Noise-induced resonant acceleration of a charge in an intermittent magnetic field: an exact solution for ergodic and non-ergodic fluctuations

TL;DR

This work derives an exact analytical framework for a charged particle diffusing in an intermittent magnetic field B(t) = B0 + B1 ξ(t), where ξ(t) is a dichotomous fluctuation. By combining a trajectory-based approach with a Poissonian density method, the authors obtain closed-form results for the mean-square displacement ⟨r^2(t)⟩ and show a sharp transition: confinement for ω0 = qB0/m = 0 and exponential (hyper-ballistic) diffusion for ω0 > 0, even under slow or non-Poissonian fluctuations. The mechanism is explained via resonance bands in the periodic case, which can be excited by stochastic fluctuations, supporting a robust “noise-induced resonant acceleration” that may surpass traditional Fermi-type acceleration in certain regimes. The framework provides exact solutions valid for arbitrary waiting-time distributions and highlights the pivotal role of the induced electric field in driving energy gain, with potential implications for space and laboratory plasmas exhibiting intermittent magnetic fields.

Abstract

We study the diffusion of a charged particle in a magnetic field subject to stochastic dichotomous fluctuations. The associated induced electric field gives rise to non-trivial dynamical regimes. In particular, when the mean magnetic field vanishes, the particle remains confined within a finite radius, regardless of the fluctuation statistics. For a non-zero mean field, we shows, using a density approach for Poissonian fluctuations, that the particle undergoes an exponential regime of accelerated diffusion. Crucially and more generally, adopting a trajectory-based formalism, we derive an exact analytical solution valid for arbitrary waiting-time distributions, including non-Poissonian and non-ergodic cases. Even rare, abrupt field reversal are shown to trigger exponential acceleration of the particle's diffusion. We demonstrate that this behaviour stems from noise exciting resonance bands present for periodic fluctuations, and we propose noise-induced resonant acceleration as a robust and efficient charge acceleration mechanism, potentially more effective than Fermi's classic model for cosmic acceleration.

Figures (6)

• Figure 1: Examples of trajectories generated with $\omega_0=1$ and $\omega_1=0.7$ at $t=10000$dt with dt$=0.1$. Top runs are generated with Poisson distribution $\psi(t)=\gamma e^{-\gamma t}$ with $\gamma=0.1$ (left) and $\gamma=0.5$ (right). Bottom runs with power-law distribution $\psi(t )\sim t^{-\alpha-1}$ with $\alpha=0.25$ (left) and $\alpha=0.75$ (right)
• Figure 2: Diffusion of a charged particle in a fluctuating dichotomous magnetic field $B(t)$ in the Poissonian case. The green line is $\langle r^2 \rangle$ from solution (\ref{['exactT']}), blue and red line are $\langle x \rangle$, $\langle y \rangle$ from solution (\ref{['quarto2']}). Black diamonds, triangles and circles are the respective simulated results, averaged over 50k runs, which show perfect agreement. Parameters are $v_0=1,\gamma=20,\,\omega _0 = 1,\,\omega _1= 0.7$ and initial conditions $x(0)=y(0)=\dot{ x}(0)=0$ and $\dot{y}(0)=v_0$ corresponding to ${\bf A}^{\pm}_{0}=\frac{1}{2}(\frac{1}{\omega_{\pm}},-\frac{1}{|\omega_{\pm}|})$. Inset: corresponding solutions from the density approach Eqs.(\ref{['mauror2']}), (\ref{['quartoZ']}). Setting $\omega_0$=0 in (\ref{['mauror2']}) and (\ref{['exactT']}) (brown line) coincides with prediction of Eq. (\ref{['confino']}) and with simulation (triangles), showing confinement at any time.
• Figure 3: $\langle r^2(t)\rangle$ for the diffusion of a charged particle in a fluctuating magnetic field $B(t)$, with parameters $\omega_0=1$ and $\omega_1=0.5$ and for field fluctuations with power-law distribution $\psi(t)_{t\gg1}\sim (t/ \tau)^{-\alpha-1}$ at large times. (Left panel) non-ergodic case of a Mittag-Leffler distribution with parameters $\tau=10$ and $\alpha=1/2$ (diverging mean time). (Right panel) Power-law distribution with finite mean time and diverging second moment, as in Eq. (\ref{['manneville']}) with $\alpha=3/2$ and $\tau=1$. In both panels, the black lines indicate the result obtained from inverting the analytical solution (\ref{['smartLAP']}). All other lines: numerical simulations. Higher number of runs show closer match with the analytical solution, with slower convergence for the non-ergodic case.
• Figure 4: Modulus of eigenvalues $\lambda_-$ (red line) and $\lambda_+$ (blue dotted line) of $G_{\pm}=M_{\mp}(T)M_{\pm}(T)$ for the periodic case as a function of period $T$ for $\omega_0=1$ and $\omega_1=0.7$. For most values of $T$ it is $|\lambda_{\pm}|=1$, corresponding to confined motion, but for T within 'resonance bands' (e.g. for $2<T<3.5$ or $4.9<T<7.1$), one of $|\lambda_{\pm}|$ is larger than one, i.e. resonance occurs between the orbital motion and the field, the charge is accelerated.
• Figure A1: Examples of trajectories generated with $\omega_0=1$ and $\omega_1=0.7$ at t=10000 for the periodic case with period $T=1$ (left, showing confinement) and $T=3$ (right, showing resonance).