A review of linear response theory for general differentiable dynamical systems

The classical theory of linear response applies to statistical mechanics close to equilibrium. Away from equilibrium, one may describe the microscopic time evolution by a general differentiable dynamical system, identify nonequilibrium steady states (NESS), and study how these vary under perturbations of the dynamics. Remarkably, it turns out that for uniformly hyperbolic dynamical systems (those satisfying the "chaotic hypothesis"), the linear response away from equilibrium is very similar to the linear response close to equilibrium: the Kramers-Kronig dispersion relations hold, and the fluctuation-dispersion theorem survives in a modified form (which takes into account the oscillations around the "attractor" corresponding to the NESS). If the chaotic hypothesis does not hold, two new phenomena may arise. The first is a violation of linear response in the sense that the NESS does not depend differentiably on parameters (but this nondifferentiability may be hard to see experimentally). The second phenomenon is a violation of the dispersion relations: the susceptibility has singularities in the upper half complex plane. These "acausal" singularities are actually due to "energy nonconservation": for a small periodic perturbation of the system, the amplitude of the linear response is arbitrarily large. This means that the NESS of the dynamical system under study is not "inert" but can give energy to the outside world. An "active" NESS of this sort is very different from an equilibrium state, and it would be interesting to see what happens for active states to the Gallavotti-Cohen fluctuation theorem.Comment: 19 pages, 2 figure

Oseledets' Splitting of Standard-like Maps

For the class of differentiable maps of the plane and, in particular, for standard-like maps (McMillan form), a simple relation is shown between the directions of the local invariant manifolds of a generic point and its contribution to the finite-time Lyapunov exponents (FTLE) of the associated orbit. By computing also the point-wise curvature of the manifolds, we produce a comparative study between local Lyapunov exponent, manifold's curvature and splitting angle between stable/unstable manifolds. Interestingly, the analysis of the Chirikov-Taylor standard map suggests that the positive contributions to the FTLE average mostly come from points of the orbit where the structure of the manifolds is locally hyperbolic: where the manifolds are flat and transversal, the one-step exponent is predominantly positive and large; this behaviour is intended in a purely statistical sense, since it exhibits large deviations. Such phenomenon can be understood by analytic arguments which, as a by-product, also suggest an explicit way to point-wise approximate the splitting.Comment: 17 pages, 11 figure

Fast numerical test of hyperbolic chaos

The effective numerical method is developed performing the test of the hyperbolicity of chaotic dynamics. The method employs ideas of algorithms for covariant Lyapunov vectors but avoids their explicit computation. The outcome is a distribution of a characteristic value which is bounded within the unit interval and whose zero indicate the presence of tangency between expanding and contracting subspaces. To perform the test one needs to solve several copies of equations for infinitesimal perturbations whose amount is equal to the sum of numbers of positive and zero Lyapunov exponents. Since for high-dimensional system this amount is normally much less then the full phase space dimension, this method provide the fast and memory saving way for numerical hyperbolicity test of such systems.Comment: 4 pages and 4 figure

Weak chaos detection in the Fermi-Pasta-Ulam-α\alpha system using qq-Gaussian statistics

We study numerically statistical distributions of sums of orbit coordinates, viewed as independent random variables in the spirit of the Central Limit Theorem, in weakly chaotic regimes associated with the excitation of the first ($k=1$) and last ($k=N$) linear normal modes of the Fermi-Pasta-Ulam-$\alpha$ system under fixed boundary conditions. We show that at low energies ($E=0.19$), when the $k=1$ linear mode is excited, chaotic diffusion occurs characterized by distributions that are well approximated for long times ($t>10^9$) by a $q$-Gaussian Quasi-Stationary State (QSS) with $q\approx1.4$. On the other hand, when the $k=N$ mode is excited at the same energy, diffusive phenomena are \textit{absent} and the motion is quasi-periodic. In fact, as the energy increases to $E=0.3$, the distributions in the former case pass through \textit{shorter} $q$-Gaussian states and tend rapidly to a Gaussian (i.e. $q\rightarrow 1$) where equipartition sets in, while in the latter we need to reach to E=4 to see a \textit{sudden transition} to Gaussian statistics, without any passage through an intermediate QSS. This may be explained by different energy localization properties and recurrence phenomena in the two cases, supporting the view that when the energy is placed in the first mode weak chaos and "sticky" dynamics lead to a more gradual process of energy sharing, while strong chaos and equipartition appear abruptly when only the last mode is initially excited.Comment: 12 pages, 3 figures, submitted for publication to International Journal of Bifurcation and Chaos. In honor of Prof. Tassos Bountis' 60th birthda

