Understanding Kappa Distributions: A Toolbox for Space Science and Astrophysics

Livadiotis, G.; McComas, D. J.

doi:10.1007/s11214-013-9982-9

Cited by 338 publications

(305 citation statements)

References 78 publications

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“…In order to proceed from equilibrium to the non-equilibrium plasmas, it is critical to understand the concept of temperature for systems in stationary states that are out of thermal equilibrium. Recently the temperature was shown to be well-defined for these nonequilibrium systems described by kappa distributions (for details, see Livadiotis and McComas 2009, 2010a, 2011b, 2013b; see also the early work of Treumann 1999;Treumann et al 2004;Treumann and Jaroschek 2008). Sections 4.6 and 4.7 show in detail the connection of temperature with the Debye length.…”

Section: R(x)mentioning

confidence: 99%

“…A well-defined temperature must be uniquely defined, and the fact of the equivalence of the two definition leads to a meaningful temperature for systems out of thermal equilibrium that are described by kappa distributions. (For more details on this topic, see Livadiotis and McComas 2009, 2010a, 2011b, 2012, 2013b; see also Livadiotis 2009;Livadiotis et al 2011Livadiotis et al , 2013 4.7. Isotropic Debye shielding In all the above, we have assumed that the Debye length is isotropic, namely, it is the same for any direction Ω ≡ (ϑ, ϕ).…”

Section: No Correlations Between Ions and Electronsmentioning

confidence: 99%

“…These distributions describe systems that are in stationary states but out of thermal equilibrium. The one-particle kappa distribution, defined in the 2-D phase space spanned by (r, u), is given by (Livadiotis and McComas 2013b) P (r, u; u b ; κ, T ) = A H (κ 0 , T ) · 1 + 1 κ 0 · H (r, u; u b ) k B T…”

Section: Appendix a Kappa Distribution Ion And Electron Densitiesmentioning

confidence: 99%

“…Stationary states refer to the distribution function of phase space of a system, meaning that must be -at least temporarily -invariant, even though is not given by the classical Boltzmann-Gibbs distribution of energy or the equivalent Maxwell distribution of velocities. Instead, the distribution function for stationary states out of thermal equilibrium are described by kappa distributions, for which the temperature is well-defined and given by the mean kinetic energy (Livadiotis and McComas 2009, 2010a, 2011b, 2012, 2013b, while the kappa index (that governs these distributions) is a thermodynamic parameter inversely proportional to the correlation between the phase space of any two particles (Livadiotis andMcComas 2011b, 2013d).…”

Section: Introductionmentioning

confidence: 99%

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Electrostatic shielding in plasmas and the physical meaning of the Debye length

Livadiotis

McComas

2014

J. Plasma Phys.

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This paper examines the electrostatic shielding in plasmas, and resolves inconsistencies about what the Debye length really is. Two different interpretations of the Debye length are currently used: (1) The potential energy approximately equals the thermal energy, and (2) the ratio of the shielded to the unshielded potential drops to 1/e. We examine these two interpretations of the Debye length for equilibrium plasmas described by the Boltzmann distribution, and non-equilibrium plasmas (e.g. space plasmas) described by kappa distributions. We study three dimensionalities of the electrostatic potential: 1-D potential of linear symmetry for planar charge density, 2-D potential of cylindrical symmetry for linear charge density, and 3-D potential of spherical symmetry for a point charge. We resolve critical inconsistencies of the two interpretations, including: independence of the Debye length on the dimensionality; requirement for small charge perturbations that is equivalent to weakly coupled plasmas; correlations between ions and electrons; existence of temperature for nonequilibrium plasmas; and isotropic Debye shielding. We introduce a third Debye length interpretation that naturally emerges from the second statistical moment of the particle position distribution; this is analogous to the kinetic definition of temperature, which is the second statistical moment of the velocity distribution. Finally, we compare the three interpretations, identifying what information is required for theoretical/experimental plasma-physics research: Interpretation 1 applies only to kappa distributions; Interpretation 2 is not restricted to any specific form of the ion/electron distributions, but these forms have to be known; Interpretation 3 needs only the second statistical moment of the positional distribution.

