The role of magnetization in phase-ordering kinetics of the short-range and long-range Ising model

The kinetics of phase ordering has been investigated for numerous systems via the growth of the characteristic length scale ℓ(t)∼tα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell (t) \sim t^{\alpha }$$\end{document} quantifying the size of ordered domains as a function of time t,  where α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document} is the growth exponent. The behavior of the squared magnetization ⟨m(t)2⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle m(t)^{2}\rangle$$\end{document} has mostly been ignored, even though it is one of the most fundamental observables for spin systems. This is most likely due to its vanishing for quenches in the thermodynamic limit. For finite systems, on the other hand, we show that the squared magnetization does not vanish and may be used as an easier to extract alternative to the characteristic length. In particular, using analytical arguments and numerical simulations, we show that for quenches into the ordered phase, one finds ⟨m(t)2⟩∼m02tdα/V,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle m(t)^{2} \rangle \sim m_0^2 t^{d\alpha }/V,$$\end{document} where m0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_0$$\end{document} is the equilibrium magnetization, d the spatial dimension, and V the volume of the system.