arXiv subjects

Pierre-Henri Chavanis

Publications and source records attributed to Pierre-Henri Chavanis.

At least 73 records · Page 4Linked to original sources

arXiv2017

Functional integral approach to the kinetic theory of inhomogeneous systems

We present a derivation of the kinetic equation describing the secular evolution of spatially inhomogeneous systems with long-range interactions, the so-called inhomogeneous Landau equation, by relying on a functional integral formalism. We start from the BBGKY hierarchy derived from the Liouville equation. At the order ${1/N}$, where $N$ is the number of particles, the evolution of the system is characterised by its 1-body distribution function and its 2-body correlation function. Introducing associated auxiliary fields, the evolution of these quantities may be rewritten as a traditional functional integral. By functionally integrating over the 2-body autocorrelation, one obtains a new constraint connecting the 1-body DF and the auxiliary fields. When inverted, this constraint allows us to obtain the closed non-linear kinetic equation satisfied by the 1-body distribution function. This derivation provides an alternative to previous methods, either based on the direct resolution of the truncated BBGKY hierarchy or on the Klimontovich equation. It may turn out to be fruitful to derive more accurate kinetic equations, e.g., accounting for collective effects, or higher order correlation terms.

astro-ph.GA↗

arXiv2017

Functional integral derivation of the kinetic equation of two-dimensional point vortices

We present a brief derivation of the kinetic equation describing the secular evolution of point vortices in two-dimensional hydrodynamics, by relying on a functional integral formalism. We start from Liouville's equation which describes the exact dynamics of a two-dimensional system of point vortices. At the order ${1/N}$, the evolution of the system is characterised by the first two equations of the BBGKY hierarchy involving the system's 1-body distribution function and its 1-body correlation function. Thanks to the introduction of auxiliary fields, these two evolution constraints may be rewritten as a functional integral. When functionally integrated over the 2-body correlation function, this rewriting leads to a new constraint coupling the 1-body distribution function and the two auxiliary fields. Once inverted, this constraint provides, through a new route, the closed non-linear kinetic equation satisfied by the 1-body distribution function. Such a method sheds new lights on the origin of these kinetic equations complementing the traditional derivation methods.

cond-mat.stat-mech↗

arXiv2017

Jeans type instability of a complex self-interacting scalar field in general relativity

We study the gravitational instability of a general relativistic complex scalar field with a quartic self-interaction in an infinite homogeneous static background. This quantum relativistic Jeans problem provides a simplified framework to study the formation of the large-scale structures of the Universe in the case where dark matter is made of a scalar field. The scalar field may represent the wave function of a relativistic self-gravitating Bose-Einstein condensate. Exact analytical expressions for the dispersion relation and Jeans length $λ_J$ are obtained from a hydrodynamical representation of the Klein-Gordon-Einstein equations. When relativistic effects are fully accounted for, we find that the perturbations are stabilized at very large scales of the order of the Hubble length $λ_H$. Numerical applications are made for ultralight bosons without self-interaction (fuzzy dark matter), for bosons with a repulsive self-interaction, and for bosons with an attractive self-interaction (QCD axions and ultralight axions). We show that the Jeans instability is inhibited in the ultrarelativistic regime (early Universe and radiationlike era) because the Jeans length is of the order of the Hubble length ($λ_J\simλ_H$), except when the self-interaction of the bosons is attractive. By contrast, structure formation can take place in the nonrelativistic regime (matterlike era) for $λ_J\le λ\le λ_H$. Since the scalar field has a nonzero Jeans length due to its quantum nature (Heisenberg uncertainty principle or quantum pressure due to the self-interaction of the bosons), it appears that the wave properties of the scalar field can stabilize gravitational collapse at small scales, providing halo cores and suppressing small-scale linear power. This may solve the CDM crisis such as the cusp problem and the missing satellite problem.

