Quantum Dissipation: A Primer

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1 Quantum Dissipation: A Primer P. Hänggi Institut für Physik Universität Augsburg

4 NOISE-INDUCED ESCAPE

11 Reaction-rate theory: fifty years after Kramers CONTENTS Peter Hanggi Lehrstuhl fur Theoretische Physik, University of Augsburg, Augsburg, Federal Republic of Germany Peter Talkner* Department of Physics, University of Basel, CH-4056 Basel, Switzerland Michal Borkovec lnstitut fur Lebensmittelwissenschaft, ETH-Zentrum, CH-8092 Zurich, Switzerland The calculation of rate coefficients is a discipline of nonlinear science of importance to much of physics, chemistry, engineering, and biology. Fifty years after Kramers' seminal paper on thermally activated barrier crossing, the authors report, extend, and interpret much of our current understanding relating to theories of noise-activated escape, for w-hich many of the notable contributions are originating from the communities both of physics and of physical chemistry. Theoretical as well as numerical approaches are discussed for single- and many-dimensional metastable systems (including fields) in gases and condensed phases. The role of many-dimensional transition-state theory is contrasted with Kramers' reaction-rate theory for moderate-to-strong friction; the authors emphasize the physical situation and the close connection between unimolecular rate theory and Kramers' work for weakly damped systems. The rate theory accounting for memory friction is presented, together with a unifying theoretical approach which covers the whole regime of weak-to-moderate-to-strong friction on the same basis (turnover theory). The peculiarities of noise-activated escape in a variety of physically different metastable potential configurations is elucidated in terms of the mean-first-passage-time technique. Moreover, the role and the complexity of escape in driven systems exhibiting possibly multiple, metastable stationary nonequilibrium states is identified. At lower temperatures, quantum tunneling effects start to dominate the rate mechanism. The early quantum approaches as well as the latest quantum versions of Kramers' theory are discussed, thereby providing a description of dissipative escape events at all temperatures. In addition, an attempt is made to discuss prominent experimental work as it relates to Kramers' reaction-rate theory and to indicate the most important areas for future research in theory and experiment. List of Symbols I. Introduction 11. Roadway To Rate Calculations A. Separation of time scales B. Equation of motion for the reaction coordinate C. Theoretical concepts for rate calculations 1. The flux-over-population method 2. Method of reactive flux 3. Method of lowest eigenvalue, mean first-passage time, and the like 111. Classical Transition-State Theory A. Simple transition-state theory B. Canonical multidimensional transition-state theory 1. Multidimensional transition-state rate for a collection of A' vibrational bath modes 2. Atom-transfer reaction D. Energy-diffusion-limited rate E. Spatial-diffusion-limited rate: the Smoluchowski limit F. Spatial-diffusion-limited rate in many dimensions and fields 1. The model 2. Stationary current-carrying probability density 3. The rate of nucleation G. Regime of validity for Kramers' rate theory V. Unimolecular Rate Theory A. Strong collision limit B. Weak collision limit C. Between strong and weak collisions D. Beyond simple unimolecular rate theory VI. Turnover between Weak and Strong Friction A. Interpolation formulas B. Turnover theory: a normal-mode approach C. Peculiarities of Kramers' theory with memory fric- 3. Dissociation reaction 264 tion 4. Recombination reaction 265 VII. Mean-First-Passage-Time Approach C. Model case: particle coupled bilinearly to a bath of A. The mean first-passage time and the rate harmonic oscillators 266 B. The general Markovian case. 1. The model 266 C. Mean first-passage time for a one-dimensional Smo- 2. Normal-mode analysis 3. The rate of escape IV. Kramers Rate Theory A. The model B. Stationary fiux and rate of escape C. Energy of injected particles *Present address: Paul Scherrer Institut, CH-5232 Villigen, Switzerland. luchowski equation 1. The transition rate in a double-well potential 2. Transition rates and effective diffusion in periodic potentials 3. Transition rates in random potentials 4. Diffusion in spherically symmetric potentials D. Mean first-passage times for Fokker-Planck process. es in many dimensions E. Sundry topics from contemporary mean-firstpassage-time theory 1. Escape over a quartic (- x') barrier Reviews of Modern Physics, Vol. 62, No. 2, April 1990 Copyright The American Physical Society

