Noise in Nonlinear Dynamical Systems, Vol. 2: Theory of Noise Induced Processes in Special Applications PDF
Preview Noise in Nonlinear Dynamical Systems, Vol. 2: Theory of Noise Induced Processes in Special Applications
Noise in nonlinear dynamical systems Volume 2 Theory of noise induced processes in special applications Noise in nonlinear dynamical systems Volume 2 Theory of noise induced processes in special applications Edited by Professor of Physics, Frank Moss, University of Missouri at St. Louis and Reader in Physics, P. V. E. McClintock, University of Lancaster rl6o1r/tt M UniTk'lnl iotfJCamb ridf� lop rint """'"''ond .r�ll •II 1roonf botb�y olc.r H�won.rr y n UnlHY/r1l// lly/Sp Jr./.i nt'd � pub/1$/ifdC'OtlM tsi nflOtu/y and llltCt 1$81. CAMBRIDGE UNIVERSITY PRESS Cambridge New York New Rochelle Melbourne Sydney CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo, Delhi Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521118521 Cambridge University Press 1989 © This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1989 This digitally printed version 2009 catalogue record for this publication is available from the British Library A Library of Congress Cataloguing in Publication data Noise in nonlinear dynamical systems. Includes indexes. Contents: v. 1. Theory of continuous Fokker-Planck systems -v. 2. Theory of noise induced processes in special applications. 1. Fluctuations (Physics) -Collected works. 2. Nonlinear theories -Collected works. I. Moss, Frank, 1934- . II. McClintock, P.V .E. QC6.4.F58N64 003 87-34856 ISBN 978-0-521-35229-1 hardback ISBN 978-0-521-11852-1 paperback Contents List of contributors ix Preface xiii Introduction to Volume xv 2 Stochastic processes in quantum mechanical settings RONALD F. FOX 1.1 Introduction 1 1.2 The Anderson-Kubo oscillator 2 1.3 General multiplicative stochastic processes 3 1.4 Operator cumulants and characteristic functionals 5 1.5 The stochastic Schrodinger equation 10 1.6 Pauli master equations 15 1.7 Magnetic relaxation and Redfield's equation 18 1.8 A variety of physical applications 20 2 Self-diffusion in non-Markovian condensed-matter systems TOYONORI MUNAKATA 24 2.1 Introduction 24 2.2 Resonant activation in non-Markovian processes 25 2.3 Self-diffusion and the generalized Langevin equation (GLE) 31 2.4 Crossover from thermal to quantum hopping 37 2.5 Summary and remarks 42 3 Escape from the underdamped potential well M. BUTTIKER 45 3.1 Introduction 45 3.2 Kramers' results 48 3.3 Refined low damping result 51 3.4 Escape energies 55 3.5 Non-Markovian damping and quantum corrections 57 3.6 Discussion of related work 59 v Contents 4 Effect of noise on discrete dynamical systems with multiple attractors EDGAR KNOBLOCH and JEFFREY B. WEISS 65 4.1 Introduction 65 4.2 Stochastic differential equations and iterated maps 66 4.3 Reduction to a stochastic map 71 4.4 Noisy maps 73 4.5 The cubic map 77 4.6 Conclusions 85 5 Discrete dynamics perturbed by weak noise PETER TALKNER and PETER HANGGI 87 5.1 Perspectives 87 5.2 Discrete dynamics driven by white noise 88 5.3 Stationary probability for weak Gaussian noise 89 5.4 Circle map: lifetime of metastable states 93 6 Bifurcation behavior under modulated control parameters M. LUCKE 100 6.1 Introduction 100 6.2 Parametrically driven Duffing oscillator 102 6.3 Linear stability analysis of the basic state 105 6.4 The bifurcating solution 112 6.5 Parametric modulation of a hysteretic bifurcation 126 6.6 Parametric modulation in a discrete dynamical system 132 6.7 Conclusion 143 7 Period doubling bifurcations: what good are they? KURT WIESENFELD 145 7.1 Prologue 145 7.2 Introduction and background 145 7.3 Noisy precursors 150 7.4 Moving bumps explained: the virtual Hopf phenomenon 158 7.5 Amplification of coherent signals 161 7.6 Nonlinear effects 168 7.7 Origin of noise