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We study the dynamics of Fermi-Pasta-Ulam chains with both harmonic and anharmonic power-law long-range interactions. We show that the dynamics is described in the continuum limit by a generalized fractional Boussinesq differential equation, whose derivation is performed in full detail. We also discuss a version of the model where couplings are alternating in sign., Comment: Corrected coefficients of the nonlinear terms of the generalized fractional Boussinesq equations, minor typos amended

We study inequivalence of canonical and microcanonical ensembles in the mean-field Blume-Emery-Griffiths model. This generalizes previous results obtained for the Blume-Capel model. The phase diagram strongly depends on the value of the biquadratic exchange interaction K, the additional feature present in the Blume-Emery-Griffiths model. At small values of K, as for the Blume-Capel model, lines of first and second order phase transitions between a ferromagnetic and a paramagnetic phase are present, separated by a tricritical point whose location is different in the two ensembles. At higher values of K the phase diagram changes substantially, with the appearance of a triple point in the canonical ensemble which does not find any correspondence in the microcanonical ensemble. Moreover, one of the first order lines that starts from the triple point ends in a critical point, whose position in the phase diagram is different in the two ensembles. This line separates two paramagnetic phases characterized by a different value of the quadrupole moment. These features were not previously studied for other models and substantially enrich the landscape of ensemble inequivalence, identifying new aspects that had been discussed in a classification of phase transitions based on singularity theory. Finally, we discuss ergodicity breaking, which is highlighted by the presence of gaps in the accessible values of magnetization at low energies: it also displays new interesting patterns that are not present in the Blume-Capel model., Comment: Small additions in the Conclusions

We present a detailed analysis of the modulational instability of the zone-boundary mode for one and higher-dimensional Fermi--Pasta--Ulam (FPU) lattices. The growth of the instability is followed by a process of relaxation to equipartition, which we have called the Anti-FPU problem because the energy is initially fed into the highest frequency part of the spectrum, while in the original FPU problem low frequency excitations of the lattice were considered. This relaxation process leads to the formation of chaotic breathers in both one and two space dimensions. The system then relaxes to energy equipartition, on time scales that increase as the energy density is decreased. We supplement this study by considering the nonconservative case, where the FPU lattice is homogeneously driven at high frequencies. Standing and travelling nonlinear waves and solitonic patterns are detected in this case. Finally we investigate the dynamics of the FPU chain when one end is driven at a frequency located above the zone boundary. We show that this excitation stimulates nonlinear bandgap transmission effects., Comment: arXiv admin note: substantial text overlap with arXiv:nlin/0409049, arXiv:cond-mat/0407134

Understanding the relaxation of a system towards equilibrium is a longstanding problem in statistical mechanics. Here we address the role of long-range interactions in this process by considering a class of two-dimensional or geophysical flows where the interaction between fluid particles varies with the distance as $\sim$ r^($\alpha$--2) with $\alpha$ \textgreater{} 0. Previous studies in the Euler case $\alpha$ = 2 had shown convergence towards a variety of quasi-stationary states by changing the initial state. Unexpectedly, all those regimes are recovered by changing $\alpha$ with a prescribed initial state. For small $\alpha$, a coarsening process leads to the formation of a sharp interface between two regions of homogenized $\alpha$-vorticity; for large $\alpha$, the flow is attracted to a stable dipolar structure through a filamentation process.

We have studied needle shaped three-dimensional classical spin systems with purely dipolar interactions in the microcanonical ensemble, using both numerical simulations and analytical approximations. We have observed spontaneous magnetization for different finite cubic lattices. The transition from the paramagnetic to the ferromagnetic phase is shown to be first-order. For two lattice types we have observed magnetization flips in the phase transition region. In some cases, gaps in the accessible values of magnetization appear, a signature of the ergodicity breaking found for systems with long-range interactions. We analytically explain these effects by performing a nontrivial mapping of the model Hamiltonian onto a one-dimensional Ising model with competing antiferromagnetic nearest-neighbor and ferromagnetic mean-field interactions. These results hint at performing experiments on isolated dipolar needles in order to verify some of the exotic properties of systems with long-range interactions in the microcanonical ensemble.

