## M/G/1-TYPE MARKOV CHAINS

*D. A. Bini, G. Latouche, and B. Meini*

### in Numerical Methods for Structured Markov Chains

- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198527688
- eISBN:
- 9780191713286
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198527688.003.0004
- Subject:
- Mathematics, Numerical Analysis

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 ... More

## Markov Processes

*M. Vidyasagar*

### in Hidden Markov Processes: Theory and Applications to Biology

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691133157
- eISBN:
- 9781400850518
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691133157.003.0004
- Subject:
- Mathematics, Probability / Statistics

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 ... More

## Dependence

*James Davidson*

### in Stochastic Limit Theory: An Introduction for Econometricians

- Published in print:
- 1994
- Published Online:
- November 2003
- ISBN:
- 9780198774037
- eISBN:
- 9780191596117
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198774036.003.0013
- Subject:
- Economics and Finance, Econometrics

A number of probabilistic approaches to the concept of dependence in stochastic sequences are contrasted. The fundamental idea is a shift transformation. The notions of stationarity and (serial) ... More

## Econometric Tools and Techniques

*David F. Hendry*

### in Dynamic Econometrics

- Published in print:
- 1995
- Published Online:
- November 2003
- ISBN:
- 9780198283164
- eISBN:
- 9780191596384
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198283164.003.0003
- Subject:
- Economics and Finance, Econometrics

Least squares and recursive methods for estimating the values of unknown parameters and the logic of testing in empirical modelling, are discussed. The tools needed for investigating the properties ... More

## INTRODUCTION TO THE QUANTUM MECHANICAL TIME-INDEPENDENT SCATTERING THEORY I: ONE-DIMENSIONAL SCATTERING

*Pier A. Mello and Narendra Kumar*

### in Quantum Transport in Mesoscopic Systems: Complexity and Statistical Fluctuations. A Maximum Entropy Viewpoint

- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198525820
- eISBN:
- 9780191712234
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525820.003.0002
- Subject:
- Physics, Condensed Matter Physics / Materials

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 ... More

## Hangzhou Lectures on Eigenfunctions of the Laplacian (AM-188)

*Christopher D. Sogge*

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691160757
- eISBN:
- 9781400850549
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691160757.001.0001
- Subject:
- Mathematics, Numerical Analysis

Based on lectures given at Zhejiang University in Hangzhou, China, and Johns Hopkins University, this book introduces eigenfunctions on Riemannian manifolds. The book gives a proof of the sharp Weyl ... More

## Ergodic Theory

*Robert Alicki and Mark Fannes*

### in Quantum Dynamical Systems

- Published in print:
- 2001
- Published Online:
- February 2010
- ISBN:
- 9780198504009
- eISBN:
- 9780191708503
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198504009.003.0007
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

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 ... More

## A Matter of Degree: Making Sense of Phase Structure

*Laura Ruetsche*

### in Interpreting Quantum Theories

- Published in print:
- 2011
- Published Online:
- September 2011
- ISBN:
- 9780199535408
- eISBN:
- 9780191728525
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199535408.003.0012
- Subject:
- Philosophy, Philosophy of Science

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 ... More

## Classical and quantum ergodicity

*Christopher D. Sogge*

### in Hangzhou Lectures on Eigenfunctions of the Laplacian (AM-188)

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691160757
- eISBN:
- 9781400850549
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691160757.003.0006
- Subject:
- Mathematics, Numerical Analysis

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 ... More

## Measures and flows on Γ\𝔽 d

*Jacques Franchi and Yves Le Jan*

### in Hyperbolic Dynamics and Brownian Motion: An Introduction

- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780199654109
- eISBN:
- 9780191745676
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199654109.003.0005
- Subject:
- Mathematics, Mathematical Physics

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 ... More

## Statistical Stability of Chaos

*David P. Feldman*

### in Chaos and Fractals: An Elementary Introduction

- Published in print:
- 2012
- Published Online:
- December 2013
- ISBN:
- 9780199566433
- eISBN:
- 9780191774966
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199566433.003.0014
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

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 ... More

## Energetics, Energy Flow, and Scaling in Life

*William H. Woodruff*

### in Protocells: Bridging Nonliving and Living Matter

- Published in print:
- 2008
- Published Online:
- August 2013
- ISBN:
- 9780262182683
- eISBN:
- 9780262282093
- Item type:
- chapter

- Publisher:
- The MIT Press
- DOI:
- 10.7551/mitpress/9780262182683.003.0021
- Subject:
- Biology, Microbiology

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 ... More

## Homogenisation

*G. F. Roach, I. G. Stratis, and A. N. Yannacopoulos*

### in Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691142173
- eISBN:
- 9781400842650
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691142173.003.0014
- Subject:
- Mathematics, Applied Mathematics

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. ... More

## Minority game

*František Slanina*

### in Essentials of Econophysics Modelling

- Published in print:
- 2013
- Published Online:
- January 2014
- ISBN:
- 9780199299683
- eISBN:
- 9780191747038
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199299683.003.0006
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

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. ... More

## Statistical Mechanics: Entropy, Order Parameters, and Complexity: Second Edition

*James P. Sethna*

- Published in print:
- 2021
- Published Online:
- April 2021
- ISBN:
- 9780198865247
- eISBN:
- 9780191897610
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198865247.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This text distills the core ideas of statistical mechanics to make room for new advances important to information theory, complexity, active matter, and dynamical systems. Chapters address random ... More

## Mixing Flows on Moduli Spaces of Flat Bundles over Surfaces

*Giovanni Forni and William M. Goldman*

### in Geometry and Physics: Volume II: A Festschrift in honour of Nigel Hitchin

- Published in print:
- 2018
- Published Online:
- December 2018
- ISBN:
- 9780198802020
- eISBN:
- 9780191869068
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198802020.003.0021
- Subject:
- Mathematics, Geometry / Topology

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 ... More

## Introduction to Stochastic Processes

*Florence Merlevède, Magda Peligrad, and Sergey Utev*

### in Functional Gaussian Approximation for Dependent Structures

- Published in print:
- 2019
- Published Online:
- April 2019
- ISBN:
- 9780198826941
- eISBN:
- 9780191865961
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198826941.003.0001
- Subject:
- Mathematics, Probability / Statistics

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 ... More

## Statistical Mechanics and Information Theory for Complex Systems

*Stefan Thurner, Rudolf Hanel, and Peter Klimekl*

### in Introduction to the Theory of Complex Systems

- Published in print:
- 2018
- Published Online:
- November 2018
- ISBN:
- 9780198821939
- eISBN:
- 9780191861062
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198821939.003.0006
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

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, ... More

## Solutions of Mean Field Models

*A. Campa, T. Dauxois, D. Fanelli, and S. Ruffo*

### in Physics of Long-Range Interacting Systems

- Published in print:
- 2014
- Published Online:
- October 2014
- ISBN:
- 9780199581931
- eISBN:
- 9780191787140
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199581931.003.0004
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

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 ... More

## Beyond Mean-Field Models

*A. Campa, T. Dauxois, D. Fanelli, and S. Ruffo*

### in Physics of Long-Range Interacting Systems

- Published in print:
- 2014
- Published Online:
- October 2014
- ISBN:
- 9780199581931
- eISBN:
- 9780191787140
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199581931.003.0005
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

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 ... More