arXiv subjects

Pierre-Henri Chavanis

Publications and source records attributed to Pierre-Henri Chavanis.

At least 109 records · Page 6Linked to original sources

arXiv2013

Kinetic theory of spatially homogeneous systems with long-range interactions: III. Application to power-law potentials, plasmas, stellar systems, and to the HMF model

We apply the general results of the kinetic theory of systems with long-range interactions to particular systems of physical interest. We consider repulsive and attractive power-law potentials of interaction r^{-γ} with γ γ_c= (d-1)/2, strong collisions must be taken into account and the evolution of the system is governed by the Boltzmann equation or by a modified Landau equation; for γ<γ_c, strong collisions are negligible and the evolution of the system is governed by the Lenard-Balescu equation. In the marginal case γ=γ_c, we can use the Landau equation (with appropriately justified cut-offs) as a relevant approximation of the Boltzmann and Lenard-Balescu equations. The divergence at small scales that appears in the original Landau equation is regularized by the effect of strong collisions. In the case of repulsive interactions with a neutralizing background (e.g. plasmas), the divergence at large scales that appears in the original Landau equation is regularized by collective effects accounting for Debye shielding. In the case of attractive interactions (e.g. gravity), it is regularized by the spatial inhomogeneity of the system and its finite extent. We provide explicit analytical expressions of the diffusion and friction coefficients, and of the relaxation time, depending on the value of the exponent γand on the dimension of space d. We treat in a unified framework the case of Coulombian plasmas and stellar systems in various dimensions of space, and the case of the attractive and repulsive HMF models.

cond-mat.stat-mech↗

arXiv2012

Kinetic theory of spatially inhomogeneous stellar systems without collective effects

We review and complete the kinetic theory of spatially inhomogeneous stellar systems when collective effects (dressing of the stars by their polarization cloud) are neglected. We start from the BBGKY hierarchy issued from the Liouville equation and consider an expansion in powers of 1/N in a proper thermodynamic limit. For $N\rightarrow +\infty$, we obtain the Vlasov equation describing the evolution of collisionless stellar systems like elliptical galaxies. At the order 1/N, we obtain a kinetic equation describing the evolution of collisional stellar systems like globular clusters. This equation does not suffer logarithmic divergences at large scales since spatial inhomogeneity is explicitly taken into account. Making a local approximation, and introducing an upper cut-off at the Jeans length, it reduces to the Vlasov-Landau equation which is the standard kinetic equation of stellar systems. Our approach provides a simple and pedagogical derivation of these important equations from the BBGKY hierarchy which is more rigorous for systems with long-range interactions than the two-body encounters theory. Making an adiabatic approximation, we write the generalized Landau equation in angle-action variables and obtain a Landau-type kinetic equation that is valid for fully inhomogeneous stellar systems and is free of divergences at large scales. This equation is less general than the Lenard Balescu-type kinetic equation recently derived by Heyvaerts (2010) since it neglects collective effects, but it is substantially simpler and could be useful as a first step. We discuss the evolution of the system as a whole and the relaxation of a test star in a bath of field stars. We derive the corresponding Fokker-Planck equation in angle-action variables and provide expressions for the diffusion coefficient and friction force.

astro-ph.GA↗

arXiv2012

Linear response theory for hydrodynamic and kinetic equations with long-range interactions

We apply the linear response theory to systems with long-range interactions described by hydrodynamic equations such as the Euler, Smoluchowski, and damped Euler equations. We analytically determine the response of the system submitted to a pulse and to a step function. We compare these results with those obtained for collisionless systems described by the Vlasov equation. We show that, in the linear regime, the evolution of a collisionless system (Vlasov) with the waterbag distribution is the same as the evolution of a collision-dominated gas without dissipation (Euler). In this analogy, the maximum velocity of the waterbag distribution plays the role of the velocity of sound in the corresponding barotropic gas. When submitted to a step function, these systems exhibit permanent oscillations. Other distributions exhibit Landau damping and relax towards a steady state. We illustrate this behaviour with the Cauchy distribution which can be studied analytically. We apply our results to the HMF model and obtain a generalized Curie-Weiss law for the magnetic susceptibility. Finally, we compare the linear response theory to the initial value problem for the linearized Vlasov equation and report a case of algebraic damping of the initial perturbation.

cond-mat.stat-mech↗

arXiv2012

Models of universe with a polytropic equation of state: III. The phantom universe

