Reductionism, or more generally the relationship connecting the different sciences, is perhaps one of the few issues of the scientific culture still capable of stirring peremptory and robust discussions among dedicated scholars. In spite of the famous Dirac' claim on the reduction of chemistry to physics [{\it The fundamental laws necessary for the mathematical treatment of a large part of physics and the whole of chemistry are thus completely known, and the difficulty lies only in the fact that application of these laws leads to equations that are too complex to be solved}] even with the help of powerful computers and efficient numerical methods chemistry is not merely applied quantum mechanics. A critical analysis shows that classical mechanics is not just the $\hbar \to 0$ limiting case of quantum mechanics:for a chaotic classical system, the proper classical limit can be obtained from quantum mechanics only taking into account the influence of the outer environment. It is correct to say that quantum mechanics plus (semi)classical physics is able to describe the observable features of molecules. The study of pyramidal molecules shows that the (semi)classical limit does not follow in a straightforward way from quantum mechanics but is a consequence of the interaction of single molecules with an external environment.

The free energy at zero temperature of Coulomb gas systems in generic dimension is considered as a function of a volume constraint, i.e. by varying the total volume available to the system. The transition between the 'pulled' and the 'pushed' phases of the gas is characterised as a third-order phase transition, in any dimension and for a rather large class of isotropic potentials. This suggests that the critical behaviour of the free energy at the 'pulled-to-pushed' transition may be universal, i.e., to some extent independent of the dimension and the details of the pairwise interaction. In particular, the logarithmic pairwise interaction, typical of random matrix problems where a similar third-order transition was previously detected, does not play any special role.

The onset of thermalization in a closed system of randomly interacting bosons, at the level of a single eigenstate, is discussed. We focus on the emergence of Bose-Einstein distribution of single-particle occupation numbers, and we give a local criterion for thermalization dependent on the eigenstate energy. We show how to define the temperature of an eigenstate, provided that it has a chaotic structure in the basis defined by the single-particle states. The analytical expression for the eigenstate temperature as a function of both inter-particle interaction and energy is complemented by numerical data.

Cell functionality requires continuous internalization of proteins and other molecular factors that are transported to specialized compartments (endosomes) by means of vesicles. Vesicle transport is characterized by an accurate selection of only the desired molecules into the vesicles and their subsequent transport towards the correct target. We describe the first stage of this process (“molecular sorting”) using a minimal lattice gas model for distillation of a single molecular factor on a lipid membrane. The affinity between similar molecular factors favors the formation of enriched domains that grow over time following classical coarsening laws. Enriched domains are then extracted by means of vesicles. At the steady-state, the efficiency of the sorting process depends on the incoming flux of molecules and on their affinity. An optimal behavior is identified, corresponding to the existence of a unique enriched domain. Preliminary experimental validations of the theoretical (and numerical) results are provided.

The decay of temporal correlation function represents a remarkable property of chaotic dynamical systems. In the case of a mixed phase space the mixing speed is typically polynomial, and the presence of transients and statistical errors makes a direct numerical investigation almost impossible. We illustrate theoretical virtues and numerical usefulness of indirect methods, based on time statistics and large deviations estimates.

We present different mean-field-like approximations for stochastic dynamics on graphs, within the framework of a cluster-variational approach. In analogy with its equilibrium counterpart, this approach allows one to give a unified view of various (previously known) approximation schemes, and suggests quite a systematic way to improve the level of accuracy. We compare the different approximations with Monte Carlo simulations on a recurrent (susceptible-infected-susceptible) discrete-time epidemic-spreading model on random graphs.

For Hamiltonian systems, the Statistical Mechanics tools for studying the large scale emerging phenomena from a underlying chaotic and complex dynamics, are quite ripe and well assessed. Among them, the Zwanzig projection approach is one of the most powerful and used, leading to important results, as the microscopic foundation of the Fluctuation Dissipation relation, and the Canonical equilibrium Density Function (see, for example, M. Bianucci et al., Phys. Rev. E 51, 3002 (1995)). Actually, the projection approach leads at a calculus with differential operators that is usually almost intractable, but that is drastically simplified when the basic dynamics of the system of interest is Hamiltonian, for which stadard Fluctuation Dissipation Relation holds true. In many real physical cases, however, the fundamental equations are not Hamiltonian. This happens, for example, in Fluid Dynamics, where the physics is described by the Navier Stokes Equations, or in Biology in general, just to cite a couple among many research fields. Here we show how it is still possible to get, in the non Hamiltonian case, a Generalized Fokker Planck Equation describing the time large scale statistics of a part of interest of the whole complex system. As an example, we focus our attention on the El Ni\~no Southern Oscillation, where a non standard Fluctuation Dissipation process plays a fundamental role.

Dynamics of epithelial monolayers has recently been interpreted in terms of a jamming or rigidity transition [1,2]. How cells control the proximity to that transition is, however, unknown. Here we show that elevation of the endocytic protein RAB5A, frequently hijacked by different epithelial-like tumors to promote their dissemination, is sufficient to induce large-scale, coordinated motility and flocking in otherwise kinetically-arrested monolayers. Our findings [3] suggest that the increased fluctuations and reawakening of motility are the result of globally enhanced endosomal trafficking and macropinocytotic internalization. These variations lead to an increase in junctional tension and traction forces exerted on the substrate and promote the extension of persistent cell protrusions which align with local velocity. To rationalize our findings, we propose a simple model where, in addition to cell-cell adhesion and cortical tension, we consider an active reorientation mechanism for the velocity of self-propelled cells. The model explains the observed reawakening of RAB5A motility in terms of a combination of large-scale directed migration and a local unjamming [3]. These changes in multicellular dynamics allow collectives to migrate under physical constraints and may be exploited by tumors for interstitial dissemination. References: [1] T. Angelini et al., PNAS 108 (12), 4714 (2011) [2] JA. Park et al., Nature Materials 14, 1040 (2015) [3] C. Malinverno et al, Nature Materials, Advance Online Publication (2017)