Self‐attracting Fermi–Dirac particles in canonical and microcanonical setting

Stańczy, Robert

doi:10.1002/mma.605

Cited by 11 publications

(8 citation statements)

References 16 publications

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“…Other standard examples are the polytropic distributions, with f (u) = u p , and Bose-Einstein or Fermi-Dirac distributions which result in non-linearities involving special functions. Existence and non-existence results were obtained for instance in [7] and [71,72], respectively, for Maxwell-Boltzmann and Fermi-Dirac statistics.…”

Section: Vol 10 (2010) Uniqueness Results In Supercritical Equationsmentioning

confidence: 99%

“…An evolution model compatible with Fermi-Dirac statistics and the convergence of its solutions towards steady states has been thoroughly examined in [9], while the steady state problem was considered by Stańczy [71,72,74]. See [22] and references therein for a model improved with respect to thermodynamics, [72] and references therein for more elaborate models, and [23] for a derivation of an evolution equation involving a mean field term, which also provides a relevant, stationary model studied in [13,73].…”

Section: Vol 10 (2010) Uniqueness Results In Supercritical Equationsmentioning

confidence: 99%

“…This can be guaranteed if Ω is bounded and if we additionally require that the function f is increasing, as in [71] for the Fermi-Dirac model. This is also true for exponential and power-like non-linearities.…”

Section: The Local Bifurcation Problemmentioning

confidence: 99%

“…As a consequence, we can state the following theorem, which generalizes known results on exponential and Fermi-Dirac distributions, cf. [7] and [71,72], respectively.…”

Section: The Non-local Bifurcation Problemmentioning

confidence: 99%

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Non-Existence and Uniqueness Results for Supercritical Semilinear Elliptic Equations

Dolbeault

Stańczy

2009

Ann. Henri Poincaré

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Non-existence and uniqueness results are proved for several local and non-local supercritical bifurcation problems involving a semilinear elliptic equation depending on a parameter. The domain is star-shaped and such that a Poincaré inequality holds but no other symmetry assumption is required. Uniqueness holds when the bifurcation parameter is in a certain range. Our approach can be seen, in some cases, as an extension of non-existence results for non-trivial solutions. It is based on Rellich-Pohožaev type estimates. Semilinear elliptic equations naturally arise in many applications, for instance in astrophysics, hydrodynamics or thermodynamics. We simplify the proof of earlier results by K. Schmitt and R. Schaaf in the so-called local multiplicative case, extend them to the case of a non-local dependence on the bifurcation parameter and to the additive case, both in local and non-local settings.

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Section: Vol 10 (2010) Uniqueness Results In Supercritical Equationsmentioning

confidence: 99%

Section: Vol 10 (2010) Uniqueness Results In Supercritical Equationsmentioning

confidence: 99%

Section: The Local Bifurcation Problemmentioning

confidence: 99%

“…As a consequence, we can state the following theorem, which generalizes known results on exponential and Fermi-Dirac distributions, cf. [7] and [71,72], respectively.…”

Section: The Non-local Bifurcation Problemmentioning

confidence: 99%

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Non-Existence and Uniqueness Results for Supercritical Semilinear Elliptic Equations

Dolbeault

Stańczy

2009

Ann. Henri Poincaré

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“…It should be noted that this form of the P function leads to a nonlinear diffusion in the first equation from (1.2). The problem of the existence of steady states has been considered in [15] and [16]. The limit case γ 0 corresponds to the Boltzmann density-pressure relation.…”

Section: Examplesmentioning

confidence: 99%

The existence of equilibria of many-particle systems

Stańczy¹

2009

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Existence and continuous dependence on parameters of radially symmetric solutions to astrophysical model of self‐gravitating particles

Bors

Stańczy

2019

Math Methods in App Sciences

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In this paper, we obtain the existence of a radial solution for some elliptic nonlocal problem with constraints. The problem is described as stationary state of some evolutionary models provided the pressure function conveys some form arising from statistical mechanics. To be more specific, we consider the following forms of the statistics: Michie-King, Maxwell-Boltzmann, Fermi-Dirac, Bose-Einstein, and polytrope. The most recent models were suggested by H. J.de Vega at al. and P. H. Chavanis et al.. We prove the continuous dependence of stationary solutions on parameters for the given statistics. KEYWORDS Fermi-Dirac, Michie-King statistics, nonlocal elliptic equation MSC CLASSIFICATION 35Q85; 47H10; 85A05 ASTROPHYSICAL MOTIVATIONAs it is well established, in the universe, there is vast preponderance of the dark matter and the dark energy. To describe it at the galactic level, numerous models were proposed.The classical Michie-King model was introduced by Michie 1 and King 2 as the natural extension of the Maxwell-Boltzmann case. The classical Michie-King model is a modification of the Maxwell-Boltzmann distribution that takes into account the escape of high-energy stars in a stellar system.Following a long list of works on the subject, de Vega et al 3-5 modeled dark matter halos as self-gravitating gas of fermions at finite temperature and confronted it with observations that correspond to warm dark matter, eg, composed of sterile neutrinos. In the approach by de Vega et al, 3-5 the system does not confine within a box, and halos extend to infinity and have infinite mass. They do not need to compute the total mass in their study, 3-5 so they do not have to deal with this problem.The study that considers self-gravitating fermions in the box is Chavanis, 6 which is concerned with phase transition in the self-gravitating Fermi gas. If we want to study phase transitions between different equilibrium states, it is necessary to have configurations with a finite mass, and this is why Chavanis 6 had to close the system in the artificial box. In more recent papers, Chavanis et al 7,8 studied phase transitions of self-gravitating fermions without the artifice of the material box by using the fermionic King distribution. This approach alleviates the problem by extension of the model with the pressure governed by the Fermi-Dirac statistics to the fermionic Michie-King model. The ad hoc considerations of Ruffini and Stella 9 can be also justified by the statistical mechanics and the kinetic theory once the evaporation of the high energy particles is taken into account as was done by Chavanis et al. 7,8 Although the fermionic King model was introduced heuristically by Ruffini and Stella, 9 a derivation from kinetic theory was first made by Chavanis. 10 To alleviate drawbacks Math Meth Appl Sci. 2019;42:7381-7394.wileyonlinelibrary.com/journal/mma

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Self‐attracting Fermi–Dirac particles in canonical and microcanonical setting

Abstract: SUMMARYThe existence of stationary solutions for the system describing the interaction of gravitationally attracting particles that obey the Fermi-Dirac statistics is studied depending on the relations between mass, the energy and the temperature of the particles.

Cited by 11 publications

References 16 publications

Non-Existence and Uniqueness Results for Supercritical Semilinear Elliptic Equations

Non-Existence and Uniqueness Results for Supercritical Semilinear Elliptic Equations

The existence of equilibria of many-particle systems

Existence and continuous dependence on parameters of radially symmetric solutions to astrophysical model of self‐gravitating particles

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