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Reaction-Rate Theory: Fifty Years After Kramers

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Reaction-Rate Theory: Fifty Years After Kramers Reaction-rate theory: fifty years after Kramers Peter Hanggi Peter Talkner* Department of Physics, University of Basel, CH-4056 Basel, Switzerland Michal Borkovec lnstitut fur Lehensmittelwissenschaft, ETH-Zentrum, CH-80922urich, 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 bar- rier crossing, the authors report, extend, and interpret much of our current understanding relating to theories of noise-activated escape, for which 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 connec- tion 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 pecu- liarities 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 es- cape 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, there- by 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 indi- cate the most important areas for future research in theory and experiment. CONTENTS D. Energy-diffusion-limited rate 273 E. Spatial-diffusion-limited rate: the Smoluchowski List of Symbols 252 limit 274 Introduction 253 F. Spatial-diffusion-limited rate in many dimensions Roadway To Rate Calculations 257 and fields 275 A. Separation of time scales 257 1 ~ The model 275 B. Equation of motion for the reaction coordinate 257 2. Stationary current-carrying probability density 276 C. Theoretical concepts for rate calculations 258 3. The rate of nucleation 277 1. The flux-over-population method 258 G. Regime of validity for Kramers' rate theory 278 2. Method of reactive flux 259 V. Unimolecular Rate Theory 279 3. Method of lowest eigenvalue, mean first-passage A. Strong collision limit 281 time, and the like 261 B. Weak collision limit 282 Classical Transition-State Theory 261 C. Between strong and weak collisions 284 A. Simple transition-state theory 262 D. Beyond simple unimolecular rate theory 285 B. Canonical multidimensional transition-state theory 263 VI. Turnover between Weak and Strong Friction 286 1. Multidimensional transition-state rate for a col- A. Interpolation formulas 286 lection of N vibrational bath modes 264 B. Turnover theory: a normal-mode approach 287 2. Atom-transfer reaction 264 C. Peculiarities of Kramers' theory with memory fric- 3. Dissociation reaction 264 tion 289 4. Recombination reaction 265 VII. Mean-First-Passage- Time Approach 290 C. Model case: particle coupled bilinearly to a bath of A. The mean first-passage time and the rate 290 harmonic oscillators 266 B. The general Markovian case 290 1. The model 266 C. Mean first-passage time for a one-dimensional Smo- 2. Normal-mode analysis 266 luchowski equation 291 3. The rate of escape 267 1. The transition rate in a double-well potential 291 IV. Kramers Rate Theory 268 2. Transition rates and effective diffusion in period- A. The model 268 ic potentials 292 B. Stationary flux and rate of escape 270 3. Transition rates in random potentials 293 C. Energy of injected particles 272 4. Diffusion in spherically symmetric potentials 294 D. Mean first-passage times for Fokker-Planck process- es in many dimensions 295 E. Sundry topics from contemporary mean-first- *Present address: Paul Scherrer Institut, CH-5232 Villigen, passage-time theory 297 Switzerland. 1. Escape over a quartic ( —x ) barrier 297 Reviews of Modern Physics, Vol. 62, No. 2, April 1990 Copyright 1990 The American Physical Society 252 H5nngi, Talkner, and Borkovec: Reaction-rate theory 2. Escape over a cusp-shaped barrier 298 K(x,x') transition probability kernel 3. Mean first-passage time for shot noise 299 M mass of reactive particle 4. First-passage-time problems for non-Markovian &(&) period of oscillation in the classically al- processes 300 lowed region VIII. Transition Rates in Nonequilibrium Systems 300 Q') classical conditional probability of finding A. Two examples of one-dimensional nonequilibrium E' rate problems 301 the energy E, given initially the energy 1. Bistable tunnel diode 301 quantum correction to the classical prefac- 2. Nonequilibrium chemical reaction 302 tor B. Brownian motion in biased periodic potentials 302 S~ dissipative bounce action C. Escape driven by