arXiv subjects

Stefano Ruffo

Publications and source records attributed to Stefano Ruffo.

At least 91 records · Page 5Linked to original sources

arXiv2012

Gravitation in the Microcanonical Ensemble: Appropriate Scaling Leading to Extensivity and Thermalization

We introduce a simple model of hard spheres with gravitational interactions, for which we study a suitable scaling limit. Usual extensive properties are maintained notwithstanding the long range of gravitational interaction. We show that a local thermalization spontaneously emerges within a microcanonical description of the stationary state. In the considered scaling limit, the resulting density profile can be determined in a hydrostatic approach.

cond-mat.stat-mech↗

arXiv2012

One-dimensional lattice of oscillators coupled through power-law interactions: Continuum limit and dynamics of spatial Fourier modes

We study synchronization in a system of phase-only oscillators residing on the sites of a one-dimensional periodic lattice. The oscillators interact with a strength that decays as a power law of the separation along the lattice length and is normalized by a size-dependent constant. The exponent $α$ of the power law is taken in the range $0 \le α<1$. The oscillator frequency distribution is symmetric about its mean (taken to be zero), and is non-increasing on $[0,\infty)$. In the continuum limit, the local density of oscillators evolves in time following the continuity equation that expresses the conservation of the number of oscillators of each frequency under the dynamics. This equation admits as a stationary solution the unsynchronized state uniform both in phase and over the space of the lattice. We perform a linear stability analysis of this state to show that when it is unstable, different spatial Fourier modes of fluctuations have different stability thresholds beyond which they grow exponentially in time with rates that depend on the Fourier modes. However, numerical simulations show that at long times, all the non-zero Fourier modes decay in time, while only the zero Fourier mode (i.e., the "mean-field" mode) grows in time, thereby dominating the instability process and driving the system to a synchronized state. Our theoretical analysis is supported by extensive numerical simulations.

nlin.CD↗

arXiv2012

Linear response theory for long-range interacting systems in quasistationary states

Long-range interacting systems, while relaxing to equilibrium, often get trapped in long-lived quasistationary states which have lifetimes that diverge with the system size. In this work, we address the question of how a long-range system in a quasistationary state (QSS) responds to an external perturbation. We consider a long-range system that evolves under deterministic Hamilton dynamics. The perturbation is taken to couple to the canonical coordinates of the individual constituents. Our study is based on analyzing the Vlasov equation for the single-particle phase space distribution. The QSS represents stable stationary solution of the Vlasov equation in the absence of the external perturbation. In the presence of small perturbation, we linearize the perturbed Vlasov equation about the QSS to obtain a formal expression for the response observed in a single-particle dynamical quantity. For a QSS that is homogeneous in the coordinate, we obtain an explicit formula for the response. We apply our analysis to a paradigmatic model, the Hamiltonian mean-field model, that involves particles moving on a circle under Hamilton dynamics. Our prediction for the response of three representative QSSs in this model (the water-bag QSS, the Fermi-Dirac QSS, and the Gaussian QSS) is found to be in good agreement with $N$-particle simulations for large $N$. We also show the long-time relaxation of the water-bag QSS to the Boltzmann-Gibbs equilibrium state.

cond-mat.stat-mech↗

arXiv2012

Kinetic theory for non-equilibrium stationary states in long-range interacting systems

We study long-range interacting systems perturbed by external stochastic forces. Unlike the case of short-range systems, where stochastic forces usually act locally on each particle, here we consider perturbations by external stochastic fields. The system reaches stationary states where external forces balance dissipation on average. These states do not respect detailed balance and support non-vanishing fluxes of conserved quantities. We generalize the kinetic theory of isolated long-range systems to describe the dynamics of this non-equilibrium problem. The kinetic equation that we obtain applies to plasmas, self-gravitating systems, and to a broad class of other systems. Our theoretical results hold for homogeneous states, but may also be generalized to apply to inhomogeneous states. We obtain an excellent agreement between our theoretical predictions and numerical simulations. We discuss possible applications to describe non-equilibrium phase transitions.

