# Rescaled state equation

Besides the relations between critical exponents, Widom's static scaling theory allows us to make predictions on the shape of the state equation of a given system. Let us now see how, again for a magnetic system.

We begin from:

${\displaystyle M(t,h)=\lambda ^{p_{2}-1}M(\lambda ^{p_{1}}t,\lambda ^{p_{2}}h)}$
If we set ${\displaystyle \lambda =|t|^{-1/p_{1}}}$:
${\displaystyle M(t,h)=|t|^{\frac {1-p_{2}}{p_{1}}}M\left({\frac {t}{|t|}},{\frac {h}{|t|^{p_{2}/p_{1}}}}\right)}$
and using ${\displaystyle \Delta =\beta \delta }$ and ${\displaystyle \alpha =2-{\frac {1}{p_{1}}}}$:
${\displaystyle {\frac {M(t,h)}{|t|^{\beta }}}=M\left({\frac {t}{|t|}},{\frac {h}{|t|^{\Delta }}}\right)}$
We can therefore define the rescaled magnetization and the rescaled magnetic field:
${\displaystyle {\tilde {m}}:=|t|^{-\beta }M(t,h)\quad \qquad {\tilde {h}}:=|t|^{-\Delta }h}$
so that becomes:
${\displaystyle {\tilde {m}}=M(\pm 1,{\tilde {h}})}$
where ${\displaystyle +1}$ corresponds to ${\displaystyle t>0}$ (namely ${\displaystyle T>T_{c}}$) and ${\displaystyle -1}$ to ${\displaystyle t<0}$ (i.e. ${\displaystyle T).

The meaning of equation ${\textstyle {\tilde {m}}=M(\pm 1,{\tilde {h}})}$ is that if we measure ${\displaystyle M}$ and ${\displaystyle h}$ and rescale them as we have just seen, all the experimental data should fall on the same curve independently of the temperature ${\displaystyle T}$; there are of course two possible curves (not necessarily equal), one for ${\displaystyle T>T_{c}}$ and one for ${\displaystyle T (which correspond to ${\displaystyle M(1,h)}$ and ${\displaystyle M(-1,h)}$). These predictions are in perfect agreement with experimental results[1], and are one of the greatest successes of Widom's static scaling theory.

1. See for example J. M. Yeomans, Statistical Mechanics of Phase Transitions, Clarendon Press (1992), pg. 119.