| 1. |
- Anderson, Johan, 1973, et al.
(författare)
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A fractional Fokker-Planck model for anomalous diffusion
- 2014
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Ingår i: Physics of Plasmas. - : AIP Publishing. - 1089-7674 .- 1070-664X. ; 21:12, s. aricle no: 122109-
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we present a study of anomalous diffusion using a Fokker-Planck description withfractional velocity derivatives. The distribution functions are found using numerical means forvarying degree of fractionality of the stable Lévy distribution. The statistical properties of thedistribution functions are assessed by a generalized normalized expectation measure and entropyin terms of Tsallis statistical mechanics. We find that the ratio of the generalized entropy andexpectation is increasing with decreasing fractionality towards the well known so-called sub-diffusivedomain, indicating a self-organising behavior.
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| 2. |
- Anderson, Johan, 1973, et al.
(författare)
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Anomalous Diffusion by the Fractional Fokker-Planck Equation and Lévy Stable Processes
- 2018
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Ingår i: Fractional Dynamics and Anomalous Transport in Plasma Science. - Cham : Springer International Publishing. - 9783030044824 ; , s. 77-92
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Bokkapitel (övrigt vetenskapligt/konstnärligt)abstract
- The work presented here is a review of current developments in modelling anomalous diffusion using a Fokker-Planck description with fractional velocity derivatives and Langevin dynamics where L´evy fluctuations are introduced to model the effect of non-local transport due to fractional diffusion in velocity space. Distribution functions are found using numerical means for varying degree of fractionality of the stable L´evy distribution as solutions to the Fokker-Planck equation and is compared to results from Langevin simulations. The statistical properties of the distribution functions are assessed by a generalized normalized expectation measure and entropy in terms of Tsallis statistical mechanics.
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| 3. |
- Anderson, Johan, 1973, et al.
(författare)
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Fractional Fokker-Planck Equation vs Tsallis’ Statistical Mechanics
- 2013
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Ingår i: Festival-de-Theorie. ; 7, s. 4-
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Konferensbidrag (refereegranskat)abstract
- In this paper we present a study of anomalous diffusion using a Fokker-Planck descriptionwith fractional velocity derivatives. The distribution functions are foundusing numerical means for varying degree of fractionality observing the transitionfrom a Gaussian distribution to a L´evy distribution. The statistical properties of thedistribution functions are assessed by a generalized expectation measure and entropyin terms of Tsallis statistical mechanics. We find that the ratio of the generalizedentropy and expectation is increasing with decreasing fractionality towards the wellknown so-called sub-diffusive domain, indicating a self-organising behavior.
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| 4. |
- Anderson, Johan, 1973, et al.
(författare)
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Modelling phase locking of large-scale modes
- 2021
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Ingår i: Bulletin of the Americal Physical Society. ; 63
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Konferensbidrag (refereegranskat)abstract
- Turbulence is often characterized by energetic couplings between different scales of a flow. However, in the context of turbulence driven transport, such as the case of magnetically confined fusion plasmas or the diffusion of cosmic rays, typical flow structures are identified by dominant modes and the global turbulent state is approximated by a superposition of linear contributions (waves in general). These theoretical studies consider the amplitudes of the fluctuating quantities but disregard the dynamics of the phases by using the so-called random-phase approximation (RPA) for which the existence of a Chirikov-like criterion for the onset of wave stochasticity is assumed. In this approximation one assumes that the dynamical amplitudes have a slow variation compared to the rapid change of the phases. It has been observed that the phase dynamic shows significant departure from the well-known RPA assumptions, with phases locking occasionally (but not in the dissipative high-k range). In non-linear turbulent flow however, three-body interactions between the phases of the various modes is of importance. We will consider examples of synchronization in different fluid system such as Burgers and Navier-Stokes turbulence and in more advanced models such as those for Edge Localized Modes (ELMs) in tokamaks which remain a critical issue for plasma stability and the lifetime of fusion reactors such as ITER.
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| 5. |
- Anderson, Johan, 1973, et al.
(författare)
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Non-Linear Langevin and Fractional Fokker-Planck Equations for Anomalous Diffusion by Levy Stable~Processes
- 2018
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Ingår i: Entropy. - : MDPI AG. - 1099-4300. ; 20:10
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Tidskriftsartikel (refereegranskat)abstract
- The~numerical solutions to a non-linear Fractional Fokker--Planck (FFP) equation are studied estimating the generalized diffusion coefficients. The~aim is to model anomalous diffusion using an FFP description with fractional velocity derivatives and Langevin dynamics where L\'{e}vy fluctuations are introduced to model the effect of non-local transport due to fractional diffusion in velocity space. Distribution functions are found using numerical means for varying degrees of fractionality of the stable L\'{e}vy distribution as solutions to the FFP equation. The~statistical properties of the distribution functions are assessed by a generalized normalized expectation measure and entropy and modified transport coefficient. The~transport coefficient significantly increases with decreasing fractality which is corroborated by analysis of experimental data.
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| 6. |
- Anderson, Johan, 1973, et al.
