Finite-barrier Corrections Research Articles

Activated quantum diffusion in a periodic potential above the crossover temperature.

The recently improved Pollak, Grabert, and Hänggi (PGH) turnover theory for activated surface diffusion, including finite barrier effects, is extended and studied in the quantum domain. Analytic expressions are presented for the diffusion coefficient, escape rate, hopping distribution, and mean squared path length of particles initially trapped in one of the wells of a periodic potential, moving under the influence of a frictional and Gaussian random force. Tunneling is included by assuming incoherent quantum hopping at temperatures which are above the crossover temperature between deep tunneling and thermal activation. In the improved version of PGH theory as applied to activated surface diffusion, the potential governing the motion of the unstable mode remains periodic but with a scaled mass which increases with the friction strength. Application of the theory to a periodic cosine potential demonstrates that in the weak damping regime quantum diffusion is slower than classical diffusion due to above barrier quantum reflection which significantly shortens the mean squared path length as compared to the classical result. Finite barrier corrections increase this quantum suppression of diffusion or, equivalently, the inverse isotope effect, whereby the diffusion is faster for a heavier mass.

Finite-barrier corrections for multidimensional barriers in colored noise.

The usual identification of reactive trajectories for the calculation of reaction rates requires very time-consuming simulations, particularly if the environment presents memory effects. In this paper, we develop a method that permits the identification of reactive trajectories in a system under the action of a stochastic colored driving. This method is based on the perturbative computation of the invariant structures that act as separatrices for reactivity. Furthermore, using this perturbative scheme, we have obtained a formally exact expression for the reaction rate in multidimensional systems coupled to colored noisy environments.

Finite-barrier correction for the ferromagnetic resonance frequency of nanomagnets with various magnetocrystalline anisotropies

Finite-barrier corrections to the ferromagnetic resonance (FMR) frequency of nanomagnets are obtained in closed integral form from the undamped deterministic equation of motion of the magnetization by averaging the precession frequency as expressed by elliptic functions over all possible precessional trajectories inside a well. The method is illustrated by determining the FMR frequency for nanomagnets with both uniaxial and biaxial anisotropy subjected to a uniform external field and for nanomagnets with mixed cubic and uniaxial anisotropy. The results agree with exact numerical calculations obtained from the magnetic Langevin equation via matrix continued fractions.

Kramers' theory for diffusion on a periodic potential.

Kramers' turnover theory, based on the dynamics of the collective unstable normal mode (also known as PGH theory), is extended to the motion of a particle on a periodic potential interacting bilinearly with a dissipative harmonic bath. This is achieved by considering the small parameter of the problem to be the deviation of the collective bath mode from its value along the reaction coordinate, defined by the unstable normal mode. With this change, the effective potential along the unstable normal mode remains periodic, albeit with a renormalized mass, or equivalently a renormalized lattice length. Using second order classical perturbation theory, this not only enables the derivation of the hopping rates and the diffusion coefficient, but also the derivation of finite barrier corrections to the theory. The analytical results are tested against numerical simulation data for a simple cosine potential, ohmic friction, and different reduced barrier heights.

A study of Kramers' turnover theory in the presence of exponential memory friction.

Originally, the challenge of solving Kramers' turnover theory was limited to Ohmic friction, or equivalently, motion of the escaping particle governed by a Langevin equation. Mel'nikov and Meshkov [J. Chem. Phys. 85, 1018 (1986)] (MM) presented a solution valid for Ohmic friction. The turnover theory was derived more generally and for memory friction by Pollak, Grabert, and Hänggi [J. Chem. Phys. 91, 4073 (1989)] (PGH). Mel'nikov proceeded to also provide finite barrier corrections to his theory [Phys. Rev. E 48, 3271 (1993)]. Finite barrier corrections were derived only recently within the framework of PGH theory [E. Pollak and R. Ianconescu, J. Chem. Phys. 140, 154108 (2014)]. A comprehensive comparison between MM and PGH theories including finite barrier corrections and using Ohmic friction showed that the two methods gave quantitatively similar results and were in quantitative agreement with numerical simulation data. In the present paper, we extend the study of the turnover theories to exponential memory friction. By comparing with numerical simulation, we find that PGH theory is rather accurate, even in the strong friction long memory time limit, while MM theory fails. However, inclusion of finite barrier corrections to PGH theory leads to failure in this limit. The long memory time invalidates the fundamental assumption that consecutive traversals of the well are independent of each other. Why PGH theory without finite barrier corrections remains accurate is a puzzle.

