[Paper Review] A kinetic equation for linear fractional stable motion with applications to space plasma physics

N. W. Watkins, D. Credgington+2 more|ArXiv.org|Mar 19, 2008

Fractional Differential Equations Solutions26 references3 citations

TL;DR

This paper derives a kinetic equation for Linear Fractional Stable Motion (LFSM), a model combining heavy-tailed Lévy jumps and long-range memory, showing it features a power-law time-dependent diffusion coefficient rather than a fractional time derivative. The key contribution is identifying a missing kinetic equation in the literature for LFSM, distinguishing it from the fully fractional CTRW and offering a physically motivated alternative for modeling space plasma time series with anomalous scaling.

ABSTRACT

Levy flights and fractional Brownian motion (fBm) have become exemplars of the heavy tailed jumps and long-ranged memory seen in space physics and elsewhere. Natural time series frequently combine both effects, and Linear Fractional Stable Motion (LFSM) is a model process of this type, combining alpha-stable jumps with a memory kernel. In contrast complex physical spatiotemporal diffusion processes where both the above effects compete-dubbed "ambivalent" by Brockmann et al (2006}-have for many years been modelled using the fully fractional (FF) kinetic equation for the continuous time random walk (CTRW), with power laws in the pdfs of both jump size and waiting time. We derive the analogous kinetic equation for LFSM and show that it has a diffusion coefficient with a power law in time rather than having a fractional time derivative like the CTRW. We develop earlier comments by Lutz (2001) on how fBm differs from its fractional time process counterpart. We go on to argue more physically why LFSM and the FFCTRW might indeed be expected to differ, and discuss some preliminary results on the scaling of burst "sizes" and "durations" in LFSM time series, with applications to modelling existing observations in space physics.

Motivation & Objective

• To address the lack of a kinetic equation for Linear Fractional Stable Motion (LFSM) in the literature, despite its relevance to space plasma physics.
• To clarify the distinction between LFSM and the fully fractional CTRW (FFCTRW), particularly in their diffusion behavior and scaling properties.
• To explore whether LFSM can explain observed burst size and duration scaling in magnetospheric and solar wind time series, suggesting a mechanism for 'apparent' self-organized criticality.
• To investigate the nature of ambivalent behavior in LFSM, showing that the self-similarity exponent H is additive rather than rational, contrasting with CTRW models.
• To evaluate the suitability of LFSM versus CTRW models for persistent time series data in space physics, challenging assumptions about model equivalence.

Proposed method

• Derives the kinetic equation for LFSM via direct differentiation of its well-known characteristic function, establishing a time-dependent diffusion coefficient with power-law scaling.
• Compares the resulting LFSM kinetic equation with the fully fractional CTRW equation, highlighting that LFSM lacks a fractional time derivative and instead exhibits a power-law diffusion coefficient.
• Applies scaling arguments based on results from Kearney & Majumdar (2005) to model burst size and duration statistics in LFSM time series.
• Analyzes the self-similarity exponent H in LFSM, showing it is an additive function of the Lévy index μ and the Hurst exponent β, rather than a rational combination.
• Uses numerical simulations to compare predicted burst statistics from LFSM with empirical observations from magnetospheric and solar wind data.
• Contrasts the physical mechanisms of LFSM (convolution of memory kernel and jumps) with CTRW (factorized jump and waiting time distributions), explaining their differing scaling behaviors.

Experimental results

Research questions

• RQ1 Why is there no kinetic equation for Linear Fractional Stable Motion (LFSM) in the existing literature, despite its relevance to anomalous diffusion in space plasmas?
• RQ2 How does the kinetic equation for LFSM differ from that of the fully fractional CTRW, particularly in terms of time dependence and memory structure?
• RQ3 Can LFSM explain the observed scaling of burst sizes and durations in space plasma time series, such as those from auroral indices or solar wind data?
• RQ4 Why does LFSM exhibit additive ambivalent scaling (H = J + L - 1/2) rather than rational ambivalent scaling, and what are the physical implications?
• RQ5 Are models like LFSM more appropriate than CTRW for persistent time series derived from space physics observations, given their differing mathematical structures?

Key findings

• A kinetic equation for LFSM is derived for the first time, showing it has a time-dependent diffusion coefficient with a power-law dependence, not a fractional time derivative.
• The LFSM kinetic equation differs fundamentally from the fully fractional CTRW equation, as LFSM's path arises from a convolution in time between a memory kernel and a Lévy process, while CTRW factorizes in Fourier space.
• LFSM can explain the observed scaling of burst sizes and durations in magnetospheric and solar wind time series through a scaling argument based on Kearney & Majumdar (2005), suggesting it as a candidate for 'apparent' self-organized criticality.
• The self-similarity exponent H in LFSM is an additive function of the Lévy index μ and the Hurst exponent β, specifically H = [1/μ] + [β/2 - 1/2] - 1/2, indicating additive ambivalent behavior.
• The difference in scaling behavior arises because LFSM's path is defined as a convolution (factorizing in Fourier space), whereas CTRW's jump and waiting time distributions are independent.
• The results challenge the assumption that the limiting cases of CTRW (FFCTRW) and LFSM are equivalent, suggesting LFSM may be more suitable for modeling persistent time series in space physics.