We construct models of universe with a generalized equation of state $p=(αρ+kρ^{1+1/n})c^2$ having a linear component and a polytropic component. The linear equation of state $p=αρc^2$ with $-1\le α\le 1$ describes radiation ($α=1/3$), pressureless matter ($α=0$), stiff matter ($α=1$), and vacuum energy ($α=-1$). The polytropic equation of state $p=kρ^{1+1/n} c^2$ may be due to Bose-Einstein condensates with repulsive ($k>0$) or attractive ($k<0$) self-interaction, or have another origin. In this paper, we consider the case where the density increases as the universe expands. This corresponds to a "phantom universe" for which $w=p/ρc^2<-1$ (this requires $k<0$). We complete previous investigations on this problem and analyze in detail the different possibilities. We describe the singularities using the classification of [S. Nojiri, S.D. Odintsov, S. Tsujikawa, Phys. Rev. D {\bf 71}, 063004 (2005)]. We show that for $α>-1$ there is no Big Rip singularity although $w\le -1$. For $n=-1$, we provide an analytical model of phantom bouncing universe "disappearing" at $t=0$. We also determine the potential of the phantom scalar field and phantom tachyon field corresponding to the generalized equation of state $p=(αρ+kρ^{1+1/n})c^2$.

astro-ph.CO↗

arXiv2012

A simple model of universe with a polytropic equation of state

We construct a simple model of universe with a generalized equation of state $p=(α+kρ^{1/n})ρc^2$ having a linear component $p=αρc^2$ and a polytropic component $p=kρ^{1+1/n}c^2$. For $α=1/3$, $n=1$ and $k=-4/(3ρ_P)$, where $ρ_P=5.16 10^{99} g/m^3$ is the Planck density, this equation of state provides a model of the early universe without singularity describing the transition between the pre-radiation era and the radiation era. The universe starts from $t=-\infty$ but, when $t<0$, its size is less than the Planck length $l_P=1.62 10^{-35} m$. The universe undergoes an inflationary expansion that brings it to a size $a_1=2.61 10^{-6} m$ on a timescale of a few Planck times $t_P=5.39 10^{-44} s$. When $t\gg t_P$, the universe decelerates and enters in the radiation era. For $α=0$, $n=-1$ and $k=-ρ_Λ$, where $ρ_Λ=7.02 10^{-24} g}/m^3$ is the cosmological density, this equation of state describes the transition from a decelerating universe dominated by baryonic and dark matter to an accelerating universe dominated by dark energy (second inflation). The transition takes place at a size $a_2=8.95 10^{25} m$ corresponding to a time of the order of the cosmological time $t_Λ=1.46 10^{18} s$. This polytropic model reveals a nice "symmetry" between the early and late evolution of the universe, the cosmological constant $Λ$ in the late universe playing a role similar to the Planck constant $\hbar$ in the early universe. We interpret the cosmological constant as a fundamental constant of nature describing the "cosmophysics" just like the Planck constant describes the microphysics. The Planck density and the cosmological density represent fundamental upper and lower bounds differing by ${122}$ orders of magnitude. The cosmological constant "problem" may be a false problem.

astro-ph.CO↗

arXiv2012

Models of universe with a polytropic equation of state: I. The early universe

We construct models of universe with a generalized equation of state $p=(αρ+kρ^{1+1/n})c^2$ having a linear component and a polytropic component. In this paper, we consider positive indices $n>0$. In that case, the polytropic component dominates in the early universe where the density is high. For $α=1/3$, $n=1$ and $k=-4/(3ρ_P)$, we obtain a model of early universe describing the transition from a pre-radiation era to the radiation era. The universe exists at any time in the past and there is no singularity. However, for $t<0$, its size is less than the Planck length $l_P=1.62 10^{-35} m$. In this model, the universe undergoes an inflationary expansion with the Planck density $ρ_P=5.16 10^{99} g/m^3$ that brings it to a size $a_1=2.61 10^{-6} m$ at $t_1=1.25 10^{-42} s$ (about 20 Planck times $t_P$). For $α=1/3$, $n=1$ and $k=4/(3ρ_P)$, we obtain a model of early universe with a new form of primordial singularity: The universe starts at t=0 with an infinite density and a finite radius $a=a_1$. Actually, this universe becomes physical at a time $t_i=8.32 10^{-45} s$ from which the velocity of sound is less than the speed of light. When $a\gg a_1$, the universe evolves like in the standard model. We describe the transition from the pre-radiation era to the radiation era by analogy with a second order phase transition where the Planck constant $\hbar$ plays the role of finite size effects (the standard Big Bang theory is recovered for $\hbar=0$).

