Regular And Stochastic Motion

This book treats stochastic motion in nonlinear oscillator systems. It describes a rapidly growing field of nonlinear mechanics with applications to a number of areas in science and engineering, including astronomy, plasma physics, statistical mechanics and hydrodynamics. The main em phasis is on intrinsic stochasticity in Hamiltonian systems, where the stochastic motion is generated by the dynamics itself and not by external noise. However, the effects of noise in modifying the intrinsic motion are also considered. A thorough introduction to chaotic motion in dissipative systems is given in the final chapter. Although the roots of the field are old, dating back to the last century when Poincare and others attempted to formulate a theory for nonlinear perturbations of planetary orbits, it was new mathematical results obtained in the 1960's, together with computational results obtained using high speed computers, that facilitated our new treatment of the subject. Since the new methods partly originated in mathematical advances, there have been two or three mathematical monographs exposing these developments. However, these monographs employ methods and language that are not readily accessible to scientists and engineers, and also do not give explicit tech niques for making practical calculations. In our treatment of the material, we emphasize physical insight rather than mathematical rigor. We present practical methods for describing the motion, for determining the transition from regular to stochastic behavior, and for characterizing the stochasticity. We rely heavily on numerical computations to illustrate the methods and to validate them.

This book treats nonlinear dynamics in both Hamiltonian and dissipative systems. The emphasis is on the mechanics for generating chaotic motion, methods of calculating the transitions from regular to chaotic motion, and the dynamical and statistical properties of the dynamics when it is chaotic. The new edition brings the subject matter in a rapidly expanding field up to date, and has greatly expanded the treatment of dissipative dynamics to include most important subjects.

A graduate-course text, written for readers familiar with measure-theoretic probability and discrete-time processes, wishing to explore stochastic processes in continuous time. The vehicle chosen for this exposition is Brownian motion, which is presented as the canonical example of both a martingale and a Markov process with continuous paths. In this context, the theory of stochastic integration and stochastic calculus is developed, illustrated by results concerning representations of martingales and change of measure on Wiener space, which in turn permit a presentation of recent advances in financial economics. The book contains a detailed discussion of weak and strong solutions of stochastic differential equations and a study of local time for semimartingales, with special emphasis on the theory of Brownian local time. The whole is backed by a large number of problems and exercises.

This book presents a concise treatment of stochastic calculus and its applications. It gives a simple but rigorous treatment of the subject including a range of advanced topics, it is useful for practitioners who use advanced theoretical results. It covers advanced applications, such as models in mathematical finance, biology and engineering.Self-contained and unified in presentation, the book contains many solved examples and exercises. It may be used as a textbook by advanced undergraduates and graduate students in stochastic calculus and financial mathematics. It is also suitable for practitioners who wish to gain an understanding or working knowledge of the subject. For mathematicians, this book could be a first text on stochastic calculus; it is good companion to more advanced texts by a way of examples and exercises. For people from other fields, it provides a way to gain a working knowledge of stochastic calculus. It shows all readers the applications of stochastic calculus methods and takes readers to the technical level required in research and sophisticated modelling.This second edition contains a new chapter on bonds, interest rates and their options. New materials include more worked out examples in all chapters, best estimators, more results on change of time, change of measure, random measures, new results on exotic options, FX options, stochastic and implied volatility, models of the age-dependent branching process and the stochastic Lotka-Volterra model in biology, non-linear filtering in engineering and five new figures.Instructors can obtain slides of the text from the author.

This “lucid, masterfully written introduction to an often difficult subject . . . belongs on the bookshelf of every student of statistical physics” (Dr. Brian J. Albright, Applied Physics Division, Los Alamos National Laboratory). This book provides an accessible introduction to stochastic processes in physics and describes the basic mathematical tools of the trade: probability, random walks, and Wiener and Ornstein-Uhlenbeck processes. With an emphasis on applications, it includes end-of-chapter problems. Physicist and author Don S. Lemons builds on Paul Langevin’s seminal 1908 paper “On the Theory of Brownian Motion” and its explanations of classical uncertainty in natural phenomena. Following Langevin’s example, Lemons applies Newton’s second law to a “Brownian particle on which the total force included a random component.” This method builds on Newtonian dynamics and provides an accessible explanation to anyone approaching the subject for the first time. This volume contains the complete text of Paul Langevin’s “On the Theory of Brownian Motion,” translated by Anthony Gythiel.

Stochastic Partial Differential Equations analyzes mathematical models of time-dependent physical phenomena on microscopic, macroscopic and mesoscopic levels. It provides a rigorous derivation of each level from the preceding one and examines the resulting mesoscopic equations in detail. Coverage first describes the transition from the microscopic equations to the mesoscopic equations. It then covers a general system for the positions of the large particles.

We have investigated the dynamics of a three-dimensional classical Rydberg atom driven by a sequence of pulses. Both the deterministic system with periodic pulses and the closely related ''noisy'' system with random pulses have been studied in parallel. The Lyapunov exponent is calculated as a function of pulse height and the angular momentum of the initial state. We find differences between noisy and deterministic perturbations to be most pronounced for small pulse heights. Low angular momentum orbits show enhanced diffusion in agreement with recent experimental data for ion-solid interaction. 22 refs., 6 figs.