases these relations are valid at any time but most often they emerge in the long-time limit. This second case is mathematically more interesting because the machinery of large deviations theory comes into play. The question is then to show that these relations hold true in specific models of physical relevance and more in general to understand what are necessary and sufficient conditions for their validity.
Similar modeling techniques also exist in the quantum regime. Indeed, the dynamics of an open quantum system with monitored environment (photon counting) is well-described by a random density matrix satisfying a quantum jump process (the state of the system is conditioned on the measurement outcomes in the environment). The state averaged over the different realizations of the process obeys a Markovian master equation in Lindblad form. While interesting questions still need to be answered for the average dynamics of complex quantum systems, especially in the interacting case, a much less explored research line regards the statistics of the conditioned state, that contains more information. In particular, one could also study the large deviations of relevant (process dependent) observables.
The project ConNEqtions consisted of two parts, related to the investigation of non-equilibrium classical and quantum systems, respectively. On the classical side, the focus was on fluctuation relations, especially those induced by symmetries other than time-reversal. The initial goal was to generalize the results available in the context of time-homogeneous Markov jump processes to the case of time-periodic rates. On the quantum side, the aim was to improve our comprehension of complex systems out-of-equilibrium, focusing on the dynamics of interacting closed systems and open quantum systems. In particular, the most ambitious goal was to characterize the fluctuations in quantum jump processes with time-periodic rates, computing the level 2.5 large deviation functional (large deviations of the joint distribution of empirical measure and empirical current).