Finite barrier corrections to the PGH solution of Kramers' turnover theory

Kramers [Physica 7, 284 (1940)], in his seminal paper, derived expressions for the rate of crossing a barrier in the underdamped limit of weak friction and the moderate to strong friction limit. The challenge of obtaining a uniform expression for the rate, valid for all damping strengths is known as Kramers turnover theory. Two different solutions have been presented. Mel'nikov and Meshkov [J. Chem. Phys. 85, 1018 (1986)] (MM) considered the motion of the particle, treating the friction as a perturbation parameter. Pollak, Grabert, and Hänggi [J. Chem. Phys. 91, 4073 (1989)] (PGH), considered the motion along the unstable mode which is separable from the bath in the barrier region. In practice, the two theories differ in the way an energy loss parameter is estimated. In this paper, we show that previous numerical attempts to resolve the quality of the two approaches were incomplete and that at least for a cubic potential with Ohmic friction, the quality of agreement of both expressions with numerical simulation is similar over a large range of friction strengths and temperatures. Mel'nikov [Phys. Rev. E 48, 3271 (1993)], in a later paper, improved his theory by introducing finite barrier corrections. In this paper we note that previous numerical tests of the finite barrier corrections were also incomplete. They did not employ the exact rate expression, but a harmonic approximation to it. The central part of this paper, is to include finite barrier corrections also within the PGH formalism. Tests on a cubic potential demonstrate that finite barrier corrections significantly improve the agreement of both MM and PGH theories when compared with numerical simulations.

Simulation of prefactor anomalies in superionic conductors in the case of arbitrary energy losses. III. Finite-barrier corrections

In previous publications (Samgin and Ezin, Ionics 14:345–348, 2008; Samgin and Ezin, Ionics 15:717–723, 2009), we presented a numerical technique to study the prefactor in the ion diffusion coefficient as a function of the average energy losses of ions to the bath. In the present paper, we model the prefactor in a perturbation theoretical fashion when energy losses are not constant with the ion energy. Using cubic and quartic potentials as examples, we show that including the energy dependence to the energy losses to be converted in finite-barrier corrections may be expected to reduce further the value of the prefactor. We give an extension of the Huberman–Boyce formula for conductivity prefactors from constant to varying energy losses. The finite-barrier corrections in a protonic conductor are estimated to be of the order of 10 % to a leading-order term in the prefactor expansion. These corrections must be built into an accurate analysis of ion diffusion mechanism.

Kramers’ turnover theory for diffusion of Na atoms on a Cu(001) surface measured by He scattering

The diffusion of adatoms and molecules on a surface at low coverage can be measured by helium scattering. The experimental observable is the dynamic structure factor. In this article, we show how Kramers’ turnover theory can be used to infer physical properties of the diffusing particle from the experiment. Previously, Chudley and Elliot showed, under reasonable assumptions, that the dynamic structure factor is determined by the hopping distribution of the adsorbed particle. Kramers’ theory determines the hopping distribution in terms of two parameters only. These are an effective frequency and the energy loss of the particle to the bath as it traverses from one barrier to the next. Kramers’ theory, including finite barrier corrections, is tested successfully against numerical Langevin equation simulations, using both separable and nonseparable interaction potentials. Kramers’ approach, which really is a steepest descent estimate for the rate, based on the Langevin equation, involves closed analytical expressions and so is relatively easy to implement. Diffusion of Na atoms on a Cu(001) surface has been chosen as an example to illustrate the application of Kramers’ theory.

Reaction rate theory: weak- to strong-friction turnover in Kramers' Fokker-Planck model

We present a unified treatment for both weak and strong friction — including the turnover regime — of Kramers' Fokker-Planck model for activated rate processes. No recourse to a specific microscopic model (à la Grabert et al.) or Langevin dynamics (à la Graham) will be made. Upon introducing some new theoretical concepts such as a constrained Gaussian transformation (CGT) and a dynamical extension of phase space (EPS), the analysis proceeds in terms of the unstable-mode energy and is systematic in a small parameter. We also introduce a nonlinear toy model for barrier recrossing for a simplified discussion of aspects like finite-barrier corrections, the boundary layer approximation, and exact solution in the absence of potential conditions.