gr-qc↗

arXiv2017

Phase transitions between dilute and dense axion stars

We study the nature of phase transitions between dilute and dense axion stars interpreted as self-gravitating Bose-Einstein condensates. We develop a Newtonian model based on the Gross-Pitaevskii-Poisson equations for a complex scalar field with a self-interaction potential $V(|ψ|^2)$ involving an attractive $|ψ|^4$ term and a repulsive $|ψ|^6$ term. Using a Gaussian ansatz for the wave function, we analytically obtain the mass-radius relation of dilute and dense axion stars for arbitrary values of the self-interaction parameter $λ\le 0$. We show the existence of a critical point $|λ|_c\sim (m/M_P)^2$ above which a first order phase transition takes place. We qualitatively estimate general relativistic corrections on the mass-radius relation of axion stars. For weak self-interactions $|λ|<|λ|_c$, a system of self-gravitating axions forms a stable dilute axion star below a general relativistic maximum mass $M_{\rm max,GR}^{\rm dilute}\sim M_P^2/m$ and collapses into a black hole above that mass. For strong self-interactions $|λ|>|λ|_c$, a system of self-gravitating axions forms a stable dilute axion star below a Newtonian maximum mass $M_{\rm max,N}^{\rm dilute}=5.073 M_P/\sqrt{|λ|}$, collapses into a dense axion star above that mass, and collapses into a black hole above a general relativistic maximum mass $M_{\rm max,GR}^{\rm dense}\sim \sqrt{|λ|}M_P^3/m^2$. Dense axion stars explode below a Newtonian minimum mass $M_{\rm min,N}^{\rm dense}\sim m/\sqrt{|λ|}$ and form dilute axion stars of large size or disperse away. We determine the phase diagram of self-gravitating axions and show the existence of a triple point $(|λ|_*,M_*/(M_P^2/m))$ separating dilute axion stars, dense axion stars, and black holes. We make numerical applications for QCD axions and ultralight axions.

gr-qc↗

arXiv2017

Secular diffusion in discrete self-gravitating tepid discs III. Resonant thickening in the tightly wound limit

The secular thickening of a discrete self-gravitating galactic disc is investigated using the inhomogeneous multi-component Balescu-Lenard equation. The thick WKB limit for the diffusion and drift coefficients is found using the epicyclic approximation, while assuming that only radially tightly wound transient spirals are sustained by the disc. This yields a simple double quadrature for the drift and diffusion coefficients, providing a clear understanding of the positions of maximum vertical orbital diffusion within the disc induced by the effects of a finite number of particles. When applied to a tepid stable tapered disc, the Balescu-Lenard formalism predicts the formation of ridges of resonant orbits towards larger vertical actions, as found in direct numerical simulations, but over-estimates the timescale involved in their appearance. Swing amplication is likely needed to resolve this discrepancy, as demonstrated in the case of razor-thin discs. The joint evolution of a population of giant molecular clouds within the disc may accelerate the secular disc's thickening induced by finite${-N}$ effects, but the observed number of clouds in the Milky Way does not seem to be sufficient to explain its thick disc.

astro-ph.GA↗

arXiv2017

Resonant thickening of self-gravitating discs: imposed or self-induced orbital diffusion in the tightly wound limit

The secular thickening of a self-gravitating stellar galactic disc is investigated using the dressed collisionless Fokker-Planck equation and the inhomogeneous multicomponent Balescu-Lenard equation. The thick WKB limits for the diffusion fluxes are found using the epicyclic approximation, while assuming that only radially tightly wound transient spirals are sustained by the disc. This yields simple quadratures for the drift and diffusion coefficients, providing a clear understanding of the positions of maximum vertical orbital diffusion within the disc, induced by fluctuations either external or due to the finite number of particles. These thick limits also offer a consistent derivation of a thick disc Toomre parameter, which is shown to be exponentially boosted by the ratio of the vertical to radial scale heights. Dressed potential fluctuations within the disc statistically induce a vertical bending of a subset of resonant orbits, triggering the corresponding increase in vertical velocity dispersion. When applied to a tepid stable tapered disc perturbed by shot noise, these two frameworks reproduce qualitatively the formation of ridges of resonant orbits towards larger vertical actions, as found in direct numerical simulations, but overestimates the time-scale involved in their appearance. Swing amplification is likely needed to resolve this discrepancy, as demonstrated in the case of razor-thin discs. Other sources of thickening are also investigated, such as fading sequences of slowing bars, or the joint evolution of a population of giant molecular clouds within the disc.