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Section: R(x)mentioning

confidence: 99%

Section: No Correlations Between Ions and Electronsmentioning

confidence: 99%

Section: Appendix a Kappa Distribution Ion And Electron Densitiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Electrostatic shielding in plasmas and the physical meaning of the Debye length

Livadiotis

McComas

2014

J. Plasma Phys.

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“…On the Gibbsian level the two versions are identical because raising an exponential to power n is the same as multiplying its argument by n. Now we violate the assumption of pure stochasticity by introducing the Lorentzian replacing exp(ax) → (1 − ax/κ) −(κ+r) , where 0 < κ ∈ R is some free parameter, and 0 < r is a fixed number that has to be adjusted to satisfying the thermodynamic relations. Determination of r, for the classical case, has been done in several places (e.g., Yoon et al, 2012;Livadiotis and McComas, 2013;Treumann and Baumjohann, 2014b). Then we obtain two new versions of the partition function…”

Section: Formulationmentioning

confidence: 99%

Generalised partition functions: inferences on phase space distributions

Treumann

Baumjohann

2016

Ann. Geophys.

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Abstract. It is demonstrated that the statistical mechanical partition function can be used to construct various different forms of phase space distributions. This indicates that its structure is not restricted to the Gibbs-Boltzmann factor prescription which is based on counting statistics. With the widely used replacement of the Boltzmann factor by a generalised Lorentzian (also known as the q-deformed exponential function, where κ = 1/|q − 1|, with κ, q ∈ R) both the kappa-Bose and kappa-Fermi partition functions are obtained in quite a straightforward way, from which the conventional Bose and Fermi distributions follow for κ → ∞. For κ = ∞ these are subject to the restrictions that they can be used only at temperatures far from zero. They thus, as shown earlier, have little value for quantum physics. This is reasonable, because physical κ systems imply strong correlations which are absent at zero temperature where apart from stochastics all dynamical interactions are frozen. In the classical large temperature limit one obtains physically reasonable κ distributions which depend on energy respectively momentum as well as on chemical potential. Looking for other functional dependencies, we examine Bessel functions whether they can be used for obtaining valid distributions. Again and for the same reason, no Fermi and Bose distributions exist in the low temperature limit. However, a classical Bessel-Boltzmann distribution can be constructed which is a Bessel-modified Lorentzian distribution. Whether it makes any physical sense remains an open question. This is not investigated here. The choice of Bessel functions is motivated solely by their convergence properties and not by reference to any physical demands. This result suggests that the Gibbs-Boltzmann partition function is fundamental not only to Gibbs-Boltzmann but also to a large class of generalised Lorentzian distributions as well as to the corresponding nonextensive statistical mechanics.

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Large‐scalequantization from local correlations in space plasmas

Livadiotis

McComas

2014

JGR Space Physics

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This study examines the large-scale quantization that can characterize the phase space of certain physical systems. Plasmas are such systems where large-scale quantization, ħ * , is caused by Debye shielding that structures correlations between particles. The value of ħ * is constant-some 12 orders of magnitude larger than the Planck constant-across a wide range of space plasmas, from the solar wind in the inner heliosphere to the distant plasma in the inner heliosheath and the local interstellar medium. This paper develops the foundation and advances the understanding of the concept of plasma quantization; in particular, we (i) show the analogy of plasma to Planck quantization, (ii) show the key points of plasma quantization, (iii) construct some basic quantum mechanical concepts for the large-scale plasma quantization, (iv) investigate the correlation between plasma parameters that implies plasma quantization, when it is approximated by a relation between the magnetosonic energy and the plasma frequency, (v) analyze typical space plasmas throughout the heliosphere and show the constancy of plasma quantization over many orders of magnitude in plasma parameters, (vi) analyze Advanced Composition Explorer (ACE) solar wind measurements to develop another measurement of the value of ħ * , and (vii) apply plasma quantization to derive unknown plasma parameters when some key observable is missing.

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Understanding Kappa Distributions: A Toolbox for Space Science and Astrophysics

Cited by 338 publications

References 78 publications

Electrostatic shielding in plasmas and the physical meaning of the Debye length

Electrostatic shielding in plasmas and the physical meaning of the Debye length

Generalised partition functions: inferences on phase space distributions

Large‐scalequantization from local correlations in space plasmas

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