[Paper Review] Mean square displacement and instantaneous diffusion coefficient of charged particles in stochastic motion

Gabriela Raluca Mocanu|arXiv (Cornell University)|Jun 25, 2019

High-Energy Particle Collisions Research4 citations

TL;DR

This paper proposes a numerical method to compute the mean square displacement (MSD) and instantaneous diffusion coefficient (D(t)) for charged particles undergoing stochastic motion in various astrophysical configurations. By solving stochastic differential equations (SDEs) numerically—using Euler, Runge-Kutta, and hybrid methods—it reveals that D(t) exhibits strong, non-monotonic behavior in the intermediate time regime, with peaks up to an order of magnitude above the long-time limit, enabling differential diagnosis of physical conditions in plasma environments.

ABSTRACT

The mean square displacement and instantaneous diffusion coefficient for different configurations of charged particles in stochastic motion are calculated by numerically solving the associated equations of motion. The method is suitable for obtaining accurate descriptions of diffusion in both intermediate and long time regimes. It is also appropriate for studying a variety of astrophysical configurations since it may incorporate microscopic physics that analytical methods cannot cope with. The results show that, in the intermediary time regime, the diffusion coefficient has an irregular behavior, which can be described in terms of the complex interplay appearing between the physical parameters describing the configuration. The main conclusion is that such an approach may serve at differential diagnosis of different astrophysical configurations.

Motivation & Objective

• To develop a numerical framework for computing the instantaneous diffusion coefficient D(t) in complex, non-equilibrium astrophysical plasma systems where analytical solutions are intractable.
• To investigate how physical parameters—such as electric field strength, friction, harmonic potential, and magnetic field—affect the time evolution of D(t) and MSD.
• To explore the intermediate-time behavior of D(t), which is often overlooked in analytical treatments but may carry critical diagnostic information for transient astrophysical events.
• To assess the potential of D(t) as a diagnostic tool for distinguishing between different astrophysical configurations based on their unique diffusion signatures.
• To demonstrate that numerical integration of SDEs can capture complex, non-trivial dynamics—including transient diffusion enhancement—beyond the scope of standard analytical approximations.

Proposed method

• Numerically solve the stochastic differential equation (SDE) of motion for charged particles, expressed as $ \frac{d^2x}{dt^2} = a_{\text{det}} + a_{\text{stoch}} $, where $ a_{\text{det}} $ is deterministic and $ a_{\text{stoch}} $ is a zero-mean white noise acceleration with $ \langle \xi_A(t_1)\xi_A(t_2) \rangle = A\delta(t_1 - t_2) $.
• Decompose particle motion into mean $ \bar{x}(t) $ and fluctuating $ \delta x(t) $ components, with initial conditions $ \delta x(0) = \delta v(0) = 0 $, to compute diffusion relative to the mean trajectory.
• Apply different numerical integrators per configuration: Euler for constant electric field (A) and harmonic potential with constant friction (B), second-order Runge-Kutta for memory-dependent friction (C), and a hybrid Euler-Lemons & Kaufman method for magnetic field (D).
• Make all equations dimensionless to improve numerical stability and allow comparison across physical scales.
• Compute the mean square displacement (MSD) $ \langle x^2(t) \rangle $ from ensemble-averaged trajectories, then derive the instantaneous diffusion coefficient via $ D(t) = \frac{1}{2} \partial_t \langle x^2(t) \rangle $.
• Analyze the time evolution of D(t), particularly its intermediate regime and long-time limit $ D = \lim_{t \to \infty} D(t) $, across four distinct physical configurations.

Experimental results

Research questions

• RQ1 How does the instantaneous diffusion coefficient D(t) evolve over time in charged particle systems with stochastic motion under different physical conditions?
• RQ2 To what extent does the intermediate-time behavior of D(t) deviate from the long-time limit, and what physical parameters govern these deviations?
• RQ3 Can the shape and peak amplitude of D(t) in the intermediate regime serve as a diagnostic signature for distinguishing between different astrophysical plasma configurations?
• RQ4 How do the ratios of stochastic acceleration to deterministic forces (e.g., electric field, harmonic potential, friction) influence the transient diffusion dynamics?
• RQ5 In what ways does memory in friction or the presence of a magnetic field alter the time evolution of D(t) compared to Markovian or non-magnetic cases?

Key findings

• In the intermediate time regime, the instantaneous diffusion coefficient D(t) exhibits non-trivial, irregular behavior, with peaks up to an order of magnitude higher than the long-time limit D.
• For particles in a harmonic potential with memory-dependent friction (case C), the maximum D(t) and the time at which it occurs both decrease with increasing $ W^2 $, indicating stronger trapping by the potential.
• For fixed $ W^2 $, the maximum D(t) and its peak time also decrease with increasing $ \bar{\alpha} $, confirming that higher friction suppresses transient diffusion.
• In a constant magnetic field (case D), the 3D mean square displacement scales linearly with time ($ \langle q^2(n) \rangle_0 \sim n $) due to unimpeded motion along the field and confined motion perpendicular to it.
• The long-time limit D is always constant and finite, but the intermediate behavior of D(t) is highly sensitive to the configuration’s physical parameters, enabling differential diagnosis.
• The distinct, non-overlapping curves of D(t) across different parameter sets confirm that the time-resolved diffusion profile can be used to infer underlying physical conditions in astrophysical plasmas.