Algebraic damping in the one-dimensional Vlasov equation

We investigate the asymptotic behavior of a perturbation around a spatially non homogeneous stable stationary state of a one-dimensional Vlasov equation. Under general hypotheses, after transient exponential Landau damping, a perturbation evolving according to the linearized Vlasov equation decays algebraically with the exponent -2 and a well defined frequency. The theoretical results are successfully tested against numerical $N$-body simulations, corresponding to the full Vlasov dynamics in the large $N$ limit, in the case of the Hamiltonian mean-field model. For this purpose, we use a weighted particles code, which allows us to reduce finite size fluctuations and to observe the asymptotic decay in the $N$-body simulations.Comment: 26 pages, 8 figures; text slightly modified, references added, typos correcte

Macroscopic detection of the strong stochasticity threshold in Fermi-Pasta-Ulam chains of oscillators

The largest Lyapunov exponent of a system composed by a heavy impurity embedded in a chain of anharmonic nearest-neighbor Fermi-Pasta-Ulam oscillators is numerically computed for various values of the impurity mass $M$. A crossover between weak and strong chaos is obtained at the same value $\epsilon_{_T}$ of the energy density $\epsilon$ (energy per degree of freedom) for all the considered values of the impurity mass $M$. The threshold \epsi lon_{_T} coincides with the value of the energy density $\epsilon$ at which a change of scaling of the relaxation time of the momentum autocorrelation function of the impurity ocurrs and that was obtained in a previous work ~[M. Romero-Bastida and E. Braun, Phys. Rev. E {\bf65}, 036228 (2002)]. The complete Lyapunov spectrum does not depend significantly on the impurity mass $M$. These results suggest that the impurity does not contribute significantly to the dynamical instability (chaos) of the chain and can be considered as a probe for the dynamics of the system to which the impurity is coupled. Finally, it is shown that the Kolmogorov-Sinai entropy of the chain has a crossover from weak to strong chaos at the same value of the energy density that the crossover value $\epsilon_{_T}$ of largest Lyapunov exponent. Implications of this result are discussed.Comment: 6 pages, 5 figures, revtex4 styl

Large deviation techniques applied to systems with long-range interactions

We discuss a method to solve models with long-range interactions in the microcanonical and canonical ensemble. The method closely follows the one introduced by Ellis, Physica D 133, 106 (1999), which uses large deviation techniques. We show how it can be adapted to obtain the solution of a large class of simple models, which can show ensemble inequivalence. The model Hamiltonian can have both discrete (Ising, Potts) and continuous (HMF, Free Electron Laser) state variables. This latter extension gives access to the comparison with dynamics and to the study of non-equilibri um effects. We treat both infinite range and slowly decreasing interactions and, in particular, we present the solution of the alpha-Ising model in one-dimension with $0\leq\alpha<1$

Generalized algebra within a nonextensive statistics

By considering generalized logarithm and exponential functions used in nonextensive statistics, the four usual algebraic operators : addition, subtraction, product and division, are generalized. The properties of the generalized operators are investigated. Some standard properties are preserved, e.g., associativity, commutativity and existence of neutral elements. On the contrary, the distributivity law and the opposite element is no more universal within the generalized algebra.Comment: 11 pages, no figure, TeX. Reports on Mathematical Physics (2003), in pres

Consequences of the ion beam irradiation on the chemical durability of thorium phosphate diphosphate – kinetics study

RADIOCHIn the field of the long-term specific immobilization of actinides, thorium phosphate diphosphate (β-TPD), as potential candidate, must respond to several criteria. Among them, the material must present a good resistance to irradiation and keep its initial good properties such as resistance to aqueous alteration. In order to check this later point, sintered samples of β-TUPD solid solutions were pre-irradiated with ion beams with various conditions (fluence, stopping power) then submitted to leaching tests in different media (pH, temperature, complexing reagents, flow rate, ...). The normalized dissolution rates depend significantly on the amorphous fraction (increase by a factor of 10–100 between unirradiated and fully amorphized materials). On the contrary, the pre-irradiation of the samples does not affect the kinetic parameters of the dissolution such as the partial order relative to the proton concentration (n = 0.37 ± 0.01 and n = 0.34 ± 0.01 for unirradiated and fully amorphized samples, respectively) and the activation energy of the reaction of dissolution (Eapp = 49 ± 4 kJ mol−1 and Eapp = 42 ± 4 kJ mol−1 for unirradiated and partly amorphized samples (fA < 1), respectively)

Lyapunov exponent of many-particle systems: testing the stochastic approach

The stochastic approach to the determination of the largest Lyapunov exponent of a many-particle system is tested in the so-called mean-field XY-Hamiltonians. In weakly chaotic regimes, the stochastic approach relates the Lyapunov exponent to a few statistical properties of the Hessian matrix of the interaction, which can be calculated as suitable thermal averages. We have verified that there is a satisfactory quantitative agreement between theory and simulations in the disordered phases of the XY models, either with attractive or repulsive interactions. Part of the success of the theory is due to the possibility of predicting the shape of the required correlation functions, because this permits the calculation of correlation times as thermal averages.Comment: 11 pages including 6 figure

Ensemble Inequivalence in the Spherical Spin Glass Model with Nonlinear Interactions

We investigate the ensemble inequivalence of the spherical spin glass model with nonlinear interactions of polynomial order $p$. This model is solved exactly for arbitrary $p$ and is shown to have first-order phase transitions between the paramagnetic and spin glass or ferromagnetic phases for $p \geq 5$. In the parameter region around the first-order transitions, the solutions give different results depending on the ensemble used for the analysis. In particular, we observe that the microcanonical specific heat can be negative and the phase may not be uniquely determined by the temperature.Comment: 15 pages, 10 figure

The Non--Ergodicity Threshold: Time Scale for Magnetic Reversal

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 figure