Linear theory and violent relaxation in long-range systems: a test case

Abstract

In this article, several aspects of the dynamics of a toy model for longrange Hamiltonian systems are tackled focusing on linearly unstable unmagnetized (i.e. force-free) cold equilibria states of the Hamiltonian Mean Field (HMF). For special cases, exact finite-N linear growth rates have been exhibited, including, in some spatially inhomogeneous case, finite-N corrections. A random matrix approach is then proposed to estimate the finite-N growth rate for some random initial states. Within the continuous, $N\to \infty$, approach, the growth rates are finally derived without restricting to spatially homogeneous cases. All the numerical simulations show a very good agreement with the different theoretical predictions. Then, these linear results are used to discuss the large-time nonlinear evolution. A simple criterion is proposed to measure the ability of the system to undergo a violent relaxation that transports it in the vicinity of the equilibrium state within some linear e-folding times.

References in corpus (6)

• Statistical mechanics and dynamics of solvable models with long-range interactions
• Lynden-Bell and Tsallis distributions for the HMF model
• Quasi-stationary states and the range of pair interactions
• Relaxation to thermal equilibrium in the self-gravitating sheet model
• Out of Equilibrium Solutions in the $XY$-Hamiltonian Mean Field model
• Unveiling the nature of out-of-equilibrium phase transitions in a system with long-range interactions

Cited by in corpus (6)

• Critical behaviour of the XY -rotors model on regular and small world networks
• Action diffusion and lifetimes of quasistationary states in the Hamiltonian Mean Field model
• Emergence of a non trivial fluctuating phase in the XY model on regular networks
• Crafting networks to achieve, or not achieve, chaotic states
• Stochastic treatment of finite-N effects in mean-field systems and its application to the lifetimes of coherent structures
• Quantum fluctuations inhibit symmetry breaking in the HMF model