rise in SUPARAMPS 171 7.8 Future vistas 175 8 Noise-induced transitions WERNER HORSTHEMKE and RENE LEFEVER 179 8.1 Introduction 179 8.2 Transition phenomena in fluctuating environments 180 8.3 Dynamics of noise-driven systems 193 8.4 The effect of colored noise 197 8.5 Poisson white noise 199 8.6 The effect of periodic dichotomous fluctuations 202 vi Contents 8.7 Applications to nonlinear systems and experimental evidence of noise-induced transitions 205 9 Mechanisms for noise-induced transitions in chemical systems RAYMOND KAPRAL and EDWARD CELARIER 209 9.1 Introduction 209 9.2 Chemical reactions and external noise 211 9.3 Basins and their boundaries 216 9.4 Discrete-time models 221 9.5 Mechanisms for noise-induced transitions 224 9.6 Discussion 247 10 State selection dynamics in symmetry-breaking transitions DILIP K. KONDEPUDI 251 10.l Introduction 251 10.2 Symmetry-breaking transitions 252 10.3 The process of state selection 254 10.4 State selection in two-and three-mode symmetry breaking 263 10.5 Concluding remarks 267 11 Noise in a ring-laser gyroscope K. VOGEL, H. RISKEN and W. SCHLEICH 271 11.l Introduction and overview 271 11.2 Ring-laser theory: a pico-review 273 11.3 Fokker-Planck equation for ring-laser gyro with white noise 279 11.4 Colored noise in the ring-laser gyroscope 281 11.5 Summary 289 12 Control of noise by noise and applications to optical systems L.A. LUG IA TO, G. BROGGI, M. MERRI and M.A. PERNIGO 293 l2.1 Introduction 293 12.2 Two especially relevant cases 296 12.3 Linearized treatment: modulation 300 12.4 Linearized treatment: noise 302 12.5 Dispersive optical bistability, thermal noise in the material 304 12.6 Noise-induced transition from colored noise 309 12.7 Dispersive bistability, input intensity noise: steady-state behavior 313 12.8 Dispersive optical bistability, frequency noise: destruction of steady-state bimodality 315 12.9 Absorptive optical bistability, white noise: transient bimodality 321 12.10 Dispersive optical bistability, colored noise: noise switching and transient bimodality 330 vii Contents 12.11 A case of nonlinear noise 334 12.12 Delayed bifurcations in swept parameter systems: noise effects 336 13 Transition probabilities and spectral density of fluctuations of noise driven bistable systems M. I. DYKMAN, M.A. KRIVOGLAZ and M. SOSKIN 347 s. 13.1 Introduction 347 13.2 Probabilities of transitions between stable states of a nonequilibrium system 350 13.3 Spectral density of fluctuations in bistable systems at low noise intensities 361 13.4 Spectral density of fluctuations in underdamped systems 367 Index 381 viii Contributors G. Broggi Physik Institut der Universitiit Schonberggasse 9 CH-8001 Zurich Switzerland M. Biittiker IBM Thomas J. Watson Research Center POB 218 Yorktown Heights NY 10598 USA Edward Celarier Chemical Physics Theory Group Department of Chemistry University of Toronto Toronto Ontario M5S lAl Canada M. I. Dykman Institute of Semiconductors Academy of Sciences of the UkrSSR pr. Nauki, 115 Kiev, 252028 USSR Ronald F. Fox School of Physics Georgia Institute of Technology Atlanta GA 30332 USA ix Contributors Peter Hanggi Lehrstuhl fiir Theoretische Physik Universitat Augsburg Memminger Strasse 6 D-8900 Ausburg FRG Werner Horsthemke Center for Studies in Statistical Mechanics Department of Physics University of Texas at Austin Austin TX 78712 USA Raymond Kapral Chemical Physics Theory Group Department of Chemistry University of Toronto Toronto Ontario M5S lAl Canada Edgar Knobloch Department of Physics University of California Berkeley CA 94720 USA Dilip K. Kondepudi Department of Chemistry Wake Forest University PO Box 7486 Winston-Salem NC 27109 USA M. A. Krivoglaz Institute of Metal Physics Academy of Sciences of the UkrSSR pr. Nauki, 115 Kiev, 252028 USSR x
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