Statistical mechanics and dynamics of solvable models with long-range interactions

The two-body potential of systems with long-range interactions decays at large distances as $V(r)\sim 1/r^\alpha$, with $\alpha\leq d$, where $d$ is the space dimension. Examples are: gravitational systems, two-dimensional hydrodynamics, two-dimensional elasticity, charged and dipolar systems. Although such systems can be made extensive, they are intrinsically non additive. Moreover, the space of accessible macroscopic thermodynamic parameters might be non convex. The violation of these two basic properties is at the origin of ensemble inequivalence, which implies that specific heat can be negative in the microcanonical ensemble and temperature jumps can appear at microcanonical first order phase transitions. The lack of convexity implies that ergodicity may be generically broken. We present here a comprehensive review of the recent advances on the statistical mechanics and out-of-equilibrium dynamics of systems with long-range interactions. The core of the review consists in the detailed presentation of the concept of ensemble inequivalence, as exemplified by the exact solution, in the microcanonical and canonical ensembles, of mean-field type models. Relaxation towards thermodynamic equilibrium can be extremely slow and quasi-stationary states may be present. The understanding of such unusual relaxation process is obtained by the introduction of an appropriate kinetic theory based on the Vlasov equation.Comment: 118 pages, review paper, added references, slight change of conten

Time evolution of wave-packets in quasi-1D disordered media

We have investigated numerically the quantum evolution of a wave-packet in a quenched disordered medium described by a tight-binding Hamiltonian with long-range hopping (band random matrix approach). We have obtained clean data for the scaling properties in time and in the bandwidth b of the packet width and its fluctuations with respect to disorder realizations. We confirm that the fluctuations of the packet width in the steady-state show an anomalous scaling and we give a new estimate of the anomalous scaling exponent. This anomalous behaviour is related to the presence of non-Gaussian tails in the distribution of the packet width. Finally, we have analysed the steady state probability profile and we have found finite band corrections of order 1/b with respect to the theoretical formula derived by Zhirov in the limit of infinite bandwidth. In a neighbourhood of the origin, however, the corrections are $O(1/\sqrt{b})$.Comment: 19 pages, 9 Encapsulated Postscript figures; submitted to ``European Physical Journal B'

Finite times to equipartition in the thermodynamic limit

We study the time scale T to equipartition in a 1D lattice of N masses coupled by quartic nonlinear (hard) springs (the Fermi-Pasta-Ulam beta model). We take the initial energy to be either in a single mode gamma or in a package of low frequency modes centered at gamma and of width delta-gamma, with both gamma and delta-gamma proportional to N. These initial conditions both give, for finite energy densities E/N, a scaling in the thermodynamic limit (large N), of a finite time to equipartition which is inversely proportional to the central mode frequency times a power of the energy density E/N. A theory of the scaling with E/N is presented and compared to the numerical results in the range 0.03 <= E/N <= 0.8.Comment: Plain TeX, 5 `eps' figures, submitted to Phys. Rev.

Chaotic dynamics and superdiffusion in a Hamiltonian system with many degrees of freedom

We discuss recent results obtained for the Hamiltonian Mean Field model. The model describes a system of N fully-coupled particles in one dimension and shows a second-order phase transition from a clustered phase to a homogeneous one when the energy is increased. Strong chaos is found in correspondence to the critical point on top of a weak chaotic regime which characterizes the motion at low energies. For a small region around the critical point, we find anomalous (enhanced) diffusion and L\'evy walks in a transient temporal regime before the system relaxes to equilibrium.Comment: 7 pages, Latex, 6 figures included, Contributed paper to the Int. Conf. on "Statistical Mechanics and Strongly Correlated System", 2nd Giovanni Paladin Memorial, Rome 27-29 September 1999, submitted to Physica

Inhomogeneous Quasi-stationary States in a Mean-field Model with Repulsive Cosine Interactions

The system of N particles moving on a circle and interacting via a global repulsive cosine interaction is well known to display spatially inhomogeneous structures of extraordinary stability starting from certain low energy initial conditions. The object of this paper is to show in a detailed manner how these structures arise and to explain their stability. By a convenient canonical transformation we rewrite the Hamiltonian in such a way that fast and slow variables are singled out and the canonical coordinates of a collective mode are naturally introduced. If, initially, enough energy is put in this mode, its decay can be extremely slow. However, both analytical arguments and numerical simulations suggest that these structures eventually decay to the spatially uniform equilibrium state, although this can happen on impressively long time scales. Finally, we heuristically introduce a one-particle time dependent Hamiltonian that well reproduces most of the observed phenomenology.Comment: to be published in J. Phys.