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arXiv2011

Self-consistent inhomogeneous steady states in Hamiltonian mean field dynamics

Long-lived quasistationary states, associated with stationary stable solutions of the Vlasov equation, are found in systems with long-range interactions. Studies of the relaxation time in a model of $N$ globally coupled particles moving on a ring, the Hamiltonian Mean Field model (HMF), have shown that it diverges as $N^γ$ for large $N$, with $γ\simeq 1.7$ for some initial conditions with homogeneously distributed particles. We propose a method for identifying exact inhomogeneous steady states in the thermodynamic limit, based on analysing models of uncoupled particles moving in an external field. For the HMF model, we show numerically that the relaxation time of these states diverges with $N$ with the exponent $γ\simeq 1$. The method, applicable to other models with globally coupled particles, also allows an exact evaluation of the stability limit of homogeneous steady states. In some cases it provides a good approximation for the correspondence between the initial condition and the final steady state.

cond-mat.stat-mech↗

arXiv2011

Phase transitions of quasistationary states in the Hamiltonian Mean Field model

The out-of-equilibrium dynamics of the Hamiltonian Mean Field (HMF) model is studied in presence of an externally imposed magnetic field h. Lynden-Bell's theory of violent relaxation is revisited and shown to adequately capture the system dynamics, as revealed by direct Vlasov based numerical simulations in the limit of vanishing field. This includes the existence of an out-of-equilibrium phase transition separating magnetized and non magnetized phases. We also monitor the fluctuations in time of the magnetization, which allows us to elaborate on the choice of the correct order parameter when challenging the performance of Lynden-Bell's theory. The presence of the field h removes the phase transition, as it happens at equilibrium. Moreover, regions with negative susceptibility are numerically found to occur, in agreement with the predictions of the theory.

cond-mat.stat-mech↗

arXiv2011

Vlasov equation for long-range interactions on a lattice

We show that, in the continuum limit, the dynamics of Hamiltonian systems defined on a lattice with long-range couplings is well described by the Vlasov equation. This equation can be linearized around the homogeneous state and a dispersion relation, that depends explicitly on the Fourier modes of the lattice, can be derived. This allows one to compute the stability thresholds of the homogeneous state, which turn out to depend on the mode number. When this state is unstable, the growth rates are also function of the mode number. Explicit calculations are performed for the $α$-HMF model with $0 \leq α<1$, for which the zero mean-field mode is always found to dominate the exponential growth. The theoretical predictions are successfully compared with numerical simulations performed on a finite lattice.

nlin.CD↗

arXiv2010

Scaling with system size of the Lyapunov exponents for the Hamiltonian Mean Field model

The Hamiltonian Mean Field (HMF) model is a prototype for systems with long-range interactions. It describes the motion of $N$ particles moving on a ring, coupled through an infinite-range potential. The model has a second order phase transition at the energy $U_c=3/4$ and its dynamics is exactly described by the Vlasov equation in the $N \to \infty$ limit. Its chaotic properties have been investigated in the past, but the determination of the scaling with $N$ of the Lyapunov Spectrum (LS) of the model remains a challenging open problem. We here show that the $N^{-1/3}$ scaling of the Maximal Lyapunov Exponent (MLE), found in previous numerical and analytical studies, extends to the full LS; not only, scaling is "precocious" for the LS, meaning that it becomes manifest for a much smaller number of particles than the one needed to check the scaling for the MLE. Besides that, the $N^{-1/3}$ scaling appears to be valid not only for $U>U_c$, as suggested by theoretical approaches based on a random matrix approximation, but also below a threshold energy $U_t \approx 0.2$. Using a recently proposed method (GALI) devised to rapidly check the chaotic or regular nature of an orbit, we find that $U_t$ is also the energy at which a sharp transition from {\it weak} to {\it strong} chaos is present in the phase-space of the model. Around this energy the phase of the vector order parameter of the model becomes strongly time dependent, inducing a significant untrapping of particles from a nonlinear resonance.

