# Critical exponents and universality

We know that when a system is in the neighbourhood of a critical point its thermodynamic quantities have peculiar behaviours (they change with the temperature, eventually diverging). We can however ask ourselves how this quantities depend on the temperature, and for this we introduce the concept of critical exponents; as we will see their study will give rise to surprising and fundamental observations, and we will also understand why they are so important in the study of critical phenomena.

Since we are only interested in the proximity of the critical point we define the reduced temperature:

${\displaystyle t={\frac {T-T_{c}}{T_{c}}}}$
so that when ${\displaystyle t=0}$ the system is at its critical temperature. If ${\displaystyle F(t)}$ is a generic function of ${\displaystyle t}$, the critical exponent ${\displaystyle \lambda }$ associated to this function is defined as:
${\displaystyle \lambda :=\lim _{t\to 0}{\frac {\ln |F(t)|}{\ln |t|}}}$
In other words, we can write:
${\displaystyle F(t){\stackrel {t\approx 0}{\sim }}|t|^{\lambda }}$
Note, however, that we are unjustifiably supposing that the critical exponent ${\displaystyle \lambda }$ is the same for both the limits ${\displaystyle t\to 0^{+}}$ and ${\displaystyle t\to 0^{-}}$; to be explicit, more in general write we should have written:
${\displaystyle F(t){\stackrel {t\approx 0}{\sim }}|t|^{\lambda }(1+at^{\lambda _{1}})}$
where ${\displaystyle a}$ is a generic constant. At this level, we are assuming that the two limits give the same result only for the sake of simplicity: however, in the framework of the scaling theory (see Scaling theory) and of the Renormalization Group (see The Renormalization Group) this can be rigorously proved, so we are a posteriori justified to make this assumption.

Depending on the thermodynamic quantity that we are considering, the relative critical exponent has a different name. For example, considering a fluid we know that its specific heat at constant volume diverges at the critical point, and we call its relative critical exponent ${\displaystyle \alpha }$ so that:

${\displaystyle C_{V}{\stackrel {t\approx 0}{\sim }}|t|^{-\alpha }}$
(where obviously ${\displaystyle \alpha >0}$, as will always be in all the other cases). On the other hand, the critical exponent associated to the order parameter of the system is called ${\displaystyle \beta }$, so again for a fluid we will have:
${\displaystyle \rho _{\text{liq}}-\rho _{\text{gas}}\sim (-t)^{\beta }}$
(where the minus sign is due to the fact that the order parameter is non null only for ${\displaystyle T, and for example looking at the ${\displaystyle (T,M)}$ phase diagram we could guess ${\displaystyle \beta \approx 1/2}$).

Of course all these considerations can be extended to other types of systems. For completeness, in the following tables we have written the most used and important critical exponents for fluid and magnetic systems:

Critical exponents for a fluid
Thermodynamic quantity Critical exponent
Specific heat at constant volume ${\displaystyle C_{V}\sim |t|^{-\alpha }}$
Density difference (order parameter) ${\displaystyle \rho _{\text{liq}}-\rho _{\text{gas}}\sim (-t)^{\beta }}$
Isothermal compressibility ${\displaystyle K_{T}\sim |t|^{-\gamma }}$
Pressure at critical isotherm (${\displaystyle t=0}$) ${\displaystyle P-P_{c}\sim |\rho _{\text{liq}}-\rho _{\text{gas}}|^{\delta }\operatorname {sgn} (\rho _{\text{liq}}-\rho _{\text{gas}})}$
Correlation length ${\displaystyle \xi \sim |t|^{-\nu }}$
Correlation function ${\displaystyle G({\vec {r}})\sim 1/|{\vec {r}}|^{d-2+\eta }}$
Critical exponents for a magnetic system
Thermodynamic quantity Critical exponent
Zero-field specific heat ${\displaystyle C_{H}\sim |t|^{-\alpha }}$
Zero-field magnetization (order parameter) ${\displaystyle M\sim (-t)^{\beta }}$
Zero-field isothermal susceptibility ${\displaystyle \chi _{T}\sim |t|^{-\gamma }}$
Field at critical isotherm (${\displaystyle t=0}$) ${\displaystyle H\sim |M|^{\delta }\operatorname {sgn} M}$
Correlation length ${\displaystyle \xi \sim |t|^{-\nu }}$
Correlation function ${\displaystyle G({\vec {r}})\sim 1/|{\vec {r}}|^{d-2+\eta }}$

Note that we have also included two quantities, the correlation length and the correlation function[1], which haven't been previously introduced. We will encounter them later on (see Long range correlations), and we have inserted them here only for the sake of completeness.

Now that we have defined critical exponents, we must justify their importance in the study of critical phenomena. What makes them really interesting is that they appear to be universal: what we mean by this is that their values are independent of the system chosen, and depend only on some very general properties of the system; for example, if the components of the system interact via short-range potentials, it turns out[2] that critical exponents depend only on the dimensionality ${\displaystyle d}$ of the system and on the symmetry of the order parameter. In other words, critical exponents are universal characteristics of thermodynamic systems, while some other properties (like the value of the critical temperature) depend strongly on its microscopic details. As an example, let us consider the results of a famous experiment done in 1945 by Guggenheim. In this experiment several different chemical compounds have been taken, and their liquid-gaseous phase coexistence curves have been measured; plotting the values of the reduced temperatures ${\displaystyle T/T_{c}}$ and densities ${\displaystyle \rho /\rho _{c}}$ of these coexistence curves for the different compounds, it turns out that all the data lie on the same curve in the proximity of the critical point, and surprisingly also far from it:

Law of corresponding states

This fact is also known as law of corresponding states (we will encounter it again in Mean field theories for fluids). All these systems can therefore be described with the same critical exponent, which fitting the data results approximately equal to ${\displaystyle 1/3}$. Note that the systems studied in this experiment are microscopically different from each other (there are monoatomic and diatomic gases, and even CH${\displaystyle {}_{4}}$ which has a much more complex structure compared to them). Now, the very interesting fact is that (compatibly with experimental errors) the same critical exponents appears in completely different systems: the measure of the ${\displaystyle \beta }$ critical exponent for the magnetization of MnFe${\displaystyle {}_{2}}$ gives ${\displaystyle \beta =0.335(5)}$ as a result[3], while the same exponent for the phase separation in a mixture of CCl${\displaystyle _{4}}$ and C${\displaystyle {}_{7}}$F${\displaystyle {}_{16}}$ gives[4]${\displaystyle \beta =0.33(2)}$.

This property of thermodynamic systems is called universality, and systems with the same set of critical exponents are said to belong to the same universality class (we will be able to actually explain this phenomenon only when we will be studying the Renormalization Group, see chapter The Renormalization Group).

This fact justifies the use of very simple and minimal models, with the right general properties, in order to describe the behaviour of thermodynamic systems in the neighbourhood of critical points, regardless of their microscopic details.
1. This is a function of ${\displaystyle {\vec {r}}}$, the distance between two points of the system; furthermore, ${\displaystyle d}$ is the dimensionality of the system itself.
2. From the study of several exactly solvable models, or also from numerical simulations.
3. Heller and Benedek, Physical Review Letters, 13, 253, 1962.
4. Thompson and Rice, Journal of the American Chemical Society, 86, 3547, 1964.