Thermodynamic calculation of spin scaling functions

Critical phenomena theory centers on the scaled thermodynamic potential per spin $\phi(\beta, h)=|t|^{p}Y(h|t|^{-q})$, 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.