The Potts model is a generalization of the Ising model, with q possible states for every lattice site. Whereas for q<5 the system has a second order transition as the Ising model, for q ≥ 5 the Potts model exhibits a first order phase transition. Moreover, the system has q degenerate ground state to collapse in, and the competition between these ground states can inhibit the condensation into a single phase. We study the out of equilibrium properties of the Potts model. We show some preliminary results of numerical simulation on a square lattice regarding how the system evolves after a quench to a subcritical temperature. A special focus is on the domain growth law, whose characteristics are analyzed as a function of the initial correlation length (finite vs infinite) and whose early stage presents peculiar behaviour. The Ising case (q=2) is shown for comparison and used as a starting point for the analysis.

We consider cooperative reactions and we study the effects of the interaction strength among the system components on the reaction rate, hence realizing a connection between microscopic and macroscopic observables. Our approach is based on statistical mechanics models and it is developed analytically via mean-field techniques. We show that cooperative and anticooperative behaviors naturally emerge from the model when the coupling strength is set to be respectively positive and negative, so that varying a simple effective parameter a plethora of different phenomena are recovered.

We present a study of submonolayer He adsorbed on two derivatives of graphene: graphene-fluoride (GF) and graphane (GH). A semiempirical interaction with the substrate is used in state of the art quantum simulations. We predict that both isotopes ³He and ⁴He form anisotropic fluid states at low coverage. The commensurate state analogous to the standard √×√ R30° phase that preempts fluid states on graphite turns out to be unstable relative to a fluid state. The commensurate insulating ground state on GF and GH is disfavored by the much smaller inter-site distance (below 1.5 Å) compared to graphite (2.46 Å) implying a large energy penalty for localizing He atoms. The 4He ground state on both substrates is a self-bound anisotropic superfluid with anisotropic roton excitations and with a superfluid density ρ_s reduced from 100% due to the corrugation of the adsorption potential. In the case of GF such corrugation is so large that ρ_s = 57% at T=0K and the superfluid is essentially restricted to move in a multiconnected space, along the bonds of a honeycomb lattice. We predict a superfluid transition temperature T ≈ 0.25(1.1)K for 4He on GF (GH). At higher coverages we find two kinds of solids, an incommensurate triangular one as well as a novel commensurate state at filling factor 2/7 with 4 atoms in the unit cell. We have evidence that this 2/7 state is supersolid. We conclude that these new platforms for adsorption studies offer the possibility of studying novel phases of quantum condensed matter like an anisotropic Fermi fluid, possibly superfluid, an anisotropic Bose superfluid and a commensurate supersolid.

I will present the main features of nonequilibrium steady states of the one-dimensional discrete nonlinear Schroedinger (DNLS) equation. Such equation has important applications in many domains of physics. A classical example is electronic transport in biomolecules. In the context of optics or acoustics it describes the propagation of nonlinear waves in a layered photonic or phononic system. On the other hand, in the realm of the physics of cold atomic gases, the model is an approximate semiclassical description of bosons trapped in periodic optical lattices. While a vast literature has been devoted to the dynamical behavior of the DNLS equation, much less is known about finite-temperature properties and almost nothing about nonequilibrium properties. Due to the presence of two conserved quantities, energy and norm (or number of particles), the model displays coupled transport in the sense of linear irreversible thermodynamics. Suitable models of thermostat are implemented to impose a given temperature and chemical potential at the chain ends. As a result, we find that the system exhibits normal transport, ruled by the Fourier law. However, for large differences between the thermostat parameters, density and temperature profiles may display an unusual nonmonotonic shape. This is due to the strong dependence of the transport coefficients on the thermodynamic variables.

The dynamics of sparse networks is investigated both at the microscopic and macroscopic level, upon varying the connectivity. In all cases (chaotic maps, Stuart-Landau oscillators, and leaky integrate-and-fire neuron models), we find that a few tens of random connections are sufficient to sustain a nontrivial (and possibly irregular) collective dynamics. At the same time, the microscopic evolution turns out to be extensive, both in the presence and absence of a macroscopic evolution. This result is quite remarkable, considered the non-additivity of the underlying dynamical rule.

Ref: S. Luccioli, S. Olmi, A. Politi and A. Torcini, "Collective dynamics in sparse networks", submitted to Phys. Rev. Lett.

