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Introduced by Shannon as a ``rate of actual transmission,'' mutual information rate (MIR) is an extension of mutual information to a pair of dynamical processes. We show a delay-independence theorem, according to which MIR is not sensitive to a time shift between the two processes. Numerical studies of several benchmark situations confirm that this theoretical asymptotic property remains valid for realistic finite sequences. Estimations based on block entropies and a causal state machine algorithm perform better than an estimation based on a Lempel-Ziv compression algorithm provided that block length and maximum history length, respectively, can be chosen larger than the delay. MIR is thus a relevant index for measuring nonlinear correlations between two experimental or simulated sequences when the transmission delay (in input-output devices) or dephasing (in coupled systems) is variable or unknown.

This chapter is dedicated to the study of the superconductor–insulator transition in high temperature superconductors. We saw, in Sect.1.3.3, that metal superconductors were not so interesting from the point of view of critical phenomena. This is because their coherence length at T=0 is of the order of a micron, so the critical region is around 10−14K, which is obviously impossible to observe. However this is not true for high temperature superconductors, which exhibit a critical temperature of the order of 100 K. The most important high temperature superconductor family is that of the superconducting cuprates discovered in 1986 by Bednorz and Muller. Since then other families have been discovered for example C60, MgB2 and AsFe. However the most important family is still the one that was discovered first, the cuprate family, which is the only one with superconducting properties above 100K. This is also the most documented family both experimentally and theoretically. For these reasons, in this chapter we will focus only on superconducting cuprates. Their coherence length at T=0 being of the order of \(15\r{A}\), the critical region can reach a few tens of kelvins! So these materials represent, in principle, an ideal case in which to study critical phenomena, even more so given that the transition can be induced by a magnetic field (as for all superconductors) or by doping (a unique property of high temperature superconductors). In principle, these two parameters, can induce quantum transitions at T=0. This physical system certainly seems interesting to test new descriptions of critical phenomena.

Our daily lives are full of interfaces of various kinds. Be it through our skin or across the membranes of our cells, we are continually exchanging matter and energy with the environments we pass through. The shape of these interfaces is crucial for exchange processes, as is for example the fine structure of our lungs for exchanging oxygen, or the surface of glass for its transparence to light. In general their morphology depends on the scale at which we look, e.g. the smooth surface of Mars seen from the Earth is very different from the roughness of its soil seen by a hypothetical Martian.

After the wide panorama presented in this book, we can draw a few general conclusions on scaling approaches. We have seen that they apply in situations when the system no longer has a characteristic scale, as for example observed for a temperature T→T c in the case of phase transitions, for a number of monomers N→∞ in polymers, for a filling density p→p c in percolation, for a time span t→∞ in growth or diffusion, at the onset of chaos in the transition to chaos by period doubling or for a Reynolds number Re→∞ in turbulence.

We continue our exploration of systems without characteristic scales and specific methods into the field of dynamical systems, with the analysis of chaotic and turbulent behaviours. Due to the sheer magnitude of this field, our presentation will be deliberately selective, focusing only on critical aspects and behaviours and associated scaling laws. A brief introduction to the theory of dynamical systems takes us first of all to the important concept of bifurcation, a qualitative change in the asymptotic dynamics.

Although their synthesis takes place in chemistry, or even biology (for example DNA), polymers are widely studies by physicists, mainly because they form materials with remarkable properties; just think of the variety and usefulness of plastics in our day to day environment. However, it is not so much their interest and importance that preoccupies us here, but the fact that polymer physics is a domain in which scaling laws are omnipresent, as well as being conceptually fundamental and extremely useful in practice [5]. Polymers possess a natural scale variable: their degree of polymerisation N, that is to say the (very large) number of monomers making up each polymer. N is directly related to the mass of the polymer, which is the quantity measured experimentally. In addition, the large size of polymers legitimises their observation, description and simulation at supermolecular scales, where details of atomic interactions are only indirectly involved in terms of apparent geometric constraints and characteristic averages.

It is a fact of life in our daily experience and was a real enigma for a century– pure matter dramatically changes at extremely precise temperatures. In the nineteenth century, pioneering scientists, from Gay-Lussac to Van der Waals, carried out meticulous measurements of the fluid state, paving the way for microscopic descriptions which underlie our natural sciences today. The study of properties of gases at low density enabled the introduction of absolute temperature as a measure of the kinetic energy of molecules. The striking generality of thermal behaviour and mechanical properties of gases with vastly varying chemical properties was thereby elucidated. Thermodynamics and its microscopic interpretations was born on the wave of this success. However the pioneers of fluid observations also tackled liquid–vapour transformations and discovered another elegant generality which was far from evident a priori: In a dilute gas, molecules are almost isolated and therefore one thinks their chemical properties are unimportant, but what about in a liquid where the molecules are in constant interaction?

