Can the work done between two non-equilibrium states be calculated?

Question

The work done during a process between two equilibrium states can be described by thermodynamics. Even when process itself is out of equilibrium, the thermodynamic laws can still be used, though calculating the work is much more difficult. But if the initial or final states, or both, are not in equilibrium, can the work done in driving the system from one to the other be calculated? How?

Edit: @Roy @genneth @Marek I do mean far out of equilibrium. There wasn't a nonequilibrium tag, and I don't have enough reputation points to create one! I can't seem to post comments today, so I'll write this here instead: I'm familiar with Jarzynski's equality and while it is very useful, it is still only valid between equilibrium initial and final states (though the final state being in equilibrium can be relaxed). I'd like to know if there's any way work (or heat) can be defined or calculated when the initial and final states are out of equilibrium, possibly very out of equilibrium.

Edit2: @Roy I'd like to know if it's possible in a system with an initial state which is not in equilibrium which is then driven to a final value which is still not in equilibrium. I don't want to make any other assumptions if possible. So I don't necessarily expect local equilibrium to hold, although I'd still be interested to know if work between the 2 nonequilibrium states can be found in that case.

Answer 1

This is a topic in Non-Equilibrium Thermodynamics. There is a standard concept in Thermodynamics of "Thermodynamic Force", "Thermodynamic Flux" and so on. In the Physical Chemistry context you might be familiar with "Affinity" and "Chemical Potential". These are the mechanisms used to explain chemical reaction directions, etc.

So to summarise this large area with an example two substances S1 (with Temperature T1) and S2 (with Temperature $T2 < T1$) are in thermal contact, the combined system is not in equilibrium. The Thermodynamic Force here is: $F=(1/T2 - 1/T1)$. This is derivable from the formula for change in Entropy in this situation:$dS = -dS1/T1 + dS2/T2$. In general a "thermodynamic force" causes a change in Entropy - a situation that can only arise in non-Equilibrium situations. The thermodynamic force will become zero when T1=T2 and the system S1+S2 is in equilibrium. So the idea is that Thermodynamic force models Entropy change (in a mechanics-like manner).

Associated with this Force is the Flux $J_Q$: the time dependent construct
$dQ/dt = J_Q = \alpha(T1-T2)$ where $\alpha$ is the Fourier coefficient of heat conductivity.

In general the First Law of Thermodynamics holds the key to the constructions.

$dU=TdS - PdV + \Sigma \mu_i dN_i $

(here the $\mu_i$ are the chemical potentials of the ith particle species) - useful if they are created or destroyed (as in a chemical or nuclear reaction.) This equation could be written as:

$dU=TdS - \Sigma X_i dx_i$

where the $X_i$ are the generalised forces and $x_i$ are the generalised conjugate variables.

When the extra variables include electrical potentials we can have thermo-electric equations, etc. These can describe the flows in electric-chemical batteries and the like.

One point of debate is whether and to what extent thermodynamic variables are really "local" as would be required in a true continuum based theory. Thus how valid is "Temperature at a point" etc.

These topics are covered in Thermodynamics texts e.g. Callen: "Introduction to Thermodynamics and Thermostatics" or Prigogine "Modern Thermodynamics".

Answer 2

If you want to treat a non equilibrium problem, you need to introduce a distribution function for each component and to solve the corresponding kinetic equations. That is currently done in fusion plasma physics. However, keep in mind that it is mathematically involved and that the thermodynamic temperature is not even defined for non equilibrium processes. The thermodynamical temperature can indeed be identified with a parameter which appears in the equilibrium solution the kinetic equation (Boltzmann's distribution), out of equilibrium the distribution function is no longer Maxwellian and the temperature becomes undefined. It is usual to keep identifying the temperature as the average particle kinetic energy, but to compute this average you need the distribution function that may depend on more than one parameter... And of course one can compute the work done calculate the work done between two non equilibrium states, assuming you have solved the kinetic equations.