nlin.CD↗

arXiv2010

Statistical mechanics of collisionless relaxation in a noninteracting system

We introduce a model of uncoupled pendula, which mimics the dynamical behavior of the Hamiltonian Mean Field (HMF) model. This model has become a paradigm for long-range interactions, like Coulomb or dipolar forces. As in the HMF model, this simplified integrable model is found to obey the Vlasov equation and to exhibit Quasi Stationary States (QSS), which arise after a "collisionless" relaxation process. Both the magnetization and the single particle distribution function in these QSS can be predicted using Lynden-Bell's theory. The existence of an extra conserved quantity for this model, the energy distribution function, allows us to understand the origin of some discrepancies of the theory with numerical experiments. It also suggests us an improvement of Lynden-Bell's theory, which we fully implement for the zero field case.

cond-mat.stat-mech↗

arXiv2010

Models with short and long-range interactions: phase diagram and reentrant phase

We study the phase diagram of two different Hamiltonians with competiting local, nearest-neighbour, and mean-field couplings. The first example corresponds to the HMF Hamiltonian with an additional short-range interaction. The second example is a reduced Hamiltonian for dipolar layered spin structures, with a new feature with respect to the first example, the presence of anisotropies. The two examples are solved in both the canonical and the microcanonical ensemble using a combination of the min-max method with the transfer operator method. The phase diagrams present typical features of systems with long-range interactions: ensemble inequivalence, negative specific heat and temperature jumps. Moreover, in a given range of parameters, we report the signature of phase reentrance. This can also be interpreted as the presence of azeotropy with the creation of two first order phase transitions with ensemble inequivalence, as one parameter is varied continuously.

cond-mat.stat-mech↗

arXiv2009

Finite Size Effects for the Ising Model on Random Graphs with Varying Dilution

We investigate the finite size corrections to the equilibrium magnetization of an Ising model on a random graph with $N$ nodes and $N^γ$ edges, with $1 < γ\leq 2$. By conveniently rescaling the coupling constant, the free energy is made extensive. As expected, the system displays a phase transition of the mean-field type for all the considered values of $γ$ at the transition temperature of the fully connected Curie-Weiss model. Finite size corrections are investigated for different values of the parameter $γ$, using two different approaches: a replica-based finite $N$ expansion, and a cavity method. Numerical simulations are compared with theoretical predictions. The cavity based analysis is shown to agree better with numerics.

cond-mat.stat-mech↗

arXiv2009

Abundance of regular orbits and out-of-equilibrium phase transitions in the thermodynamic limit for long-range systems

We investigate the dynamics of many-body long-range interacting systems, taking the Hamiltonian Mean Field model as a case study. We show that an abundance of regular trajectories, associated with invariant tori of the single-particle dynamics, exists. The presence of such tori provides a dynamical interpretation of the emergence of long-lasting out-of-equilibrium regimes observed generically in long-range systems. This is alternative to a previous statistical mechanics approach to such phenomena which was based on a maximum entropy principle. Previously detected out-of-equilibrium phase transitions are also reinterpreted within this framework.

cond-mat.stat-mech↗

arXiv2008

Stochastic Resonance in the Fermi-Pasta-Ulam Chain

We consider a damped $β$-Fermi-Pasta-Ulam chain, driven at one boundary subjected to stochastic noise. It is shown that, for a fixed driving amplitude and frequency, increasing the noise intensity, the system's energy resonantly responds to the modulating frequency of the forcing signal. Multiple peaks appear in the signal to noise ratio, signalling the phenomenon of stochastic resonance. The presence of multiple peaks is explained by the existence of many stable and metastable states that are found when solving this boundary value problem for a semi-continuum approximation of the model. Stochastic resonance is shown to be generated by transitions between these states.

