1st order phase transition, superheating/supercooling, metastable state I read that superheating and supercooling characterize 1st order phase transitions in papers. Some of them also use the metastable state at the same time as the superheating/supercooling. Are metastable states and superheating/supercooling the same thing, or if not, how do they relate?
Are they just features of a 1st order phase transition? 
Can anyone share any hints or some references to explain this in detail?
 A: You are right, these terms are related. Metastability usually comes about in systems which are described by a Landau free energy which contains a cubic or power 6 term on top of the usual $\phi^4$-theory. E.g. the Landau free energy with a cubic term is shown below.
There are three special temperatures:


*

*$T^{**}$ at which an additional local minimum forms at finite $\phi$.

*$T_c$ at which this local minimum becomes the global minimum.

*$T^*$ at which the curvature at $\phi=0$ becomes negative.


Let the system start at $\phi=0$ (disordered phase) for $T>T^{**}$. Upon lowering $T$ below $T_c$ the finite $\phi$ local minimum would be the favorable one. But the curvature at $\phi=0$ is still positive, so it stays in the disordered phase; it is said to be supercooled. Lowering the temperature further, the curvature eventually becomes negative and the $\phi=0$ phase becomes unstable, so $\phi$ jumps at the finite value of the local minimum. When raising the temperature again, the (ordered) finite $\phi$ phase will stay stable (positive curvature at the local minimum) even when the temperature exceeds $T_c$; the system is superheated, then. Once $T>T^{**}$, again, the local minimum at finite $\phi$ disappears and $\phi$ jumps back to zero.
As $\phi$ jumps, this is a discontinuous, a.k.a. first-order, phase transition. Metastability is the umbrella term describing both supercooling and superheating. In magnetic systems, this is also referred to as hystersis.
More information on this can be found e.g. in N. Goldenfeld: Lectures on phase transitions and critical phenomena, chapter 5.5. This is also a good introduction to Landau theory in general if you are not familiar with it.