In a coupled ferroelectric/ferromagnetic system, i.e. a composite multiferroic, the propagation of magnetic or ferroelectric excitations across the whole structure is a key issue for applications. Of a special interest is the dynamics of localized magnetic or ferroelectric modes (LM) across the ferroelectric-ferromagnetic interface, particularly when the LM's carrier frequency is in the band of the ferroelectric and in the band gap of the ferromagnet. For a proper choice of the system's parameters, we find that there is a threshold amplitude above which the interface becomes transparent and a band gap ferroelectric LM penetrates the ferromagnetic array. Below that threshold, the LM is fully reflected. Slightly below this transmission threshold, the addition of noise may lead to energy transmission, provided that the noise level is not too low nor too high, an effect that resembles stochastic resonance. These findings represent an important step towards the application of ferroelectric and/or ferromagnetic LM-based logic., Comment: 5 pages, 4 figures. Phys. Rev. Lett, (2013)

The Vlasov equation is well known to provide a good description of the dynamics of mean-field systems in the $N \to \infty$ limit. This equation has an infinity of stationary states and the case of {\it homogeneous} states, for which the single-particle distribution function is independent of the spatial variable, is well characterized analytically. On the other hand, the inhomogeneous case often requires some approximations for an analytical treatment: the dynamics is then best treated in action-angle variables, and the potential generating inhomogeneity is generally very complex in these new variables. We here treat analytically the linear stability of toy-models where the inhomogeneity is created by an external field. Transforming the Vlasov equation into action-angle variables, we derive a dispersion relation that we accomplish to solve for both the growth rate of the instability and the stability threshold for two specific models: the Hamiltonian Mean-Field model with additional asymmetry and the mean-field $\phi^4$ model. The results are compared with numerical simulations of the $N$-body dynamics. When the {\it inhomogeneous} state is stationary, we expect to observe in the $N$-body dynamics Quasi-Stationary-States (QSS), whose lifetime diverge algebraically with $N$.

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 content

We calculate the thermodynamic entropy of the mean-field $\phi^4$ spin model in the microcanonical ensemble as a function of the energy and magnetization of the model. The entropy and its derivative are obtained from the theory of large deviations, as well as from Rugh's microcanonical formalism, which is implemented by computing averages of suitable observables in microcanonical molecular dynamics simulations. Our main finding is that the entropy is a concave function of the energy for all values of the magnetization, but is nonconcave as a function of the magnetization for some values of the energy. This last property implies that the magnetic susceptibility of the model can be negative when calculated microcanonically for fixed values of the energy and magnetization. This provides a magnetization analog of negative heat capacities, which are well-known to be associated in general with the nonequivalence of the microcanonical and canonical ensembles. Here, the two ensembles that are nonequivalent are the microcanonical ensemble in which the energy and magnetization are held fixed and the canonical ensemble in which the energy and magnetization are fixed only on average by fixing the temperature and magnetic field., Comment: 14 pages, 4 figures, 2 appendices, REVTeX4

The thermodynamic and dynamical properties of an Ising model with both short range and long range, mean field like, interactions are studied within the microcanonical ensemble. It is found that the relaxation time of thermodynamically unstable states diverges logarithmically with system size. This is in contrast with the case of short range interactions where this time is finite. Moreover, at sufficiently low energies, gaps in the magnetization interval may develop to which no microscopic configuration corresponds. As a result, in local microcanonical dynamics the system cannot move across the gap, leading to breaking of ergodicity even in finite systems. These are general features of systems with long range interactions and are expected to be valid even when the interaction is slowly decaying with distance., Comment: 4 pages, 5 figures

