A local fluctuation theorem for large systems

The fluctuation theorem characterizes the distribution of the dissipation in nonequilibrium systems and proves that the average dissipation will be positive. For a large system with no external source of fluctuation, fluctuations in properties will become unobservable and details of the fluctuation theorem are unable to be explored. In this letter, we consider such a situation and show how a fluctuation theorem can be obtained for a small open subsystem within the large system. We find that a correction term has to be added to the large system fluctuation theorem due to correlation of the subsystem with the surroundings. Its analytic expression can be derived provided some general assumptions are fulfilled, and its relevance it checked using numerical simulations.Comment: 5 pages, 5 figures; revised and supplementary material include

Variations of Hausdorff Dimension in the Exponential Family

In this paper we deal with the following family of exponential maps $(f_\lambda:z\mapsto \lambda(e^z-1))_{\lambda\in [1,+\infty)}$. Denoting $d(\lambda)$ the hyperbolic dimension of $f_\lambda$. It is known that the function $\lambda\mapsto d(\lambda)$ is real analytic in $(1,+\infty)$, and that it is continuous in $[1,+\infty)$. In this paper we prove that this map is C$^1$ on $[1,+\infty)$, with $d'(1^+)=0$. Moreover, depending on the value of $d(1)$, we give estimates of the speed of convergence towards 0.Comment: 32 pages. A para\^itre dans Annales Academi{\ae} Scientiarum Fennic{\ae} Mathematic

Global well-posedness of a conservative relaxed cross diffusion system

We prove global existence in time of solutions to relaxed conservative cross diffusion systems governed by nonlinear operators of the form $u_i\to \partial_tu_i-\Delta(a_i(\tilde{u})u_i)$ where the $u_i, i=1,...,I$ represent $I$ density-functions, $\tilde{u}$ is a spatially regularized form of $(u_1,...,u_I)$ and the nonlinearities $a_i$ are merely assumed to be continuous and bounded from below. Existence of global weak solutions is obtained in any space dimension. Solutions are proved to be regular and unique when the $a_i$ are locally Lipschitz continuous

Stabilization and controllability of first-order integro-differential hyperbolic equations

In the present article we study the stabilization of first-order linear integro-differential hyperbolic equations. For such equations we prove that the stabilization in finite time is equivalent to the exact controllability property. The proof relies on a Fredholm transformation that maps the original system into a finite-time stable target system. The controllability assumption is used to prove the invertibility of such a transformation. Finally, using the method of moments, we show in a particular case that the controllability is reduced to the criterion of Fattorini

Bifurcations of a large scale circulation in a quasi-bidimensional turbulent flow

We report the experimental study of the bifurcations of a large-scale circulation that is formed over a turbulent flow generated by a spatially periodic forcing. After shortly describing how the flow becomes turbulent through a sequence of symmetry breaking bifurcations, we focus our study on the transitions that occur within the turbulent regime. They are related to changes in the shape of the probability density function (PDF) of the amplitude of the large scale flow. We discuss the nature of these bifurcations and how to model the shape of the PDF.Comment: 6 pages, 9 figure