Handbook of Dynamics and Probability, 1st ed. 2022

Our time is characterized by an explosive growth in the use of ever more complicated and sophisticated (computer) models. These models rely on dynamical systems theory for the interpretation of their results and on probability theory for the quantification of their uncertainties. A conscientious and intelligent use of these models requires that both these theories are properly understood.

This book is to provide such understanding. It gives a unifying treatment of dynamical systems theory and probability theory. It covers the basic concepts and statements of these theories, their interrelations, and their applications to scientific reasoning and physics. The book stresses the underlying concepts and mathematical structures but is written in a simple and illuminating manner without sacrificing too much mathematical rigor.

The book is aimed at students, post-docs, and researchers in the applied sciences who aspire to better understand the conceptual and mathematical underpinnings of the models that they use. Despite the peculiarities of any applied science, dynamics and probability are the common and indispensable tools in any modeling effort. The book is self-contained, with many technical aspects covered in appendices, but does require some basic knowledge in analysis, linear algebra, and physics.

Peter Müller, now a professor emeritus at the University of Hawaii, has worked extensively on ocean and climate models and the foundations of complex system theories.

Introduction

I Dynamics

Deterministic dynamical systems

Linear and integrable systems

Stability and long-time behavior

Ergodic theory

Numerical Algorithms

II Probability

Probability Theory

Discrete-time stochastic processes

Continuous-time stochastic processes

Information entropy

III Probability in dynamical evolution

Time evolution of broadened initial states

Stochastic processes generated by observables

Stochastic dynamical systems

IV Probability in scientific reasoning

Interpretations of probability

Parameter estimation and hypothesis testing

Bayesian inferences

V Probability in physics

Equilibrium statistical mechanics

Non-equilibrium statistical mechanics

Foundational issues of statistical mechanics

Quantum Mechanics

A Classical Mechanics

B Thermostatics

D Some proofs and explicit computations

E Mathematical tools, conventions and notation

References

Index

Presents unified treatment of dynamical systems and probability theory, all in one place

Stresses the inherent assumptions in applying these abstract theories to the analysis of real-world systems

Includes a chapter on the interpretation of probability and the many faces of randomness