# Inequalities between critical exponents

Critical exponents are not all completely independent: as we shall now show, from relations that we have previously found we can deduce some inequalities that involve critical exponents. The two main such relations are the Rushbrooke and Griffiths inequalities. We will limit ourselves to show explicitly only the derivation of the first one.

In Response functions we have shown the following relation for magnetic systems that involve the specific heats at constant field and magnetization:

${\displaystyle C_{H}-C_{M}={\frac {T}{\chi _{T}}}\left({\frac {\partial M}{\partial T}}\right)^{2}}$
We thus have (since ${\displaystyle C_{M}}$ is surely positive, because giving heat to a system at constant magnetization will always increase its temperature):
${\displaystyle C_{H}=C_{M}+{\frac {T}{\chi _{T}}}\left({\frac {\partial M}{\partial T}}\right)^{2}\geq {\frac {T}{\chi _{T}}}\left({\frac {\partial M}{\partial T}}\right)^{2}\quad \Rightarrow \quad C_{H}\geq {\frac {T}{\chi _{T}}}\left({\frac {\partial M}{\partial T}}\right)^{2}}$
If we now suppose ${\displaystyle H=0}$ and ${\displaystyle T}$ slightly lower than the critical temperature ${\displaystyle T_{c}}$, from the definitions of critical exponents we have:
${\displaystyle C_{H}\sim |t|^{-\alpha }\quad \qquad \chi _{t}\sim |t|^{-\gamma }\quad \qquad {\frac {\partial M}{\partial T}}\sim (-t)^{\beta -1}}$
Thus substituting and remembering that ${\displaystyle T:
${\displaystyle (T_{c}-T)^{-\alpha }\geq A\cdot T(T_{c}-T)^{\gamma -2(\beta -1)}\quad \Rightarrow \quad (T_{c}-T)^{-\alpha -\gamma +2-2\beta }\geq A\cdot T}$
where ${\displaystyle A}$ is a (positive) constant. Taking the limit ${\displaystyle T\to T_{c}^{-}}$:
${\displaystyle \lim _{T\to T_{c}^{-}}(T_{c}-T)^{-\alpha -\gamma +2-2\beta }\geq A\cdot T_{c}}$
and this inequality can be satisfied only if the left hand side doesn't tend to zero. This happens if ${\displaystyle -\alpha -\gamma +2+2\beta \leq 0}$, namely:
${\displaystyle \alpha +2\beta +\gamma \geq 2}$
This is the Rushbrooke inequality.

The Griffiths inequality, on the other hand, is:

${\displaystyle \alpha +\beta (1+\delta )\geq 2}$
and can be obtained from the fact that the free energy is a convex function.

At this level they are both inequalities; however, when the first critical transitions were studied, numerical simulations showed that the critical exponents satisfied them as exact equalities. We will show much later on (see Relations between critical exponents), that once assumed the static scaling hypothesis Rushbrooke and Griffiths inequalities will become exact equalities.