astro-ph.GA↗

arXiv2017

Dressed diffusion and friction coefficients in inhomogeneous multicomponent self-gravitating systems

General self-consistent expressions for the coefficients of diffusion and dynamical friction in a stable, bound, multicomponent self-gravitating and inhomogeneous system are derived. They account for the detailed dynamics of the colliding particles and their self-consistent dressing by collective gravitational interactions. The associated Fokker-Planck equation is shown to be fully consistent with the corresponding inhomogeneous Balescu-Lenard equation and, in the weak self-gravitating limit, to the inhomogeneous Landau equation. Hence it provides an alternative derivation to both and demonstrates their equivalence. The corresponding stochastic Langevin equations are presented: they can be a practical alternative to numerically solving the inhomogeneous Fokker-Planck and Balescu-Lenard equations. The present formalism allows for a self-consistent description of the secular evolution of different populations covering a spectrum of masses, with a proper accounting of the induced secular mass segregation, which should be of interest to various astrophysical contexts, from galactic centers to protostellar discs.

astro-ph.GA↗

arXiv2017

Comparison between the Logotropic and $Λ$CDM models at the cosmological scale

We perform a detailed comparison between the Logotropic model [P.H. Chavanis, Eur. Phys. J. Plus 130 (2015) 130] and the $Λ$CDM model. These two models behave similarly at large (cosmological) scales up to the present. Differences will appear only in the far future, in about $25\, {\rm Gyrs}$, when the Logotropic Universe becomes phantom while the $Λ$CDM Universe enters in the de Sitter era. However, the Logotropic model differs from the $Λ$CDM model at small (galactic) scales, where the latter encounters serious problems. Having a nonvanishing pressure, the Logotropic model can solve the cusp problem and the missing satellite problem of the $Λ$CDM model. In addition, it leads to dark matter halos with a constant surface density $Σ_0=ρ_0 r_h$, and can explain its observed value $Σ_0=141 \, M_{\odot}/{\rm pc}^2$ without adjustable parameter. This makes the logotropic model rather unique among all the models attempting to unify dark matter and dark energy. In this paper, we compare the Logotropic and $Λ$CDM models at the cosmological scale where they are very close to each other in order to determine quantitatively how much they differ. This comparison is facilitated by the fact that these models depend on only two parameters, the Hubble constant $H_0$ and the present fraction of dark matter $Ω_{\rm m0}$. Using the latest observational data from Planck 2015+Lensing+BAO+JLA+HST, we find that the best fit values of $H_0$ and $Ω_{\rm m0}$ are $H_0=68.30\, {\rm km}\, {\rm s}^{-1}\,{\rm Mpc}^{-1}$ and $Ω_{\rm m0}=0.3014$ for the Logotropic model, and $H_0=68.02\, {\rm km}\, {\rm s}^{-1}\,{\rm Mpc}^{-1}$ and $Ω_{\rm m0}=0.3049$ for the $Λ$CDM model. The difference between the two models appears at the percent level.

astro-ph.CO↗

arXiv2017

The secular evolution of discrete quasi-Keplerian systems. II. Application to a multi-mass axisymmetric disc around a supermassive black hole

The drift and diffusion coefficients of the inhomogeneous multi-mass degenerate Landau equation are computed to describe the self-induced resonant relaxation of a discrete self-gravitating quasi-Keplerian razor-thin axisymmetric disc orbiting a massive black hole while relying on Gauss' method. For a disc-like configuration in our Galactic centre, secular diffusion induces an adiabatic distortion of orbits. When considering a disc composed of multiple masses similarly distributed, the population of lighter stars will gain eccentricity, driving it closer to the central black hole provided the distribution function increases with angular momentum. The quenching of the diffusion of a test star in the vicinity of the black hole due to the divergence of the relativistic precessions (the "Schwarzschild barrier") is correctly recovered by the kinetic equation. The dual stochastic Langevin formulation yields consistent results and provides a versatile framework in which to incorporate other stochastic processes.

astro-ph.GA↗

arXiv2017

Cosmological evolution of a complex scalar field with repulsive or attractive self-interaction