We study the distribution of the resistance fluctuations of biased resistor networks in nonequilibrium steady states. The stationary conditions arise from the competition between two stochastic and biased processes of breaking and recovery of the elementary resistors. The fluctuations of the network resistance are calculated by Monte Carlo simulations which are performed for different values of the applied current, for networks of different size and shape and by considering different levels of intrinsic disorder. The distribution of the resistance fluctuations generally exhibits relevant deviations from Gaussianity, in particular when the current approaches the threshold of electrical breakdown. For two-dimensional systems we have shown that this non-Gaussianity is in general related to finite size effects, thus it vanishes in the thermodynamic limit, with the remarkable exception of highly disordered networks. For these systems, close to the critical point of the conductor-insulator transition, non-Gaussianity persists in the large size limit and it is well described by the universal Bramwell-Holdsworth-Pinton distribution. In particular, here we analyze the role of the shape of the network on the distribution of the resistance fluctuations. Precisely, we consider quasi-one-dimensional networks elongated along the direction of the applied current or trasversal to it. A significant anisotropy is found for the properties of the distribution. These results apply to conducting thin films or wires with granular structure stressed by high current densities., Comment: 8 pages, 4 figures. Invited talk at the 18-th International Conference on Noise and Fluctuations, 19-23 September 2005, Salamanca

We prove the existence of a non-ergodicity threshold for an anisotropic classical Heisenberg model with all-to-all couplings. Below the threshold, the energy surface is disconnected in two components with positive and negative magnetizations respectively. Above, in a fully chaotic regime, magnetization changes sign in a stochastic way and its behavior can be fully characterized by an average magnetization reversal time. We show that statistical mechanics predicts a phase--transition at an energy higher than the non-ergodicity threshold. We assess the dynamical relevance of the latter for finite systems through numerical simulations and analytical calculations. In particular, the time scale for magnetic reversal diverges as a power law at the ergodicity threshold with a size-dependent exponent, which could be a signature of the phenomenon., Comment: 4 pages 4 figures

We study the distribution of resistance fluctuations of conducting thin films with different levels of internal disorder. The film is modeled as a resistor network in a steady state determined by the competition between two biased processes, breaking and recovery of the elementary resistors. The fluctuations of the film resistance are calculated by Monte Carlo simulations which are performed under different bias conditions, from the linear regime up to the threshold for electrical breakdown. Depending on the value of the external current, on the level of disorder and on the size of the system, the distribution of the resistance fluctuations can exhibit significant deviations from Gaussianity. As a general trend, a size dependent, non universal distribution is found for systems with low and intermediate disorder. However, for strongly disordered systems, close to the critical point of the conductor-insulator transition, the non-Gaussianity persists when the size is increased and the distribution of resistance fluctuations is well described by the universal Bramwell-Holdsworth-Pinton distribution., Comment: 6 pages, 7 figures, 2th. Int. SPIE Symp. on Fluctuations and Noise, Maspalomas (Spain) 25-28 May, 2004

We show that recent observations of fractal dimensions in the $\mu$-space of $N$-body Hamiltonian systems with long-range interactions are due to finite $N$ and finite resolution effects. We provide strong numerical evidence that, in the continuum (Vlasov) limit, a set which initially is not a fractal (e.g. a line in 2D) remains such for all finite times. We perform this analysis for the Hamiltonian Mean Field (HMF) model, which describes the motion of a system of $N$ fully coupled rotors. The analysis can be indirectly confirmed by studying the evolution of a large set of initial points for the Chirikov standard map., Comment: Proceedings of the Vlasovia Workshop, Nancy, 26-28 November 2003. to be published on Transport Theory and Statistical Mechanics (Kluwer)

The distribution of resistance fluctuations of conducting thin films with granular structure near electrical breakdown is studied by numerical simulations. The film is modeled as a resistor network in a steady state determined by the competition between two biased processes, breaking and recovery. Systems of different sizes and with different levels of internal disorder are considered. Sharp deviations from a Gaussian distribution are found near breakdown and the effect increases with the degree of internal disorder. However, we show that in general this non-Gaussianity is related to the finite size of the system and vanishes in the large size limit. Nevertheless, near the critical point of the conductor-insulator transition, deviations from Gaussianity persist when the size is increased and the distribution of resistance fluctuations is well fitted by the universal Bramwell-Holdsworth-Pinton distribution., Comment: 8 pages, 6 figures; accepted for publication on Physica A

