Dynamics and physical interpretation of quasistationary states in systems with long-range interactions

Abstract

Although the Vlasov equation is used as a good approximation for a sufficiently large N , Braun and Hepp have showed that the time evolution of the one particle distribution function of a N particle classical Hamiltonian system with long range interactions satisfies the Vlasov equation in the limit of infinite N . Here we rederive this result using a different approach allowing a discussion of the role of inter-particle correlations on the system dynamics. Otherwise for finite N collisional corrections must be introduced. This has allowed the a quite comprehensive study of the Quasi Stationary States (QSS) but many aspects of the physical interpretations of these states remain unclear. In this paper a proper definition of timescale for long time evolution is discussed and several numerical results are presented, for different values of N . Previous reports indicates that the lifetimes of the QSS scale as N 1.7 or even the system properties scales with exp(N ). However, preliminary results presented here shows indicates that time scale goes as N 2 for a different type of initial condition. We also discuss how the form of the inter-particle potential determines the convergence of the N -particle dynamics to the Vlasov equation. The results are obtained in the context of following models: the Hamiltonian Mean Field, the Self Gravitating Ring Model, and a 2-D Systems of Gravitating Particles. We have also provided information of the validity of the Vlasov equation for finite N , i. e. how the dynamics converges to the mean-field (Vlasov) description as N increases and how inter-particle correlations arise.

Dynamics, thermodynamics, and phase transitions of classical spins interacting through the magnetic field

“…This indicates that a thermodynamically large system, N → ∞, will take an infinite amount of time to relax to equilibrium, remaining trapped in the initial paramagnetic state. This is similar to what one encounters in the HMF model where the equilibration time t grows algebraically with the system size as [15] t ∼ N 2 and thus a thermodynamically large HMF will never relax to the Boltzmann-Gibbs (BG) equilibrium state. Note that such dynamical slowing down does not arise if the parameter ρ is not scaled in the manner described above.…”

Ensemble inequivalence and Maxwell construction in the self-gravitating ring model

“…Many properties of the model were studied in previous works [28][29][30][31][32][33]. It has a phasetransition from a low energy ferromagnetic phase to a high energy homogeneous (nonmagnetic) phase.…”

“…The system is initially prepared in a waterbag non-equilibrium state with a uniform distribution in the intervals −p 0 < p < p 0 and −θ 0 < θ < θ 0 , with the constants p 0 and θ 0 chosen for the system to have the required energy. Since the relaxation time to reach equilibrium is typically very long in long-range interacting systems, and scales with N for non-homogeneous or N 2 for homogeneous states [33,39,40], very long computer runs are required. The left panel of Fig.…”

Um estudo de escala da dinâmica de estados homogêneos do sistema gravitacional unidimensional

“…Deve ser natural esperar que no presente caso as correções colisionais predominantes para a equação cinética vêm de termos de alta ordem proporcionais a 1/N 2 , sugerindo uma escala de relaxação proporcional a N 2 . Em um trabalho prévio [9] mostra-se que para os modelos HMF e…”

“…Neste trabalho propomos a extensão dos cálculos teóricos de [9] para sistemas gravitacionais 1D para encontrar a equação cinética destes sistemas.…”