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Title: Lifetime of quasi-stationary states in the Hamiltonian mean-field model
Plain: In the simplest model of particles on a circle that all attract each other, the system gets stuck in non-equilibrium states for a time that grows with the number of particles N. Simulations and kinetic theory disagree on how fast it grows.
Precise: For the Hamiltonian mean-field model $H = \sum_i p_i^2/2 + (1/2N) \sum_{(i,j)} [1 - \cos(\theta_i - \theta_j)]$ started in a spatially homogeneous, Vlasov-stable quasi-stationary state, determine the exponent $\delta$ in the relaxation time $\tau \sim N^{\delta}$. Simulations report $\delta$ approximately 1.7, while kinetic theory (the Lenard-Balescu collision term vanishes for 1D homogeneous systems, leaving $1/N^2$ terms) predicts $\delta = 2$. An answer is $\delta$ and its dependence on the initial distribution and energy.
Settled by: Simulations at N large enough to separate N^1.7 from $N^2$ scaling, together with a kinetic theory that reproduces the measured prefactor.
Kind: well-posed
Answer type: value
References: (empty)
Literature status: contested
Status note: (empty)
Posed since: 2004