In the introductory chapter (Chap.1) we showed several examples of critical behaviours near second order phase transitions. Let us remind ourselves of the key observations. When the temperature is near the critical temperature, most physical quantities obey a power law(T−T c ) x , where xis called a critical exponent. In this chapter we will see that this sort of behaviour is the signature of the scale invariance of the system close to the critical point.

What physical concept can unite the flow of liquid through ground coffee, electrical conduction in a conductor–insulator mixture, target collapse on shooting, the birth of a continent, polymer gelification and the spread of epidemics or forest fires? One question is common to all these processes: how is “something” produced at large scales by contributions at small scales? In all these situations, a quantity (liquid, electric charges, target fracture, dry land, molecular crosslinks, disease or fire) may or may not propagate from one element to its neighbour. As in the previous chapters, we are interested in asymptotic properties resulting at large scale, that is to say in a system that is large compared with the size of the individual elements. The analogue of temperature T here is the inverse of the relative population density p of elements (pores, conducting regions, impacts, etc) which varies between 0 and 1 for the maximum population density. p ∼ 0 corresponds to a large amount of disorder for a dilute population and p ∼ 1 corresponds to a large amount of order established. In these situations we can ask ourselves the same questions as for changes of states of matter in thermal systems.

In this chapter we will show that biology cannot escape the omnipresence of scaling laws and self-similar structures underlying them. Some authors go as far as to suggest that there are scaling laws that could be specific to living systems and their particular multiscale organisation due to the result of evolution.

During this book we have encountered many examples of critical phenomenon and have highlighted their common characteristics of divergence of the range of correlations, absence of characteristic scales, fluctuations of all sizes and anomalous response (at all scales in amplitude, spatial extent and duration) to even a tiny perturbation. These characteristics are reflected in many scaling laws, expressing quantitatively the scale invariance of the phenomena. Typically, criticality occurs for a particular value of the control parameter which is adjusted externally, for example the critical temperature in second order phase transitions, percolation threshold p c or bifurcation point in a dynamic system. However, it turns out that certain systems, maintained in a nonequilibrium state by a continuous supply of material or energy, can evolve spontaneously to a critical state, without external regulation. This is the concept of self-organised criticality, which we will present in this chapter, discussing various examples.

A reactive lattice-gas cellular automaton model is used to simulate a chemical wave front in two dimensions. The computed value of the front propagation velocity agrees with the one-dimensional (1D) theoretical value. In contrast, the front width is half the predicted 1D value. This result is explained by the fractal character of the interface. The fractal structure, described through several fractal dimensions, is shown to be independent of the reaction and diffusion parameter values.

International audience; A model of early metastatic process is based on the role of the protein PAI-1, which at high enough extracellular concentration promotes the transition of cancer cells to a state prone to migration. This transition is described at the single cell level as a bi-stable switch associated with a subcritical bifurcation. In a multilevel reaction-diffusion scenario, the micro-environment of the tumor is modified by the proliferating cell population so as to push the concentration of PAI-1 above the bifurcation threshold. The formulation in terms of partial differential equations fails to capture spatio-temporal heterogeneity. Cellular-automata and agent-based simulations of cell populations support the hypothesis that a randomly localized accumulation of PAI-1 can arise and trigger the escape of a few isolated cells. Far away from the primary tumor, these cells experience a reverse transition back to a proliferative state and could generate a secondary tumor. The suggested role of PAI-1 in controlling this metastatic cycle is candidate to explain its role in the progression of cancer.

Complexity of biological systems originates not only from their emergent properties but also from the feedbacks exerted by these properties on elementary structures and mechanisms. A systemic approach is required to capture this multiscale organization and its among-level feedback loops which have resulted from selective pressure during evolution: the scientific challenge is not to take into account all the relevant details but rather to consider jointly several integration levels. New mathematical tools are necessary to express how the consistency and interrelations among these different levels control biological functions.

Nicotinic acetylcholine receptors (nAChRs) are widely expressed throughout the central nervous system and modulate neuronal function in most mammalian brain structures. The contribution of defined nAChR subunits to a specific behavior is thus difficult to assess. Mice deleted for ß2-containing nAChRs (ß2−/−) have been shown to be hyperactive in an open-field paradigm, without determining the origin of this hyperactivity. We here develop a quantitative description of mouse behavior in the open field based upon first order Markov and variable length Markov chain analysis focusing on the time-organized sequence that behaviors are composed of. This description reveals that this hyperactivity is the consequence of the absence of specific inactive states or “stops”. These stops are associated with a scanning of the environment in wild-type mice (WT), and they affect the way that animals organize their sequence of behaviors when compared with stops without scanning. They characterize a specific “decision moment” that is reduced in ß2−/− mutant mice, suggesting an important role of ß2-nAChRs in the strategy used by animals to explore an environment and collect information in order to organize their behavior. This integrated analysis of the displacement of an animal in a simple environment offers new insights, specifically into the contribution of nAChRs to higher brain functions and more generally into the principles that organize sequences of behaviors in animals., Author Summary Understanding mechanisms underlying complex behaviors and the abnormalities that accompany most neuropathologies is a current challenge in biomedical research. A number of approaches is primarily based on the identification of genes and their associated molecular pathways implicated in complex motor or cognitive pathologies. However, optimal use of the large body of genetic, molecular, electro-physiological, and imaging data is hampered by the practical and theoretical limitations of currently available behavioral analysis methods. Complex behaviors consist of a finite number of actions combined in a variety of spatial and temporal patterns. In this paper we develop a sequential analysis of mouse displacement in an open-field paradigm and demonstrate that a description based on a Markov model can be used to describe quantitatively patterns of behaviors and to detect changes in the way that animals organize their displacement, especially in mice lacking nicotinic acetylcholine receptor subunits. This paper would be of broad interest not only to those concerned with this particular mice model but also generally to those interested in modeling complex behavior traits in mice.