We study the cosmological evolution of a complex scalar field with a self-interaction potential $V(|φ|^2)$, possibly describing self-gravitating Bose-Einstein condensates, using a fully general relativistic treatment. We generalize the hydrodynamic representation of the Klein-Gordon-Einstein equations in the weak field approximation developed in our previous paper. We establish the general equations governing the evolution of a spatially homogeneous complex scalar field in an expanding background. We show how they can be simplified in the fast oscillation regime and derive the equation of state of the scalar field in parametric form for an arbitrary potential. We explicitly consider the case of a quartic potential with repulsive or attractive self-interaction and determine the phase diagram of the scalar field. We show that the transition between the weakly self-interacting regime and the strongly self-interacting regime depends on how the scattering length of the bosons compares with their effective Schwarzschild radius. We also constrain the parameters of the scalar field from astrophysical and cosmological observations. Numerical applications are made for ultralight bosons without self-interaction (fuzzy dark matter), for bosons with repulsive self-interaction, and for bosons with attractive self-interaction (QCD axions and ultralight axions).

gr-qc↗

arXiv2016

Dissipative self-gravitating Bose-Einstein condensates with arbitrary nonlinearity as a model of dark matter halos

We develop a general formalism applying to Newtonian self-gravitating Bose-Einstein condensates. This formalism may find application in the context of dark matter halos. We introduce a generalized Gross-Pitaevskii equation including a source of dissipation (damping) and an arbitrary nonlinearity. Using the Madelung transformation, we derive the hydrodynamic representation of this generalized Gross-Pitaevskii equation and obtain a damped quantum Euler equation involving a friction force proportional and opposite to the velocity and a pressure force associated with an equation of state determined by the nonlinearity present in the generalized Gross-Pitaevskii equation. In the strong friction limit, we obtain a quantum Smoluchowski equation. These equations satisfy an $H$-theorem for a free energy functional constructed with a generalized entropy. We specifically consider the Boltzmann and Tsallis entropies associated with isothermal and polytropic equations of state. We also consider the entropy associated with the logotropic equation of state. We derive the virial theorem corresponding to the generalized Gross-Pitaevskii equation, damped quantum Euler equation, and quantum Smoluchowski equation. Using a Gaussian ansatz, we obtain a simple equation governing the dynamical evolution of the size of the condensate. We highlight a specific model of dark matter halos corresponding to a generalized Gross-Pitaevskii equation with a logarithmic nonlinearity and a cubic nonlinearity. It leads to dark matter halos with an equation of state $P=ρk_B T_{\rm eff}/m+2πa_s\hbar^2ρ^{2}/m^3$ presenting a condensed core (BEC/soliton) and an isothermal halo with an effective temperature $T_{\rm eff}$. We propose that this model provides an effective coarse-grained parametrization of dark matter halos experiencing gravitational cooling.

gr-qc↗

arXiv2016

Derivation of a generalized Schrödinger equation from the theory of scale relativity

Using Nottale's theory of scale relativity relying on a fractal space-time, we derive a generalized Schrödinger equation taking into account the interaction of the system with the external environment. This equation describes the irreversible evolution of the system towards a static quantum state. We first interpret the scale-covariant equation of dynamics stemming from Nottale's theory as a hydrodynamic viscous Burgers equation for a potential flow involving a complex velocity field and an imaginary viscosity. We show that the Schrödinger equation can be directly obtained from this equation by performing a Cole-Hopf transformation equivalent to the WKB transformation. We then introduce a friction force proportional and opposite to the complex velocity in the scale-covariant equation of dynamics in a way that preserves the local conservation of the normalization condition. We find that the resulting generalized Schrödinger equation, or the corresponding fluid equations obtained from the Madelung transformation, involve not only a damping term but also an effective thermal term. The friction coefficient and the temperature are related to the real and imaginary parts of the complex friction coefficient in the scale-covariant equation of dynamics. This may be viewed as a form of fluctuation-dissipation theorem. We show that our generalized Schrödinger equation satisfies an $H$-theorem for the quantum Boltzmann free energy. As a result, the probability distribution relaxes towards an equilibrium state which can be viewed as a Boltzmann distribution including a quantum potential. We propose to apply this generalized Schrödinger equation to dark matter halos in the Universe, possibly made of self-gravitating Bose-Einstein condensates.

