arXiv:cond-mat/0002168 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Canonical solution of a system of long-range interacting rotators on a lattice

Alessandro Campa, Andrea Giansanti, Daniele Moroni

Published Jul 1, 2000• 82 citations

Abstract

The canonical partition function of a system of rotators (classical X-Y spins) on a lattice, coupled by terms decaying as the inverse of their distance to the power alpha, is analytically computed. It is also shown how to compute a rescaling function that allows us to reduce the model, for any d-dimensional lattice and for any alpha<d, to the mean-field (alpha=0) model.

DOI: 10.1103/physreve.62.303

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arXiv:cond-mat/0002168 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Canonical solution of a system of long-range interacting rotators on a lattice

Alessandro Campa, Andrea Giansanti, Daniele Moroni

Published Jul 1, 2000• 82 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physreve.62.303

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Universal behavior in the static and dynamic properties of the $α$ -XY modelUniversal behavior in the static and dynamic properties of the $α$-XY model

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Universal behavior in the static and dynamic properties of the

α

-XY model

The

α

-XY model generalizes, through the introduction of a power-law decaying potential, a well studied mean-field hamiltonian model with attractive long-range interactions. In the

α

-model, the interaction between classical rotators on a lattice is gauged by the exponent

α

in the couplings decaying as

r^α

, where

r

are distances between sites. We review and comment here a few recent results on the static and dynamic properties of the

α

-model. We discuss the appropriate

α

dependent rescalings that map the canonical thermodynamics of the

α

-model into that of the mean field model. We also show that the chaotic properties of the model, studied as a function of

α

display an universal behaviour.

We analyze the equilibrium properties of a chain of ferromagnetically coupled rotators which interact through a force that decays as ${r}^{\ensuremath{-}\ensuremath{\alpha}}$ where

r

is the interparticle distance and $\ensuremath{\alpha}\ensuremath{\ge}0$. By integrating the equations of motion we obtain the microcanonical time averages of both the magnetization and the kinetic energy. We detect three different regimes depending on whether $\ensuremath{\alpha}$ belongs to the intervals

[0,1)

(1,2)

, or $(2,\ensuremath{\infty})$. For $0<\ensuremath{\alpha}<1$, the microcanonical averages agree, after a scaling, with those obtained in the canonical ensemble for the mean-field case $(\ensuremath{\alpha}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0)$. This correspondence offers a mathematically tractable way of dealing with systems governed by slowly decaying long-range interactions.