We perform a detailed study of the relaxation towards equilibrium in the Hamiltonian Mean-Field (HMF) model, a prototype for long-range interactions in $N$-particle dynamics. In particular, we point out the role played by the infinity of stationary states of the associated $N ~$ Vlasov dynamics. In this context, we derive a new general criterion for the stability of any spatially homogeneous distribution, and compare its analytical predictions with numerical simulations of the Hamiltonian, finite $N$, dynamics. We then propose and verify numerically a scenario for the relaxation process, relying on the Vlasov equation. When starting from a non stationary or a Vlasov unstable stationary initial state, the system shows initially a rapid convergence towards a stable stationary state of the Vlasov equation via non stationary states: we characterize numerically this dynamical instability in the finite $N$ system by introducing appropriate indicators. This first step of the evolution towards Boltzmann-Gibbs equilibrium is followed by a slow quasi-stationary process, that proceeds through different stable stationary states of the Vlasov equation. If the finite $N$ system is initialized in a Vlasov stable homogenous state, it remains trapped in a quasi-stationary state for times that increase with the nontrivial power law $N^{1.7}$. Single particle momentum distributions in such a quasi-stationary regime do not have power-law tails, and hence cannot be fitted by the $q$-exponential distributions derived from Tsallis statistics., Comment: To appear in Physica A

We study the resistance fluctuation distribution of a thin film near electrical breakdown. The film is modeled as a stationary resistor networkunder biased percolation. Depending on the value of the external current,on the system sizes and on the level of internal disorder, the fluctuation distribution can exhibit a non-Gaussian behavior. We analyze this non-Gaussianity in terms of the generalized Gumbel distribution recently introduced in the context of highly correlated systems near criticality. We find that when the average fraction of defects approaches the random percolation threshold, the resistance fluctuation distribution is well described by the universal behavior of the Bramwell-Holdsworth-Pinton distribution., Comment: 3 figures, accepted for publication on Semicond Sci Tech

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. A

Ensemble inequivalence has been observed in several systems. In particular it has been recently shown that negative specific heat can arise in the microcanonical ensemble in the thermodynamic limit for systems with long-range interactions. We display a connection between such behaviour and a mean-field like structure of the partition function. Since short-range models cannot display this kind of behaviour, this strongly suggests that such systems are necessarily non-mean field in the sense indicated here. We illustrate our results showing an application to the Blume-Emery-Griffiths model. We further show that a broad class of systems with non-integrable interactions are indeed of mean-field type in the sense specified, so that they are expected to display ensemble inequivalence as well as the peculiar behaviour described above in the microcanonical ensemble., Comment: 12 pages, no figures

Ensemble inequivalence has been observed in several systems. In particular it has been recently shown that negative specific heat can arise in the microcanonical ensemble in the thermodynamic limit for systems with long-range interactions. We display a connection between such behaviour and a mean-field like structure of the partition function. Since short-range models cannot display this kind of behaviour, this strongly suggests that such systems are necessarily non-mean field in the sense indicated here. We further show that a broad class of systems with non-integrable interactions are indeed of mean-field type in the sense specified, so that they are expected to display ensemble inequivalence as well as the peculiar behaviour described above in the microcanonical ensemble., Comment: 4 pages, no figures, given at the NEXT2001 conference on non-extensive thermodynamics

We consider the class of long-range Hamiltonian systems first introduced by Anteneodo and Tsallis and called the alpha-XY model. This involves N classical rotators on a d-dimensional periodic lattice interacting all to all with an attractive coupling whose strength decays as r^{-alpha}, r being the distances between sites. Using a recent geometrical approach, we estimate for any d-dimensional lattice the scaling of the largest Lyapunov exponent (LLE) with N as a function of alpha in the large energy regime where rotators behave almost freely. We find that the LLE vanishes as N^{-kappa}, with kappa=1/3 for alpha/d between 0 and 1/2 and kappa=2/3(1-alpha/d) for alpha/d between 1/2 and 1. These analytical results present a nice agreement with numerical results obtained by Campa et al., including deviations at small N., Comment: 10 pages, 3 eps figures

