We study the dynamics of a system of N classical spins with infinite-range interaction. We show that, if the thermodynamic limit is taken before the infinite-time limit, the system does not relax to the Boltzmann-Gibbs equilibrium, but exhibits different equilibrium properties, characterized by stable non-Gaussian velocity distributions, Lévy walks and dynamical correlation in phase-space. [17,18] and complex systems [19,20] found in the last years, provide further motivation for a generalization of thermodynamics.In this paper we consider a simple model of classical spins with infinite range interactions [21][22][23][24], and we show that, if the thermodynamic limit is performed before the infinite time limit, the system does not relax to the BoltzmannGibbs (BG) equilibrium, but exhibits different equilibrium properties characterized by non-Gaussian velocity distributions, Lévy walks and dynamical correlation in phase-space, and the validity of the zeroth principle of thermodynamics. Our results show some consistency with the predictions of a generalized non-extensive thermodynamics recently proposed [25,26]. The Hamiltonian Mean Field (HMF) model describes a system of N planar classical spins interacting through an infinite-range potential [21]. The Hamiltonian can be written as:where θ i is the ith angle and p i the conjugate variable representing the angular momentum (or the rotational velocity since unit mass is assumed). The interaction is the same as in the ferromagnetic X-Y model [2], though the summation is extended to all couples of spins and not restricted to first neighbors. Following tradition, the coupling constant in the potential is divided by N. This makes H only formally extensive (V ∼ N when N → ∞) [25][26][27][28], since the energy remains non-additive, i.e. the system cannot be trivially divided in two independent sub-systems. The canonical analytical solution of the model predicts a second-order phase transition from a low-energy ferromagnetic phase with magnetization M ∼ 1 (M is the modulus of, to a high-energy one where the spins are homogeneously oriented on the unit circle and M ∼ 0. The caloric curve, i.e. the dependence of the energy density U = E/N on the temperature T , is given by U =2 ) and shown in Fig.1(a). The critical point is at energy density U c = 0.75 corresponding to a critical temperature T c = 0.5 [21]. The dynamical behavior of HMF can be investigated in the microcanonical ensemble by starting the system with water bag initial conditions (WBIC), i.e. θ i = 0 for all i (M = 1) and velocities uniformly distributed, and integrating numerically the equations of motion [22]. As shown in Fig.1(a), microcanonical simulations are in general in good agreement with the canonical ensemble, except for a region below U c , where it has also been found a dynamics characterized by Lévy walks, anomalous diffusion [23] and a negative specific heat [24]. Ensemble inequivalence and negative specific heat have also been found in self-gravitating systems [8], nuclei and atomic clusters [1...

Communication/transportation systems are often subjected to failures and attacks. Here we represent such systems as networks and we study their ability to resist failures (attacks) simulated as the breakdown of a group of nodes of the network chosen at random (chosen accordingly to degree or load). We consider and compare the results for two di erent network topologies: the Erd os-Rà enyi random graph and the Barabà asi-Albert scale-free network. We also discuss brie y a dynamical model recently proposed to take into account the dynamical redistribution of loads after the initial damage of a single node of the network.

The concept of network e ciency, recently proposed to characterize the properties of smallworld networks, is here used to study the e ects of errors and attacks on scale-free networks. Two di erent kinds of scale-free networks, i.e., networks with power law P(k), are considered: (1) scale-free networks with no local clustering produced by the Barabasi-Albert model and (2) scale-free networks with high clustering properties as in the model by Klemm and Eguà luz, and their properties are compared to the properties of random graphs (exponential graphs). By using as mathematical measures the global and the local e ciency we investigate the e ects of errors and attacks both on the global and the local properties of the network. We show that the global e ciency is a better measure than the characteristic path length to describe the response of complex networks to external factors. We ÿnd that, at variance with random graphs, scale-free networks display, both on a global and on a local scale, a high degree of error tolerance and an extreme vulnerability to attacks. In fact, the global and the local e ciency are una ected by the failure of some randomly chosen nodes, though they are extremely sensitive to the removal of the few nodes which play a crucial role in maintaining the network's connectivity.

We study the largest Lyapunov exponent l and the finite size effects of a system of N fully coupled classical particles, which shows a second order phase transition. Slightly below the critical energy density U c , l shows a peak which persists for very large N values ͑N 20 000͒. We show, both numerically and analytically, that chaoticity is strongly related to kinetic energy fluctuations. In the limit of small energy, l goes to zero with an N-independent power law: l ϳ p U. In the continuum limit the system is integrable in the whole high temperature phase. More precisely, the behavior l ϳ N 21͞3 is found numerically for U . U c and justified on the basis of a random matrix approximation.[S0031-9007 (97)05121-1] PACS numbers: 05.45. + b, 03.20. + i, 05.70.Fh

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-ofequilibrium 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. PACS numbers: 05.45.Pq, 05.60.cd, 05.70.Fh Recently, there has been an increasing interest in physical phenomena which violate the central limit theorem such as anomalous diffusion and Lévy walks. These violations are not an exception in Nature and have been observed in many different fields and also in connection with deterministic chaos in low-dimensional systems [1][2][3][4][5]. The availability of more powerful computers has made it possible to study deterministic chaos and subdiffusive motion in systems with many degrees of freedom using nearest-neighbor coupled symplectic maps [6]. In a very recent paper, superdiffusive motion has been found in an N-body Hamiltonian system with long-range couplings [7]. The mechanism underlying this anomalous diffusion is similar to the one proposed by Geisel et al. [1] in "eggcrate" two-dimensional potentials.In this Letter we present a novel study of superdiffusion and Lévy walks in a Hamiltonian system of N fully coupled rotors [called Hamiltonian mean field (HMF)] [8,9]. The new interesting result is that, in HMF, superdiffusion is connected to the presence of quasistationary nonequilibrium states, rather than to the mechanism proposed by Geisel et al. [1] and found also in [7]. Hamiltonian mean field has been used to investigate relaxation to thermodynamical equilibrium for systems with long-range interactions. It has been studied both at a macroscopic level, by means of the canonical formalism, and at a microscopic dynamical level. The canonical ensemble predicts a second-order phase transition from a clustered phase to a homogeneous one [8,9]. On the other hand, microcanonical simulations show strong chaotic behavior in the region below the critical energy; Lyapunov exponents and Kolmogorov-Sinai entropy reach a maximum at the critical point [9]. These results have been confirmed also for long but finite-range interactions [10]. Of particular importance for this Letter are the results obtained in Ref. [9] concerning the discrepancies between microcanonical results and canonical predictions. In fact, numerical simulations performed at constant energy reveal the existence of out-of-equilibrium quasistationary states (QSS) with an extremely slow relaxation to equilibrium. In Ref.[11] these QSS are shown to become stationary solutions in the continuum limit.The main results of this letter are the following: (1) We find evidence of an anomalo...