Thermodynamic calculation of spin scaling functions

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