# Are state variables defined outside equilibrium?

State functions seem to always be described as relating state variables in equilibrium, wikipedia about state functions:

In thermodynamics, a state function, [...] is a function defined for a system relating several state variables [...] that depends only on the current equilibrium thermodynamic state of the system (e.g. gas, liquid, solid, crystal, or emulsion), not the path which the system took to reach its present state.

The German wikipedia about state variables states that if all state variables are constant in time, the system is in thermodynamic equilibrium. Leading me to believe, they may be at least defined outside of equilibrium.

Am I understanding it correctly if I say, that they are only well defined in thermodynamic equilibrium, at least partial equilibrium, meaning thermal, mechanical or chemical respectively, allowing at least the corresponding state variables to be well defined?

Would this be sensible definitions, based on this?

State variable: A Measurable macroscopic quantity that is only well defined, in the sense of describing the state of a system, if it is constant in time. And that is not subject to hysteresis.

State function: A function between state variables, that are well defined and thus in equilibrium.

Where equilibrium just means the state variables in consideration are constant in time and no macroscopic fluxes are occurring. I hesitate to use the thermodynamic equilibrium in the state functions definition, because the zeroth law of thermodynamics only requires thermal equilibrium to measure temperature, so full thermodynamic equilibrium can not always be required.

Clarification: Maybe the question really is: We can measure variables like pressure at any point in time, but does it become a state variables only, when they it is constant in time, so we can be sure, that they represent the average in an equilibrium sense, and not just a short snapshot in time?

## 2 Answers

Purists only accept the idea that state variables are only defined at thermodynamic equilibrium, and do not vary with spatial position or time, even in a Lagrangian sense. But, when we write down the open system version of the first law of thermodynamics, we are already accepting the idea that internal energy can be varying with time within the control volume, and with spatial position from inlet to outlet of the control volume. The extension of this further assumes that it is valid to regard state functions per unit mass as able to vary spatially and temporally in a local sense, with no conceptual issues. We do this, for example, when we solve the transient heat conduction equation. Even if this is an approximation, in engineering and continuum mechanics, it has been found to be exceedingly accurate in virtually all practical situations.

See Problem 11.D.1 in Chapter 11 of Transport Phenomena by Bird, Stewart, and Lightfoot, part (b).

• Tanks for the answer, I think it's a bit advanced for me. I am trying to understand the basic definitions, which are on the first three pages of my thermodynamics script. Do I understand you correctly, that my understanding, that state variables are only defined for equlibrated systems, but we can stretch the definition of what that means quite far? Oct 17 '20 at 17:34
• Yes, that is correct. That enters into the realm of non equilibrium thermodynamics (which is not really a major stretch). Oct 17 '20 at 18:33

Take an isolated system for the sake of argument. In this case the internal energy and the volume are well-defined ideas whether or not the system is in internal equilibrium. Quantities such as pressure and temperature are not necessarily well-defined out of equilibrium. In some states they are, in some they are not. Entropy can be defined even for out-of-equilibrium states with a bit of care. One models the system as many parts, each small enough to be in internal equilibrium but large enough to be distinguishable from each other and have a thermodynamic entropy $$S_i$$. Then the total entropy is $$S = \sum_i S_i$$ Having done all that, you could now take an interest in $$\partial S/\partial U$$ even for out-of-equilibrium states, and you could argue that this fits the the bill for $$1/T$$, but usually we reserve the name 'temperature' for equilibrium states.