Decay of metastable states with discrete dynamics

Reimann, Peter; Müller, Reinhard; Talkner, Peter

doi:10.1103/physreve.49.3670

Cited by 17 publications

(12 citation statements)

References 26 publications

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“…The time averaged rate (116) displays a remarkable structural similarity with the rate-expressions obtained in [50] for one-dimensional discrete-time systems in the presence of weak Gaussian white noise. While a general qualitative connection between these two different types of escape problems via some kind of stroboscopic mapping is quite suggestive, the quantitative details are not so simple.…”

Section: Discussionmentioning

confidence: 53%

Surmounting oscillating barriers: Path-integral approach for weak noise

2000

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We consider the thermally activated escape of an overdamped Brownian particle over a potential barrier in the presence of periodic driving. A time-dependent path-integral formalism is developed which allows us to derive asymptotically exact weak-noise expressions for both the instantaneous and the time-averaged escape rate. Our results comprise a conceptually different, systematic treatment of the rate prefactor multiplying the exponentially leading Arrhenius factor. Moreover, an estimate for the deviations at finite noise strengths is provided and a supersymmetry-type property of the time-averaged escape rate is verified. For piecewise parabolic potentials, the rate expression can be evaluated in closed analytical form, while in more general cases, as exemplified by a cubic potential, an action-integral remains to be minimized numerically. Our comparison with very accurate numerical results demonstrates an excellent agreement with the theoretical predictions over a wide range of driving strengths and driving frequencies.

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Section: Discussionmentioning

confidence: 53%

Surmounting oscillating barriers: Path-integral approach for weak noise

2000

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“…9͒ and systems without detailed balance. [10][11][12][13] In all these investigations, however, the height of the potential barrier ⌬U is assumed to be sufficiently large when compared to the energy of thermal motion ␤ Ϫ1 . Yet another fundamental assumption which is almost always made is that the inverse transition rate over the barrier is the largest relaxation time of the considered system.…”

Section: Introductionmentioning

confidence: 99%

Thermally activated escape processes in a double well coupled to a slow harmonic mode

Дроздов

Talkner

1996

The Journal of Chemical Physics

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Thermally activated escape rate for the Brownian motion of a fixed axis rotator in an asymmetrical double-well potential for all values of the dissipation Thermally activated escape rate for the Brownian motion of a fixed axis rotator in a double well potential for all values of the dissipation J. Chem. Phys. 120, 9199 (2004); 10.1063/1.1703525Thermally activated traversal of an energy barrier of arbitrary shape

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“…Decay rates of metastable states were determined for special two-dimensional Fokker-Planck cases for which the generalized potential is sufficiently smooth 122 and also for simple one-dimensional noisy maps. 123,124 In the latter case, as a consequence of the singularities of the generalized potential, the prefactor of the rate depends on the noise strength in a nonanalytical way.…”

Section: G Rates In Nonequilibrium Systemsmentioning

confidence: 99%

Reaction rate theory: What it was, where is it today, and where is it going?

Pollak

Talkner

2005

Chaos: An Interdisciplinary Journal of Nonlinear Science

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A brief history is presented, outlining the development of rate theory during the past century. Starting from Arrhenius [Z. Phys. Chem. 4, 226 (1889)], we follow especially the formulation of transition state theory by Wigner [Z. Phys. Chem. Abt. B 19, 203 (1932)] and Eyring [J. Chem. Phys. 3, 107 (1935)]. Transition state theory (TST) made it possible to obtain quick estimates for reaction rates for a broad variety of processes even during the days when sophisticated computers were not available. Arrhenius' suggestion that a transition state exists which is intermediate between reactants and products was central to the development of rate theory. Although Wigner gave an abstract definition of the transition state as a surface of minimal unidirectional flux, it took almost half of a century until the transition state was precisely defined by Pechukas [Dynamics of Molecular Collisions B, edited by W. H. Miller (Plenum, New York, 1976)], but even this only in the realm of classical mechanics. Eyring, considered by many to be the father of TST, never resolved the question as to the definition of the activation energy for which Arrhenius became famous. In 1978, Chandler [J. Chem. Phys. 68, 2959 (1978)] finally showed that especially when considering condensed phases, the activation energy is a free energy, it is the barrier height in the potential of mean force felt by the reacting system. Parallel to the development of rate theory in the chemistry community, Kramers published in 1940 [Physica (Amsterdam) 7, 284 (1940)] a seminal paper on the relation between Einstein's theory of Brownian motion [Einstein, Ann. Phys. 17, 549 (1905)] and rate theory. Kramers' paper provided a solution for the effect of friction on reaction rates but left us also with some challenges. He could not derive a uniform expression for the rate, valid for all values of the friction coefficient, known as the Kramers turnover problem. He also did not establish the connection between his approach and the TST developed by the chemistry community. For many years, Kramers' theory was considered as providing a dynamic correction to the thermodynamic TST. Both of these questions were resolved in the 1980s when Pollak [J. Chem. Phys. 85, 865 (1986)] showed that Kramers' expression in the moderate to strong friction regime could be derived from TST, provided that the bath, which is the source of the friction, is handled at the same level as the system which is observed. This then led to the Mel'nikov-Pollak-Grabert-Hanggi [Mel'nikov and Meshkov, J. Chem. Phys. 85, 1018 (1986); Pollak, Grabert, and Hanggi, ibid. 91, 4073 (1989)] solution of the turnover problem posed by Kramers. Although classical rate theory reached a high level of maturity, its quantum analog leaves the theorist with serious challenges to this very day. As noted by Wigner [Trans. Faraday Soc. 34, 29 (1938)], TST is an inherently classical theory. A definite quantum TST has not been formulated to date although some very useful approximate quantum rate theories have been invented. The s...

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Decay of metastable states with discrete dynamics

Cited by 17 publications

References 26 publications

Surmounting oscillating barriers: Path-integral approach for weak noise

Surmounting oscillating barriers: Path-integral approach for weak noise

Thermally activated escape processes in a double well coupled to a slow harmonic mode

Reaction rate theory: What it was, where is it today, and where is it going?

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