Microcanonical solution of the mean-field ϕ4\phi^4 model: comparison with time averages at finite size

We solve the mean-field $\phi^4$ model in an external magnetic field in the microcanonical ensemble using two different methods. The first one is based on Rugh's microcanonical formalism and leads to express macroscopic observables, such as temperature, specific heat, magnetization and susceptibility, as time averages of convenient functions of the phase-space. The approach is applicable for any finite number of particles $N$. The second method uses large deviation techniques and allows us to derive explicit expressions for microcanonical entropy and for macroscopic observables in the $N \to\infty$ limit. Assuming ergodicity, we evaluate time averages in molecular dynamics simulations and, using Rugh's approach, we determine the value of macroscopic observables at finite $N$. These averages are affected by a slow time evolution, often observed in systems with long-range interactions. We then show how the finite $N$ time averages of macroscopic observables converge to their corresponding $N\to\infty$ values as $N$ is increased. As expected, finite size effects scale as $N^{-1}$.Comment: 18 pages, 1 figur

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.Comment: 18 pages, 7 figures, (revised version, several minor typos fixed - accepted for publication in Transport Theory and Statistical Physics

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.Comment: 10 pages, 3 eps figure

Coexistence of Josephson oscillations and novel self-trapping regime in optical waveguide arrays

Considering the coherent nonlinear dynamics between two weakly linked optical waveguide arrays, we find the first example of coexistence of Josephson oscillations with a novel 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, moreover, illustrated and confirmed by numerical simulations. This property allows to conceive an optical switch based on the variation of the refractive index of the linking central waveguide.Comment: 4 pages, 4 figure

Controversy about the applicability of Tsallis statistics to the HMF model

Comment to "Nonextensive Thermodynamics and Glassy Behaviour in Hamiltonian Systems" by A. Rapisarda and A. Pluchino, Europhysics News 36, 202 (2005)

Violation of ensemble equivalence in the antiferromagnetic mean-field XY model

It is well known that long-range interactions pose serious problems for the formulation of statistical mechanics. We show in this paper that ensemble equivalence is violated in a simple mean-field model of N fully coupled classical rotators with repulsive interaction (antiferromagnetic XY model). While in the canonical ensemble the rotators are randomly dispersed over all angles, in the microcanonical ensemble a bi-cluster of rotators separated by angle $\pi$, forms in the low energy limit. We attribute this behavior to the extreme degeneracy of the ground state: only one harmonic mode is present, together with N-1 zero modes. We obtain empirically an analytical formula for the probability density function for the angle made by the rotator, which compares extremely well with numerical data and should become exact in the zero energy limit. At low energy, in the presence of the bi-cluster, an extensive amount of energy is located in the single harmonic mode, with the result that the energy temperature relation is modified. Although still linear, $T = \alpha U$, it has the slope $\alpha \approx 1.3$, instead of the canonical value $\alpha =2$.Comment: 12 pages, Latex, 7 Figure

Modulational Estimate for Fermi-Pasta-Ulam Chain Lyapunov Exponents

In the framework of the Fermi-Pasta-Ulam (FPU) model, we show a simple method to give an accurate analytical estimation of the maximal Lyapunov exponent at high energy density. The method is based on the computation of the mean value of the modulational instability growth rates associated to unstable modes. Moreover, we show that the strong stochasticity threshold found in the $\beta$-FPU system is closely related to a transition in tangent space: the Lyapunov eigenvector being more localized in space at high energy.Comment: 4 pages, revtex, 4 ps figures, submitted to PR