physics.gen-ph↗

arXiv2016

The quasilinear theory in the approach of long-range systems to quasi-stationary states

We develop a quasilinear theory of the Vlasov equation in order to describe the approach of systems with long-range interactions to quasi-stationary states. We derive a diffusion equation governing the evolution of the velocity distribution of the system towards a steady state. This steady state is expected to correspond to the angle-averaged quasi-stationary distribution function reached by the Vlasov equation as a result of a violent relaxation. We compare the prediction of the quasilinear theory to direct numerical simulations of the Hamiltonian Mean Field model, starting from an unstable spatially homogeneous distribution, either Gaussian or semi-elliptical. We find that the quasilinear theory works reasonably well for weakly unstable initial conditions and that it is able to predict the energy marking the out-of-equilibrium phase transition between unmagnetized and magnetized quasi-stationary states. At energies lower than the out-of-equilibrium transition the quasilinear theory works less well, the disagreement with the numerical simulations increasing by decreasing the energy. In that case, we observe, in agreement with our previous numerical study [A. Campa and P.-H. Chavanis, Eur. Phys. J. B 86, 170 (2013)], that the quasi-stationary states are remarkably well fitted by polytropic distributions (Tsallis distributions) with index $n=2$ (Gaussian case) or $n=1$ (semi-elliptical case). In particular, these polytropic distributions are able to account for the region of negative specific heats in the out-of-equilibrium caloric curve, unlike the Boltzmann and Lynden-Bell distributions.

cond-mat.stat-mech↗

arXiv2016

Covariant theory of Bose-Einstein condensates in curved spacetimes with electromagnetic interactions: the hydrodynamic approach

We develop a hydrodynamic representation of the Klein-Gordon-Maxwell-Einstein equations. These equations combine quantum mechanics, electromagnetism, and general relativity. We consider the case of an arbitrary curved spacetime, the case of weak gravitational fields in a static or expanding background, and the nonrelativistic (Newtonian) limit. The Klein-Gordon-Maxwell-Einstein equations govern the evolution of a complex scalar field, possibly describing self-gravitating Bose-Einstein condensates, coupled to an electromagnetic field. They may find applications in the context of dark matter, boson stars, and neutron stars with a superfluid core.

gr-qc↗

arXiv2016

Collapse of a self-gravitating Bose-Einstein condensate with attractive self-interaction

We study the collapse of a self-gravitating Bose-Einstein condensate with attractive self-interaction. Equilibrium states in which the gravitational attraction and the attraction due to the self-interaction are counterbalanced by the quantum pressure exist only below a maximum mass $M_{\rm max}=1.012\hbar/\sqrt{Gm|a_s|}$ where $a_s<0$ is the scattering length of the bosons and $m$ is their mass. For $M>M_{\rm max}$ the system is expected to collapse and form a black hole. We study the collapse dynamics by making a Gaussian ansatz for the wave function. We find that the collapse time scales as $(M/M_{\rm max}-1)^{-1/4}$ for $M\rightarrow M_{\rm max}^+$ and as $M^{-1/2}$ for $M\gg M_{\rm max}$. We apply our results to standard axions with mass $m=10^{-4}\, {\rm eV}/c^2$ and scattering length $a_s=-5.8\times 10^{-53}\, {\rm m}$ for which $M_{\rm max}=6.5\times 10^{-14}M_{\odot}$ and $R=3.3\times 10^{-4}\, R_{\odot}$. We confirm our previous claim that bosons with attractive self-interaction, such as standard axions, may form low mass stars but cannot form dark matter halos of relevant mass and size. These mini axions stars could be the constituents of dark matter. They can collapse into mini black holes of mass $\sim 10^{-14}\, M_{\odot}$ in a few hours. In that case, dark matter halos would be made of mini black holes. We also apply our results to ultralight axions with mass $m=1.93\times 10^{-20}\, {\rm eV}/c^2$ and scattering length $a_s=-8.29\times 10^{-60}\, {\rm fm}$ for which $M_{\rm max}=0.39\times 10^6\, M_{\odot}$ and $R=33\, {\rm pc}$. These ultralight axions could cluster into dark matter halos. Axionic dark matter halos with attractive self-interaction can collapse into supermassive black holes of mass $\sim 10^{6}\, M_{\odot}$ in about one million years.

