Hamiltonian and Brownian systems with long-range interactions: III. The BBGKY hierarchy for spatially inhomogeneous systems

Abstract

We study the growth of correlations in systems with weak long-range interactions. Starting from the BBGKY hierarchy, we determine the evolution of the two-body correlation function by using an expansion of the solutions of the hierarchy in powers of 1/N in a proper thermodynamic limit $N\to +\infty$. These correlations are responsible for the ``collisional'' evolution of the system beyond the Vlasov regime due to finite $N$ effects. We obtain a general kinetic equation that can be applied to spatially inhomogeneous systems and that takes into account memory effects. These peculiarities are specific to systems with unshielded long-range interactions. For spatially homogeneous systems with short memory time like plasmas, we recover the classical Landau (or Lenard-Balescu) equations. An interest of our approach is to develop a formalism that remains in physical space (instead of Fourier space) and that can deal with spatially inhomogeneous systems. This enlightens the basic physics and provides novel kinetic equations with a clear physical interpretation. However, unless we restrict ourselves to spatially homogeneous systems, closed kinetic equations can be obtained only if we ignore some collective effects between particles. General exact coupled equations taking into account collective effects are also given. We use this kinetic theory to discuss the processes of violent collisionless relaxation and slow collisional relaxation in systems with weak long-range interactions. In particular, we investigate the dependence of the relaxation time with the system size and provide a coherent discussion of all the numerical results obtained for these systems.

References in corpus (25)

• Thermodynamic Description of the Relaxation of Two-Dimensional Euler Turbulence Using Tsallis Statistics
• Non-Gaussian equilibrium in a long-range Hamiltonian system
• Stability criteria of the Vlasov equation and quasi-stationary states of the HMF model
• Dynamics and thermodynamics of a simple model similar to self-gravitating systems: the HMF model
• A maximum entropy principle explains quasi-stationary states in systems with long-range interactions: the example of the Hamiltonian Mean Field model
• Out-of-equilibrium tricritical point in a system with long-range interactions
• Long-term Evolution of Stellar Self-Gravitating System away from the Thermal Equilibrium: connection with non-extensive statistics
• Prediction of anomalous diffusion and algebraic relaxations for long-range interacting systems, using classical statistical mechanics
• Kinetic theory of point vortices in two dimensions: analytical results and numerical simulations
• Coarse-grained distributions and superstatistics
• Anomalous diffusion and collapse of self-gravitating Langevin particles in D dimensions
• Lynden-Bell and Tsallis distributions for the HMF model
• Hamiltonian and Brownian systems with long-range interactions
• On the diffusive anomalies in a long-range Hamiltonian system
• Quasi-stationary states and incomplete violent relaxation in systems with long-range interactions
• Long time behavior of quasi-stationary states of the Hamiltonian Mean-Field model
• The stochastic process of equilibrium fluctuations, of a system with long range interactions
• Nonextensive Thermodynamics and Glassy behaviour in Hamiltonian systems
• Kinetic theory with angle-action variables
• Hamiltonian and Brownian systems with long-range interactions: IV. General kinetic equations from the quasilinear theory
• Kinetic theory of two dimensional point vortices from a BBGKY-like hierarchy
• Relaxation of a test particle in systems with long-range interactions: diffusion coefficient and dynamical friction
• Nonlinear mean-field Fokker-Planck equations and their applications in physics, astrophysics and biology
• Curious behaviour of the diffusion coefficient and friction force for the strongly inhomogeneous HMF model
• Controversy about the applicability of Tsallis statistics to the HMF model

Cited by in corpus (27)

• Statistical mechanics and dynamics of solvable models with long-range interactions
• Coarse-Grained Modelling Out of Equilibrium
• Kinetic theory of long-range interacting systems with angle-action variables and collective effects
• Kinetic theory of spatially inhomogeneous stellar systems without collective effects
• Dynamical phase transitions in long-range Hamiltonian systems and Tsallis distributions with a time-dependent index
• Hamiltonian and Brownian systems with long-range interactions: V. Stochastic kinetic equations and theory of fluctuations
• Kinetic equations for systems with long-range interactions: a unified description
• Does the brain behave like a (complex) network? I. Dynamics
• Brownian particles with long and short range interactions
• Microcanonical quasi-stationarity of long-range interacting systems in contact with a heat bath
• Hamiltonian and Brownian systems with long-range interactions: IV. General kinetic equations from the quasilinear theory
• Dynamical stability of systems with long-range interactions: application of the Nyquist method to the HMF model
• Out-of-equilibrium phase re-entrance(s) in long-range interacting systems
• Inhomogeneous Tsallis distributions in the HMF model
• Statistics of the gravitational force in various dimensions of space: from Gaussian to Levy laws
• Dynamics and physical interpretation of quasi-stationary states in systems with long-range interactions
• Two-dimensional Brownian vortices
• Caloric curves fitted by polytropic distributions in the HMF model
• Functional integral derivation of the kinetic equation of two-dimensional point vortices
• Hollow cores in Warm Dark Matter halos from the Vlasov-Poisson equation
• Cored DARKexp systems with finite size: numerical results
• Cored density profiles in the DARKexp model
• Spectral BBGKY: a scalable scheme for nonlinear Boltzmann and correlation kinetics
• Vlasov dynamics of 1D models with long-range interactions
• Violent relaxation in one-dimensional self-gravitating system: deviation from the Vlasov limit due to finite-$N$ effect
• Cosmic structure from the path integral of classical mechanics and its comparison to standard perturbation theory
• Phase-space perturbation theory for cosmic large-scale structure