Thermodynamic calculation of spin scaling functions

Abstract

Critical phenomena theory centers on the scaled thermodynamic potential per spin $\varphi (\beta ,h)=|t{|}^{p}Y(h|t{|}^{-q})$, with inverse temperature $\beta =1/T$, $h=-\beta H$, ordering field $H$, reduced temperature $t=t(\beta )$, critical exponents $p$ and $q$, and function $Y(z)$ of $z=h|t{|}^{-q}$. I discuss calculating $Y(z)$ with the information geometry of thermodynamics. Scaled solutions obtain with three admissible functions $t(\beta )$: 1) $t=e^{-J\beta }$, 2) $t={\beta }^{-1}$, and 3) $t={\beta }_C-\beta$, where $J$ and ${\beta }_C$ are constants. For $p=q$, information geometry yields $Y(z)=\sqrt{1+z^2}$, consistent with the one-dimensional (1D) ferromagnetic Ising model.

10 pages, 1 figure, 22 references