Nonlinear effects, disorder and nonstandard topologies are typically at the origin of anomalous transport properties and complex collective dynamical regimes, both in condensed matter substrates and in living matter. When these ingredients are present, statistical correlations, large fluctuations and coherent effects can lead to interesting dynamical patterns, whose repertoire is still largely unexplored. The aim of this project is to investigate non-equilibrium behavior in a variety of complex systems. We will integrate analytical and numerical methods from dynamical systems theory and statistical mechanics, providing a general framework of methodological tools and physical concepts to be applied to a wide range of physical and biological instances.

We can identify three main lines of research, with strong interconnections between them: anomalous dynamics and transport in classical and quantum systems, long-range interacting systems, and emergence of collective behavior in active and living matter. We sketch here the general context and aims, in Sec.II the specific objectives.

*1) Anomalous dynamics and transport*

Anomalous dynamics and transport can arise for different reasons, such as disorder, non-trivial network topologies, the occurrence of phase transitions. In many cases, they are associated to large anomalous fluctuations, whose mathematical description is therefore of the uttermost importance. New theoretical tools, based on infinite ergodic theory, record statistics and extreme events, have been recently developed, and their role appears to be crucial in systems like networks of interacting subsystems and non-homogeneous diffusion. Some of us recently studied large fluctuations in several transport processes through complex media with fat-tailed (Lévy like) correlated disorder. Here the rare events distribution is driven by a single huge event and does not follow the standard large deviation approach, requiring novel methods of analysis.

The occurrence of transitions can originate non-trivial dynamical patterns. This is the case, for example, of the synchronization transition that we identified in neural networks dynamics, giving rise to regimes of `neuronal avalanches’ similar to those measured in several experiments, both in vivo and in vitro.

The presence of randomness and non-trivial topologies also have a crucial role in biological and socio-economic systems. Some of us have intensively worked on the dynamics of complex networks, using numerical methods, agent-based models, and mathematical approaches. This is a rich and evolving field where much work still needs to be done.

For what concerns quantum transport, there are many instances of both theoretical relevance and potential application impact. An example is the quenched dynamics of a wave packet in quantum chaotic many-body systems, where the number of principal components grows exponentially with a rate related to the initial local density of states and to the thermalization time. Thermalization can also be affected by - direct or indirect - long-range interactions, as in ion-traps and molecular chains, even in presence of localization effects (see also next point). Another interesting problem is the design of quantum walks of atoms with Bose-Einstein condensates in optical lattices, where one can observe the classical-to-quantum transition. Advanced control of complex quantum networks is a topic with key applications to Quantum Technologies. Quantum networks are in general hybrid structures, connecting atoms or devices by microwave or optical modes, and “integrated” with environments or quantum measurement setups. Understanding transport on such networks (of excitations/information/heat) is a key issue to control their functioning.

*2) Long-range models*

Systems with long-range interactions are ubiquitous in Nature. Self-gravitating systems are the paradigmatic example, but also unscreened electrostatic interactions in plasmas as well as dipolar forces in condensed matter or effective, electromagnetic field-mediated interactions between cold atoms in optical cavities are long-range.

Additivity breaks down for such systems, yielding peculiar equilibrium properties as the inequivalence of statistical ensembles or negative specific heats. Yet, a consistent thermodynamic description is possible, as recently shown by some of us, although it is peculiar too: for instance, temperature, pressure and chemical potential can be simultaneously chosen as independent variables. In statistical mechanics, the latter corresponds to the completely open, or unconstrained, ensemble.

The most striking feature of long-range-interacting systems is that the relaxation time to thermal equilibrium diverges with the number of particles *N*, so that a large system remains out of equilibrium forever, and typically relaxes towards quasi-stationary, non-thermal states. Characterizing such states as well as the (essentially collisionless) relaxation process would be crucial to understand, e.g., global properties of stellar systems or galaxies as well as many aspects of their evolution. An effective evolution equation for a coarse-grained distribution function has been very recently obtained, which may prove a very useful tool to understand the dynamics of these systems.

*3) Emergent collective dynamics in active and living matter*

Flocking or collective motion in active matter systems is a ubiquitous emergent phenomenon that occurs in many living and synthetic systems over a wide range of scales. Examples range from fish schools and bird flocks to bacteria colonies and cellular migrations, down to subcellular molecular motors and biopolymers. Flocking systems are characterized by long-lived non-equilibrium fluctuations (the Nambu-Goldstone modes of the system), determined by the non-trivial interplay between symmetry breaking and non-equilibrium activity. At the microscopic level, the ingredients giving rise to such rich phenomenology are the injection of energy at the individual level (each flocker has a `motor’ converting energy into motion), local alignment interactions between neighbors, and the presence of noise (e.g. an environmental bath, errors in decision-making etc). Understanding the role of such ingredients, and how they affect the emergent large-scale patterns is crucial. We describe here below a few general open issues that we plan to address in this project:

- A fruitful cooperation of field theoretical argument, numerical simulations and experimental observation greatly advanced our understanding of the bulk behavior of collective motion in unperturbed systems. However, much remains to be understood for what concerns response to external perturbations and motion under confinement (both crucial problems for living systems).

- Many living aggregates are large, but far from the thermodynamic limit. As a consequence, the asymptotic regimes described by many theoretical computations do not capture the observational timescales relevant for such systems. Inertial effects, for example, have been shown to be important for both natural flocks and swarms. At field-theoretic level this implies a novel focus on crossover behavior, and novel effective classes of critical dynamical behavior.

- Active systems are intrinsically non-equilibrium, as such they are elective subjects of stochastic thermodynamics, which focuses on irreversible entropy production in non-equilibrium systems, energy dissipation and loss of information. This aspect connects to a related line of our research: information processing properties of linear (multidimensional) signal-response models. Here, in general, we plan to explore the irreversibility of time series as a measure of the efficiency of signaling.

- Model building has a central role in the process of knowledge acquisition from observational data. A possible approach is to build effective Langevin equations describing the macroscopic dynamics of the system, directly from the data. We will explore methods of Bayesian statistical inference, with the aim of disentangling contributions from inertia, dissipation and noise. We will also study effective equations achieved in terms of "reaction coordinates", i.e. a set of few variables relevant for the macroscopic dynamical regimes.