238 research outputs found

### Microcanonical solution of the mean-field $\phi^4$ 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

### 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)

### 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

### Dynamics and statistics of simple models with infinite-range attractive interaction

In this paper we review a series of results obtained for 1D and 2D simple
N-body dynamical models with infinite-range attractive interactions and without
short distance singularities. The free energy of both models can be exactly
obtained in the canonical ensemble, while microcanonical results can be derived
from numerical simulations. Both models show a phase transition from a low
energy clustered phase to a high energy gaseous state, in analogy with the
models introduced in the early 70's by Thirring and Hertel. The phase
transition is second order for the 1D model, first order for the 2D model.
Negative specific heat appears in both models near the phase transition point.
For both models, in the presence of a negative specific heat, a cluster of
collapsed particles coexists with a halo of higher energy particles which
perform long correlated flights, which lead to anomalous diffusion. The
dynamical origin of the "superdiffusion" is different in the two models, being
related to particle trapping and untrapping in the cluster in 1D, while in 2D
the channelling of particles in an egg-crate effective potential is responsible
of the effect. Both models are Lyapunov unstable and the maximal Lyapunov
exponent $\lambda$ has a peak just in the region preceeding the phase
transition. Moreover, in the low energy limit $\lambda$ increases
proportionally to the square root of the internal energy, while in the high
energy region it vanishes as $N^{-1/3}$.Comment: 33 pages, Latex2 - 12 Figs - Proceedings of the Conference "The
Chaotic Universe" held in Rome-Pescara in Feb. 199

### 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

### Lyapunov instability and finite size effects in a system with long-range forces

We study the largest Lyapunov exponent $\lambda$ 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 $U_c$, $\lambda$
shows a peak which persists for very large N-values (N=20000). We show, both
numerically and analytically, that chaoticity is strongly related to kinetic
energy fluctuations. In the limit of small energy, $\lambda$ goes to zero with
a N-independent power law: $\lambda \sim \sqrt{U}$. In the continuum limit the
system is integrable in the whole high temperature phase. More precisely, the
behavior $\lambda \sim N^{-1/3}$ is found numerically for $U > U_c$ and
justified on the basis of a random matrix approximation.Comment: 5 pages, Revtex, 3 figures included. Both text and figures have been
changed. New Version accepted for publication in Physical Review Letter

### Analytical Estimation of the Maximal lyapunov Exponent in Oscillator Chains

An analytical expression for the maximal Lyapunov exponent $\lambda_1$ in
generalized Fermi-Pasta-Ulam oscillator chains is obtained. The derivation is
based on the calculation of modulational instability growth rates for some
unstable periodic orbits. The result is compared with numerical simulations and
the agreement is good over a wide range of energy densities $\epsilon$. At very
high energy density the power law scaling of $\lambda_1$ with $\epsilon$ can be
also obtained by simple dimensional arguments, assuming that the system is
ruled by a single time scale. Finally, we argue that for repulsive and hard
core potentials in one dimension $\lambda_1 \sim \sqrt{\epsilon}$ at large
$\epsilon$.Comment: Latex, 10 pages, 5 Figs - Contribution to the Conference "Disorder
and Chaos" held in memory of Giovanni Paladin (Sept. 1997 - Rome) - submitted
to J. de Physiqu

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