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- 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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- 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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- 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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- 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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- 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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- 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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- 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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