The dynamical properties of the finite-size magnetization M in the critical region T

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 suppl

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 A

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''

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. E

A detailed description and validation of a recently developed integration scheme is here reported for one- and two-dimensional reaction-diffusion models. As paradigmatic examples of this class of partial differential equations the complex Ginzburg-Landau and the Fitzhugh-Nagumo equations have been analyzed. The novel algorithm has precision and stability comparable to those of pseudo-spectral codes, but it is more convenient to employ for systems with quite large linear extention $L$. As for finite-difference methods, the implementation of the present scheme requires only information about the local enviroment and this allows to treat also system with very complicated boundary conditions., Comment: 14 page, Latex - 4 EPS Figs - Submitted to Int. J. Mod. Phys. C

We study the link between relaxation to the equilibrium and anomalous superdiffusive motion in a classical N-body hamiltonian system with long-range interaction showing a second-order phase-transition in the canonical ensemble. Anomalous diffusion is observed only in a transient out-of-equilibrium regime and for a small range of energy, below the critical one. Superdiffusion is due to L\'evy walks of single particles and is checked independently through the second moment of the distribution, power spectra, trapping and walking time probabilities. Diffusion becomes normal at equilibrium, after a relaxation time which diverges with N., Comment: 5 pages, 4 postscript figures included, revtex, revised version refs. and a few sentences added, fig.3 revised, typos corrected. Accepted for publication in Phys. Rev. Lett., september (1999)

Using the concept of the Boolean derivative we study damage spreading for one dimensional elementary cellular automata and define their maximal Lyapunov exponent. A random matrix approximation describes quite well the behavior of ``chaotic'' rules and predicts a directed percolation-type phase transition. After the introduction of a small noise elementary cellular automata reveal the same type of transition.

A new approach is applied to the 1D Anderson model by making use of a two-dimensional Hamiltonian map. For a weak disorder this approach allows for a simple derivation of correct expressions for the localization length both at the center and at the edge of the energy band, where standard perturbation theory fails. Approximate analytical expressions for strong disorder are also obtained., Comment: 10 pages in LaTex

After a brief comprehensive review of old and new results on the well known Fermi-Pasta-Ulam (FPU) conservative system of $N$ nonlinearly coupled oscillators, we present a compact linear mode representation of the Hamiltonian of the FPU system with quartic nonlinearity and periodic boundary conditions, with explicitly computed mode coupling coefficients. The core of the paper is the proof of the existence of one-mode and two-mode exact solutions, physically representing nonlinear standing and travelling waves of small wavelength whose explicit lattice representations are obtained, and which are valid also as $N \rightarrow \infty$. Moreover, and more generally, we show the presence of multi-mode invariant submanifolds. Destabilization of these solutions by a parametric perturbation mechanism leads to the establishment of chaotic in time mode interaction channels, corresponding to the formation in phase space of bounded stochastic layers on submanifolds. The full mode-space stability problem of the $N/2$ zone-boundary mode is solved, showing that this mode becomes unstable through a mechanism of the modulational Benjamin-Feir type. In the thermodynamic limit the mode is always unstable but with instability growth rate linearly vanishing with energy density. The physical significance of these solutions and of their stability properties, with respect to the previously much more studied equipartition problem for long wavelength initial excitations, is briefly discussed., Comment: Plain LaTeX, 39 pages, 6 Postscript figures available upon request from ruffo@ingfi1.ing.unifi.it

We study local diffusion rate $D$ in Chirikov standard map near the critical golden curve. Numerical simulations confirm the predicted exponent $\alpha=5$ for the power law decay of $D$ as approaching the golden curve via principal resonances with period $q_n$ ($D \sim 1/q^{\alpha}_n$). The universal self-similar structure of diffusion between principal resonances is demonstrated and it is shown that resonances of other type play also an important role., Comment: 4 pages Latex, revtex, 3 uuencoded postscript figures