Kramers problem

Berezhkovskii A M and Zitserman V Yu 1991 Activated rate processes in the multidimensional case. Consideration of recrossings in the multidimensional Kramers problem with anisotropic friction Chem. Phys. 157 141-55... 

The most general problem should be that of a particle in a nonseparable potential, linearly coupled to an oscillator heat bath, when the dynamics of the particle in the classically accessible region is subject to friction forces due to the bath. However, this multidimensional quantum Kramers problem has not been explored as yet. 

V. I. Mel nikov and S. V. Meshkov, Theory of activated rate processes exact solution of the Kramers problem, J. Chem. Phys. 85, 1018 (1986). 

EVOLUTION TIMES OF PROBABILITY DISTRIBUTIONS AND AVERAGES—EXACT SOLUTIONS OF THE KRAMERS PROBLEM... 

The original work of Kramers [11] stimulated research devoted to calculation of escape rates in different systems driven by noise. Now the problem of calculating escape rates is known as Kramers problem [1,47]. 

We are now at the point where a quantum theory of condensed phase reactions may be developed. The Zwanzig Hamiltonian Eq. (7) has a natural quantum analog that consists in treating the Hamiltonian quantum-mechanically. In the rest of this paper we shall call this quantum analog the quantum Kramers problem. 

After a lengthy calculation the correlation function for the Kramers problem Eq. (17) can be shown to be equal to... 

The instanton method takes into account only the dynamics of the lowest energy doublet. This is a valid description at low temperature or for high barriers. What happens when excitations to higher states in the double well are possible And more importantly, the equivalent of this question in the condensed phase case, what is the effect of a symmetrically coupled vibration on the quantum Kramers problem The new physical feature introduced in the quantum Kramers problem is that in addition to the two frequencies shown in Eq. (28) there is a new time scale the decay time of the flux-flux correlation function, as discussed in the previous Section after Eq. (14). We expect that this new time scale makes the distinction between the comer cutting and the adiabatic limit in Eq. (29) to be of less relevance to the dynamics of reactions in condensed phases compared to the gas phase case. 

Table I Activation energies for H and D transfer Three values are shown the activation energies calculated using a one- and two-dimensional Kramers problem and the experimental values.

The position-dependent part of the friction is manifest in the spatial dependence of the coupling function g (s). The usual quantum Kramers problem is recovered when g(s) = s. An implicit assumption in Eq. (36) is that the functional form of the coupling g(s) is the same for all modes k. 

In the standard overdamped version of the Kramers problem, the escape of a particle subject to a Gaussian white noise over a potential barrier is considered in the limit of low diffusivity—that is, where the barrier height AV is large in comparison to the diffusion constant K [14] (compare Fig.6). Then, the probability current over the potential barrier top near xmax is small, and the time change of the pdf is equally small. In this quasi-stationary situation, the probability current is approximately position independent. The temporal decay of the probability to find the particle within the potential well is then given by the exponential function [14, 22]... 

Evidently, because of its exponential dependence on the parameter ct = KVm/T, the quantity x(a) may grow virtually unboundedly. This is the specific property of the solution of any Kramers problem that the probability of transition between the potential minima (wells) depends exponentially on the barrier height related to the thermofluctuational energy. In Section III.B we revisit the problem of calculation of the overbarrier relaxation time and using a mathematically correct asymptotic expansion procedure, show how the preexponential factor in the overbarrier relaxation time may be evaluated with the accuracy far higher than that in Eq. (4.50). 

A cumulant expansion supplemented by a boson operator representation was used by Steiger and Fox to tackle the Kramers problem of Eq. (1.9) in the multidimensional case. This allowed them to obtain a multidimensional version of their earlier results. 

Very recently, Lavenda devised an interesting method of solution of the Kramers problem in the extreme low-friction limit. He was able to show that it could be reduced to a formal Schrddinger equation for the radial part of the hydrogen atom and thus be solved exactly. One particular form of the long-time behavior of the rigorous rate equation coincides with that obtained by Kramers with the quasi-stationary hypothesis and may thus clarify the implications of this hypothesis. The method of Lavenda is reminiscent of that used by van Kampen but applied to a Smoluchowski equation for the diffusion of the energy. 

This rate of energy exchange between an oscillator and the thermal environment was the focus of Chapter 13, where we have used a quantum harmonic oscillator model for the well motion. In the y -> 0 limit of the Kramers model we are dealing with energy relaxation of a classical anharmonic oscillator. One may justifiably question the use of Markovian classical dynamics in this part of the problem, and we will come to this issue later. For now we focus on the solution of the mathematical problem posed by the low friction limit of the Kramers problem. 

S. D. Schwartz, Nonequilibrium solvation and the Quantum Kramers problem proton transfer in aqueous glycine, /. Phys. Chem. B (Bill Miller festschrift), 105, 2563-2567 (2001) b) D. Antoniou, S. D. Schwartz, A Molecular Dynamics Quantum Kramers Study of Proton Transfer in Solution, /. Chem. Phys., 110, 465-472 (1999) c) D. Antoniou, S. D. Schwartz, Quantum Proton Transfer with Spatially Dependent Friction ... 

Kramers model is simplistic. It is one-dimensional and assumes that the friction is Markovian—uncorrelated in time. A multidimensional generalization of Kramers problem in the spatial diffusion limit was proposed and solved by Langer (18). The multidimensional energy diffusion limit was solved by Matkowsky, Schuss, and coworkers (19,20). A multidimensional turnover theory has been recently formulated (21). 

Ferrando, R., Spadacini, R., Tommei, G.E. Kramers problem in periodic potentials Jump rate and jump lengths, Phys. Rev. E 1993,48,2437. 

The determination of the constants Cs, t, and is a complicated problem requiring a complete solution of the Boltzmann equation, including the kinetic boundary layer. Exact solutions have been found only for certain modeled Boltzmann equations, like the BGK equation, in flows with a simple geometry (e-g- stationary shear flow along a flat plate in a semi-infinite space, the so-called Kramers problem). Approximate results have been obtained by using variational methods and moment expansions. ... 

If one replaces L in Eq. (138b) by Lbgk> the linearized Boltzmann equation can be solved for a number of interesting cases. The simplest case where Eqs. (138b) and (138c) have been solved completely is the so-called Kramers problem. Here one considers the flow of a gas in a semi-infinite space bounded by a plane wall with which the molecules make diffusive collisions. For this problem one can show that there is a kinetic boundary layer near the wall and that the Chapman-Enskog normal solution is correct for points that... 

In this connection it is worth pointing out that for the Kramers problem, an exact solution to the linearized Boltzmann equation for a BGK model gas has been constructed by Cercignani. Even for this case the answers to questions (a) and (b) are not known. ... 

Kramers paper of 1940 presents what today we call the "Kramers problem" the dynamics of a particle moving in a bi-stable external field of force, subject to the irregular forces of a... 

The Kramers Problem in the Turnover Regime The Role of the Stochastic Separatrix. 

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