At any time there is only a thin layer separating what is trivial from what is impossibly difficult. It is in that layer that discoveries are made … Andrei N. Kolmogorov An important aspect of the theory of dynamical systems is the formalization and quantitative characterization of the sensitivity to initial conditions. The Lyapunov exponents {λ i } are the indicators used to measure the average rate of exponential error growth in a system. Starting from the idea of Kolmogorov of characterizing dynamical systems by means of entropy-like quantities, following the work by Shannon in information theory, another approach to dynamical systems has been developed in the context of information theory, data compression and algorithmic complexity theory. In particular, the Kolmogorov–Sinai entropy, h ks , can be defined and interpreted as a measure of the rate of information production of a system. Since the ability to produce information is tightly linked to the exponential diversification of trajectories, it is not a surprise that a relation exists between h ks and {λ i }, the Pesin relation. One has to note that quantities such as {λ i } and h ks are properly defined only in specific asymptotic limits, that is, very long times and arbitrary accuracy. Since in realistic situations one has to deal with finite accuracy and finite time – as Keynes said, in the long run we shall all be dead – it is important to take into account these limitations. Relaxing the requirement of infinite time, one can investigate the relevance of finite time fluctuations of the “effective” Lyapunov exponent.

International audience; Family theories of anorexia nervosa point out that patients’ autobiographic speech may reflect internalized family interactions. Our study characterizes the statistical distribution and temporal organization of the narrative components describing personal relationships in anorexic and control subjects. Semantic components related to personal interactions were encoded from life narratives of 14 adolescent girls with anorexia nervosa (restrictive type) and of 13 matched healthy adolescent girls. Speech analysis was performed using both statistical methods and non-linear time-series analysis. Static indices showed an over-representation of family relations and an under-representation of love relations in anorexic patients. Dynamical indices showed the independence between relational systems in anorexic patients. Dynamical analysis reveals that interactional patterns are internalized through the temporal organization of autobiographical speech. Moreover, these results support the existence of a specific temporal organization in anorexic adolescents’ life narratives which express the internalization of stationary family patterns and indicate relative inability to disengage from active previous relational patterns and to create new ones.

While statistical mechanics describe the equilibrium state of systems with many degrees of freedom, and dynamical systems explain the irregular evolution of systems with few degrees of freedom, new tools are needed to study the evolution of systems with many degrees of freedom. This book presents the basic aspects of chaotic systems, with emphasis on systems composed by huge numbers of particles. Firstly, the basic concepts of chaotic dynamics are introduced, moving on to explore the role of ergodicity and chaos for the validity of statistical laws, and ending with problems characterized by the presence of more than one significant scale. Also discussed is the relevance of many degrees of freedom, coarse graining procedure, and instability mechanisms in justifying a statistical description of macroscopic bodies. Introducing the tools to characterize the non asymptotic behaviors of chaotic systems, this text will interest researchers and graduate students in statistical mechanics and chaos.

A.N. Kolmogorov (b. Tambov 1903, d. Moscow 1987) was one of the most brilliant mathematicians that the world has ever known. Incredibly deep and creative, he was able to approach each subject with a completely new point of view: in a few magnificent pages, which are models of shrewdness and imagination, and which astounded his contemporaries, he changed drastically the landscape of the subject. Most mathematicians prove what they can, Kolmogorov was of those who prove what they want. For this book several world experts were asked to present one part of the mathematical heritage left to us by Kolmogorov. Each chapter treats one of Kolmogorov's research themes, or a subject that was invented as a consequence of his discoveries. His contributions are presented, his methods, the perspectives he opened to us, the way in which this research has evolved up to now, along with examples of recent applications and a presentation of the current prospects. This book can be read by anyone with a master's (even a bachelor's) degree in mathematics, computer science or physics, or more generally by anyone who likes mathematical ideas. Rather than present detailed proofs, the main ideas are described. A bibliography is provided for those who wish to understand the technical details. One can see that sometimes very simple reasoning (with the right interpretation and tools) can lead in a few lines to very substantial results. The Kolmogorov Legacy in Physics was published by Springer in 2004 (ISBN 978-3-540-20307-0).