Classical Heisenberg spins with long-range interactions: Relaxation to equilibrium for finite systems

Abstract

Systems with long-range interactions often relax towards statistical equilibrium over timescales that diverge with $N$, the number of particles. A recent work [S. Gupta and D. Mukamel, J. Stat. Mech.: Theory Exp. P03015 (2011)] analyzed a model system comprising $N$ globally coupled classical Heisenberg spins and evolving under classical spin dynamics. It was numerically shown to relax to equilibrium over a time that scales superlinearly with $N$. Here, we present a detailed study of the Lenard-Balescu operator that accounts at leading order for the finite-$N$ effects driving this relaxation. We demonstrate that corrections at this order are identically zero, so that relaxation occurs over a time longer than of order $N$, in agreement with the reported numerical results.

20 pages, 3 figures; v2: minor changes, published version

References in corpus (5)

• Statistical mechanics and dynamics of solvable models with long-range interactions
• Relaxation to thermal equilibrium in the self-gravitating sheet model
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Cited by in corpus (7)

• The world of long-range interactions: A bird's eye view
• Secular dynamics of long-range interacting particles on a sphere in the axisymmetric limit
• Kinetic theory of $1D$ homogeneous long-range interacting systems sourced by $1/N^2$ effects
• Kinetic theory of one-dimensional homogeneous long-range interacting systems with an arbitrary potential of interaction
• Kinetic theory of one-dimensional inhomogeneous long-range interacting $N$-body systems at order $1/N^2$ without collective effects
• Critical exponents in mean-field classical spin systems
• Relaxation to equilibrium in models of classical spins with long-range interactions