Physical interpretation of the Padé approximation of the plasma dispersion function

2000 ◽  
Vol 64 (3) ◽  
pp. 287-296 ◽  
Author(s):  
ANDERS TJULIN ◽  
ANDERS I. ERIKSSON ◽  
MATS ANDRÉ
Keyword(s):  
Distribution Function ◽  
Physical Interpretation ◽  
Distribution Functions ◽  
Padé Approximant ◽  
General Distribution ◽  
Dispersion Function ◽  
Complex Velocity ◽  
Pade Approximant ◽  
Maxwellian Plasma ◽  
Plasma Dispersion

It is shown that using Padé approximants in the evaluation of the plasma dispersion function Z for a Maxwellian plasma is equivalent to the exact treatment for a plasma described by a ‘simple-pole distribution’, i.e. a distribution function that is a sum of simple poles in the complex velocity plane (v plane). In general, such a distribution function will have several zeros on the real v axis, and negative values in some ranges of v. This is shown to be true for the Padé approximant of Z commonly used in numerical packages such as WHAMP. The realization that an approximation of Z is equivalent to an approximation of f(v) leads the way to the study of more general distribution functions, and we compare the distribution corresponding to the Padé approximant used in WHAMP with a strictly positive and monotonically decreasing approximation of a Maxwellian.

Asymptotic behavior of Hill's estimate and applications

1986 ◽  
Vol 23 (04) ◽  
pp. 922-936
Author(s):  
Gane Samb Lo
Keyword(s):  
Distribution Function ◽  
Asymptotic Behavior ◽  
Finite Number ◽  
Domain Of Attraction ◽  
Distribution Functions ◽  
Complete Theory ◽  
General Distribution ◽  
Stable Law ◽  
Biological Phenomena ◽  
Pareto’S Law

The problem of estimating the exponent of a stable law is receiving an increasing amount of attention because Pareto's law (or Zipf's law) describes many biological phenomena very well (see e.g. Hill (1974)). This problem was first solved by Hill (1975), who proposed an estimate, and the convergence of that estimate to some positive and finite number was shown to be a characteristic of distribution functions belonging to the Fréchet domain of attraction (Mason (1982)). As a contribution to a complete theory of inference for the upper tail of a general distribution function, we give the asymptotic behavior (weak and strong) of Hill's estimate when the associated distribution function belongs to the Gumbel domain of attraction. Examples, applications and simulations are given.

scholarly journals Dispersion in a Relativistic Quantum Electron Gas. I. General Distribution Functions

1984 ◽  
Vol 37 (6) ◽  
pp. 615 ◽  
Author(s):  
Leith M Hayes ◽  
DB Melrose
Keyword(s):  
Relativistic Electron ◽  
Occupation Number ◽  
Relativistic Quantum ◽  
Electron Gas ◽  
Distribution Functions ◽  
Quantum Limit ◽  
General Distribution ◽  
Electron Positron ◽  
Quantum Electron ◽  
Plasma Dispersion

The covariant response tensor for a relativistic electron gas is calculated in two ways. One involves introducing a four-dimensional generalization of the electron-positron occupation number, and the other is a covariant generalization of a method due to Harris. The longitudinal and transverse parts are. evaluated for an isotropic electron gas in terms of three plasma dispersion functions, and the contributions from Landau damping and pair creation to the dispersion curve are identified separately. The long-wavelength limit and the non-quantum limit, with first quantum corrections, are found. The plasma dispersion functions are evaluated explicitly for a completely degenerate relativistic electron gas, and a detailed form due to Jancovici is reproduced.

