Large fluctuations and singular behavior of nonequilibrium systems

D. Pinna, A. D. Kent, and D. L. Stein

Phys. Rev. E 93, 012114 – Published 11 January 2016

Abstract

We present a general geometrical approach to the problem of escape from a metastable state in the presence of noise. The accompanying analysis leads to a simple condition, based on the norm of the drift field, for determining whether caustic singularities alter the escape trajectories when detailed balance is absent. We apply our methods to systems lacking detailed balance, including a nanomagnet with a biaxial magnetic anisotropy and subject to a spin-transfer torque. The approach described within allows determination of the regions of experimental parameter space that admit caustics.

Received 8 May 2015
Revised 29 October 2015

DOI:https://doi.org/10.1103/PhysRevE.93.012114

Physics Subject Headings (PhySH)

Fluctuations & noise Nonequilibrium statistical mechanics

Statistical Physics & Thermodynamics

Authors & Affiliations

D. Pinna^* and A. D. Kent^†

Department of Physics, New York University, New York, New York 10003, USA

D. L. Stein^‡

Department of Physics, New York University, New York, New York 10003, USA; Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA; and NYU-ECNU Institutes of Physics and Mathematical Sciences at NYU Shanghai, 3663 Zhongshan Road North, Shanghai, 200062, China

^*Current address: Unité Mixte de Physique CNRS/Thales, 1 Avenue Augustin Fresnel, 91767 Palaiseau, France; daniele.pinna@nyu.edu
^†andy.kent@nyu.edu
^‡daniel.stein@nyu.edu

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Vol. 93, Iss. 1 — January 2016

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Images

Figure 1
Diagram of momentum ellipse parametrized by $γ$ . $γ_{0}$ and $π + γ_{0}$ correspond to anti-instanton and instanton solutions, respectively.
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Figure 2
Contour plot of the norm of the drift field from Ref. [13], for $α = 3$ (left) and $α = 5$ (right). Optimal escape trajectories are shown in red (medium gray), and other instanton trajectories in green (light gray). For $α = 3$ , instanton trajectories do not cross, and the escape trajectory lies along the $x$ axis. Correspondingly, two global minima in the norm of the drift field are present at the unstable $(0, 0)$ and stable $(1, 0)$ equilibria, respectively, along with a saddle close to the midpoint of the $x$ axis. For $α = 5$ , instanton trajectories cross, indicating the presence of a caustic (not shown). Correspondingly, the saddle in the norm of the drift field for $α = 3$ is now a local maximum, with two new local minima appearing off the $x$ axis. Consequently, there are now two symmetrical off-axis optimal escape trajectories that follow the new off-axis minima in the norm of the drift field (and in so doing avoid the caustic). A standard “shooting method” was employed to compute the trajectories. The FW dynamical equations were solved directly by evolving initial states taken very close to the fixed point of the drift field. Initial momenta were chosen to satisfy a gradient-like solution $p (x) = - 2 F (x)$ . The accuracy of this approach is guaranteed by the fact that in the neighborhood of a fixed point, the deterministic drift can always be approximated by the gradient of a potential.
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Figure 3
Constant energy contours of the unit magnetization sphere. Red (medium gray) and blue (dark gray) contours correspond to $ε < 0$ and $ε > 0$ precessions, respectively. Dashed black lines are the $ε = 0$ separatrices.
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Figure 4
Escape trajectory for a macrospin model with $α = 0.01, D = 0, I = 0.3 I_{C}$ , and $ω = 0$ . Blue (solid) line shows the least-action result of numerical integration of the FW dynamics. Red (dashed) line is the analytical result Eq. (14).
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Figure 5
Escape trajectory for a macrospin model with $α = 0.01, D = 0, I = 0.3 I_{C}$ , and $ω = 0.3 θ_{C}$ (defined in text). The purple (solid) line shows the least-action result of numerical integration of the FW dynamics. The red (dashed) line is the analytical result Eq. (14).
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Figure 6
Escape trajectory for macrospin model with $α = 0.01, D = 20, I = 0.8 I_{C}$ , and varying tilts: (a) $ω = 0$ , (b) $ω = 0.1 θ_{C}$ , (c) $ω = 0.25 θ_{C}$ . Different color trajectories correspond to different initial conditions of the FW dynamics. Dashed black lines correspond to the $ε = 0$ separatrices. In all scenarios, escape trajectories never cross, indicating the absence of caustics.
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