# Is “equilibrium state” equivalent to “well-defined state variables”?

Intuitively, why is a reversible process one in which the system is always at equilibrium?

and

How slow is a reversible adiabatic expansion of an ideal gas?

Suppose you have a piston with some air in it and you perform a slow, reversible expansion. The air in the piston must be in an equilibrium state the entire time.

Now suppose you do the expansion quickly. During the expansion, the air is not in an equilibrium state. My question is: should it have well-defined state variables? Should the pressure be well-defined, for example?

Presumably there is air moving around in bulk and a pressure gauge would give different readings depending on where you put it. Similarly, there is a mean kinetic energy of the molecules that might be used to define $T$, but there is no $\beta$ exponential factor because the kinetic energies of the molecules will not follow a simple, single-parameter distribution. This would indicate that concepts like pressure and temperature are not well-defined when out of equilibrium.

Is that right, and is it always the case? Can I have a process where I know what the pressure and temperature are the entire time, but the system is not in equilibrium?

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 This is the realm of nonequilibrium thermodynamics--- the pressure and temperature follow local laws, and if the gradient is not too big compared to the mean free path, you expect the continuum approximation to hold. I don't know if you want the detailed local laws for relaxation in a gas, or if you just want a pointer to the field. – Ron Maimon Dec 12 '11 at 11:05 @Ron A detailed description would be beyond what I'm looking for. I'm just trying to understand the scope of equilibrium thermodynamics clearly. – Mark Eichenlaub Dec 12 '11 at 11:55

Strictly speaking there are no reversible processes in Nature; it is an idealization that enables one to get bounds on efficiency of nonequilibrium processes by using techniques of equilibrium thermodynamics only.

A reversible process is therefore primarily a theoretical concept for discussing what would happen in a process if dissipation were absent. It is defined as a motion in equilibrium state space, and hence presupposes that the system always remains in equilibrium.

However, empirically, fast processes generate far more excess entropy (the source of the dissipation) than slow ones, so one can treat a slow process approximately as a reversible one.

In a nonequilibrium state, the extensive variables are still well-defined, while the intensive variables (temperature, pressure, chemical potential) typically aren't. On the other hand, most nonequilibrium processes in ordinary life are well-described by local equilibrium; which means that every small region is approximately in equilibrium, and then temperature, pressure, and chemical potential can be assigned definite values. As a result one gets a temperature field, a pressure field, etc. (This is what you feel when you move through a room from the heating to a open window.)

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I've started to take Jaynes' wacky point on this subject more seriously --- the key is experimental reproducibility, of which equilibration is a helpful but neither necessary nor sufficient condition. The point is that you know you have enough "macroscopic" degrees of description when you find that it is sufficient to reproduce the phenomena you are interested in.

So for instance, with gases, if all you're interested in is the state of play from equilibrium to equilibrium, then you find that pressure, temperature, volume, entropy (and partial versions if you have a mixture) are sufficient, and one may sensibly have a theory of it. On the other hand, if you're interested in what happens in between, then you have to be more specific (a rather prosaic example of how coarse graining intellectually drives all condensed matter) --- are you interested in the fluctuation spectrum of things like pressure, or maybe localised versions of the macroscopic variables (as Ron suggested)? Or maybe you just care about the case where you set up a shock wave (in which case I can recommend the excellent book on the subject by Zel'dovich).

In the base case and extensions, the logical starting point is experiment: to reproducibly observe this phenomenon, what are things I have to control? This is not a theoretical problem, and can only be answered with experiment.

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For a non equilibrium process, pressure and average kinetic energy are still defined (force/surface) but as you have noticed they no longer necesseraly depend on one parameter. For instance, in atomic physics, it is quite common to have a two temperature system to describe the electron populations around an heavy element. Now the non equilibrium state you mention is in fact a non uniform state where the temperature and pressure varies from point to point. Basic thermodynamics introduces T, P constant within the system. Fluid thermodynamics allows T,P,n to be fonction of space and time. Fluid equations are only valid if the mean free path is shorter than the gradient scale length. One assumes that a fluid cell is a microcosm where the distribution function is a Maxwellian. Kinetic theory is the fundamental theory and is always valid but one can not simply define state variables (assuming they have a meaning).

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