Asymptotic behavior of Hill's estimate and applications

1986 ◽  
Vol 23 (4) ◽  
pp. 922-936 ◽  
Author(s):  
Gane Samb Lo
Keyword(s):  
Distribution Function ◽  
Asymptotic Behavior ◽  
Finite Number ◽  
Domain Of Attraction ◽  
Distribution Functions ◽  
Complete Theory ◽  
General Distribution ◽  
Stable Law ◽  
Biological Phenomena ◽  
Pareto’S Law

The problem of estimating the exponent of a stable law is receiving an increasing amount of attention because Pareto's law (or Zipf's law) describes many biological phenomena very well (see e.g. Hill (1974)). This problem was first solved by Hill (1975), who proposed an estimate, and the convergence of that estimate to some positive and finite number was shown to be a characteristic of distribution functions belonging to the Fréchet domain of attraction (Mason (1982)). As a contribution to a complete theory of inference for the upper tail of a general distribution function, we give the asymptotic behavior (weak and strong) of Hill's estimate when the associated distribution function belongs to the Gumbel domain of attraction. Examples, applications and simulations are given.

scholarly journals Electron velocity distribution function in a plasma with temperature gradient and in the presence of suprathermal electrons: application to incoherent-scatter plasma lines

1998 ◽  
Vol 16 (10) ◽  
pp. 1226-1240 ◽  
Author(s):  
P. Guio ◽  
J. Lilensten ◽  
W. Kofman ◽  
N. Bjørnå
Keyword(s):  
Distribution Function ◽  
Temperature Gradient ◽  
Velocity Distribution ◽  
Velocity Distribution Function ◽  
Electron Velocity ◽  
Dispersion Function ◽  
Electron Velocity Distribution ◽  
Incoherent Scatter ◽  
Suprathermal Electrons ◽  
Plasma Dispersion

Abstract. The plasma dispersion function and the reduced velocity distribution function are calculated numerically for any arbitrary velocity distribution function with cylindrical symmetry along the magnetic field. The electron velocity distribution is separated into two distributions representing the distribution of the ambient electrons and the suprathermal electrons. The velocity distribution function of the ambient electrons is modelled by a near-Maxwellian distribution function in presence of a temperature gradient and a potential electric field. The velocity distribution function of the suprathermal electrons is derived from a numerical model of the angular energy flux spectrum obtained by solving the transport equation of electrons. The numerical method used to calculate the plasma dispersion function and the reduced velocity distribution is described. The numerical code is used with simulated data to evaluate the Doppler frequency asymmetry between the up- and downshifted plasma lines of the incoherent-scatter plasma lines at different wave vectors. It is shown that the observed Doppler asymmetry is more dependent on deviation from the Maxwellian through the thermal part for high-frequency radars, while for low-frequency radars the Doppler asymmetry depends more on the presence of a suprathermal population. It is also seen that the full evaluation of the plasma dispersion function gives larger Doppler asymmetry than the heat flow approximation for Langmuir waves with phase velocity about three to six times the mean thermal velocity. For such waves the moment expansion of the dispersion function is not fully valid and the full calculation of the dispersion function is needed.Key words. Non-Maxwellian electron velocity distribution · Incoherent scatter plasma lines · EISCAT · Dielectric response function

scholarly journals Kinetic entropy-based measures of distribution function non-Maxwellianity: theory and simulations

2020 ◽  
Vol 86 (5) ◽  
Author(s):  
Haoming Liang ◽  
M. Hasan Barbhuiya ◽  
P. A. Cassak ◽  
O. Pezzi ◽  
S. Servidio ◽  
...  
Keyword(s):  
Distribution Function ◽  
Magnetic Reconnection ◽  
Entropy Density ◽  
Distribution Functions ◽  
Local Distribution ◽  
Particle In Cell ◽  
Maxwellian Plasma ◽  
Plasma Distribution ◽  
Maxwellian Distribution Function ◽  
Quantitative Understanding

We investigate kinetic entropy-based measures of the non-Maxwellianity of distribution functions in plasmas, i.e. entropy-based measures of the departure of a local distribution function from an associated Maxwellian distribution function with the same density, bulk flow and temperature as the local distribution. First, we consider a form previously employed by Kaufmann & Paterson (J. Geophys. Res., vol. 114, 2009, A00D04), assessing its properties and deriving equivalent forms. To provide a quantitative understanding of it, we derive analytical expressions for three common non-Maxwellian plasma distribution functions. We show that there are undesirable features of this non-Maxwellianity measure including that it can diverge in various physical limits and elucidate the reason for the divergence. We then introduce a new kinetic entropy-based non-Maxwellianity measure based on the velocity-space kinetic entropy density, which has a meaningful physical interpretation and does not diverge. We use collisionless particle-in-cell simulations of two-dimensional anti-parallel magnetic reconnection to assess the kinetic entropy-based non-Maxwellianity measures. We show that regions of non-zero non-Maxwellianity are linked to kinetic processes occurring during magnetic reconnection. We also show the simulated non-Maxwellianity agrees reasonably well with predictions for distributions resembling those calculated analytically. These results can be important for applications, as non-Maxwellianity can be used to identify regions of kinetic-scale physics or increased dissipation in plasmas.

