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This chapter is devoted to the theoretical analysis of M/G/1-type Markov chains. Conditions for the ergodicity are given in terms of the drift. The reduction to solving a nonlinear matrix equation is shown and the role played by canonical factorizations in this regard is pointed out. The Ramaswami formula for the computation of the stationary distribution is revisited in terms of canonical factorizations.

This chapter deals with Markov processes. It first defines the “Markov property” and shows that all the relevant information about a Markov process assuming values in a finite set of cardinality n can be captured by a nonnegative n x n matrix known as the state transition matrix, and an n-dimensional probability distribution of the initial state. It then invokes the results of the previous chapter on nonnegative matrices to analyze the temporal evolution of Markov processes. It also estimates the state transition matrix and considers the dynamics of stationary Markov chains, recurrent and transient states, hitting probability and mean hitting times, and the ergodicity of Markov chains.

This chapter is devoted to basic potential scattering theory, focusing on the case of a one-dimensional conductor and an open cavity with a one-channel lead connected to it. The contents of this chapter include potential scattering in infinite one-dimensional space; Lippmann-Schwinger equation; free Green function; reflection and transmission amplitudes; transfer matrix; T matrix; S matrix and its analytic structure; phase shifts and resonances from the analytic structure of S matrix in complex momentum and complex energy planes; parametrization of the matrices; combination of the S matrices for two scatterers in series; and invariant-imbedding approach for a one-dimensional disordered conductor.

This chapter defines and characterizes various degrees of ergodicity — both for classical and quantum systems — in terms of the Hilbert space formalism, Koopman formalism for classical systems, and GNS-representation for quantum systems. It presents mixing and asymptotic Abelianness. It then discusses a number of examples with non-trivial algebraic structures: quasi-free Fermionic automorphisms, highly anti-commutative systems, and Powers–Price shifts. It is shown that under certain ergodic assumptions the fluctuations around ergodic means can be modelled by Bose fields in quasi-free states (Gaussian distributions), the other extreme cases lead to a free probability scheme with semi-circular distributions. Expanding maps and Lyapunov exponents for classical dynamics are briefly discussed and a possible quantum analog, horocyclic actions, is presented for a quantum cat map.

This chapter presents three reasons to take quantum statistical mechanics to the thermodynamic limit, where the number of systems considered becomes infinite, in the form of three phenomena that can't be modeled short of the thermodynamic limit: ergodicity, phase structure, and broken symmetry. Focusing on phase structure, exemplified by ferromagnetism, the chapter documents a promising model of the phenomenon that makes use of unitarily inequivalent representations in a way rigid interpretations of QM_∞, such as Hilbert Space Conservatism and Algebraic Imperialism, can't endorse. Taking a virtue of interpretation to be its capacity to sustain the explanatory aspirations of the theory interpreted, the chapter scores this as a strike against rigid interpretations.

This chapter proves results involving the quantum ergodicity of certain high-frequency eigenfunctions. Ergodic theory originally arose in the work of physicists studying statistical mechanics at the end of the nineteenth century. The word ergodic has as its roots two Greek words: ergon, meaning work or energy, and hodos, meaning path or way. Even though ergodic theory's initial development was motivated by physical problems, it has become an important branch of pure mathematics that studies dynamical systems possessing an invariant measure. Thus, this chapter first presents some of the basic limit theorems that are key to the classical theory. It then turns to quantum ergodicity.

This chapter considers measures of Γ-invariant sets, and establish a mixing theorem for the action of the geodesic and horocyclic flows on square-integrable Γ-invariant functions. A Poincaré inequality (i.e., the existence of a spectral gap) for the Laplacian acting on Γ-invariant functions, Γ being a generic cofinite and geometrically finite Kleinian group is derived.

Sensitive dependence on initial conditions makes it difficult to predict chaotic systems in the long term, as two initial conditions that begin close to each other rapidly get pushed apart. This chapter reconsiders this general phenomenon and examines some other ways to visualise and characterise the behaviour of chaotic orbits. It demonstrates that chaotic systems exhibit statistical regularities despite the unpredictability of the path of a particular orbit. Before discussing the statistical stability of chaos, the chapter considers the statistical properties and chaotic behaviour of orbits with the aid of histograms. It then looks at the ergodicity of orbits and the ergodic nature of the logistic equation with r = 4.0. It also shows how a chaotic logistic equation can be statically predictable despite having unpredictable orbits.

This chapter gives an overview of the origin of life, with particular attention to transition from ergodicity to contingency. It also considers physical insights from dynamical self-organization in reversible thermal systems that serve as a model for the emergence of order in a primordial chemical reaction network. The implications of the design of artificial protocells are also examined.

