Persistent random walks of charged particles across magnetic field lines

We investigate the time evolution of the mean location and variance of a charged particle subject to random collisions that are Poisson distributed. The particle moves on a plane and is subject to a magnetic field applied perpendicular to the plane, so it is constrained to move in circles in the absence of collisions. We develop a procedure that yields analytic expressions of the mean and variance. These results are valid for arbitrary times after the start of the walk, including early on when, on average, less than one collision is expected. As an example of their applicability, we use these expressions to model experimental results and simulations of suprathermal ions propagating in a turbulent plasma in TORPEX (the TORoidal Plasma EXperiment).

Figure 1

Depiction of the system dynamics at different times $t_1<t_2<t_3$. A particle with $q<0$ moves in the $xy$ plane subject to $B_z>0$. At $t_1$ the particle has velocity $\mathbf{v}\left(t_1\right)$ and follows a circular trajectory (light gray dotted line). A collision occurs at time $t_2$ which instantaneously changes the direction of motion by a random angle $\mathrm{\Phi }$. The particle then follows a different circular trajectory (dark gray dotted line) that takes it to a new location at $t_3$.

Figure 2

Evolution of ${\sigma }_{\chi }^2\left(t\right)$ for different values of ${\gamma }_1/\mathrm{\Omega }$.

Figure 3

Evolution of ${\sigma }_{\chi }^2\left(t\right)$ for $\gamma /\mathrm{\Omega }=0.1$ and different values of the width parameter $c$. Solid lines correspond to ${\beta }_0=0$ and dotted lines to ${\beta }_0=\pi /2$ (same color coding). The case $c=0$ yields an isotropic result compatible with the model of Sec. 3c. Notice the large slope at $\left|\mathrm{\Omega }t\right|\lesssim 1$ for the curves with $c\gtrsim 10$.

Figure 4

Suprathermal ions in TORPEX. We use a magnetic configuration with open helical lines (depicted with the solid purple line). Lithium-6 ions are injected in a turbulent hydrogen plasma using a miniature source. As they propagate, the ions spread in the perpendicular direction ($xy$ plane). A detector measures the ions arriving at different $(x,y)$ locations. The toroidal distance between source and detector can be changed to resolve different propagation times.

Figure 5

PRW model (lines), experimental data (squares), and GBS simulations (bands) of ${}^6\mathrm{Li}+$ location variance (along $x$) for $30\text{eV}$ (top) and $70\text{eV}$ (bottom). Data error bars are the $1\sigma$ uncertainties corresponding to the square of the $1\sigma$ errors in Refs. [8, 15]. Dashed lines correspond to isotropic initial conditions and solid lines to injection biased in the $+x$ direction. See text for a discussion on parameter values. Initially, transport is ballistic with $\lambda \approx 2$, while at long times ($\left|\mathrm{\Omega }\right|t\gtrsim 100$) it is diffusive ($\lambda \approx 1$). A subdiffusive ($\lambda <1$) regime is observed in the $70\text{eV}$ case in the region between vertical dotted lines.

Figure 6

Comparison between simulated (markers) and theoretical (solid lines) values of (a) ${\mu }_{\chi }\left(t\right)$ and (b) ${\sigma }_{\chi }^2\left(t\right)$ for different ${\gamma }_1/\mathrm{\Omega }$ (colors as per legend). Different marker shapes represent different choices of simulation parameters as described in the text. The purple shading in (a) corresponds to the expected $1\sigma$ statistical uncertainty area around the theoretical mean $[{\mu }_{\chi }\left(t\right)=0]$ for the simulations with ${\gamma }_1/\mathrm{\Omega }=5$.