astro-ph.CO↗

arXiv2012

Models of universe with a polytropic equation of state: II. The late universe

We construct models of universe with a generalized equation of state $p=(αρ+kρ^{1+1/n})c^2$ having a linear component and a polytropic component. In this paper, we consider negative indices $n<0$. In that case, the polytropic component dominates in the late universe where the density is low. For $α=0$, $n=-1$ and $k=-ρ_Λ$, we obtain a model of late universe describing the transition from the matter era to the dark energy era. The universe exists eternally in the future and undergoes an inflationary expansion with the cosmological density $ρ_Λ=7.02 10^{-24} g/m^3$ on a timescale $t_Λ=1.46 10^{18} s$. For $α=0$, $n=-1$ and $k=ρ_Λ$, we obtain a model of cyclic universe appearing and disappearing periodically. If we were living in this universe, it would disappear in about 2.38 billion years. We make the connection between the early and the late universe and propose a simple equation describing the whole evolution of the universe. This leads to a model of universe that is eternal in past and future without singularity (aioniotic universe). This model exhibits a nice "symmetry" between an early and late phase of inflation, the cosmological constant in the late universe playing the same role as the Planck constant in the early universe. The Planck density and the cosmological density represent fundamental upper and lower bounds differing by 122 orders of magnitude. The cosmological constant "problem" may be a false problem. We determine the potential of the scalar field (quintessence, tachyon field) corresponding to the generalized equation of state $p=(αρ+kρ^{1+1/n})c^2$. We also propose a unification of pre-radiation, radiation and dark energy through the quadratic equation of state $p/c^2=-4ρ^2/3ρ_P+ρ/3-4ρ_Λ/3$.

astro-ph.CO↗

arXiv2012

Virial theorem for Onsager vortices in two-dimensional hydrodynamics

We derive the virial theorem appropriate to two-dimensional point vortices at statistical equilibrium in the microcanonical and canonical ensembles. In an unbounded domain, it relates the angular velocity to the angular momentum and the temperature. Our expression is valid for an arbitrary number of point vortices of possibly different species. In the single-species case, and in the mean field approximation, it reduces to the relation empirically obtained by J.H. Williamson [J. Plasma Physics 17, 85 (1977)].

cond-mat.stat-mech↗

arXiv2012

Statistical mechanics of quasi-geostrophic flows on a rotating sphere

Statistical mechanics provides an elegant explanation to the appearance of coherent structures in two-dimensional inviscid turbulence: while the fine-grained vorticity field, described by the Euler equation, becomes more and more filamented through time, its dynamical evolution is constrained by some global conservation laws (energy, Casimir invariants). As a consequence, the coarse-grained vorticity field can be predicted through standard statistical mechanics arguments (relying on the Hamiltonian structure of the two-dimensional Euler flow), for any given set of the integral constraints. It has been suggested that the theory applies equally well to geophysical turbulence; specifically in the case of the quasi-geostrophic equations, with potential vorticity playing the role of the advected quantity. In this study, we demonstrate analytically that the Miller-Robert-Sommeria theory leads to non-trivial statistical equilibria for quasi-geostrophic flows on a rotating sphere, with or without bottom topography. We first consider flows without bottom topography and with an infinite Rossby deformation radius, with and without conservation of angular momentum. When the conservation of angular momentum is taken into account, we report a case of second order phase transition associated with spontaneous symmetry breaking. In a second step, we treat the general case of a flow with an arbitrary bottom topography and a finite Rossby deformation radius. Previous studies were restricted to flows in a planar domain with fixed or periodic boundary conditions with a beta-effect. In these different cases, we are able to classify the statistical equilibria for the large-scale flow through their sole macroscopic features. We build the phase diagrams of the system and discuss the relations of the various statistical ensembles.

cond-mat.stat-mech↗

arXiv2012

Phase transitions and marginal ensemble equivalence for freely evolving flows on a rotating sphere

The large-scale circulation of planetary atmospheres like that of the Earth is traditionally thought of in a dynamical framework. Here, we apply the statistical mechanics theory of turbulent flows to a simplified model of the global atmosphere, the quasi-geostrophic model, leading to non-trivial equilibria. Depending on a few global parameters, the structure of the flow may be either a solid-body rotation (zonal flow) or a dipole. A second order phase transition occurs between these two phases, with associated spontaneous symmetry-breaking in the dipole phase. This model allows us to go beyond the general theory of marginal ensemble equivalence through the notion of Goldstone modes.

physics.flu-dyn↗

arXiv2011

Effective merging dynamics of two and three fluid vortices: Application to two-dimensional decaying turbulence