This chapter presents a theory of homogenisation for random bianisotropic media exhibiting an ergodic structure. It shows that for such a medium there exists a homogenised system of the Maxwell type. To begin, the chapter presents an introduction to the necessary notions from the theory of ergodicity that will be used in this treatment of homogenisation. It then showcases a model for a random complex medium, on which this chapter's analysis will be based. The chapter also details a formal two-scale approach to set ideas and understand the basic mechanisms that will lead to a homogenised system, as well as to identify the coefficients of the homogenised system. Finally, this chapter presents some rigorous results on the homogenisation of random complex electromagnetic media.

In this chapter we present the minority game, which is one of the keystone models of econophysics. It is the mathematic implementation of the idea of inductive, rather than deductive, thinking. First, we explain how the agents adapt to each other, and how the efficiency of the system as a whole is increased by such an adaptation. Then the properties of the phase transition from efficient but non-ergodic to inefficient ergodic phase is discussed. In the following the replica formalism is introduced, and the full replica solution of the thermal batch minority game is shown. Finally, it is shown what modifications make the minority game useful as a model for stock-market fluctuations.

This chapter extends Teichmüller dynamics to a flow on the total space of a flat bundle of deformation spaces of representations of the fundamental group of a fixed surface S in a Lie group G. The resulting dynamical system is a continuous version of the action of the mapping class group of S on the deformation space. It observes how ergodic properties of this action relate to this flow. When G is compact, this flow is strongly mixing over each component of the deformation space and of each stratum of the Teichmüller unit sphere bundle over the Riemann moduli space. It proves ergodicity for the analogous lift of the Weil–Petersson geodesic local flow.

We start by stating the need for a Gaussian approximation for dependent structures in the form of the central limit theorem (CLT) or of the functional CLT. To justify the need to quantify the dependence, we introduce illustrative examples: linear processes, functions of stationary sequences, recursive sequences, dynamical systems, additive functionals of Markov chains, and self-interactions. The limiting behavior of the associated partial sums can be handled with tools developed throughout the book. We also present basic notions for stationary sequences of random variables: various definitions and constructions, and definitions of ergodicity, projective decomposition, and spectral density. Special attention is given to dynamical systems, as many of our results also apply in this context. The chapter also surveys the basic theory of the convergence of stochastic processes in distribution, and introduces the reader to tightness, finite-dimensional convergence, and the need for maximal inequalities. It ends with the concepts of the moderate deviations principle and its functional form.

Most complex systems are statistical systems. Statsitical mechanics and information theory usually do not apply to complex systems because the latter break the assumptions of ergodicity, independence, and multinomial statistics. We show that it is possible to generalize the frameworks of statistical mechanics and information theory in a meaningful way, such that they become useful for understanding the statistics of complex systems.We clarify that the notion of entropy for complex systems is strongly dependent on the context where it is used, and differs if it is used as an extensive quantity, a measure of information, or as a tool for statistical inference. We show this explicitly for simple path-dependent complex processes such as Polya urn processes, and sample space reducing processes.We also show it is possible to generalize the maximum entropy principle to path-dependent processes and how this can be used to compute timedependent distribution functions of history dependent processes.

In this chapter the theory and the tools described in the previous two chapters are applied to several simple mean-field models, computing both the microcanonical and the canonical partition functions. It is shown that in the first model, the Hamiltonian Mean Field model, ensembles are equivalent, while in the others, the generalized XY model, the phi-4 model and the Self-Gravitating Ring model, ensembles are nonequivalent, with the occurrence of negative specific heat or negative susceptibility in the microcanonical ensemble. A feature that can appear in long-range systems is presented, namely the occurrence of ergodicity breaking. Mentioned in chapter 2 as a consequence of the possibility to have not connected regions in the thermodynamic parameter space, it is here explicitly shown in the XY model. Ergodicity breaking clearly manifests itself in the dynamical behaviour.

This chapter aims at showing that the features occurring in mean-field models, described in the previous chapters, can be found also in the other long-range systems. The first four sections are dedicated to generalizations of the models of chapter 4, in which either the mean-field interaction is augmented with a nearest neighbour interaction, or it is replaced by a slowly decaying interaction. It is shown that the long-range characteristics of the associated mean-field models are preserved, and in addition ensemble inequivalence, microcanonical negative specific heat and ergodicity breaking are induced in some cases. The final section introduces the dipolar interaction, a marginal long-range system. Dipolar systems are treated in details in chapter 15, and in this chapter few relevant properties are presented, focussing in particular on elongated ferromagnets and on ergodicity breaking.