nlin.PS↗

arXiv2007

Out of equilibrium phase transitions in mean field Hamiltonian dynamics

Systems with long-range interactions display a short-time relaxation towards Quasi-Stationary States (QSSs), whose lifetime increases with system size. With reference to the Hamiltonian Mean Field (HMF) model, we here review Lynden-Bell's theory of ``violent relaxation''. The latter results in a maximum entropy scheme for a water-bag initial profile which predicts the presence of out-of-equilibrium phase transitions} separating homogeneous (zero magnetization) from inhomogeneous (non-zero magnetization) QSSs. Two different parametric representations of the initial condition are analyzed and the features of the phase diagram are discussed. In both representations we find a second order and a first order line of phase transitions that merge at a tricritical point. Particular attention is payed to the condition of existence and stability of the homogenous phase.

cond-mat.stat-mech↗

arXiv2007

Nonlinear Dynamics in Double Square Well Potential

Considering the coherent nonlinear dynamics in double square well potential we find the example of coexistence of Josephson oscillations with a self-trapping regime. This macroscopic bistability is explained by proving analytically the simultaneous existence of symmetric, antisymmetric and asymmetric stationary solutions of the associated Gross-Pitaevskii equation. The effect is illustrated and confirmed by numerical simulations. This property allows to make suggestions on possible experiments using Bose-Einstein condensates in engineered optical lattices or weakly coupled optical waveguide arrays.

nlin.PS↗

arXiv2007

The Hamiltonian Mean Field model: anomalous or normal diffusion?

We consider the out-of-equilibrium dynamics of the Hamiltonian Mean Field (HMF) model, by focusing in particular on the properties of single-particle diffusion. As we shall here demonstrate analytically, if the autocorrelation of momenta in the so-called quasi-stationary states can be fitted by a q-exponential, then diffusion ought to be normal for q<2, at variance with the interpretation of the numerical experiments proposed in Refs. A. Pluchino, A. Rapisarda, Progress in Theoretical Physics Supplement 162, 18 (2006); A. Pluchino, V. Latora, A. Rapisarda, Physica A 338, 60 (2004); A. Rapisarda, A. Pluchino, Europhysics News 36, 202 (2005).

cond-mat.stat-mech↗

arXiv2007

Equilibrium and nonequilibrium properties of systems with long-range interactions

We briefly review some equilibrium and nonequilibrium properties of systems with long-range interactions. Such systems, which are characterized by a potential that weakly decays at large distances, have striking properties at equilibrium, like negative specific heat in the microcanonical ensemble, temperature jumps at first order phase transitions, broken ergodicity. Here, we mainly restrict our analysis to mean-field models, where particles globally interact with the same strength. We show that relaxation to equilibrium proceeds through quasi-stationary states whose duration increases with system size. We propose a theoretical explanation, based on Lynden-Bell's entropy, of this intriguing relaxation process. This allows to address problems related to nonequilibrium using an extension of standard equilibrium statistical mechanics. We discuss in some detail the example of the dynamics of the free electron laser, where the existence and features of quasi-stationary states is likely to be tested experimentally in the future. We conclude with some perspectives to study open problems and to find applications of these ideas to dipolar media.

cond-mat.stat-mech↗

arXiv2007

Phase space gaps and ergodicity breaking in systems with long range interactions

We study a generalized isotropic XY-model which includes both two-spin and four-spin mean-field interactions. This model can be solved in the microcanonical ensemble. It is shown that in certain parameter regions the model exhibits gaps in the magnetization at fixed energy, resulting in ergodicity breaking. This phenomenon has previously been reported in anisotropic and discrete spin models. The entropy of the model is calculated and the microcanonical phase diagram is derived, showing the existence of first order phase transitions from the ferromagnetic to a paramagnetic disordered phase. It is found that ergodicity breaking takes place both in the ferromagnetic and the paramagnetic phases. As a consequence, the system can exhibit a stable ferromagnetic phase within the paramagnetic region, and conversely a disordered phase within the magnetically ordered region.

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