# 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:

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

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