colored noise 304 T temperature D. Nucleation of driven sine-Gordon solitons 306 crossover 1. Nucleation of a single string 307 +0 temperature 2. Nucleation of interacting pairs 308 T(E) period in the classically forbidden regime IX. Quantum Rate Theory 308 U(x) metastable potential function for the reac- A. Historic background and perspectives; traditional tion coordinate quantum approaches 308 V volume of a reacting system B. The functional-integral approach 310 z partition function, inverse normalization C. The crossover temperature 311 zO~zA partition function of the locally stable state D. The dissipative tunneling rate 313 1. Flux-flux autocorrelation function expression for (A) the quantum rate 314 z" partition function of the transition rate 2. Unified approach to the quantum-Kramers rate 314 Hamiltonian function of the metastable sys- 3. Results for the quantum-Kramers rate 315 tem a. Dissipative tunneling above crossover 315 complex-valued free energy of a Inetastable b. Dissipative tunneling near crossover 316 state c. Dissipative tunneling below crossover 316 Fokker-Planck operator 4. Regime of validity of the quantum-Kramers rate 318 E. Dissipative tunneling at weak dissipation 319 backward operator of a Fokker-Planck pro- 1. Quantum escape at very weak friction 319 cess 2. Quantum turnover 320 total probability Aux of the reaction coordi- F. Sundry topics on dissipative tunneling 321 nate 1. Incoherent tunneling in weakly biased metasta- h Planck's constant ble wells 321 h (2m) 2. Coherent dissipative tunneling 322 Boltzmann 3. Tunneling with fermionic dissipation 322 k~ constant X. Numerical Methods in Rate Theory 322 k reaction rate XI. Experiments 324 k+ forward rate A. Classical activation regime 325 k backward rate B. Low-temperature quantum effects 327 kTsT transition-state rate XII. Conclusions and Outlook 327 k(E) microcanonical transition-state rate, semi- Acknowledgments 330 classical cumulative reaction probability Appendix A: Evaluation of the Gaussian Surface Integral in Eq. (4.77) ks spatial-diffusion-limited Smoluchowski rate Appendix B: A Formal Relation between the MFPT and the mi. mass of ith degree of freedom Flux-Over-Population Method p(x, t) probability density References po(x) stationary nonequilibrium probability densi- ty for the reaction coordinate LIST GF SYMBGLS Ps momentum degree of freedom configurational degree of freedom temperature-dependent quantum rate prefac- r(E) quantum reAection coe%cient tor s(x) density of sources and sinks C(t) correlation function t (E) quantum transmission coefBcient D difT'usion coeKcient t„(x) mean first-passage time to leave the domain energy function Q, with the starting point at x activation energy (=barrier energy with the t MFPT constant part of the mean first-passage time energy at the metastable state set equal to to leave a metastable domain of attraction zero) velocity of the reaction coordinate Hessian matrix of the energy function at reaction coordinate the stable state X0,Xg location of well minimum or potential E(s) Hessian matrix of the energy function minimum of state 2, respectively around the saddle-point configuration barrier location action variable of the reaction coordinate location of the transition state Jacobian inversion temperature (kz T) Rev. Mod. Phys. , Vol. 62, No. 2, April 1990 Hanngi, Talkner, and Borkovec: Reaction-rate theory 253 'V damping relaxation rate The discipline of rate theory was created when Svante y(t) memory friction Arrhenius gave an extensive discussion of various y(z) Laplace transform of the memory friction reaction-rate data which, as a function of the inverse 5 dimensionless energy loss temperature P=(k~ T) ', vary on a logarithmic scale. In parameter measuring the small noise inten- other words, the rate coefticient k, or simply the rate of sity escape, follows the Van't Hoff-Arrhenius law (Van't Hoff, exponentially correlated Gaussian noise 1884, Arrhenius, 1889)' (Ornstein-Uhlenbeck noise) k =vexp( PE— 8(x) characteristic function for the reaction &), coordinate where Eb denotes the threshold energy for activation and classical noise v is a prefactor. Figure 2 depicts the Van't HofF- C noise
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