Asymptotic behavior of Hill's estimate and applications

1986 ◽  
Vol 23 (04) ◽  
pp. 922-936 ◽  
Author(s):  
Gane Samb Lo
Keyword(s):  
Distribution Function ◽  
Asymptotic Behavior ◽  
Finite Number ◽  
Domain Of Attraction ◽  
Distribution Functions ◽  
Complete Theory ◽  
General Distribution ◽  
Stable Law ◽  
Biological Phenomena ◽  
Pareto’S Law

The problem of estimating the exponent of a stable law is receiving an increasing amount of attention because Pareto's law (or Zipf's law) describes many biological phenomena very well (see e.g. Hill (1974)). This problem was first solved by Hill (1975), who proposed an estimate, and the convergence of that estimate to some positive and finite number was shown to be a characteristic of distribution functions belonging to the Fréchet domain of attraction (Mason (1982)). As a contribution to a complete theory of inference for the upper tail of a general distribution function, we give the asymptotic behavior (weak and strong) of Hill's estimate when the associated distribution function belongs to the Gumbel domain of attraction. Examples, applications and simulations are given.

Scaling

2018 ◽  
Author(s):  
Stefan Thurner ◽  
Rudolf Hanel ◽  
Peter Klimekl
Keyword(s):  
Distribution Function ◽  
Dynamical Systems ◽  
Complex Systems ◽  
Power Law ◽  
Scaling Laws ◽  
Power Laws ◽  
Distribution Functions ◽  
Temporal Process ◽  
Approximate Power

Scaling appears practically everywhere in science; it basically quantifies how the properties or shapes of an object change with the scale of the object. Scaling laws are always associated with power laws. The scaling object can be a function, a structure, a physical law, or a distribution function that describes the statistics of a system or a temporal process. We focus on scaling laws that appear in the statistical description of stochastic complex systems, where scaling appears in the distribution functions of observable quantities of dynamical systems or processes. The distribution functions exhibit power laws, approximate power laws, or fat-tailed distributions. Understanding their origin and how power law exponents can be related to the particular nature of a system, is one of the aims of the book.We comment on fitting power laws.

K+p scattering with unitary Padé approximant

1975 ◽  
Vol 87 (3) ◽  
pp. 485-508 ◽  
Author(s):  
Sarah C.B. Andrade ◽  
Erasmo Ferreira ◽  
Luis Ye Chang
Keyword(s):  
Padé Approximant ◽  
Pade Approximant

Expansions of the mildly relativistic dispersion function for nearly perpendicular electron cyclotron ray tracing

1999 ◽  
Vol 61 (1) ◽  
pp. 121-128 ◽  
Author(s):  
I. P. SHKAROFSKY
Keyword(s):  
Ray Tracing ◽  
Computer Programs ◽  
Higher Order ◽  
Electron Cyclotron ◽  
Dispersion Function ◽  
Relativistic Dispersion ◽  
Plasma Dispersion ◽  
Derivatives Of ◽  
Cyclotron Harmonic ◽  
Experienced Difficulty

To trace rays very close to the nth electron cyclotron harmonic, we need the mildly relativistic plasma dispersion function and its higher-order derivatives. Expressions for these functions have been obtained as an expansion for nearly perpendicular propagation in a region where computer programs have previously experienced difficulty in accuracy, namely when the magnitude of (c/vt)2 (ω−nωc)/ω is between 1 and 10. In this region, the large-argument expansions are not yet valid, but partial cancellations of terms occur. The expansion is expressed as a sum over derivatives of the ordinary dispersion function Z. New expressions are derived to relate higher-order derivatives of Z to Z itself in this region of concern in terms of a finite series.

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