On the Kramers Very Low Damping Escape Rate for Point Particles and Classical Spins

Byrne, Declan J.; Coffey, W. T.; Dowling, William J.; Kalmykov, Yuri P.; Titov, S. V.

doi:10.1002/9781118949702.ch7

Cited by 4 publications

(11 citation statements)

References 32 publications

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“…As a final remark, one observes that the approach introduced here can be addressed to the well known problem of the diffusion controlled escaping from a potential well [16,17]. In this kind of problem, the calculation of the rate coefficients has a central importance [18] and the calculation developed in the present work can be used to compute these quantities.…”

Section: Resultsmentioning

confidence: 89%

Approximate solution for Fokker-Planck equation

Araujo¹,

Filho²

2015

Condens. Matter Phys.

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In this paper, an approximate solution to a specific class of the Fokker-Planck equation is proposed. The solution is based on the relationship between the Schrödinger type equation with a partially confining and symmetrical potential. To estimate the accuracy of the solution, a function error obtained from the original Fokker-Planck equation is suggested. Two examples, a truncated harmonic potential and non-harmonic polynomial, are analyzed using the proposed method. For the truncated harmonic potential, the system behavior as a function of temperature is also discussed.

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Section: Resultsmentioning

confidence: 89%

Approximate solution for Fokker-Planck equation

Araujo¹,

Filho²

2015

Condens. Matter Phys.

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“…By analogy with Kramers' derivation of the energy-controlled diffusion equation for point particles in the VLD limit [1], one may parameterize the instantaneous magnetization direction of a macrospin by the slow dimensionless energy variable E and the fast precessional variable  running uniformly along a closed Stoner-Wohlfarth orbit of energy E implying that E d f dt   [13], where E f is the precession frequency of precession in the potential well at a given energy E [13,24]. The phase  is the generalized coordinate conjugate to the magnetic action S(E), i.e., the area inside a closed region of constant energy E [13,24]. The magnetic action can be written in dimensionless form as [13] 0 ()…”

Section: Energy-controlled-diffusion Equation With Sttmentioning

confidence: 99%

“…We now consider the right-hand side of the Fokker-Planck equation (13) for the surface probability function ( , , )…”

Section: Acknowledgementsmentioning

confidence: 99%

“…As two degrees of freedom (namely, the polar and azimuthal angles ,  describing the magnetization orientation in configuration space) are involved, the spin Hamiltonian, unlike that of particles, is no longer separable and additive since the governing magnetic Langevin equation is essentially a form of the Larmor equation so that inertial effects play no role here [11,12]. Nevertheless, the role of inertia in the mechanical system is essentially mimicked in the magnetic system for noncircularly symmetric free energy potentials by the gyromagnetic term, which causes coupling or entanglement of We recall that the VLD spin escape rate both with and without STT has already been determined [12,13] either via involved vector manipulation [14] or else via long and complicated calculations with the energy and phase as variables in the magnetic Langevin equation involving multiplicative noise [13,15,16,17]. Both methods ultimately lead to the same energy-controlled-diffusion equation for giant classical spins with quasi-stationary solutions yielding the VLD rate.…”

Section: Introductionmentioning

confidence: 99%

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On a simple derivation of the very low damping escape rate for classical spins by modifying the method of Kramers

Byrne

Coffey

Kalmykov

et al. 2019

Physica A: Statistical Mechanics and its Applications

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The original perturbative Kramers' method (starting from the phase space coordinates) [H.A. Kramers, Physica 7, 384 (1940)] of determining the energy-controlled-diffusion equation for Newtonian particles with separable and additive Hamiltonians is generalized to yield the energy-controlled diffusion equation and thus the very low damping (VLD) escape rate including spin-transfer torque for classical giant magnetic spins with two degrees of freedom. These have dynamics governed by the magnetic Langevin and Fokker-Planck equations and thus are generally based on non-separable and non-additive Hamiltonians. The derivation of the VLD escape rate directly from the (magnetic) Fokker-Planck equation for the surface distribution of magnetization orientations in the configuration space of the polar and azimuthal angles ( , )

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“…(1.1) is modified when the dynamics is driven by a general multiplicative noise, modeled by a diffusion function g(x). This topic have been rarely treated in the past and there is some controversy in the literature [28][29][30][31][32][33]. In particular, we study the dependence of the escape rate on the stochastic prescription, necessary to correctly define the multiplicative noise Langevin equation.…”

Section: Introductionmentioning

confidence: 99%

State dependent diffusion in a bistable potential: conditional probabilities and escape rates

Moreno,

Barci,

Arenas

2019

Preprint

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We consider a simple model of a bistable system under the influence of multiplicative noise. We provide a path integral representation of the overdamped Langevin dynamics and compute conditional probabilities and escape rates in the weak noise approximation. The saddle-point solution of the functional integral is given by a diluted gas of instantons and anti-instantons, similarly to the additive noise problem. However, in this case, the integration over fluctuations is more involved. We introduce a local time reparametrization that allows its computation in the form of usual Gaussian integrals. We found corrections to the Kramers escape rate produced by the diffusion function which governs the state dependent diffusion for arbitrary values of the stochastic prescription parameter.

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On the Kramers Very Low Damping Escape Rate for Point Particles and Classical Spins

Cited by 4 publications

References 32 publications

Approximate solution for Fokker-Planck equation

Approximate solution for Fokker-Planck equation

On a simple derivation of the very low damping escape rate for classical spins by modifying the method of Kramers

State dependent diffusion in a bistable potential: conditional probabilities and escape rates

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