The Topological Origin of Bohm Resistivity in Magnetic Reconnection

TL;DR

The work derives Bohm-like resistivity for magnetic reconnection from a first-principles spintronic framework in which the magnetized electron fluid is an overdamped condensate governed by the Landau-Lifshitz-Gilbert equation. Reconnection arises as an Adler-Ohmic bifurcation, corresponding to a phase-slip breakdown of gyrophase synchronization and a transition in the XY universality class, culminating in an explosive, logistic onset of resistivity. The model yields a concrete anomalous resistivity form $\eta_{spin}(J) = \eta_0 \sqrt{1 - (J_c/J)^2}$ for $J>J_c$ with $\eta_0 \propto B^{-1}$ and a critical current $J_c$ set by the gyromotion scale $J_c = B/(\mu_0 \rho_L)$, while phase slips saturate at the electron thermal speed and defect densities decay as $N_d \sim t^{-0.75}$. Together, these results cast Bohm resistivity as a universal topological property of magnetized matter at the reconnection threshold, linking magnetic topology, phase transition theory, and spintronic dynamics with potential implications for both space plasmas and laboratory experiments.

Abstract

The physical origin of 'anomalous' resistivity in magnetic reconnection remains one of the longest-standing problems in space plasma physics. While the empirical Bohm diffusion scaling ($η~\propto~T/B$) is widely invoked to explain fast reconnection rates, it lacks a rigorous derivation from first principles. Here, we derive this scaling by modeling the magnetized electron fluid as an overdamped spintronic condensate governed by the Landau-Lifshitz-Gilbert equation. We demonstrate that the breakdown of the "frozen-in" condition is rigorously identified as an Adler-Ohmic bifurcation: a topological phase transition where electron gyro-axes lose synchronization with the magnetic field. By rigorously mapping the breakdown of adiabatic invariance to electron gyro axis slippage on the unit sphere, we show that the resulting resistivity naturally saturates at the Bohm limit. Numerical simulations of the $XY$ universality class confirm that the onset of this resistive state is explosive, following a logistic trigger consistent with the impulsive phase of solar flares. Furthermore, the topological defects in the condensate decay via a $t^{-0.75}$ power law, identifying magnetic island coalescence as the mechanism of anomalous transport. These results suggest that Bohm resistivity is a universal topological property of magnetized matter at the critical point of reconnection.

Figures (3)

• Figure 1: Panel a): the geometry of the two-dimensional tangent plane of the electron gyro rotor, showing how the restoring force acts to align the rotor's axis with the mean field $\Psi$. Panel b): the geometry of the full three-dimensional model, where the electron gyro axis is the unit vector on the unit sphere ($|\hat{n}_i|=1$), and where the vector torque proportional to $(\hat{n}_i \times \hat{\mathbf{\Psi}}) \times \hat{n}_i$ (Eq. \ref{['eq:eom3d']}) acts to align the rotor with the mean field lakshmanan_fascinating_2011.
• Figure 2: A statistical aggregate of 512 simulation runs, each featuring a two-dimensional $512\times512$ lattice, systematically varying $0.1<T_e<3.2$ and $0.032<K<10$ in logarithmic increments. Panel a) shows the defect count (red) and a $t^{-0.75}$ powerlaw fit (blue dashes). Panel b) shows the ensemble average phase slip speed (red, evaluated from the second half of 16 simulations with varying values of $K$), a logistic fit (blue dashes), and $J_c$, the deflection point in the logistic curve (green dots). Panel c) shows $J_c$ for all the simulations, in bins of temperature $T_e$. Errorbars denote the 95$^\text{th}$ percent confidence interval of the logistic fit in each bin. Fits of $\propto T_e^{-1.14}$ (determined through non-linear least squares minimziation) and $\propto T_e^{-1}$ are shown in blue dashes and blue solid line respectively; the shaded gray region delineates ideal magnetohydrodynamics from magnetic reconnection. Panel d) shows ensemble average phase slip speed, varying the temperature (the color of the line indicates the coupling strength $K$). A $\propto T_e$ fit is shown in blue dashes.
• Figure 3: Illustration of a reconnection event in the overdamped spintronic condensate (black and red compasses representing rotors that align with the magnetic field). The current tension electric field is indicated, which causes slipping electrons to enter the outflow jets.