We present a kinetic theory of two-dimensional decaying turbulence in the context of two-body and three-body vortex merging processes. By introducing the equations of motion for two or three vortices in the effective noise due to all the other vortices, we demonstrate analytically that a two-body mechanism becomes inefficient at low vortex density $n\ll 1$. When the more efficient three-body vortex mergings are considered {(involving vortices of different signs)}, we show that $n\sim t^{-ξ}$, with $ξ=1$. We generalize this argument to three-dimensional geostrophic turbulence, finding $ξ=5/4$, in excellent agreement with direct Navier-Stokes simulations [J.\,C. McWilliams \emph{et al.}, J. Fluid Mech. {\bf 401}, 1 (1999)].

cond-mat.stat-mech↗

arXiv2011

Kinetic theory of long-range interacting systems with angle-action variables and collective effects

We develop a kinetic theory of systems with long-range interactions taking collective effects and spatial inhomogeneity into account. Starting from the Klimontovich equation and using a quasilinear approximation, we derive a Lenard-Balescu-type kinetic equation written in angle-action variables. We confirm the result obtained by Heyvaerts [Mon. Not. R. Astron. Soc. {\bf 407}, 355 (2010)] who started from the Liouville equation and used the BBGKY hierarchy truncated at the level of the two-body distribution function. When collective effects are neglected, we recover the Landau-type kinetic equation obtained in our previous papers [P.H. Chavanis, Physica A {\bf 377}, 469 (2007); J. Stat. Mech., P05019 (2010)]. We also consider the relaxation of a test particle in a bath of field particles. Its stochastic motion is described by a Fokker-Planck equation written in angle-action variables. We determine the diffusion tensor and the friction force by explicitly calculating the first and second order moments of the increment of action of the test particle from its equations of motion, taking collective effects into account. This generalizes the expressions obtained in our previous works. We discuss the scaling with $N$ of the relaxation time for the system as a whole and for a test particle in a bath.

cond-mat.stat-mech↗

arXiv2011

Kinetic theory of spatially homogeneous systems with long-range interactions: I. General results

We review and complete the existing literature on the kinetic theory of spatially homogeneous systems with long-range interactions taking collective effects into account. The evolution of the system as a whole is described by the Lenard-Balescu equation which is valid in a weak coupling approximation. When collective effects are neglected it reduces to the Landau equation and when collisions (correlations) are neglected it reduces to the Vlasov equation. The relaxation of a test particle in a bath is described by a Fokker-Planck equation involving a diffusion term and a friction term. For a thermal bath, the diffusion and friction coefficients are connected by an Einstein relation. General expressions of the diffusion and friction coefficients are given, depending on the potential of interaction and on the dimension of space. We also discuss the scaling with $N$ (number of particles) or with $Λ$ (plasma parameter) of the relaxation time towards statistical equilibrium. Finally, we consider the effect of an external stochastic forcing on the evolution of the system and compute the corresponding term in the kinetic equation.

cond-mat.stat-mech↗

arXiv2011

Kinetic theory of Onsager's vortices in two-dimensional hydrodynamics

Starting from the Liouville equation, and using a BBGKY-like hierarchy, we derive a kinetic equation for the point vortex gas in two-dimensional (2D) hydrodynamics, taking two-body correlations and collective effects into account. This equation is valid at the order 1/N where N>>1 is the number of point vortices in the system (we assume that their individual circulation scales like γ~ 1/N). It gives the first correction, due to graininess and correlation effects, to the 2D Euler equation that is obtained for $N\rightarrow +\infty$. For axisymmetric distributions, this kinetic equation does not relax towards the Boltzmann distribution of statistical equilibrium. This implies either that (i) the "collisional" (correlational) relaxation time is larger than Nt_D, where t_D is the dynamical time, so that three-body, four-body... correlations must be taken into account in the kinetic theory, or (ii) that the point vortex gas is non-ergodic (or does not mix well) and will never attain statistical equilibrium. Non-axisymmetric distributions may relax towards the Boltzmann distribution on a timescale of the order Nt_D due to the existence of additional resonances, but this is hard to prove from the kinetic theory. On the other hand, 2D Euler unstable vortex distributions can experience a process of "collisionless" (correlationless) violent relaxation towards a non-Boltzmannian quasistationary state (QSS) on a very short timescale of the order of a few dynamical times. This QSS is possibly described by the Miller-Robert-Sommeria (MRS) statistical theory which is the counterpart, in the context of two-dimensional hydrodynamics, of the Lynden-Bell statistical theory of violent relaxation in stellar dynamics.

cond-mat.stat-mech↗

arXiv2011

Mass-radius relation of Newtonian self-gravitating Bose-Einstein condensates with short-range interactions: I. Analytical results

