# Widom's static scaling theory

We have seen in the first part of Statistical mechanics of phase transitions that when a phase transition occurs the free energy of the system is such that the response functions exhibit singularities, often in the form of divergences. To make a concrete example (but of course all our statements are completely general) if we consider a magnetic system we can suppose to write its free energy density as:

${\displaystyle f(T,H)=f_{\text{r}}(T,H)+f_{\text{s}}(t,h)}$
where ${\displaystyle t=(T-T_{c})/T_{c}}$, ${\displaystyle h=(H-H_{c})/k_{B}T}$, ${\displaystyle f_{\text{r}}}$ is the "regular" part of the free energy (which does not significantly change near a critical point), while ${\displaystyle f_{\text{s}}}$ is the "singular" one, which contains the non-analytic behaviour of the system near a critical point (i.e. ${\displaystyle t\approx 0}$ and ${\displaystyle h\approx 0}$).

Widom's static scaling hypothesis consists in assuming that the singular part ${\displaystyle f_{\text{s}}}$ of the free energy is a generalized homogeneous function, i.e.:

${\displaystyle f_{\text{s}}(\lambda ^{p_{1}}t,\lambda ^{p_{2}}h)=\lambda f_{\text{s}}(t,h)\quad \quad \forall \lambda \in \mathbb {R} }$
in the appendix Homogeneous functions we discuss the main properties of such functions. Note that assuming that one thermodynamic potential is a generalized homogeneous function implies that all the other thermodynamic potentials are so.

The exponents ${\displaystyle p_{1}}$ and ${\displaystyle p_{2}}$ are not specified by the scaling hypothesis; however, we are shortly going to show that all the critical exponents of a system can be expressed in terms of ${\displaystyle p_{1}}$ and ${\displaystyle p_{2}}$; this also implies that if two critical exponents are known, we can write ${\displaystyle p_{1}}$ and ${\displaystyle p_{2}}$ in terms of them (since in general we will have a set of two independent equations in the two variables ${\displaystyle p_{1}}$ and ${\displaystyle p_{2}}$) and therefore determine all the critical exponents of the system. In other words, we just need to know two critical exponents to obtain all the others.

As shown in the appendix Homogeneous functions, an important property of generalized homogeneous functions is that with a proper choice of ${\displaystyle \lambda }$ we can remove the dependence on one of their arguments; for example, if in our case we choose ${\displaystyle \lambda =h^{-1/p_{2}}}$ then:

${\displaystyle f_{\text{s}}(t,h)=h^{1/p_{2}}f_{\text{s}}\left({\frac {t}{h^{p_{1}/p_{2}}}},1\right)}$
The ratio ${\displaystyle \Delta :=p_{2}/p_{1}}$ is sometimes called gap exponent.