(författare)
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Non-local transport based on the fractional Fokker–Planck Equation model
- 2018
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Konferensbidrag (övrigt vetenskapligt/konstnärligt)abstract
- Recently a non-local (non-diffusive) heat flux model based on a fractional derivative of plasma pressure was proposed for the heat transport in the JET tokamak plasmas [1], where the degree $\alpha$ of the fractional derivative i.e. non-locality (non-diffusivity), of the heat flux was defined though a simple power balance analysis of the steady state. The findings showed that the fractional degree in all of the analysed plasmas was $\alpha < 2$ for both ion and electron channels, suggesting that the heat transport in these plasmas is likely to be of a non-local (non-diffusive) nature. Thus, a study of anomalous diffusion of heat transport using a Fokker-Planck description with fractional velocity derivatives while keeping the non-linear terms is strongly called for. The distribution functions are found using numerical means for varying degree of fractionality of the stable L\'{e}vy distribution. The statistical properties of the distribution functions are assessed by a generalized normalized expectation measure and entropy. We find that the ratio of the generalized entropy and expectation is increasing with decreasing fractionality towards the well known so-called sub-diffusive domain, indicating a self-organising behavior. Here it is pertinent to keep in mind that the success of a fractional or non-local diffusion model indicates that there is lack of physics in current transport models, namely the super-diffusive character of heat transport, as such it is not only a simplified transport model. When the experimentally found values of the fractional derivatives are used in the model, within a good agreement the experimental heat fluxes were reproduced.
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| 7. |
- Anderson, Johan, 1973, et al.
(författare)
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Self-organisation of random oscillators
- 2015
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Ingår i: Festival-de-Theorie.
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Konferensbidrag (refereegranskat)abstract
- A model for the stochastic passive advection - diffusion of a scalar with external forcing is further developed by introducing a non-linear phase coupling dynamic between the phases of the stochastic flow and the forcing. The model for the phase coupling dynamic follows the well known Kuramoto model of the limit cycle oscillators with an additional linear coupling term between the phasesthe two stochastic fields. The aim is to study the impact of a collective phase synchronization or self-organisation on the fluctuation level of the scalar through a simple stochastic passive advection - diffusion relation. The results shown here, present a significant impact of the collective phase synchronization on the correlation time of the fluctuations, and on the suppression of the fluctuation amplitudes. The model predicts that in the presence of an additional linear coupling between the phases of the two stochastic fields, the phase synchronizations leads to a localisation as well as strong suppression of the fluctuation amplitudes. While, in the a-synchronized state we observe a predator-prey behavior between the correlations of the two fields and time auto-correlation of the fluctuations decay with an oscillatory trend.
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| 8. |
- Anderson, Johan, 1973, et al.
(författare)
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The Information Length Concept Applied to Plasma Turbulence
- 2024
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Ingår i: Entropy. - 1099-4300. ; 26:6
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Tidskriftsartikel (refereegranskat)abstract
- A methodology to study statistical properties of anomalous transport in fusion plasma is investigated. Three time traces generated by the full-f gyrokinetic code GKNET are analyzed for this purpose. The time traces consist of heat flux as a function of the radial position, which is studied in a novel manner using statistical methods. The simulation data exhibit transport processes with both medium and long correlation length along the radius. A typical example of a phenomenon with long correlation length is avalanches. In order to investigate the evolution of the turbulent state, two basic configurations are studied, one flux-driven and one gradient-driven with decaying turbulence. The information length concept in tandem with Boltzmann–Gibbs and Tsallis entropy is used in the investigation. It is found that the dynamical states in both flux-driven and gradient-driven cases are surprisingly similar, but the Tsallis entropy reveals differences between them. This indicates that the types of probability distribution function are nevertheless quite different since the higher moments are significantly different.
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| 9. |
- Gürcan, O, et al.
(författare)
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Phase and amplitude evolution in the network of triadic interactions of the Hasegawa-Wakatani system
- 2022
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Ingår i: Physics of Plasmas. - : AIP Publishing. - 1089-7674 .- 1070-664X. ; 29:5
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Tidskriftsartikel (refereegranskat)abstract
- The Hasegawa-Wakatani system, commonly used as a toy model of dissipative drift waves in fusion devices, is revisited with considerations of phase and amplitude dynamics of its triadic interactions. It is observed that a single resonant triad can saturate via three way phase locking, where the phase differences between dominant modes converge to constant values as individual phases increase in time. This allows the system to have approximately constant amplitude solutions. Non-resonant triads show similar behavior only when one of its legs is a zonal wave number. However, when an additional triad, which is a reflection of the original one with respect to the y axis is included, the behavior of the resulting triad pair is shown to be more complex. In particular, it is found that triads involving small radial wave numbers (large scale zonal flows) end up transferring their energy to the subdominant mode which keeps growing exponentially, while those involving larger radial wave numbers (small scale zonal flows) tend to find steady chaotic or limit cycle states (or decay to zero). In order to study the dynamics in a connected network of triads, a network formulation is considered, including a pump mode, and a number of zonal and non-zonal subdominant modes as a dynamical system. It was observed that the zonal modes become clearly dominant only when a large number of triads are connected. When the zonal flow becomes dominant as a "collective mean field,"individual interactions between modes become less important, which is consistent with the inhomogeneous wave-kinetic picture. Finally, the results of direct numerical simulation are discussed for the same parameters, and various forms of the order parameter are computed. It is observed that nonlinear phase dynamics results in a flattening of the large scale phase velocity as a function of scale in direct numerical simulations.
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| 10. |
- Moradi, Sara, 1981, et al.
(författare)
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A theory of non-local linear drift wave transport
- 2011
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Ingår i: Physics of Plasmas. - : AIP Publishing. - 1089-7674 .- 1070-664X. ; 18:6, s. 062106-
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Tidskriftsartikel (refereegranskat)abstract
- Transport events in turbulent tokamak plasmas often exhibit non-local or non-diffusive action at a distance features that so far have eluded a conclusive theoretical description. In this paper a theory of non-local transport is investigated through a Fokker-Planck equation with fractional velocity derivatives. A dispersion relation for density gradient driven linear drift modes is derived including the effects of the fractional velocity derivative in the Fokker-Planck equation. It is found that a small deviation (a few percent) from the Maxwellian distribution function alters the dispersion relation such that the growth rates are substantially increased and thereby may cause enhanced levels of transport.
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