We provide an approximate analytical expression of the mass-radius relation of a Newtonian self-gravitating Bose-Einstein condensate (BEC) with short-range interactions described by the Gross-Pitaevskii-Poisson system. These equations model astrophysical objects such as boson stars and, presumably, dark matter galactic halos. Our study connects the non-interacting case studied by Ruffini & Bonazzola (1969) to the Thomas-Fermi limit studied by Böhmer & Harko (2007). For repulsive short-range interactions (positive scattering lengths), there exists configurations of arbitrary mass but their radius is always larger than a minimum value. For attractive short-range interactions (negative scattering lengths), equilibrium configurations only exist below a maximum mass. Above that mass, the system is expected to collapse and form a black hole. We also study the radius versus scattering length relation for a given mass. We find that stable configurations only exist above a (negative) minimum scattering length. Our approximate analytical solution, based on a Gaussian ansatz, provides a very good agreement with the exact solution obtained by numerically solving a nonlinear differential equation representing hydrostatic equilibrium. Our treatment is, however, easier to handle and permits to study the stability problem, and derive an analytical expression of the pulsation period, by developing an analogy with a simple mechanical problem.

astro-ph.CO↗

arXiv2011

Instability of a uniformly collapsing cloud of classical and quantum self-gravitating Brownian particles

We study the growth of perturbations in a uniformly collapsing cloud of self-gravitating Brownian particles. This problem shares analogies with the formation of large-scale structures in a universe experiencing a "big-crunch" or with the formation of stars in a molecular cloud experiencing gravitational collapse. Starting from the barotropic Smoluchowski-Poisson system, we derive a new equation describing the evolution of the density contrast in the comoving (collapsing) frame. This equation can serve as a prototype to study the process of self-organization in complex media with structureless initial conditions. We solve this equation analytically in the linear regime and compare the results with those obtained by using the "Jeans swindle" in a static medium. The stability criteria, as well as the laws for the time evolution of the perturbations, are different. The Jeans criterion is expressed in terms of a critical wavelength $λ_J$ while our criterion is expressed in terms of a critical polytropic index $γ_{4/3}$. We also study the fragmentation process in the nonlinear regime. We determine the growth of the skewness, the long-wavelength tail of the power spectrum and find a self-similar solution to the nonlinear equations valid for large times. Finally, we consider dissipative self-gravitating Bose-Einstein condensates with short-range interactions and show that, in a strong friction limit, the dissipative Gross-Pitaevskii-Poisson system is equivalent to the quantum barotropic Smoluchowski-Poisson system. This yields a new type of nonlinear mean field Fokker-Planck equations including quantum effects.

cond-mat.stat-mech↗

arXiv2011

Exact analytical solution of the collapse of self-gravitating Brownian particles and bacterial populations at zero temperature

We provide an exact analytical solution of the collapse dynamics of self-gravitating Brownian particles and bacterial populations at zero temperature. These systems are described by the Smoluchowski-Poisson system or Keller-Segel model in which the diffusion term is neglected. As a result, the dynamics is purely deterministic. A cold system undergoes a gravitational collapse leading to a finite time singularity: the central density increases and becomes infinite in a finite time t_coll. The evolution continues in the post collapse regime. A Dirac peak emerges, grows and finally captures all the mass in a finite time t_end, while the central density excluding the Dirac peak progressively decreases. Close to the collapse time, the pre and post collapse evolution is self-similar. Interestingly, if one starts from a parabolic density profile, one obtains an exact analytical solution that describes the whole collapse dynamics, from the initial time to the end, and accounts for non self-similar corrections that were neglected in previous works. Our results have possible application in different areas including astrophysics, chemotaxis, colloids and nanoscience.

cond-mat.stat-mech↗

arXiv2011

Kinetic theory of two-dimensional point vortices with collective effects

We develop a kinetic theory of point vortices in two-dimensional hydrodynamics taking collective effects into account. We first recall the approach of Dubin & O'Neil [Phys. Rev. Lett. 60, 1286 (1988)] that leads to a Lenard-Balescu-type kinetic equation for axisymmetric flows. When collective effects are neglected, it reduces to the Landau-type kinetic equation obtained independently in our previous papers [P.H. Chavanis, Phys. Rev. E 64, 026309 (2001); Physica A 387, 1123 (2008)]. We also consider the relaxation of a test vortex in a "sea" (bath) of field vortices. Its stochastic motion is described in terms of a Fokker-Planck equation. We determine the diffusion coefficient and the drift term by explicitly calculating the first and second order moments of the radial displacement of the test vortex from its equations of motion, taking collective effects into account. This generalizes the expressions obtained in our previous works. We discuss the scaling with N of the relaxation time for the system as a whole and for a test vortex in a bath.

cond-mat.stat-mech↗