Algebraic Correlation Function and Anomalous Diffusion in the HMF model

Abstract

In the quasi-stationary states of the Hamiltonian Mean-Field model, we numerically compute correlation functions of momenta and diffusion of angles with homogeneous initial conditions. This is an example, in a N-body Hamiltonian system, of anomalous transport properties characterized by non exponential relaxations and long-range temporal correlations. Kinetic theory predicts a striking transition between weak anomalous diffusion and strong anomalous diffusion. The numerical results are in excellent agreement with the quantitative predictions of the anomalous transport exponents. Noteworthy, also at statistical equilibrium, the system exhibits long-range temporal correlations: the correlation function is inversely proportional to time with a logarithmic correction instead of the usually expected exponential decay, leading to weak anomalous transport properties.

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• Thermodynamics and dynamics of systems with long-range interactions
• Kinetic theory of jet dynamics in the stochastic barotropic and 2D Navier-Stokes equations
• Limit distributions of scale-invariant probabilistic models of correlated random variables with the q-Gaussian as an explicit example
• A note on q-Gaussians and non-Gaussians in statistical mechanics
• Kinetic equations for systems with long-range interactions: a unified description
• Relaxation times of unstable states in systems with long range interactions
• Dynamical stability of systems with long-range interactions: application of the Nyquist method to the HMF model
• Nonlinear response for external field and perturbation in the Vlasov system
• Thermodynamics of the HMF model with a magnetic field
• Kinetic theory of point vortex systems from the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy
• Unveiling the nature of out-of-equilibrium phase transitions in a system with long-range interactions
• Non-diagonalizable and non-divergent susceptibility tensor in the Hamiltonian mean-field model with asymmetric momentum distributions
• Lifetime of locally stable states near a phase transition in the Thirring model
• Lyapunov stability of Vlasov Equilibria using Fourier-Hermite modes
• Equilibrium and out of equilibrium phase transitions in systems with long range interactions and in 2D flows
• Classical Goldstone modes in Long-Range Interacting Systems