Approach on Tsallis statistical interpretation of hydrogen-atom by adopting the generalized radial distribution function

Livadiotis, G.

doi:10.1007/s10910-009-9524-6

Cited by 39 publications

(33 citation statements)

References 9 publications

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“…This is invariant under variations of the system's particles N and the total degrees of freedom f N = d·N, where d denotes now the kinetic degrees of freedom per particle. (For details on the derivations, see Appendices; for more details on the kappa distribution formulation, see Livadiotis andMcComas 2011b, 2013b.) Namely,…”

Section: R(x)mentioning

confidence: 99%

“…H (r, u) = 1 2 m u 2 − e Φ C (r). This is a case of negative potential energy (Livadiotis andMcComas 2009, 2013b) and the Hamiltonian is not always positive; still, the quantity 1 + H (r, u)/(κ 0 k B T e ) in (13) must be always non-negative (because it represents the kinetic energy ε), hence, ε ≡ 1 2 m u 2 > e Φ C (r) − κ 0 k B T e . If the quantity at the right-hand side of this inequality is positive, it gives the smallest possible kinetic energy, ε M (r) ≡ e Φ C (r) − κ 0 k B T e .…”

Section: Cut-off Of the Electron And Ion Densitiesmentioning

confidence: 99%

“…This is because in all the derivations of the Debye length, the one-particle formulation of kappa distributions has been used, instead of the ) more complicated N-particle formulation (Livadiotis andMcComas 2011b, 2013b). At least, a two-particle distribution must be used if we would like to subscribe a correlation between electrons and ions.…”

Section: No Correlations Between Ions and Electronsmentioning

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%

“…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: Cut-off Of the Electron And Ion Densitiesmentioning

confidence: 99%

Section: No Correlations Between Ions and Electronsmentioning

confidence: 99%

Section: No Correlations Between Ions and Electronsmentioning

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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Kappa distribution in the presence of a potential energy

Livadiotis

2015

JGR Space Physics

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The present paper develops the theory and formulations of the kappa distributions that describe particle systems characterized by a nonzero potential energy. As yet, kappa distributions were used for the statistical description of the velocity or kinetic energy of particles but not of the potential energy. With the results provided here, it is straightforward to use the developed kappa distributions to describe any particle population of space plasmas subject to a nonnegligible potential energy. Starting from the kappa distribution of the Hamiltonian function, we develop the distributions that describe either the complete phase space or the marginal spaces of positions and velocities. The study shows, among others: (a) The kappa distributions of velocities that describe space plasmas can be vastly different from the standard formulation of the kappa distribution, because of the presence of a potential energy; the correct formulation should be given by the marginal kappa distribution of velocities by integrating the distribution of the Hamiltonian over the potential energy. (b) The long-standing problem of the divergence of the Boltzmannian exponential distribution for bounded radial potentials is solved using kappa distributions of negative kappa index. (c) Anisotropic distributions of velocities can exist in the presence of a velocity-dependent potential. (d) A variety of applications, including derivations/verifications of the following: (i) the Jeans' , the most frequent, and the maximum radii in spherical/linear gravitational potentials; (ii) the Virial theorem for power law potentials; (iii) the generalized barometric formula, (iv) the plasma density profiles in Saturnian magnetosphere, and (v) the average electron magnetic moment in Earth's magnetotail.

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Introduction to special section on Origins and Properties of Kappa Distributions: Statistical Background and Properties of Kappa Distributions in Space Plasmas

2015

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Empirical kappa distributions provide a straightforward replacement of the Maxwell distribution for systems out of thermal equilibrium such as space plasmas. Kappa distributions have become increasingly widespread across space physics with the number of relevant publications following, remarkably, an exponential growth rate. However, a breakthrough in the field came with the connection of kappa distributions with the framework of nonextensive statistical mechanics. This introductory paper clarifies fundamental physical concepts and provides mathematical formulations of the theory of kappa distributions, which are a consequence of the connection of kappa distributions with a solid statistical background. Among others, the paper presents the existence of a consistent definition of temperature in systems out of thermal equilibrium described by kappa distributions, the physical meaning of the kappa index, and the formulation of the kappa distribution of a Hamiltonian. In addition, the paper examines the most frequent values of kappa indices in space plasmas. Statistical analysis reveals trends between the characteristic values of density, temperature, and kappa index of space plasmas. Finally, understanding the kinetic interpretation of the temperature as the mean kinetic energy, and of the kappa index as the correlation of kinetic energies, helps to develop all the possible formulations of isotropic/anisotropic kappa distributions.

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Approach on Tsallis statistical interpretation of hydrogen-atom by adopting the generalized radial distribution function

Cited by 39 publications

References 9 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

Kappa distribution in the presence of a potential energy

Introduction to special section on Origins and Properties of Kappa Distributions: Statistical Background and Properties of Kappa Distributions in Space Plasmas

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