2014/12/26 by C. R. Lourenço, Lourenço, C. R., T. M. Rocha Filho +1 · 1 citation
Economics, Econometrics and Finance · Physics and Astronomy · #Complex Systems and Time Series Analysis #FOS: Physical sciences #Statistical Mechanics (cond-mat.stat-mech) #Statistical Mechanics and Entropy #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1412.7886
openalex publication_date 2014/12/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The dynamics of quasi-stationary states of long-range interacting systems with N particles can be described by kinetic equations such as the Balescu-Lenard and Landau equations. In the case of one-dimensional homogeneous systems, two-body contributions vanish as two-body collisions in one dimension only exchange momentum and thus cannot change the one-particle distribution. Using a Kac factor in the interparticle potential implies a scaling of the dynamics proportional to Nδ with δ=1 except for one-dimensional homogeneous systems. For the latter different values for δ were reported for a few models. Recently it was show by Rocha Filho and collaborators [Phys. Rev. E \bf 90, 032133 (2014)] for the Hamiltonian mean-field model that δ=2 provided that N is sufficiently large, while small N effects lead to δ≈1.7. More recently Gupta and Mukamel [J. Stat. Mech. P03015 (2011)] introduced a classical spin model with an anisotropic interaction with a scaling in the dynamics proportional to N1.7 for a homogeneous state. We show here that this model reduces to a one-dimensional Hamiltonian system and that the scaling of the dynamics approaches N2 with increasing N. We also explain from theoretical consideration why usual kinetic theory fails for small N values, which ultimately is the origin of non-integer exponents in the scaling.