The stability of the dynamical states characterised by a uniform firing rate ("splay states") is analysed in a network of N globally pulse-coupled rotators (neurons) subjected to a generic velocity field. This is done by reducing the set of differential equations to an event driven map that is investigated in the limit of large network size. We show that the Floquet spectrum characterising the stability of the splay state can be decomposed in two components: (i) a long-wavelength component which is the only one usually considered in mean-field analysis [Abbott-van Vreesvijk, 1993]; (ii) a short-wavelenght component measuring the instability of ``finite-frequency" modes. By developing a perturbative technique, we have found analytically that, in the limit of large N, the short-wavelenght spectrum scales as 1/N^2 for generic discontinuous velocity fields. Moreover, the stability of this component is determined by the sign of the jump at the discontinuity. Altogether, the form of the spectrum depends on the pulse shape but is independent of the velocity field. Furthermore, numerical results indicate that in the case of continuous velocity fields, the Floquet exponents scale faster than 1/N^2 (namely, as 1/N^4) and we even find strictly neutral directions in a wider class than the sinusoidal velocity fields considered by Watanabe and Strogatz in Physica D 74 (1994) 197-253.

References
R. Zillmer, R. Livi, A. Politi, and A. Torcini, Phys. Rev. E 76 (2007) 046102 M. Calamai, A. Politi, and A. Torcini, Phys. Rev. E 80, 036209 (2009)
S. Olmi, A.Politi, and A. Torcini, "Stability of the splay state in networks of pulse-coupled neurons", submitted to J. Mathematical Neuroscience (2012)

The form of the correct equations describing brownian motion with state-dependent diffusion is subject to long-standing debates. In this poster I discuss one proposal which is founded on the principle of general covariance, by which I mean: Covariance under coordinate transformations; Invariance under internal "gauge" transformations. I derive a generally covariant Langevin equation from first principles; the overdamping limit yields a first-oder SDE which is neither Ito, Straonovich, Klimontovich nor else. The corresponding Fokker-Planck equation affords an equilibrium (detailed-balanced) steady state, differing from other proposals. Among the experimentally testable consequences of the theory, the spatial density of the steady state is not uniform, dispensing with the equal-a-priori postulate.

We consider a lattice polymer model of the 2-tolerant type (i.e., a random walk allowed to visit lattice bonds at most twice), in which doubly-visited bonds yield an attractive energy term (pairing energy). Such a model has been previously proposed as a rough, non-specific description of the RNA folding mechanism. In fact, the model predicts, besides the usual theta-collapse, an extra transition to a low-temperature fully-paired state. In the current work, we propose an extension of the model, in which an additional (micromolecular) chemical species can bind the polymer and locally forbid base-pairing. This special kind of interaction is meant to mimic a generic mechanism by which micro-RNA molecules could hamper messenger RNA folding, and ultimately exert their down-regulating effect within a gene-expression network. We investigate equilibrium thermodynamics in the grand-canonical picture, at the level of a Bethe approximation, which is, a refined mean-field technique, equivalent to the exact solution on a random-regular graph. The general trend we observe is that expected from the mechanism implemented in the model (increasing micro-RNA concentration favors denaturation and lowers the folding temperature), but the resulting phase diagram turns out to be unexpectedly interesting and rich.

In this work we study the effects of waves on the motion of inertial particles in an incompressible fluid. We performed analytical calculations, using also multiple-scale techniques, to predict the behaviour of such a particle, firstly in deep water waves and then in an arbitrarily deep regime. All these analytical results were checked by numerical simulations of the problem. We found that the presence of inertia leads to corrections to the well-known Stokes Drift. These corrections become relevant in certain ranges of the parameters of the system. Furthermore, with the same methods, we observed that, even in the absence of gravity (g = 0), inertial particles drift downwards.

Stochastic processes with memory are a common way to model systems ranging from physics to ecology and biology.
In particular, self-attracting and self-avoiding random walks provide basic, yet non trivial, examples.
Here, we first review the main definitions of random walks with memory and then we focus on the case of (true) reinforced random walk where memory effects are implemented at each time step, differently from the static case, where memory effects are accounted for globally.
We investigate the model in dimension $d=1$, also accounting for the presence of a field of strength $s$, which biases the walker toward a target site.
We analyze in details the asymptotic long-time behavior of the walker through the main statistical quantities (e.g. distinct sites visited, end-to-end distance) and we discuss a possible mapping between such dynamic self-attracting model and the trapping problem for a simple random walk, in analogy with the static model. Moreover, we find that, for any $s>0$, the random walk behavior switches to ballistic and that field effects always prevail on memory effects without any singularity, already in $d=1$; this is in contrast with the behavior observed in the static model.