Lévy Flights and Leaky Boxes: Anomalous Diffusion of Cosmic Rays

TL;DR

This work probes whether cosmic ray transport in turbulent galactic environments can be described by anomalous diffusion (sub- or superdiffusion) and how finite galactic sizes truncate Lévy flights, forcing a reversion to standard diffusion on the escape timescale.Using Monte-Carlo particle-tracking, the authors implement fractional diffusion with stable distributions (parameters α for space and β for time) and examine the impact of finite pathlengths, absorbing boundaries, and advection/streaming, revealing that gaussianization is a robust outcome except in systems with very long confinement times.Key findings show that in bounded or wind-influenced settings, CR profiles converge to standard diffusion with an effective κ_eff ≈ κα l_max^{2−α}, while in unbounded cases, pure fractional diffusion yields distinctive CR distributions such as P_CR ∝ r^{α} in 3D spheres and heavy-tailed PDFs in 1D/2D.Practically, the results imply that conventional diffusion modeling remains broadly valid for most CR applications, but environments with long escape times (e.g., CGM/ICM) or unusual trapping structures could retain anomalous diffusion signatures, motivating a shift in both interpretation and diagnostics, such as using width scaling γ ∝ t^{1/α} instead of ⟨x^2⟩ when truncation is present.

Abstract

In classical diffusion, particle step-sizes have a Gaussian distribution. However, in superdiffusion, they have power-law tails, with transport dominated by rare, long Lévy flights. Similarly, if the time interval between scattering events has power-law tails, subdiffusion occurs. Both forms of anomalous diffusion are seen in cosmic ray (CR) particle tracking simulations in turbulent magnetic fields. They also likely occur if CRs are scattered by discrete intermittent structures. Anomalous diffusion mimics a scale-dependent diffusion coefficient, with potentially wide-ranging consequences. However, the finite size of galaxies implies an upper bound on step-sizes before CRs escape. This truncation results in eventual convergence to Gaussian statistics by the central limit theorem. Using Monte-Carlo simulations, we show that this occurs in both standard finite-thickness halo models, or when CR diffusion transitions to advection or streaming-dominated regimes. While optically thick intermittent structures produce power-law trapping times and thus subdiffusion, gaussianization also eventually occurs on timescales longer than the maximum trapping time. Anomalous diffusion is a transient, and converges to standard diffusion on the (usually short) timescale of particle escape, either from confining structures (subdiffusion), or the system as a whole (superdiffusion). Thus, standard assumptions of classical diffusion are physically justified in most applications, despite growing simulation evidence for anomalous diffusion. However, if escape times are long, this is no longer true. For instance, anomalous diffusion in the CGM or ICM would change CR pressure profiles. Finally, we show the standard diagnostic for anomalous diffusion, $\langle d^2 \rangle \propto t^α$ with $α\neq 1$, is not justified for truncated Lévy flights, and propose an alternative robust measure.

Figures (20)

• Figure 1: Stable distributions on (a) linear scale and (b) log scale, with $\alpha=1,1.5,2$ and same scale factor $\gamma=1$. While $\alpha=2$ gives the standard Gaussian distribution, $\alpha=1.5$ and $\alpha=1$ (the Cauchy distribution) have a stronger central peak (visible on the linear plot) and broad power law tails (visible on the log plot). In particular, they are significantly different at large $|x|$.
• Figure 3: Monte-Carlo simulation of the trajectories (black) of a particle diffusing at $\alpha=1, 1.5, 2$ and the step-sizes (blue). For $\alpha<2$, the particle undergoes Lévy flights in each time step, where the pathlength distribution has infinite variance, and the displacement of the particle is dominated by rare large steps. At $\alpha=2$, the pathlengths follow a Gaussian distribution, and the steps contribute more evenly to the overall trajectory.
• Figure 4: The CR radial distributions $N(r)$ from simulations with different stability parameter $\alpha$. The CRs are injected at a fixed rate as a delta point source at the origin. They undergo 3D diffusion in $x,y,$and $z$, and are binned into $r$. The simulations are run until the number of CRs within $r=4$ plateaus and the profiles reach steady states. The scaling of the total number of cosmic rays is consistent with $N(r)\propto r^{\alpha}$, as predicted in Equation \ref{['eqn:Pcr_alpha']}.
• Figure 5: (a) The 1D diffusion profile with CRs constantly injected at a delta source at the origin and absorbing boundaries placed at $z=\pm 5$ kpc for different stability parameters $\alpha$. (b) The CR confinement time in the central disk of thickness 0.15 kpc. With absorbing boundaries, the confinement time has an exponential distribution and is independent of the simulation run time. (c) The CR total age distribution. Both of the CR age distributions are extracted from the particles in the central disk, where observational data is available.
• Figure 6: (a) The 3D diffusion (R,z) profiles of CRs injected constantly along R following an exponential distribution of sources $Q(\rho)=Q_{0}\rho^{1.2}\text{exp}(-6.44\rho)$, as in stecker77 We adopt stability parameter $\alpha=1,1.5,2$, with the mean confinement time in the disk $z= \pm 0.15$ kpc controlled to be similar. Absorbing boundaries are placed both in R and z, with $L_{z}=5$ kpc, and $L_{R}=16$ kpc. The distributions are normalized to unity at $r=8$ kpc. Particles are injected along a fixed radial line (i.e., at a fixed polar angle). Due to cylindrical symmetry, this gives the same result as injecting over the entire disk. (b) The time CRs spent in the central disk for different alphas. The result is similar to 1D diffusion (Fig. \ref{['fig:1d_abs']}). (c) The CR total age distribution, measured from the particles in the disk.