### Abstract

From the very beginning of its development statistical mechanics, as a mathematical approach to the generic behavior of complex systems, has been one of the most fruitful cross-fertilization areas between physics and mathematics. Nowadays its use has pervaded a large spectrum of scientific activities, far beyond its original scope in physics. It provides invaluable tools for the analysis of a variety of phenomena, from the microscopic to the cosmological scale, and from the social sciences to nanotechnology. Besides its many achievements in such applications, statistical mechanics is still under active development. During the last decades researchers have faced new challenges, looking towards a mathematically rigorous approach to nonequilibrium phenomena and to the emerging thermodynamics of out of equilibrium processes. From their works, new fundamental concepts have emerged: current carrying nonequilibrium steady states, entropy production, fluctuation relations, large deviations in the time domain and their relation to linear and non-linear response theory... There is no doubt that these and future advances in our understanding of statistical properties of nonequilibrium processes will have considerable impact on various scientific domains outside of physics and foster the development of new research directions in mathematics.

NonStops is a project in mathematical physics on the dynamics of open and stochastically driven classical and quantum systems. The project focuses on some challenging mathematical problems raised by recent advances in non-equilibrium statistical mechanics. From the technical point of view, these problems are mainly concerned with the large time asymptotics and ergodic properties of classical and quantum dynamical systems, with special emphasis on the large deviations of some functionals related to transport phenomena. From a more conceptual perspective, the main purpose is a critical investigation of the universal nature of entropic fluctuation relations which are at the forefront of these new developments, extending them to a larger class of dynamical systems. We will consider three directions which appear very promising given the present state of the art: classical systems with an infinite number of degrees of freedom described by stochastic PDEs, open quantum systems driven by thermodynamic forces and quantum walks, the non-commutative counterparts of classical random walks.

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