What's the point with equilibrium in thermodynamics? All the thermodynamics books I saw until now state that in thermodynamics we are mainly concerned with equilibrium states (I know there's "non-equilibrium thermodynamics", but I'm interested on the standard viewpoint).
Now, why is that? Books try to explain this in terms of the implications in measurements, but I don't feel comfortable with experimental physics, so that I just get more confusing with this kind of argument.
Why do we need to consider states of equilibrium? And also, what really is that equilibrium? I think it's different from the classical mechanics notion of equilibrium where we demand the net force to be zero at an equilibrium state. Does equilibrium in thermodynamics have any relationship with this notion of equilibrium in mechanics?
 A: First of all, from a computation point of view, thermal equilibrium is much simpler than out-of-equilibrium. In equilibrium, the state of the system is (more or less) straightforwardly defined by its Hamiltonian and a few thermodynamic parameters. Out-of-equilibrium contains everything that is not equilibrium and ranges from dropping a pencil to heavy ion collisions at CERN.
Equilibrium is, however, an important part of physics, because it is the asymptotic state systems normally assume if we wait long enough. Even though there are infinitely many more out-of-equilibrium states, most of them evolve to thermal equilibrium after a while and the simple calculation provides useful information.
But that's the easy part of your question. The real question is: "What is thermal equilibrium?" First, you are right; thermal equilibrium is not the same as mechanical equilibrium. Being in mechanical equilibrium just means that none of your parts are moving. In thermal equilibrium however, it is only the thermodynamic parameters (average energy, temperature, pressure, volume, average number of particles, etc.) that do not change. Individual particles move around wildly.
First note that being in thermal equilibrium is a relative property. A system is equilibrated with another system. For example, when you pour a cup of tea, the water starts by swirling around. You can say that when the swirling has died out your tea is equilibrated (although I would call that mechanical equilibrium). On the other hand, you could wait an hour until you tea is a room temperature and then say that it's at equilibrium.
However, if you open the window, turn off the heating and come back a couple hours later your tea will be colder than room temperature and you will say that the tea has equilibrated with the local weather system. You can go on like this almost forever.
The truth is that there is always a (space or/and time) scale at which your system is not equilibrated. If you leave your tea for a century on the moon, you might notice that your the mug has changed shape. Glass is actually a very complicated system that has a (very slow) evolution of its own which is comparable to hyper-viscous fluids. Before you talk about equilibrium you must first restrict your observation range.
A super cool example of this is the free falling Bose condensate. Experimental physicists are studying the equilibrium properties of an ultra-cold Bose gas in zero gravity. To do so, they prepare their gas in an electromagnetic container and drop the whole apparatus down a high (110 m) tower. During the fall, the gas equilibrates and they make their measurements. After a few (4.5) seconds, the apparatus reaches the ground and the gas is dispersed. Even though the whole system is clearly out-of-equilibrium the gas is at equilibrium during the time of the fall.
But this still does not really answer your question. In practice, theoretical physicists define thermal equilibrium with statistical physics as the state of matter that has the largest entropy consistent with the macroscopic constraints of the system (fixed energy, volume, numbers of particles, etc.). This is the micro-canonical ensemble. Then, computing equilibrium properties is a well-defined mathematical procedure.
This definition is valid for a closed system and can be used to define an "absolute" thermal equilibrium. It says that the system is equilibrated when its dynamics visits all the micro-states that are consistent with the macroscopic constraints with the same probability.
The dynamics of the microscopic degrees of freedom are maximally disordered. In practice however, no system is really closed. Moreover, if you can find a closed system, it will take an infinite time to equilibrate because energy conserving dynamics never really forgets its initial conditions.
Another definition from Wikipedia is:
"Two physical systems are in thermal equilibrium if no heat flows between them when they are connected by a path permeable to heat."
This definition is a much more practical, but it is only relative. You first must pick a system and declare it to be equilibrated. Then you can compare it to other systems and check weather they are equilibrated as well. Moreover, it needs a definition of heat (the non-mechanical part of the energy that is spontaneously transferred in between two non equilibrated systems) to be useful and heat requires a definition of equilibrium to be useful.
Ok, I can't think of anything more to say. I hope that I didn't confuse you even more. This is a deep and difficult question. Intuitively, it is clear when a system is equilibrated. In theory, however, the definition can be quite tricky. Note however that physicists to not spend all their time thinking about the definition of thermal equilibrium. In experiments you can look at you system, nod and declare it to be equilibrated (sometimes you are wrong and that's when the interesting things start). In theory, the computation of equilibrium properties is a well-defined mathematical procedure which may seem a little remote from reality, but it works really well.
