Ergodicité, collage et transport anomal

Abstract

International audienceNous nous intéressons à la convergence vers sa moyenne spatiale ergodique de la moyenne temporelle d'une observable d'un flow hamiltonien à un degré et demi de liberté avec espace des phases mixte. L'analyse est faite au travers de l'évolution de la distribution des moyennes en temps fini d'un ensemble de conditions initiales sur la même composante ergodique. Un exposant caractérisant la vitesse de convergence est défini. Les résultats indiquent que pour le système considéré la convergence évolue en $t^{\alpha}$, avec $\alpha=0.45$ pour alors qu'elle évolue en $t^{1/2}$ lorsque la dynamique est globalement chaotique dans l'espace des phases. De même une loi $\alpha=1-\beta/2$ reliant cet exposant $\alpha$ à l'exposant caractéristique du deuxième moment associé aux propriétés de transport $\beta$ est proposée et est vérifiée pour les cas considérés

Anomalous transport and observable average in the standard map

“…Finally, we investigate the relation between α, the characteristic exponent of the evolution of the maximum of the distribution of finite-time observable averages, and µ, the characteristic exponent of the second moment of transport associated 50 to the observable. In [8], a simple law was proposed, namely α = 1 − µ/2, our findings lead to good agreements for two out of the three cases. As a consequence, a slightly more general law is then proposed which captures all features; and then we conclude.…”

“…Measuring this exponent is therefore a good indicator of the nature of transport, but how is it related to the measured transport exponent. In fact in all of our previous computations, as well as in the results presented in [8] to an exponent whose value can be obtained by simply extrapolating the linear behavior of the function µ(q) for small values of q, i.e the small moments linear behavior. This feature was also true in the results presented in [8], where the 200 expression Eq.…”

“…If the 65 system is ergodic we may naturally expect that the average will converge to the phase space average of v with the ergodic measure. In order to assess this statement we shall consider initial conditions in the stochastic sea, and follow trajectories for large times; and following the ideas developed in [8,9], we consider the distribution of finite-time averages. This means that we shall 70 compute averages of v over finite times, namely for the given initial condition…”

“…This paper inscribes itself in this series and tries to tackle the prob- 15 lem of transport using distributions of finite time observable averages and their evolution as the average is computed over larger and larger times. A first attempt using this approach was performed in [8], using a perturbed pendulum as a case study, and it found its roots in the study of advection described in [9], where finite time averages were used to detect sticky parts of trajectories. In 20 the present case we simply use the standard or Taylor-Chirikov map [1].…”

Cross-field chaotic transport of electrons by E × B electron drift instability in Hall thruster

“…Its observed values are D = (0.1, 0.03) for no-reflection and D = (0.18, 0.14) for reflecting boundary. The change in slope around t = 2×10 5 ω −1 pe is related to the different structure formation of the stochastic web, controlling the velocity transport [21,22].…”