Self-consistent inhomogeneous steady states in Hamiltonian mean field dynamics

Abstract

Long-lived quasistationary states, associated with stationary stable solutions of the Vlasov equation, are found in systems with long-range interactions. Studies of the relaxation time in a model of $N$ globally coupled particles moving on a ring, the Hamiltonian Mean Field model (HMF), have shown that it diverges as $N^{\gamma }$ for large $N$, with $\gamma \simeq 1.7$ for some initial conditions with homogeneously distributed particles. We propose a method for identifying exact inhomogeneous steady states in the thermodynamic limit, based on analysing models of uncoupled particles moving in an external field. For the HMF model, we show numerically that the relaxation time of these states diverges with $N$ with the exponent $\gamma \simeq 1$. The method, applicable to other models with globally coupled particles, also allows an exact evaluation of the stability limit of homogeneous steady states. In some cases it provides a good approximation for the correspondence between the initial condition and the final steady state.

5 pages, 4 figures, revised, accepted for publication on Phys. Rev. E

References in corpus (14)

• Statistical mechanics and dynamics of solvable models with long-range interactions
• Out-of-equilibrium tricritical point in a system with long-range interactions
• Collisionless relaxation in non-neutral plasmas
• Core-halo distribution in the Hamiltonian Mean-Field Model
• On the diffusive anomalies in a long-range Hamiltonian system
• Slow relaxation in long-range interacting systems with stochastic dynamics
• Statistical Mechanics of Unbound Two Dimensional Self-Gravitating Systems
• Simpler Variational Problem for Statistical Equilibria of the 2D Euler Equation and Other Systems with Long Range Interactions
• Relaxation to thermal equilibrium in the self-gravitating sheet model
• Relaxation times of unstable states in systems with long range interactions
• Out of Equilibrium Solutions in the $XY$-Hamiltonian Mean Field model
• Statistical mechanics of collisionless relaxation in a noninteracting system
• Stability of inhomogeneous states in mean-field models with a local potential
• Unveiling the nature of out-of-equilibrium phase transitions in a system with long-range interactions

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