Nonlinear supratransmission and bistability in the Fermi-Pasta-Ulam model

The recently discovered phenomenon of nonlinear supratransmission consists in a sudden increase of the amplitude of a transmitted wave triggered by the excitation of nonlinear localized modes of the medium. We examine this process for the Fermi-Pasta-Ulam chain, sinusoidally driven at one edge and damped at the other. The supratransmission regime occurs for driving frequencies above the upper band-edge and originates from direct moving discrete breather creation. We derive approximate analytical estimates of the supratransmission threshold, which are in excellent agreement with numerics. When analysing the long-time behavior, we discover that, below the supratransmission threshold, a conducting stationary state coexists with the insulating one. We explain the bistable nature of the energy flux in terms of the excitation of quasi-harmonic extended waves. This leads to the analytical calculation of a lower-transmission threshold which is also in reasonable agreement with numerical experiments.Comment: 8 pages, 9 figures. Phys. Rev. E (accepted

Chaos in the thermodynamic limit

We study chaos in the Hamiltonian Mean Field model (HMF), a system with many degrees of freedom in which $N$ classical rotators are fully coupled. We review the most important results on the dynamics and the thermodynamics of the HMF, and in particular we focus on the chaotic properties.We study the Lyapunov exponents and the Kolmogorov--Sinai entropy, namely their dependence on the number of degrees of freedom and on energy density, both for the ferromagnetic and the antiferromagnetic case.Comment: 10 pages, Latex, 4 figures included, invited talk to the Int. school/Conf. on "Let's face Chaos Through Nonlinear Dynamics" Maribor (Slovenia) 27 june - 11 july 1999, submitted to Prog. Theor. Physics supp

Landau-Zener Tunnelling in Waveguide Arrays

Landau-Zener tunnelling is discussed in connection with optical waveguide arrays. Light injected in a specific band of the Bloch spectrum in the propagation constant can be transmitted to another band, changing its physical properties. This is achieved using two waveguide arrays with different refractive indices, which amounts to consider a Schr\"odinger equation in a periodic potential with a step. The step causes wave "acceleration" and thus induces Landau-Zener tunnelling. The region of physical parameters where this phenomenon can occur is analytically determined and a realistic experimental setup is suggested. Its application could allow the realization of light filters.Comment: 4 pages, 6 figure

PyPlutchik: Visualising and comparing emotion-annotated corpora

The increasing availability of textual corpora and data fetched from social networks is fuelling a huge production of works based on the model proposed by psychologist Robert Plutchik, often referred simply as the “Plutchik Wheel”. Related researches range from annotation tasks description to emotions detection tools. Visualisation of such emotions is traditionally carried out using the most popular layouts, as bar plots or tables, which are however sub-optimal. The classic representation of the Plutchik’s wheel follows the principles of proximity and opposition between pairs of emotions: spatial proximity in this model is also a semantic proximity, as adjacent emotions elicit a complex emotion (a primary dyad) when triggered together; spatial opposition is a semantic opposition as well, as positive emotions are opposite to negative emotions. The most common layouts fail to preserve both features, not to mention the need of visually allowing comparisons between different corpora in a blink of an eye, that is hard with basic design solutions. We introduce PyPlutchik the Pyplutchik package is available as a Github repository (http://github.com/alfonsosemeraro/pyplutchik) or through the installation commands pip or conda. For any enquiry about usage or installation feel free to contact the corresponding author, a Python module specifically designed for the visualisation of Plutchik’s emotions in texts or in corpora. PyPlutchik draws the Plutchik’s flower with each emotion petal sized after how much that emotion is detected or annotated in the corpus, also representing three degrees of intensity for each of them. Notably, PyPlutchik allows users to display also primary, secondary, tertiary and opposite dyads in a compact, intuitive way. We substantiate our claim that PyPlutchik outperforms other classic visualisations when displaying Plutchik emotions and we showcase a few examples that display our module’s most compelling features

Energetics of critical oscillators in active bacterial baths

We investigate the nonequilibrium energetics near a critical point of a non-linear oscillator immersed in an active bacterial bath. At the critical point, we reveal a scaling exponent of the average power exerted by a constant non-conservative torque 〈W‧ 〉 ∼ (Da/τ)1/4, where Da is the effective diffusivity and τ the correlation time of the bacterial bath described by a Gaussian colored noise. Other features that we investigate are the average stationary power and the variance of the work both below and above the saddle-node bifurcation. Above the bifurcation, the average power attains an optimal, minimum value for finite τ that is below its zero-temperature limit. Furthermore, we reveal a finite-time uncertainty relation for active matter which leads to values of the Fano factor of the work that can be below 2kBTeff, with Teff the effective temperature of the oscillator in the bacterial bath. We analyze different Markovian approximations to describe the nonequilibrium stationary state of the system. Finally, we illustrate our results in the experimental context by considering the example of driven colloidal particles in periodic optical potentials within an E. Coli bacterial bath