12 HBnngi, Talkner, and Borkovec: Reaction-rate theory 2. Escape over a cusp-shaped barrier 3. Mean first-passage time for shot noise 4. First-passage-time problems for non-markovian processes VIII. Transition Rates in Nonequilibrium Systems A. Two examples of one-dimensional nonequilibrium rate problems 1. Bistable tunnel diode 2. Nonequilibrium chemical reaction B. Brownian motion in biased periodic poteiitials C. Escape driven by colored noise D. Nucleation of driven sine-gordon solitolls 1. Nucleation of a single string 2. Nucleation of interacting pairs IX. Quantum Rate Theory A. Historic background and perspectives; traditional quantum approaches B. The functional-integral approach C. The crossover temperature D. The dissipative tunneling rate 1. Flux-flux autocorrelation function expression for the quantum rate 2. Unified approach to the quantum-~;amers rate 3. Results for the quantum-kramers rate a. Dissipative tunneling above crossover b. Dissipative tunneling near crossover c. Dissipative tunneling below crossover 4. Regime of validity of the quantum-kramers rate E. Dissipative tuntieling at weak dissipation 1. Quantum escape at very weak friction 2. Quantum turnover F. Sundry topics on dissipative tunneling 1. Incoherent tunneling in weakly biased metastable wells 2. Coherent dissipative tunneling 3. Tunneling with fermionic dissipation X. Numerical Methods in Rate Theory XI. Experiments A. Classical activation regime B. Low-temperature quantum effects XII. Conclusions and Outlook Acknowledgments Appendix A: Evaluation of the Gaussian Surface Integral in Eq. (4.77) Appendix B: A Formal Relation between the MFPT and the Flux-Over-Population Method References LIST OF SYMBOLS temperature-dependent quantum rate prefactor correlation function diffusion coefficient energy function activation energy (=barrier energy with the energy at the metastable state set equal to zero) Hessian matrix of the energy function at the stable state Hessian matrix of the energy function around the saddle-point configuration action variable of the reaction coordinate Jacobian transition probability kernel mass of reactive particle period of oscillation in the classically allowed region classical conditional probability of finding the energy E, given initially the energy E' quantum correction to the classical prefactor dissipative bounce action temperature crossover temperature period in the classically forbidden regime metastable potential function for the reaction coordinate volume of a reacting system partition function, inverse normalization partition function of the locally stable state (A) partition function of the transition rate Hamiltonian function of the metastable system complex-valued free energy of a metastable state Fokker-Planck operator backward operator of a Fokker-Planck process total probability flux of the reaction coordinate Planck's constant h ( 2 ~ ) ~ ' Boltzmann constant reaction rate forward rate backward rate transition-state rate microcanonical transition-state rate, semiclassical cumulative reaction probability spatial-diffusion-limited Smoluchowski rate mass of ith degree of freedom probability density stationary nonequilibrium probability density for the reaction coordinate momentum degree of freedom configurational degree of freedom quantum reflection coefficient density of sources and sinks quantum transmission coefficient mean first-passage time to leave the domain fl, with the starting point at x constant part of the mean first-passage time to leave a metastable domain of attraction velocity of the reaction coordinate reaction coordinate location of well minimum or potential minimum of state A, respectively barrier location location of the transition state inversion temperature (kg T)- ' Rev. Mod. Phys., Vol. 62, No. 2, April 1990

15 QUANTUM NOISE

26 J.B. Johnson Thermal agitation of electricity in conductors. Phys. Rev. (1928) 32 (July) H. Nyquist Thermal agitation of electric charge in conductors. Phys. Rev. (1928) 32 (July) L. Onsager Reciprocal relation in irreversible process. Phys. Rev. (1931) 32 (February) H.B. Callen, T.A. Welton Irreversibility and Generalized Noise. Phys. Rev. (1951) 83 (1) 34-40

44 HOMEPAGE HANGGI GO TO : FEATURE ARTICLES Quantum Dissipation and Quantum Transport theo1/hanggi/quantum.html

Preface. Preface to the Third Edition. Preface to the Second Edition. Preface to the First Edition. 1 Introduction 1

Preface. Preface to the Third Edition. Preface to the Second Edition. Preface to the First Edition. 1 Introduction 1 xi Contents Preface Preface to the Third Edition Preface to the Second Edition Preface to the First Edition v vii viii ix 1 Introduction 1 I GENERAL THEORY OF OPEN QUANTUM SYSTEMS 5 Diverse limited approaches: