arXiv subjects

Pierre-Henri Chavanis

Publications and source records attributed to Pierre-Henri Chavanis.

At least 55 records · Page 3Linked to original sources

arXiv2019

Mass-radius relation of self-gravitating Bose-Einstein condensates with a central black hole

We determine the mass-radius relation of self-gravitating Bose-Einstein condensates with an attractive $-1/r$ external potential created by a central mass. Following our previous work [P.H. Chavanis, Phys. Rev. D 84, 043531 (2011)], we use an analytical approach based on a Gaussian ansatz. We consider the case of noninteracting bosons as well as the case of self-interacting bosons with a repulsive or an attractive self-interaction. These results may find application in the context of dark matter halos made of self-gravitating Bose-Einstein condensates. In that case, the central mass may mimic a supermassive black hole. We apply our results to ultralight axions with an attractive self-interaction. We determine how the central black hole affects the mass-radius relation and the maximum mass of axionic halos found in our previous papers. Our approximate analytical results based on the Gaussian ansatz are compared with exact analytical results obtained in particular limits.

gr-qc↗

arXiv2019

Supernova implosion-explosion in the light of catastrophe theory

The present understanding of supernova explosion of massive stars as a two-step process, with an initial gravitational collapse toward the center of the star followed by an expansion of matter after a bouncing on the core, meets several difficulties. We show that it is not the only possible one: a simple model based on fluid mechanics, catastrophe theory, and stability properties of the equilibrium state shows that one can have also a simultaneous inward/outward motion in the early stage of the instability of the supernova described by a dynamical saddle-center bifurcation. The existence of this simultaneous inward/outward motion is sensitive to the model in such systems with long-range interactions. If a constant temperature is assumed (canonical ensemble), an overall inward motion occurs, but if one imposes with the same equation of state the constraint of energy conservation (microcanonical ensemble) there is an inward velocity field near the center of the star together with an outward velocity field in the rest of the star. We discuss the expansion stage of the remnants away from the collapsed core, and propose a new explanation for the formation of shock waves in the ejecta which differs from the usual Sedov-Taylor self-similar description.

astro-ph.SR↗

arXiv2019

Implosion-explosion in supernovae

Supernovae explosions of massive stars are nowadays believed to result from a two-step process, with an initial gravitational core collapse followed by an expansion of matter after a bouncing on the core. This scenario meets several difficulties. We show that it is not the only possible one: a simple model based on fluid mechanics and stability properties of the equilibrium state shows that one can have also a simultaneous inward/outward motion in the early stage of the instability of the supernova. This shows up in the slow sweeping across a saddle-center bifurcation found when considering equilibrium states associated to the constraint of energy conservation. We first discuss the weakly nonlinear regime in terms of a Painlevé I equation. We then show that the strongly nonlinear regime displays a self-similar behavior of the core collapse. Finally, the expansion of the remnants is revisited as an isentropic process leading to shocks formation.

astro-ph.SR↗

arXiv2019

Statistical mechanics of self-gravitating systems in general relativity: I. The quantum Fermi gas

We develop a general formalism to determine the statistical equilibrium states of self-gravitating systems in general relativity and complete previous works on the subject. Our results are valid for an arbitrary form of entropy but, for illustration, we explicitly consider the Fermi-Dirac entropy for fermions. The maximization of entropy at fixed mass-energy and particle number determines the distribution function of the system and its equation of state. It also implies the Tolman-Oppenheimer-Volkoff equations of hydrostatic equilibrium and the Tolman-Klein relations. Our paper provides all the necessary equations that are needed to construct the caloric curves of self-gravitating fermions in general relativity as done in recent works. We consider the nonrelativistic limit $c\rightarrow +\infty$ and recover the equations obtained within the framework of Newtonian gravity. We also discuss the inequivalence of statistical ensembles as well as the relation between the dynamical and thermodynamical stability of self-gravitating systems in Newtonian gravity and general relativity.

gr-qc↗

arXiv2019

Statistical mechanics of self-gravitating systems in general relativity: II. The classical Boltzmann gas

We study the statistical mechanics of classical self-gravitating systems confined within a box of radius $R$ in general relativity. It has been found that the caloric curve $T_{\infty}(E)$ has the form of a double spiral whose shape depends on the compactness parameter $ν=GNm/Rc^2$. The double spiral shrinks as $ν$ increases and finally disappears when $ν_{\rm max}=0.1764$. Therefore, general relativistic effects render the system more unstable. On the other hand, the cold spiral and the hot spiral move away from each other as $ν$ decreases. Using a normalization $Λ=-ER/GN^2m^2$ and $η=GNm^2/R k_B T_{\infty}$ appropriate to the nonrelativistic limit, and considering $ν\rightarrow 0$, the hot spiral goes to infinity and the caloric curve tends towards a limit curve (determined by the Emden equation) exhibiting a single cold spiral, as found in former works. Using another normalization ${\cal M}=GM/Rc^2$ and ${\cal B}={Rc^4}/{GNk_B T_{\infty}}$ appropriate to the ultrarelativistic limit, and considering $ν\rightarrow 0$, the cold spiral goes to infinity and the caloric curve tends towards a limit curve (determined by the general relativistic Emden equation) exhibiting a single hot spiral. This result is new. We discuss the analogies and the differences between this asymptotic caloric curve and the caloric curve of the self-gravitating black-body radiation. Finally, we compare box-confined isothermal models with heavily truncated isothermal distributions in Newtonian gravity and general relativity.

gr-qc↗

arXiv2019

Gravitational phase transitions and instabilities of self-gravitating fermions in general relativity

We discuss the occurrence of gravitational phase transitions and instabilities in a gas of self-gravitating fermions within the framework of general relativity. In the classical (nondegenerate) limit, the system undergoes a gravitational collapse at low energies $E N_{\rm OV}$, there can be a subsequent gravitational collapse below a lower critical energy $E<E''_c$ or a lower critical temperature $T<T'_c$ leading presumably to the formation of a black hole. The evolution of the system is different in the microcanonical and canonical ensembles. In the microcanonical ensemble, the system takes a "core-halo" structure. The core consists in a compact quantum object or a black hole while the hot halo is expelled at large distances. This is reminiscent of the red giant structure of low-mass stars or the implosion-explosion of massive stars (supernova). In the canonical ensemble, the system collapses as a whole towards a compact object or a black hole. This is reminiscent of the implosion of supermassive stars (hypernova).

gr-qc↗

arXiv2019

Caloric curves of classical self-gravitating systems in general relativity

We determine the caloric curves of classical self-gravitating systems at statistical equilibrium in general relativity. In the classical limit, the caloric curves of a self-gravitating gas depend on a unique parameter $ν=GNm/Rc^2$, called the compactness parameter, where $N$ is the particle number and $R$ the system's size. Typically, the caloric curves have the form of a double spiral. The "cold spiral", corresponding to weakly relativistic configurations, is a generalization of the caloric curve of nonrelativistic classical self-gravitating systems. The "hot spiral'", corresponding to strongly relativistic configurations, is similar (but not identical) to the caloric curve of the ultrarelativistic self-gravitating black-body radiation. We introduce two types of normalization of energy and temperature in order to obtain asymptotic caloric curves describing respectively the cold and the hot spirals in the limit $ν\rightarrow 0$. As the number of particles increases, the cold and the hot spirals approach each other, merge at $ν'_S=0.128$, form a loop above $ν_S=0.1415$, reduce to a point at $ν_{\rm max}=0.1764$, and finally disappear. Therefore, the double spiral shrinks when the compactness parameter $ν$ increases, implying that general relativistic effects render the system more unstable. We discuss the nature of the gravitational collapse at low and high energies with respect to a dynamical (fast) or a thermodynamical (slow) instability.

gr-qc↗

arXiv2019

Kinetic theory of ${1D}$ homogeneous long-range interacting systems sourced by ${1/N^{2}}$ effects

The long-term dynamics of long-range interacting $N$-body systems can generically be described by the Balescu-Lenard kinetic equation. However, for ${1D}$ homogeneous systems, this collision operator exactly vanishes by symmetry. These systems undergo a kinetic blocking, and cannot relax as a whole under ${1/N}$ resonant effects. As a result, these systems can only relax under ${1/N^{2}}$ effects, and their relaxation is drastically slowed down. In the context of the homogeneous Hamiltonian Mean Field model, we present a new, closed and explicit kinetic equation describing self-consistently the very long-term evolution of such systems, in the limit where collective effects can be neglected, i.e. for dynamically hot initial conditions. We show in particular how that kinetic equation satisfies an $H$-Theorem that guarantees the unavoidable relaxation to the Boltzmann equilibrium distribution. Finally, we illustrate how that kinetic equation quantitatively matches with the measurements from direct $N$-body simulations.

cond-mat.stat-mech↗

arXiv2019

Generalized Euler, Smoluchowski and Schrödinger equations admitting self-similar solutions with a Tsallis invariant profile

The damped isothermal Euler equations, the Smoluchowski equation and the damped logarithmic Schrödinger equation with a harmonic potential admit stationary and self-similar solutions with a Gaussian profile. They satisfy an $H$-theorem for a free energy functional involving the von Weizsäcker functional and the Boltzmann functional. We derive generalized forms of these equations in order to obtain stationary and self-similar solutions with a Tsallis profile. In particular, we introduce a nonlinear Schrödinger equation involving a generalized kinetic term characterized by an index $q$ and a power-law nonlinearity characterized by an index $γ$. We derive an $H$-theorem satisfied by a generalized free energy functional involving a generalized von Weizsäcker functional (associated with $q$) and a Tsallis functional (associated with $γ$). This leads to a notion of generalized quantum mechanics and generalized thermodynamics. When $q=2γ-1$, our nonlinear Schrödinger equation admits an exact self-similar solution with a Tsallis invariant profile. Standard quantum mechanics (Schrödinger) and standard thermodynamics (Boltzmann) are recovered for $q=γ=1$.

cond-mat.stat-mech↗

arXiv2019

Gravitational phase transition of self-gravitating systems of fermions in General Relativity

The Thomas-Fermi model is extended at finite temperature, to describe the gravitational phase transition occurring in massive fermionic systems in a general-relativistic framework. It is shown that, when a nondegenerate fermionic gas (for $N < N_{OV}$, where $N_{OV}$ is the Oppenheimer-Volkoff limit) is cooled down below a critical temperature, a condensed phase emerges and the gravitational collapse is prevented by quantum mechanics. If $N > N_{OV}$, by contrast, the system is destined to collapse towards a Black Hole because no equilibrium states exist.

gr-qc↗

arXiv2019

Relativistic Gravitational Phase Transitions and Instabilities of the Fermi Gas

We describe microcanonical phase transitions and instabilities of the ideal Fermi gas in general relativity at nonzero temperature confined in the interior of a spherical shell. The thermodynamic behaviour is governed by the compactness of rest mass, namely of the total rest mass over radius of the system. For a fixed value of rest mass compactness, we study the caloric curves as a function of the size of the spherical box. At low compactness values, low energies and for sufficiently big systems the system is subject to a gravothermal catastrophe, which cannot be halted by quantum degeneracy pressure, and the system collapses towards the formation of a black hole. For small systems, there appears no instability at low energies. For intermediate sizes, between two marginal values, gravothermal catastrophe is halted and a microcanonical phase transition occurs from a gaseous phase to a condensed phase with a nearly degenerate core. The system is subject to a relativistic instability at low energy, when the core gets sufficiently condensed above the Oppenheimer-Volkoff limit. For sufficiently high values of rest mass compactness the microcanonical phase transitions are suppressed. They are replaced either by an Antonov type gravothermal catastrophe for sufficiently big systems or by stable equilibria for small systems. At high energies the system is subject to the `relativistic gravothermal instability', identified by Roupas in [1], for all values of compactness and any size.

gr-qc↗

arXiv2018

Vector Resonant Relaxation of Stars around a Massive Black Hole

In the vicinity of a massive black hole, stars move on precessing Keplerian orbits. The mutual stochastic gravitational torques between the stellar orbits drive a rapid reorientation of their orbital planes, through a process called vector resonant relaxation. We derive, from first principles, the correlation of the potential fluctuations in such a system, and the statistical properties of random walks undergone by the stellar orbital orientations. We compare this new analytical approach with effective $N$-body simulations. We also provide a simple scheme to generate the random walk of a test star's orbital orientation using a stochastic equation of motion. We finally present quantitative estimations of this process for a nuclear stellar cluster such as the one of the Milky Way.

astro-ph.GA↗

arXiv2018

Secular dynamics of long-range interacting particles on a sphere in the axisymmetric limit

We investigate the secular dynamics of long-range interacting particles moving on a sphere, in the limit of an axisymmetric mean field potential. We show that this system can be described by the general kinetic equation, the inhomogeneous Balescu--Lenard equation. We use this approach to compute long-term diffusion coefficients, that are compared with direct simulations. Finally, we show how the scaling of the system's relaxation rate with the number of particles fundamentally depends on the underlying frequency profile. This clarifies why systems with a monotonic profile undergo a kinetic blocking and cannot relax as a whole under ${1/N}$ resonant effects. Because of its general form, this framework can describe the dynamics of globally coupled classical Heisenberg spins, long-range couplings in liquid crystals, or the orbital inclination evolution of stars in nearly Keplerian systems.

cond-mat.stat-mech↗

arXiv2018

New predictions from the logotropic model

In a previous paper we have introduced a new cosmological model that we called the logotropic model. The logotropic model is able to account, without free parameter, for the constant surface density of the dark matter halos, for their mass-radius relation, and for the Tully-Fisher relation. In this paper, we explore other consequences of this model. By advocating a form of "strong cosmic coincidence" we predict that the present proportion of dark energy in the Universe is $Ω_{\rm de,0}=e/(1+e)\simeq 0.731$ which is close to the observed value. We also remark that the surface density of dark matter halos and the surface density of the Universe are of the same order as the surface density of the electron. This makes a curious connection between cosmological and atomic scales. Using these coincidences, we can relate the Hubble constant, the electron mass and the electron charge to the cosmological constant. We also suggest that the famous numbers $137$ (fine-structure constant) and $123$ (logotropic constant) may actually represent the same thing. This could unify microphysics and cosmophysics. We study the thermodynamics of the logotropic model and find a connection to the Bekenstein-Hawking entropy of black holes if we assume that the logotropic fluid is made of particles of mass $m_Λ\sim \hbar\sqrtΛ/c^2=2.08\times 10^{-33}\, {\rm eV/c^2}$ (cosmons). In that case, the universality of the surface density of the dark matter halos may be related to a form of holographic principle (the fact that their entropy scales like their area). We use similar arguments to explain why the surface density of the electron and the surface density of the Universe are of the same order and justify the empirical Weinberg relation.

gr-qc↗

arXiv2018

A predictive model of BEC dark matter halos with a solitonic core and an isothermal atmosphere

We develop a model of Bose-Einstein condensate dark matter halos with a solitonic core and an isothermal atmosphere based on a generalized Gross-Pitaevskii-Poisson equation [P.H. Chavanis, Eur. Phys. J. Plus 132, 248 (2017)]. This equation provides a heuristic coarse-grained parametrization of the ordinary Gross-Pitaevskii-Poisson equation accounting for violent relaxation and gravitational cooling. It involves a cubic nonlinearity taking into account the self-interaction of the bosons, a logarithmic nonlinearity associated with an effective temperature, and a source of dissipation. It leads to superfluid dark matter halos with a core-halo structure. The quantum potential or the self-interaction of the bosons generates a solitonic core that solves the cusp problem of the cold dark matter model. The logarithmic nonlinearity generates an isothermal atmosphere accounting for the flat rotation curves of the galaxies. The dissipation ensures that the system relaxes towards an equilibrium configuration. In the Thomas-Fermi approximation, the dark matter halo is equivalent to a barotropic gas with an equation of state $P=2πa_s\hbar^2ρ^2/m^3+ρk_B T/m$, where $a_s$ is the scattering length of the bosons and $m$ is their individual mass. We numerically solve the equation of hydrostatic equilibrium and determine the corresponding density profiles and rotation curves. Our model has no free parameter so it is completely predictive. Extension of this model to noninteracting bosons and fermions will be presented in forthcoming papers.

gr-qc↗

arXiv2018

Caloric curves of self-gravitating fermions in general relativity

We study the nature of phase transitions between gaseous and condensed states in the self-gravitating Fermi gas at nonzero temperature in general relativity. The condensed states can represent compact objects such as white dwarfs, neutron stars, or dark matter fermion balls. The caloric curves depend on two parameters: the system size $R$ and the particle number $N$. When $N N_{\rm OV}$, there is no equilibrium state below a critical energy and below a critical temperature. In that case, the system is expected to collapse towards a black hole. We plot the caloric curves of the general relativistic Fermi gas, study the different types of phase transitions that occur in the system, and determine the phase diagram in the $(R,N)$ plane. The nonrelativistic results are recovered for $N\ll N_{\rm OV}$ and $R\gg R_{\rm OV}$ with $NR^3$ fixed. The classical results are recovered for $N\gg N_{\rm OV}$ and $R\gg R_{\rm OV}$ with $N/R$ fixed. We highlight a situation of physical interest where a gaseous Fermi gas, by cooling, first undergoes a phase transition towards a compact object (white dwarf, neutron star, dark matter fermion ball), then collapses into a black hole. This situation occurs in the microcanonical ensemble when $N_{\rm OV}<N<3.73\, N_{\rm OV}$. We also relate the phase transitions from a gaseous state to a core-halo state in the microcanonical ensemble to the onset of red-giant structure and to the supernova phenomenon.

gr-qc↗

arXiv2018

Derivation of a generalized Schrödinger equation for dark matter halos from the theory of scale relativity

Using Nottale's theory of scale relativity, we derive a generalized Schrödinger equation applying to dark matter halos. This equation involves a logarithmic nonlinearity associated with an effective temperature and a source of dissipation. Fundamentally, this wave equation arises from the nondifferentiability of the trajectories of the dark matter particles whose origin may be due to ordinary quantum mechanics, classical ergodic (or almost ergodic) chaos, or to the fractal nature of spacetime at the cosmic scale. The generalized Schrödinger equation involves a coefficient ${\cal D}$, possibly different from $\hbar/2m$, whose value for dark matter halos is ${\cal D}=1.02\times 10^{23}\, {\rm m^2/s}$. We suggest that the cold dark matter crisis may be solved by the fractal (nondifferentiable) structure of spacetime at the cosmic scale, or by the chaotic motion of the particles on a very long timescale, instead of ordinary quantum mechanics. The equilibrium states of the generalized Schrödinger equation correspond to configurations with a core-halo structure. The quantumlike potential generates a solitonic core that solves the cusp problem of the classical cold dark matter model. The logarithmic nonlinearity accounts for the presence of an isothermal halo that leads to flat rotation curves. The damping term ensures that the system relaxes towards an equilibrium state. This property is guaranteed by an $H$-theorem satisfied by a Boltzmann-like free energy functional. In our approach, the temperature and the friction arise from a single formalism. They correspond to the real and imaginary parts of the complex friction coefficient present in the scale covariant equation of dynamics that is at the basis of Nottale's theory of scale relativity.

gr-qc↗

arXiv2018

On the Schrodinger-Poisson--Vlasov-Poisson correspondence

The Schrödinger-Poisson equations describe the behavior of a superfluid Bose-Einstein condensate under self-gravity with a 3D wave function. As $\hbar/m\to 0$, $m$ being the boson mass, the equations have been postulated to approximate the collisionless Vlasov-Poisson equations also known as the collisionless Boltzmann-Poisson equations. The latter describe collisionless matter with a 6D classical distribution function. We investigate the nature of this correspondence with a suite of numerical test problems in 1D, 2D, and 3D along with analytic treatments when possible. We demonstrate that, while the density field of the superfluid always shows order unity oscillations as $\hbar/m\to 0$ due to interference and the uncertainty principle, the potential field converges to the classical answer as $(\hbar/m)^{2}$. Thus, any dynamics coupled to the superfluid potential is expected to recover the classical collisionless limit as $\hbar/m\to 0$. The quantum superfluid is able to capture rich phenomena such as multiple phase-sheets, shell-crossings, and warm distributions. Additionally, the quantum pressure tensor acts as a regularizer of caustics and singularities in classical solutions. This suggests the exciting prospect of using the Schrödinger-Poisson equations as a low-memory method for approximating the high-dimensional evolution of the Vlasov-Poisson equations. As a particular example we consider dark matter composed of ultra-light axions, which in the classical limit ($\hbar/m\to 0$) is expected to manifest itself as collisionless cold dark matter.

astro-ph.CO↗