Analyticity of the SRB measure of a lattice of coupled Anosov diffeomorphisms of the torus

We consider the "thermodynamic limit"of a d-dimensional lattice of hyperbolic dynamical systems on the 2-torus, interacting via weak and nearest neighbor coupling. We prove that the SRB measure is analytic in the strength of the coupling. The proof is based on symbolic dynamics techniques that allow us to map the SRB measure into a Gibbs measure for a spin system on a (d+1)-dimensional lattice. This Gibbs measure can be studied by an extension (decimation) of the usual "cluster expansion" techniques.Comment: 28 pages, 2 figure

Thermodynamic formalism for contracting Lorenz flows

We study the expansion properties of the contracting Lorenz flow introduced by Rovella via thermodynamic formalism. Specifically, we prove the existence of an equilibrium state for the natural potential $\hat\phi_t(x,y, z):=-t\log J_{(x, y, z)}^{cu}$ for the contracting Lorenz flow and for $t$ in an interval containing $[0,1]$. We also analyse the Lyapunov spectrum of the flow in terms of the pressure

Generalised dimensions of measures on almost self-affine sets

We establish a generic formula for the generalised q-dimensions of measures supported by almost self-affine sets, for all q>1. These q-dimensions may exhibit phase transitions as q varies. We first consider general measures and then specialise to Bernoulli and Gibbs measures. Our method involves estimating expectations of moment expressions in terms of `multienergy' integrals which we then bound using induction on families of trees

Escape orbits and Ergodicity in Infinite Step Billiards

In a previous paper we defined a class of non-compact polygonal billiards, the infinite step billiards: to a given decreasing sequence of non-negative numbers $\{p_{n}$, there corresponds a table \Bi := \bigcup_{n\in\N} [n,n+1] \times [0,p_{n}]. In this article, first we generalize the main result of the previous paper to a wider class of examples. That is, a.s. there is a unique escape orbit which belongs to the alpha and omega-limit of every other trajectory. Then, following a recent work of Troubetzkoy, we prove that generically these systems are ergodic for almost all initial velocities, and the entropy with respect to a wide class of ergodic measures is zero.Comment: 27 pages, 8 figure

Dynamical ensembles in stationary states

We propose as a generalization of an idea of Ruelle to describe turbulent fluid flow a chaotic hypothesis for reversible dissipative many particle systems in nonequilibrium stationary states in general. This implies an extension of the zeroth law of thermodynamics to non equilibrium states and it leads to the identification of a unique distribution \m describing the asymptotic properties of the time evolution of the system for initial data randomly chosen with respect to a uniform distribution on phase space. For conservative systems in thermal equilibrium the chaotic hypothesis implies the ergodic hypothesis. We outline a procedure to obtain the distribution \m: it leads to a new unifying point of view for the phase space behavior of dissipative and conservative systems. The chaotic hypothesis is confirmed in a non trivial, parameter--free, way by a recent computer experiment on the entropy production fluctuations in a shearing fluid far from equilibrium. Similar applications to other models are proposed, in particular to a model for the Kolmogorov--Obuchov theory for turbulent flow.Comment: 31 pages, 3 figures, compile with dvips (otherwise no pictures

Multifractal properties of return time statistics

Fluctuations in the return time statistics of a dynamical system can be described by a new spectrum of dimensions. Comparison with the usual multifractal analysis of measures is presented, and difference between the two corresponding sets of dimensions is established. Theoretical analysis and numerical examples of dynamical systems in the class of Iterated Functions are presented.Comment: 4 pages, 3 figure