Is anyone else reading this? I don't think I can convince you, but if someone else is reading this, they might be learning some astrophysics, so it's useful. If no one else is following this thread, then it's a waste of my time.
No people aren't confusing mechanical equilibrium with thermodynamic equilibrium.
What happens is that in astrophysical situations, the time scales for hydrodynamic equilibrium are considerable smaller than the thermodynamic equilibrium timescales. Typically in a star, time it takes to reach hydro equilibrium is in minutes, whereas it takes several thousand of years to reach thermo equlibrium. Therefore in modelling stars, thermodynamic equilibrium *is an incorrect assumption*, and the dynamics is driven by hydro rather than by thermo. You can assume (at least in stars) assume *local* thermo equilibrium, which allows you to use equations of state, but that's it, and that's not even true when you are talking about stellar atmospheres which are wildly out of equilibrium.
Because of time scales, you can't take thermo equations and add potentials. You have to start with hydro equations, then add in local thermo equilibrium. Stars are wildly out of thermo equilibrium because gravity pushes them out of thermo equlibrium. If they were in thermo equilibrium, the sun wouldn't shine.
Because gravity affects the behavior of atoms, you often have to rederive the equations from the atomic level using statistical mechanics. Gravity fundamentally changes the thermo behavior of gasses, so that you have to think about things at the atomic level. Equations and relationships that work in the laboratory, just don't work in self-gravitating systems.
This is important for stars. If you dump energy into ordinary gas, it will just expand. Gravity changes the thermo properties of gases so that if you dump energy into them, they will contract. This means that the extra energy has to go somewhere, which is why stars shine. If gravity *didn't* change the thermo property of gases, then stars would not exist.
You keep saying that, but it's not true. You can derive the laws of thermodynamics from statistical mechanics, so once you have to rederive the equations from basic principles, those statistical rules still hold, so that you can't get anything. So when you rewrite all of the equations to take into account of gravity, you end up with different equations, but the statistical microphysics makes sure that energy is conserved, entropy increases, and entropy goes to zero as T->0.
You keep saying that people don't understand themodynamics, but personally I think people understand it a lot better than you do. You just can't take equations out of a book and assume they are universally true. You have to understand the *principles* behind those equations, and in stars, they are very different than what you see in the laboratory, and you can get weird stuff.
If you are not willing to learn, then there is no point in me teaching it to you, but if anyone else is interested, I'll keep talking.