arXiv subjects

S. Ruffo

Publications and source records attributed to S. Ruffo.

31 records · Page 2Linked to original sources

arXiv2001

Ensemble inequivalence in systems with long-range interactions

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.

cond-mat.stat-mech↗

arXiv2001

Chaos suppression in the large size limit for long-range systems

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.

cond-mat.stat-mech↗

arXiv2001

Finite-size effects on the Hamiltonian dynamics of the XY-model

The dynamical properties of the finite-size magnetization M in the critical region T<T_{KTB} of the planar rotor model on a L x L square lattice are analyzed by means of microcanonical simulations . The behavior of the q=0 structure factor at high frequencies is consistent with field-theoretical results, but new additional features occur at lower frequencies. The motion of M determines a region of spectral lines and the presence of a central peak, which we attribute to phase diffusion. Near T_{KTB} the diffusion constant scales with system size as D ~ L^{-1.6(3)}.

cond-mat.stat-mech↗

arXiv2000

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.

cond-mat.stat-mech↗

arXiv1999

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 Lévy walks in a transient temporal regime before the system relaxes to equilibrium.

cond-mat.stat-mech↗

arXiv1999

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 $O(1/\sqrt{b})$.

cond-mat.dis-nn↗

arXiv1999

Superdiffusion and Out-of-equilibrium Chaotic Dynamics with Many Degrees of Freedoms

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 Lévy 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.

cond-mat.stat-mech↗

arXiv1999

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.

chao-dyn↗

arXiv1999

An integration scheme for reaction-diffusion models

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.

cond-mat.stat-mech↗

arXiv1998

Damage spreading and Lyapunov exponents in cellular automata

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.

cond-mat.stat-mech↗

arXiv1998

Classical Representation of the 1D Anderson Model

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.

cond-mat.dis-nn↗

arXiv1995

Exact solutions in the FPU oscillator chain

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.

chao-dyn↗

arXiv1995

Universal diffusion near the golden chaos border

We study local diffusion rate $D$ in Chirikov standard map near the critical golden curve. Numerical simulations confirm the predicted exponent $α=5$ for the power law decay of $D$ as approaching the golden curve via principal resonances with period $q_n$ ($D \sim 1/q^α_n$). 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.

chao-dyn↗