astro-ph.CO↗

arXiv2016

Euler-like modelling of dense granular flows: application to a rotating drum

General conservation equations are derived for 2D dense granular flows from the Euler equation within the Boussinesq approximation. In steady flows, the 2D fields of granular temperature, vorticity and stream function are shown to be encoded in two scalar functions only. We checked such prediction on steady surface flows in a rotating drum simulated through the Non-Smooth Contact Dynamics method. This result is non trivial because granular flows are dissipative and therefore not necessarily compatible with Euler equation. Finally, we briefly discuss some possible ways to predict theoretically these two functions using statistical mechanics.

cond-mat.stat-mech↗

arXiv2015

Hydrodynamic representation of the Klein-Gordon-Einstein equations in the weak field limit: I. General formalism and perturbations analysis

Using a generalization of the Madelung transformation, we derive the hydrodynamic representation of the Klein-Gordon-Einstein equations in the weak field limit. We consider a complex self-interacting scalar field with a $λ|φ|^4$ potential. We study the evolution of the homogeneous background in the fluid representation and derive the linearized equations describing the evolution of small perturbations in a static and in an expanding universe. We compare the results with simplified models in which the gravitational potential is introduced by hand in the Klein-Gordon equation, and assumed to satisfy a (generalized) Poisson equation. We study the evolution of the perturbations in the matter era using the nonrelativistic limit of our formalism. Perturbations whose wavelength is below the Jeans length oscillate in time while pertubations whose wavelength is above the Jeans length grow linearly with the scale factor as in the cold dark matter model. The growth of perturbations in the scalar field model is substantially faster than in the cold dark matter model. When the wavelength of the pertubations approaches the cosmological horizon (Hubble length), a relativistic treatment is mandatory. In that case, we find that relativistic effects attenuate or even prevent the growth of pertubations. This paper exposes the general formalism and provides illustrations in simple cases. Other applications of our formalism will be considered in companion papers.

gr-qc↗

arXiv2015

A cosmological model describing the early inflation, the intermediate decelerating expansion, and the late accelerating expansion by a quadratic equation of state

We develop a cosmological model based on a quadratic equation of state p/c^2=-(α+1){ρ^2}/{ρ_P}+αρ-(α+1)ρ_Λ (where ρ_P is the Planck density and ρ_Λ the cosmological density) "unifying" vacuum energy and dark energy in the spirit of a generalized Chaplygin gas model. For $ρ\rightarrow ρ_P$, it reduces to p=-ρc^2 leading to a phase of early accelerated expansion (early inflation) with a constant density equal to the Planck density ρ_P (vacuum energy). For $ρ_Λ\llρ\ll ρ_P$, we recover the standard linear equation of state p=αρc^2 describing radiation (α=1/3) or pressureless matter (α=0) and leading to an intermediate phase of decelerating expansion. For $ρ\rightarrow ρ_Λ$, we get p=-ρc^2 leading to a phase of late accelerated expansion (late inflation) with a constant density equal to the cosmological density ρ_Λ (dark energy). We show a nice symmetry between the early universe (vacuum energy + α-fluid) and the late universe (α-fluid + dark energy). In our model, they are described by two polytropic equations of state with index n=+1 and n=-1 respectively. Furthermore, the Planck density ρ_P in the early universe plays a role similar to the cosmological density ρ_Λ in the late universe. They represent fundamental upper and lower density bounds differing by 122 orders of magnitude. This quadratic equation of state leads to a fully analytical model describing the evolution of the universe from the early inflation (Planck era) to the late accelerated expansion (de Sitter era). These two phases are bridged by a decelerating algebraic expansion (α-era). This model does not present any singularity at t=0 and exists eternally in the past. It admits a scalar field interpretation based on a quintessence field or a tachyon field.

astro-ph.CO↗