Anomalous cross-field diffusion in a magnetic trap

Sergey E. Savel'ev and Fabio Marchesoni

Phys. Rev. E 90, 062117 – Published 10 December 2014

Abstract

We numerically simulated the diffusion of a charged Brownian particle confined to a plane under the action of an orthogonal magnetic field with intensity depending on the distance from a center. Despite its apparent simplicity, this system exhibits anomalous diffusion. For positive field gradients, radial and angular dynamics are asymptotically subdiffusive, with exponents given by simple analytical expressions. In contrast, when driven by a weakly decaying field, the particle attains normal diffusion only after exceedingly long superdiffusive transients. These mechanisms can be related to Bohm diffusion in magnetized plasmas.

Received 14 May 2014

DOI:https://doi.org/10.1103/PhysRevE.90.062117

Authors & Affiliations

Sergey E. Savel'ev¹ and Fabio Marchesoni²

¹Department of Physics, Loughborough University, Loughborough LE11 3TU, United Kingdom
²Dipartimento di Fisica, Università di Camerino, I-62032 Camerino, Italy

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Vol. 90, Iss. 6 — December 2014

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Images

Figure 1
Sketch of a 2D magnetic trap with orthogonal magnetic field, $B = B (r) \hat{z}$ (vertical arrows) (a), and rotational potential vector $A (r)$ (circular field lines) (b)–(f). Panel (a): a charged Brownian particle is confined to the plane region encircled by a ring traversed by constant electric current. Panel (b): trajectory sample obtained by integrating the Langevin equations (1) with $q = m = 1$ , $k T = 1$ , $γ = 0.1$ , and cyclotron frequency of Eq. (2) with $Ω_{0} = Ω_{r} = r_{0} = 1$ and $n = 2$ . The end points are marked by dots. Panels (c)–(f): deterministic trajectories, $γ = 0$ , with initial conditions (c) $r (0) = (0, 0)$ , $v (0) = (0, 3)$ , (d) $r (0) = (0, 0)$ , $v (0) = (0, 10)$ , (e) $r (0) = (2, 0)$ , $v (0) = (0, 10)$ , and (f) $r (0) = (2, 0)$ , $v (0) = (0, - 10)$ . The inner and outer radial bounds, $r_{m}$ and $r_{M}$ , coincide with the analytical estimates given in the text.
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Figure 2
Particle diffusion $〈 r^{2} (t) 〉$ vs $t$ for the radially dependent $Ω (r)$ of Eq. (2) with $Ω_{0} = Ω_{r} = 1$ and positive $n$ (see legend). Lower-right inset: anomalous diffusion exponent. $n$ dependence of the fitting exponent $α_{n}$ introduced in Eq. (3): simulation data (dots) vs our prediction, $α_{n} = 1 / (n + 1)$ , from Eq. (5) (solid curve). The dot $α_{0} = 1$ obtained in the limit $n \to 0$ (normal diffusion) has been added for completeness. Upper-left inset: the corresponding phase diffusion, $〈 Δ ϕ 〉 = 〈 ϕ^{2} 〉 - {〈 ϕ 〉}^{2}$ vs $t$ , for three choices of $n$ (see legend); the relevant asymptotic laws, Eq. (6), are drawn for a comparison (dashed lines). Other simulation parameters are $q = m = 1$ , $k T = 1$ , $γ = 0.1$ , and $r (0) = 0$ .
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Figure 3
Dependence of the asymptotic diffusion law, $〈 r^{2} (t) 〉$ vs $t$ , on $γ$ and $n$ . (a) $γ$ dependence of the fitting parameter $D (n, γ)$ of Eq. (3) for different $n$ (see legend). The power laws $D (n, γ) \propto γ^{β_{n}}$ with $β_{n} = α_{n}$ (straight lines) are drawn as a validity check of Eq. (5). (b) Rescaled diffusion curves, ${〈 r^{2} (t) 〉}^{n + 1}$ vs $γ t$ , for $n = 1 / 2, 1$ , and 2, and $γ = 0.01, 0.04, 0.16, 0.5$ , and 2. The color code is not reported as, after rescaling, the data sets are undistinguishable. For $γ t ≫ 1$ , all data sets collapse on top of the straight line with slope ${[2 v_{th} / (Ω_{r} r_{0})]}^{2}$ predicted in Eq. (5) (thick dotted-dashed line). The remaining simulation parameters are as in Fig. 2.
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Figure 4
Diffusion transients $〈 r^{2} (t) 〉$ vs $t$ in units of $t_{st} = γ / Ω_{0}^{2}$ for low (a) and large $γ$ (b), $Ω_{0} = Ω_{r} = 1$ , and $n = 1 / 2, 1$ , and 2. Color code in panel (a): $γ = 8$ (yellow), 24 (blue), and 100 (orange); for curves of the same color, $n$ decreases from top to bottom; for $n = 2$ we only drew the orange curve and not the yellow one. In panel (b) $γ$ ranges between 100 and 1600; after rescaling the data sets are hardly distinguishable. In both panels the power law (7) for short-time normal diffusion is represented by a thick dotted-dashed line with slope ${[2 v_{th} / (Ω_{0} r_{0})]}^{2}$ . In panel (a) the asymptotic laws (5) for $γ = 24$ and $n = 1$ and $1 / 2$ are drawn for reader convenience. Panel (c): negative field radial gradients, $n < 0$ , $〈 r^{2} (t) 〉$ vs $t$ , in dimensionless units, for $r (0) = 1$ , $Ω_{0} = 0$ , $Ω_{r} = 1$ , $γ = 0.1$ , and different $n$ (see legends). The transient law (8) is fitted to the data in the appropriate time domains (dashed lines); the expected asymptotic diffusion law is drawn as a solid line. The remaining simulation parameters are as in Fig. 2.
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