Stability index for chaotically driven concave maps

We study skew product systems driven by a hyperbolic base map S (e.g. a baker map or an Anosov surface diffeomorphism) and with simple concave fibre maps on an interval [0,a] like h(x)=g(\theta) tanh(x) where g(\theta) is a factor driven by the base map. The fibre-wise attractor is the graph of an upper semicontinuous function \phi(\theta). For many choices of the function g, \phi has a residual set of zeros but \phi>0 almost everywhere w.r.t. the Sinai-Ruelle-Bowen measure of S^(-1). In such situations we evaluate the stability index of the global attractor of the system, which is the subgraph of \phi, at all regular points (\theta,0) in terms of the local exponents \Gamma(\theta):=\lim_{n\to\infty} 1/n log g_n(\theta) and \Lambda(\theta):=\lim_{n\to\infty} 1/n\log|D_u S^{-n}(\theta)| and of the positive zero s_* of a certain thermodynamic pressure function associated with S^(-1) and g. (In queuing theory, an analogon of s_* is known as Loyne's exponent.) The stability index was introduced by Podvigina and Ashwin in 2011 to quantify the local scaling of basins of attraction

On Efimov spaces and Radon measures

We give a construction under CH of an infinite Hausdorff compact space having no converging sequences and carrying no Radon measure of uncountable type. Under ? we obtain another example of a compact space with no convergent sequences, which in addition has the stronger property that every nonatomic Radon measure on it is uniformly regular. This example refutes a conjecture of Mercourakis from 1996 stating that if every measure on a compact space K is uniformly regular then K is necessarily sequentially compact

Nonconcave entropies in multifractals and the thermodynamic formalism

We discuss a subtlety involved in the calculation of multifractal spectra when these are expressed as Legendre-Fenchel transforms of functions analogous to free energy functions. We show that the Legendre-Fenchel transform of a free energy function yields the correct multifractal spectrum only when the latter is wholly concave. If the spectrum has no definite concavity, then the transform yields the concave envelope of the spectrum rather than the spectrum itself. Some mathematical and physical examples are given to illustrate this result, which lies at the root of the nonequivalence of the microcanonical and canonical ensembles. On a more positive note, we also show that the impossibility of expressing nonconcave multifractal spectra through Legendre-Fenchel transforms of free energies can be circumvented with the help of a generalized free energy function, which relates to a recently introduced generalized canonical ensemble. Analogies with the calculation of rate functions in large deviation theory are finally discussed.Comment: 9 pages, revtex4, 3 figures. Changes in v2: sections added on applications plus many new references; contains an addendum not contained in published versio

Receptor for advanced glycation end products (RAGE) regulates sepsis but not the adaptive immune response

This is the publisher's version, also available electronically from http://www.jci.org/articles/view/18704While the initiation of the adaptive and innate immune response is well understood, less is known about cellular mechanisms propagating inflammation. The receptor for advanced glycation end products (RAGE), a transmembrane receptor of the immunoglobulin superfamily, leads to perpetuated cell activation. Using novel animal models with defective or tissue-specific RAGE expression, we show that in these animal models RAGE does not play a role in the adaptive immune response. However, deletion of RAGE provides protection from the lethal effects of septic shock caused by cecal ligation and puncture. Such protection is reversed by reconstitution of RAGE in endothelial and hematopoietic cells. These results indicate that the innate immune response is controlled by pattern-recognition receptors not only at the initiating steps but also at the phase of perpetuation

The large deviation approach to statistical mechanics

The theory of large deviations is concerned with the exponential decay of probabilities of large fluctuations in random systems. These probabilities are important in many fields of study, including statistics, finance, and engineering, as they often yield valuable information about the large fluctuations of a random system around its most probable state or trajectory. In the context of equilibrium statistical mechanics, the theory of large deviations provides exponential-order estimates of probabilities that refine and generalize Einstein's theory of fluctuations. This review explores this and other connections between large deviation theory and statistical mechanics, in an effort to show that the mathematical language of statistical mechanics is the language of large deviation theory. The first part of the review presents the basics of large deviation theory, and works out many of its classical applications related to sums of random variables and Markov processes. The second part goes through many problems and results of statistical mechanics, and shows how these can be formulated and derived within the context of large deviation theory. The problems and results treated cover a wide range of physical systems, including equilibrium many-particle systems, noise-perturbed dynamics, nonequilibrium systems, as well as multifractals, disordered systems, and chaotic systems. This review also covers many fundamental aspects of statistical mechanics, such as the derivation of variational principles characterizing equilibrium and nonequilibrium states, the breaking of the Legendre transform for nonconcave entropies, and the characterization of nonequilibrium fluctuations through fluctuation relations.Comment: v1: 89 pages, 18 figures, pdflatex. v2: 95 pages, 20 figures, text, figures and appendices added, many references cut, close to published versio