Numerical test of finite-barrier corrections for the hopping rate in the underdamped regime.

It is shown that the large differences, which have been found in the underdamped regime, between analytical calculations and exact numerical results for the hopping rate in a periodic potential are essentially due to finite-barrier effects. In fact, if finite-barrier corrections are taken into account, the analytical results turn out to be in quite good agreement with the exact ones in the full damping range below the turnover point.

Erratum: Activated decay rate: Finite-barrier corrections

Erratum: Activated decay rate: Finite-barrier corrections

Low-dimensional turnover theory of thermal activation.

Kramers' Fokker-Planck model for activated rate processes (noise-induced escape over a potential-energy barrier) is solved without recourse to any microscopic theory (Grabert [Phys. Rev. Lett. 61, 1683 (1988)]) or associated Langevin equations (Graham [J. Stat. Phys. 60, 675 (1990)]). The independence of the microscopy of Grabert's original result for the escape rate is thereby definitely proven and clarified. Throughout, the analysis is systematic in a small parameter and provides a unified treatment of both weak and strong friction, including the turnover regime. Inter alia we introduce some novel theoretical concepts, such as a constrained Gaussian transformation and a dynamical extension of phase space. Finite-barrier corrections are also investigated.

Finite barrier corrections for the Kramers rate problem in the spatial diffusion regime

Finite barrier height corrections for the rate of thermally activated escape of a particle over a barrier out of a metastable well are determined within the framework of Fokker—Planck processes in the spatial diffusion regime. Different existing concepts to obtain the rate are extended and compared to each other. The central role that is played by Kramers' stationary current carrying probability density is demonstrated. An improvement of this function by means of a perturbation theory allows one to calculate corrections to the rate by means fo the flux over population expression, the Rayleigh quotient and the mean first passage time to the stochastic separatrix. The Rayleigh quotient is in a sense superior as it allows one to obtain from the same stationary current carrying Kramers function higher order corrections than by means of other methods. Explicit results are obtained for the case of a cubic and a quartic potential and compared with existing results.

Activated decay rate: Finite-barrier corrections.

The activated escape of an underdamped Brownian particle out of a deep potential well is characterized by weak friction \ensuremath{\gamma}\ensuremath{\ll}\ensuremath{\omega} (\ensuremath{\gamma} is the coefficient of friction and \ensuremath{\omega} is a typical frequency of the intrawell motion) and by a large barrier height ${\mathit{U}}_{0}$\ensuremath{\gg}T (${\mathit{U}}_{0}$ is the barrier height and T is the temperature). The approach developed previously to calculate the decay rate is based on the derivation of an integral equation and enables one to sum up an infinite series in powers of the ratio \ensuremath{\gamma}${\mathit{U}}_{0}$/T\ensuremath{\omega}\ensuremath{\sim}1 contributing to the preexponential factor of the Arrhenius law. In the present paper it is shown that the leading correction to the above result comes from the slowing down of the particle motion near the top of the barrier and is of the order of (T/${\mathit{U}}_{0}$) ln(${\mathit{U}}_{0}$/T). To calculate it explicitly, one needs to find a correction to the kernel of the above-mentioned integral equation. Beyond the leading-logarithmic approximation, two different factors contribute corrections of the order of T/${\mathit{U}}_{0}$\ensuremath{\sim}\ensuremath{\gamma}/\ensuremath{\omega}. The noise-induced effects in the barrier crossing-recrossing by particles in a narrow energy range \ensuremath{\varepsilon}\ensuremath{\sim}\ensuremath{\gamma}T/\ensuremath{\omega} can be easily incorporated into the general scheme of the calculations. On the other hand, a more accurate derivation of the kernel of the integral equation is required to take into account small variations of the intrawell particle motion caused by variations of the particle energy on the scale T\ensuremath{\ll}${\mathit{U}}_{0}$ under the effects of friction and thermal noise. The proposed consistent expansion in terms of the small parameters of the problem provides an effective approach to a quantitative investigation of the turnover behavior in the Kramers problem. For the regime of an intermediate-to-strong friction, the finite-barrier corrections can be neglected, since, for typical barrier shapes, they are always small.

Numerical test of finite-barrier corrections for the hopping rate in a periodic potential.

It is demonstrated that a recent finite-barrier expansion for jump rates accounts quantitatively for the observed discrepancy between numerically determined exact rates and the Kramers estimates of these rates.