Coarsening kinetics in one-dimensional spin systems with long-range interactions: from voter to Ising

Abstract In this paper, we start reviewing the main features of the one-dimensional Ising model with long-range interactions, where the spin-spin coupling decays as a power law, $$J(r) \propto r^{-\alpha }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>J</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>r</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>∝</mml:mo> <mml:msup> <mml:mi>r</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>α</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . We then discuss the key properties of the one-dimensional voter model, in which two agents (spins) at distance r interact with a power-law probability with the same form of J ( r ). The two models are compared, and the so-called p -voter model is presented, which provides a framework to interpolate between them. Specifically, the p -voter model reduces to the voter model for $$p = 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and $$p = 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , while for $$p \ge 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> it falls into the universality class of the Ising model. Graphical abstract