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

We consider threshold Boolean gene regulatory networks, where the update function of each gene is described as a majority rule evaluated among the regulators of that gene: it is turned ON when the sum of its regulator contributions is positive (activators contribute positively whereas repressors contribute negatively) and turned OFF when this sum is negative. In case of a tie (when contributions cancel each other out), it is often assumed that the gene keeps it current state. This framework has been successfully used to model cell cycle control in yeast. Moreover, several studies consider stochastic extensions to assess the robustness of such a model.Here, we introduce a novel, natural stochastic extension of the majority rule. It consists in randomly choosing the next value of a gene only in case of a tie. Hence, the resulting model includes deterministic and probabilistic updates. We present variants of the majority rule, including alternate treatments of the tie situation. Impact of these variants on the corresponding dynamical behaviours is discussed. After a thorough study of a class of two-node networks, we illustrate the interest of our stochastic extension using a published cell cycle model. In particular, we demonstrate that steady state analysis can be rigorously performed and can lead to effective predictions; these relate for example to the identification of interactions whose addition would ensure that a specific state is absorbing.

International audienceIn this paper we study the flux through a finite Markov chain of a quantity, that we will call mass, which moves through the states of the chain according to the Markov transition probabilities. Mass is supplied by an external source and accumulates in the absorbing states of the chain. We believe that studying how this conserved quantity evolves through the transient (non-absorbing) states of the chain could be useful for the modelization of open systems whose dynamics has a Markov property

The distribution of finite time observable averages and transport in low dimensional Hamiltonian systems is studied. Finite time observable average distributions are computed, from which an exponent α characteristic of how the maximum of the distributions scales with time is extracted. To link this exponent to transport properties, the characteristic exponent µ(q) of the time evolution of the different moments of order q related to transport are computed. As a testbed for our study the standard map is used. The stochasticity parameter K is chosen so that either phase space is mixed with a chaotic sea and islands of stability or with only a chaotic sea. Our observations lead to a proposition of a law relating the slope in q = 0 of the function µ(q) with the exponent α.

We consider a diffusion model with stochastic porosity for which the average solution exhibits an abnormal transport. In this paper we investigate the relation of such an anomalous diffusive property of the mean value with the behavior of the solution corresponding to each realization of the stochastic porosity. Such a solution will correspond to the actual measurements in an experiment made on a particular tube. The most relevant result of our work is that, although the concentration corresponding to each realization diffuses normally for large times, it experiments on large deviations from the mean value during intermediate times.