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We discuss the kinetic theory of stellar systems and two-dimensional vortices and stress their analogies. We recall the derivation of the Landau and Lenard–Balescu equations from the Klimontovich formalism. These equations take into account two-body correlations and are valid at the order 1/N, where N is the number of particles in the system. They have the structure of a Fokker–Planck equation involving a diffusion term and a drift term. The systematic drift of a vortex is the counterpart of the dynamical friction experienced by a star. At equilibrium, the diffusion and the drift terms balance each other establishing the Boltzmann distribution of statistical mechanics. We discuss the problem of kinetic blocking in certain cases and how it can be solved at the order by the consideration of three-body correlations. We also consider the behaviour of the system close to the critical point following a recent suggestion by Hamilton and Heinemann (2023). We present a simple calculation, valid for spatially homogeneous systems with long-range interactions described by the Cauchy distribution, showing how the consideration of the Landau modes regularizes the divergence of the friction by polarization at the critical point. We mention, however, that fluctuations may be very important close to the critical point and that deterministic kinetic equations for the mean distribution function (such as the Landau and Lenard–Balescu equations) should be replaced by stochastic kinetic equations.

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We consider an isothermal self-gravitating system surrounding a central body. This model can represent a galaxy or a globular cluster harboring a central black hole. It can also represent a gaseous atmosphere surrounding a protoplanet. In three dimensions, the Boltzmann-Poisson equation must be solved numerically to obtain the density profile of the gas [Chavanis et al., Phys. Rev. E 109, 014118 (2024)]. In one and two dimensions, we show that the Boltzmann-Poisson equation can be solved analytically. We obtain explicit analytical expressions of the density profile around a central body which generalize the analytical solutions found by Camm (1950) and Ostriker (1964) in the absence of a central body. Our results also have applications for self-gravitating Brownian particles (Smoluchowski-Poisson system), for the chemotaxis of bacterial populations in biology (Keller-Segel model), and for two-dimensional point vortices in hydrodynamics (Onsager's model). In the case of bacterial populations, the central body could represent a supply of “food” that attracts the bacteria (chemoattractant). In the case of two-dimensional vortices, the central body could be a central vortex.

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Current fluctuations for the one-dimensional totally asymmetric exclusion process (TASEP) connected to reservoirs of particles, and their large scale limit to the KPZ fixed point in finite volume, are studied using exact methods. Focusing on the maximal current phase for TASEP, corresponding to infinite boundary slopes for the KPZ height field, we obtain for general initial condition an exact expression for the late time correction to stationarity, involving extreme value statistics of Brownian paths. In the special cases of stationary and narrow wedge initial conditions, a combination of Bethe ansatz and numerical conjectures alternatively provide fully explicit exact expressions.

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The Riemann surface associated with counting the current between two states of an underlying Markov process is hyperelliptic. We explore the consequences of this property for the time-dependent probability of that current for Markov processes with generic transition rates. When the system is prepared in its stationary state, the relevant meromorphic differential is in particular fully characterized by the precise identification of all its poles and zeroes.

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After reviewing the peculiar thermodynamics and statistical mechanics of self-gravitating systems, we consider the case of a “binary star” consisting of two particles of size a in gravitational interaction in a box of radius R. The caloric curve of this system displays a region of negative specific heat in the microcanonical ensemble, which is replaced by a first-order phase transition in the canonical ensemble. The free energy viewed as a thermodynamic potential exhibits two local minima that correspond to two metastable states separated by an unstable maximum forming a barrier of potential. By introducing a Langevin equation to model the interaction of the particles with the thermal bath, we study the random transitions of the system between a “dilute” state, where the particles are well separated, and a “condensed” state, where the particles are bound together. We show that the evolution of the system is given by a Fokker–Planck equation in energy space and that the lifetime of a metastable state is given by the Kramers formula involving the barrier of free energy. This is a particular case of the theory developed in a previous paper (Chavanis, 2005) for N Brownian particles in gravitational interaction associated with the canonical ensemble. In the case of a binary star (N=2), all the quantities can be calculated exactly analytically. We compare these results with those obtained in the mean field limit N→+∞.

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Sujets

Bose-Einstein Collective behavior Energy density Cosmological model Gravitation collapse Random walker Collective intelligence Gravitational collapse Denaturation Marcheur aléatoire Computational modeling Turbulence Galaxy Stability Bose–Einstein condensates Quantum mechanics Axion star Mass density Collective motion Chemotaxie Condensation Bose-Einstein Phase separation Nonlinear Physique statistique Gravitation self-force Current fluctuations Axion Field theory scalar Diffusion DNA Mouvement brownien Nonrelativistic 9535+d Pressure Wave function Distributed Control Asymptotic behavior 9880-k Halo Nanofiltration Computational modelling Collective behaviour Bethe ansatz Transition vitreuse Dark matter condensation Smoluchowski-Poisson Energy internal Electromagnetic Evaporation Formation Dark matter theory Fokker-Planck 9862Gq Cosmology Effect relativistic Dark matter halo Brownian motion Einstein Gravitation TASEP Thermodynamics Dark matter fuzzy Catastrophe theory Entropy Black hole Kinetic theory Critical phenomena Mass Rotation Chemotaxis Dark matter Scalar field Numerical calculations Equation of state Smoluchowski equation Collapse Keller-Segel Hydrodynamics 9530Sf Euler-Maclaurin Quantum chromodynamics axion Dark energy Statistical mechanics 9536+x Expansion acceleration Fermions Density Competition Scattering length General relativity Structure Cosmological constant Gas Chaplygin Dark matter density Fermi gas Atmosphere Collisionless stellar-systems Effondrement gravitationnel Energy high Dissipation

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