Some Questions about the Critical Point I'm currently trying to understand the physics of phase transitions and I'm having a hard time doing that. First of all, the discussions on the topic seem to be confusing and there is no methodical approach to study such systems. I will list a couple questions concerning this topic, and hope someone could clarify them to me.


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*The wikipedia definition of a critical point is:the end point of a phase equilibrium curve. This seems to be qualitatively ok to me, but is there a more rigorous, mathematical definition? I know this is related to second order derivatives of a convenient free energy but in most books the critical temperature, critical pressure etc shows up from nowhere and after that some connections between the second order derivatives are made. For example, take the Ising model in $d$ dimensions with the mean field approximation. The critical temperature is obtained from the behavior of the (Taylor expansion) of the free energy and no connection to the heat capacity (the appropriate second order derivative of the free energy to be taken account) is made. Besides, how the discontinuity of the second order derivative of the free energy imply the wikipedia definition? 

*Is it possible for a system to have more than one critical point? 

*Since childhood we learn in school about basic phase transitions such as liquid-solid, gas-liquid etc. It is funny that these phase transitions we are familiar with seem to be all of first order. Is it accurate? Is there a reason for that? The only reason I could think of is that the critical point of a given system is considerably high and we are not familiar with such systems. 

*What does the system become if we go beyond the critical point?

*If two systems have the same critical point, are their critical exponents necessarily the same?
 A: 1) The critical point of thermodynamic system is a concept which require a physical definition. Math comes after. The physical definition should convey the information that, at criticality, fluctuations over all space scales dominate the physics of the system. Space fluctuations are described by correlation functions so a signature of critical behavior is the divergence of long wave-length fluctuations. It turns out that for thermodynamic systems the  long wave-length behavior of pair correlation functions is controlled by second derivatives of a thermodynamic potential, i.e. first derivative of an equation of state. This established a bridge with the mathematical definition of critical point of a function. Which is a point where the structural behavior of a function changes. Extremum points are critical points in the mathematical sense. For instance, a function in the neighborhood of a horizontal inflection point is structurally unstable: small perturbations may change its behavior from monotonic to non-monotonic.
For example, let's take the  ubiquitous one-component fluid system with its liquid-gas critical point. The long wave-length limit of the density correlation function $S(k)$ is $\rho k_B T \chi_T$, where $\rho$ is the number density, $T$ the temperature, $k_B$ the Boltzmann's constant and $\chi_T= \frac{1}{V}\left.\frac{\partial V}{\partial P}\right|_T$ the isothermal compressibility. The divergence of $S(k)$ for $k \rightarrow 0~~$implies the divergence of $\chi_T$, i.e. the presence of a horizontal inflection point on an isotherm.
Notice that the non-analytic behavior is not a discontinuity of second derivatives of a thermodynamic potential but a divergence.
2) multi-component systems may have lines, surfaces o hyper-surfaces of critical points. A one-component system may have only isolated critical points for the same reason one cannot have more than isolated extrema in functions of one variable.
3) it is not common to find materials which have critical point at normal thermodynamic conditions. Although in laboratory it is not too complicate to have two-component liquid mixtures exhibiting citical behavior at normal pressure and temperatures in the range between $20$ and $50$ Celsius degrees.
4) it depends what one mean by beyond. At temperature higher than the critical one, a fluid system behaves like ... a homogeneous fluid.
5) two systems may have a critical point at the same thermodynamic state, but there is no reason their critical exponents would be similar. On the other side, completely different systems may have the same critical exponents. Critical exponents are controlled  by


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*the dimensionality of the space;

*the dimensionality of the order parameter;

*the symmetries of the system

A: Rather than "get lost" in pure mathematics, I will give a conceptual description of what happens to a pure substance as you approach and exceed the critical point.  Assume a pure substance in a closed container that is half full of liquid and half full of vapor, on a volume basis.  If enough time elapses whereby this system comes to thermal equilibrium, there is no net condensation or net evaporation involved, and the pressure inside the container corresponds to the Antoine equation, which specifies a one-to-one correspondence between temperature and vapor pressure (see https://en.wikipedia.org/wiki/Antoine_equation).  If you heat such a system up and let it come to a new equilibrium temperature, the higher temperature will result in a lower liquid density, and that same temperature will result in a higher pressure and vapor density because some of the liquid vaporized and increased the vapor density as a result.  As higher and higher temperatures are reached, the liquid density keeps falling and the vapor density keeps rising, until both phases acquire the same density at the critical temperature.  At that temperature and higher temperatures, only one phase exists, only one critical point exists, and the phase is usually called a super-critical gas or vapor.
Other physical properties, such as heat of vaporization (aka heat of condensation) follow a similar pattern, such that these properties become zero at the critical point, which means that at the critical temperature and higher, no condensation can occur, and other differences in physical properties associated with liquid vs. vapor are also not seen.
Two systems that have the same critical point must have the same critical temperature, the same critical pressure, the same critical volume, etc.  This many physical constraints for a given system means that critical properties are unique to each pure substance, so two systems that have the same critical properties must both contain the same pure substance.  
