Crossover between Anomalous Superdiffusion and Normal Diffusion in Oscillating Convection Flows

Ito, Kei; Miyazaki, Syuji

doi:10.1143/ptp.110.875

Cited by 13 publications

(10 citation statements)

References 11 publications

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“…Although that appears to be caused by the decrease in the efficiency of SMFR acceleration with increasing distance from the Sun (Van Eck et al 2021), partly it might be that the level of superdiffusiveness is also dropping as the parallel transport and acceleration become more diffusive as the SMFR acceleration region is advected radially outward with the solar wind flow. This is consistent with the expectation that in complex dynamic systems, anomalous diffusion processes are intermediate or transitional, crossing over into normal diffusion states for system evolution times longer than a certain critical time (e.g., Ito & Miyazaki 2003). In the solar wind, a crossover from anomalous diffusion to normal diffusion is also expected to occur when energetic particles temporarily trapped in SMFR structures escape and propagate along diffusing magnetic field lines (Ruffolo et al 2003), or when solar energetic particles escaping nearly scatter free in front of traveling shocks scatter on magnetic turbulence at later times (e.g., Effenberger & Litvinenko 2014).…”

Section: Discussionsupporting

confidence: 90%

Parallel and Momentum Superdiffusion of Energetic Particles Interacting with Small-scale Magnetic Flux Ropes in the Large-scale Solar Wind

Roux

2023

ApJ

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A recently developed time-dependent fractional Parker transport equation is solved to investigate the parallel and momentum superdiffusion of energetic charged particles in an inner heliospheric region containing dynamic small-scale flux ropes (SMFRs). Both types of superdiffusive transport are investigated with fractional transport terms containing a fractional time integral combined with normal spatial or momentum derivatives. Just as for normal diffusion, accelerated particles form spatial peaks with a maximum amplification factor that increases with particle energy. Instead of growth of the spatial peaks until a steady state is reached as for normal diffusion, parallel superdiffusion causes the peaks to dissipate into plateaus followed by a rollover at late times. The peaks dissipate at a faster rate when parallel transport is more superdiffusive. Furthermore, the accelerated particle spectral distribution function inevitably becomes an f 0 ∝ p −3 spectrum at late times in the test particle limit near the particle source despite the potential for spectral steepening from other transport terms. All this is a product of the growing domination of parallel spatial and especially momentum superdiffusion over other transport terms with time. Such extreme late time effects can be avoided by a transition to a normal diffusive state. Finally, fitting spatial peaks observed during SMFR acceleration events with the solution of the fractional Parker transport equation can potentially be used as a diagnostic for estimating the level of spatial and momentum superdiffusion in these events and how the levels of superdiffusion vary with distance from the Sun.

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Section: Discussionsupporting

confidence: 90%

Parallel and Momentum Superdiffusion of Energetic Particles Interacting with Small-scale Magnetic Flux Ropes in the Large-scale Solar Wind

Roux

2023

ApJ

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“…A fundamental concept of a non-Markovian process is the continuous-time random walk introduced already some years ago [4]. This theory has numerous important applications, studied for instance in [5] or recently in [6]. A broad variety of examples in biology is analyzed in [7].…”

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confidence: 99%

Elephants can always remember: Exact long-range memory effects in a non-Markovian random walk

2004

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We consider a discrete-time random walk where the random increment at time step t depends on the full history of the process. We calculate exactly the mean and variance of the position and discuss its dependence on the initial condition and on the memory parameter p. At a critical value p (1) c = 1/2 where memory effects vanish there is a transition from a weakly localized regime (where the walker returns to its starting point) to an escape regime. Inside the escape regime there is a second critical value where the random walk becomes superdiffusive. The probability distribution is shown to be governed by a non-Markovian Fokker-Planck equation with hopping rates that depend both on time and on the starting position of the walk. On large scales the memory organizes itself into an effective harmonic oscillator potential for the random walker with a time-dependent spring constant k = (2p − 1)/t. The solution of this problem is a Gaussian distribution with time-dependent mean and variance which both depend on the initiation of the process.

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“…As a result, the particles at late times are freely undergoing parallel superdiffusion away from the shock without significant modulation from the incoming solar wind flow. It is unlikely that such a uniform upstream particle distribution will ever form though, because there are indications that in complex systems anomalous diffusion processes do cross over into normal diffusion states for system evolution times longer than a certain correlation time (e.g., Ito & Miyazaki 2003). In the solar wind, a crossover from anomalous diffusion to normal diffusion is also expected to occur when energetic particles temporarily trapped in SMFR structures escape and propagate along diffusing magnetic field lines (Ruffolo et al 2003), or when solar energetic particles, escaping nearly scatter free in front of traveling shocks, scatter on magnetic turbulence at later times (e.g., Effenberger & Litvinenko 2014).…”

Section: Summary and Discussionmentioning

confidence: 99%

Investigating Superdiffusive Shock Acceleration at a Parallel Shock with a Fractional Parker Equation for Energetic-particle Interaction with Small-scale Magnetic Flux Ropes

Roux

2022

ApJ

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It has been suggested before that small-scale magnetic flux rope (SMFR) structures in the solar wind can temporarily trap energetic charged particles. We present the derivation of a new fractional Parker equation for energetic-particle interaction with SMFRs from our pitch-angle-dependent fractional diffusion-advection equation that can account for such trapping effects. The latter was derived previously in le Roux & Zank from the first principles starting with the standard focused transport equation. The new equation features anomalous advection and diffusion terms. It suggests that energetic-particle parallel transport occurs with a decaying efficiency of advection effects as parallel superdiffusion becomes more dominant at late times. Parallel superdiffusion can be linked back to underlying anomalous pitch-angle transport, which might be subdiffusive during interaction with quasi-helical coherent SMFRs. We apply the new equation to time-dependent superdiffusive shock acceleration at a parallel shock. The results show that the superdiffusive-shock-acceleration timescale is fractional, the net fractional differential particle flux is conserved across the shock ignoring particle injection at the shock, and the accelerated particle spectrum at the shock converges to the familiar power-law spectrum predicted by standard steady-state diffusive-shock-acceleration theory at late times. Upstream, as parallel superdiffusion progressively dominates the advection of energetic particles, their spatial distributions decay on spatial scales that grow with time. Furthermore, superdiffusive parallel shock acceleration is found to be less efficient if parallel anomalous diffusion is more superdiffusive, while perpendicular particle escape from the shock, thought to be subdiffusive during SMFR interaction, is reduced when increasingly subdiffusive.

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Crossover between Anomalous Superdiffusion and Normal Diffusion in Oscillating Convection Flows

Cited by 13 publications

References 11 publications

Parallel and Momentum Superdiffusion of Energetic Particles Interacting with Small-scale Magnetic Flux Ropes in the Large-scale Solar Wind

Parallel and Momentum Superdiffusion of Energetic Particles Interacting with Small-scale Magnetic Flux Ropes in the Large-scale Solar Wind

Elephants can always remember: Exact long-range memory effects in a non-Markovian random walk

Investigating Superdiffusive Shock Acceleration at a Parallel Shock with a Fractional Parker Equation for Energetic-particle Interaction with Small-scale Magnetic Flux Ropes

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