For general processes – reversible or irreversible – there are two complications with respect to thermostatics:

1 – The notion of "state" is generalized. A state at time $t$ may be given not just by the values of some set of thermodynamics (or thermomechanics) variables at that time – say, $\bigl(V(t), T(t)\bigr)$ – but by the full history of those values for some previous interval of time $\Delta t$. For example, informally, $$\text{state}(t) = \{V(t'), T(t') \,\vert\; t-\Delta t < t' \le t\} $$ (or more precisely an equivalence class of such histories).

This takes care of materials with memory (eg subject to hysteresis).

This means that the entropy becomes a functional rather than a function; that is, a function of a function (since the state is effectively a function).

In some cases the memory is "differential", that is, it only concerns very short time intervals. Then the history can be approximated by the time-derivative of the thermodynamic quantities. For example we can have, informally, $$\text{state}(t) = \{V(t), T(t), \dot{V}(t), \dot{T}(t)\} .$$

2 – The entropy itself can be a non-unique function(al). That is, we have a (convex) set of possible entropy-function(al)s, different from one another – and not just by a constant. Each of them does its job and leads to identical experimental consequences, so it doesn't matter which we choose.

This "causes nonmeasureability of entropy and nonexistence of entropometers" (Samohýl & Pekar 2014, p. 52). This latter fact was discovered in the 1970s. I think one of the first to point it out was W. A. Day (1977).

Despite these two complications, the concept of entropy and its mathematical use has still proved to be fundamental, also in irreversible thermodynamics. Its role changed a little though.

Earlier on, entropy and its related inequality was used, roughly speaking, for checking which processes were possible and which impossible. Its modern use, instead, is for checking which consitutive equations are possible and which impossible. This is a much bigger role, because it decides upon whole physical models, not just processes.

Very instructive simple examples of how this happens, for toy systems described eg by $\{V(t), T(t), \dot{V}(t), \dot{T}(t)\}$ variables, are given in chapter 2 of Samohýl & Pekar (2014).

Here are some references, from different perspectives (applied, mathematical) about the points above:

• Astarita: Thermodynamics: An Advanced Textbook for Chemical Engineers (Springer 1990). A truly wonderful book! It discusses all points above.

• Owen: A First Course in the Mathematical Foundations of Thermodynamics (Springer 1984). It gives examples of the non-uniqueness of entropy, and studies the structure of the set of entropy-functions.

• Samohýl, Pekar: The Thermodynamics of Linear Fluids and Fluid Mixtures (Springer 2014). I warmly recommend chapter 2 of this book for an overview of all point above with simple examples.

• Truesdell (ed.): Rational Thermodynamics (Springer 1984).

• Day: An objection to using entropy as a primitive concept in continuum thermodynamics (1977) https://doi.org/10.1007/BF01180089

For general processes – reversible or irreversible – there are two complications with respect to thermostatics:

1 – The notion of "state" is generalized. A state at time $t$ may be given not just by the values of some set of thermodynamics (or thermomechanics) variables at that time – say, $\bigl(V(t), T(t)\bigr)$ – but by the full history of those values for some previous interval of time $\Delta t$. For example, informally, $$\text{state}(t) = \{V(t'), T(t') \,\vert\; t-\Delta t < t' \le t\} $$ (or more precisely an equivalence class of such histories).

This takes care of materials with memory (eg subject to hysteresis).

This means that the entropy becomes a functional rather than a function; that is, a function of a function (since the state is effectively a function).

In some cases the memory is "differential", that is, it only concerns very short time intervals. Then the history can be approximated by the time-derivative of the thermodynamic quantities. For example we can have, informally, $$\text{state}(t) = \{V(t), T(t), \dot{V}(t), \dot{T}(t)\} .$$

2 – The entropy itself can be a non-unique function(al). That is, we have a (convex) set of possible entropy-function(al)s, different from one another – and not just by a constant. Each of them does its job and leads to identical experimental consequences, so it doesn't matter which we choose. Pedagogical examples are given by Owen (1984) for simplified elastic-plastic materials.

This "causes nonmeasureability of entropy and nonexistence of entropometers" (Samohýl & Pekar 2014, p. 52). This latter fact was discovered in the 1970s. I think one of the first to point it out was W. A. Day (1977).

Despite these two complications, the concept of entropy and its mathematical use has still proved to be fundamental, also in irreversible thermodynamics. Its role changed a little though.

Earlier on, entropy and its related inequality was used, roughly speaking, for checking which processes were possible and which impossible. Its modern use, instead, is for checking which consitutive equations are possible and which impossible. This is a much bigger role, because it decides upon whole physical models, not just processes.

Very instructive simple examples of how this happens, for toy systems described eg by $\{V(t), T(t), \dot{V}(t), \dot{T}(t)\}$ variables, are given in chapter 2 of Samohýl & Pekar (2014).

Here are some references, from different perspectives (applied, mathematical) about the points above:

• Astarita: Thermodynamics: An Advanced Textbook for Chemical Engineers (Springer 1990).
  – A truly wonderful book! It discusses all points above.

• Owen: A First Course in the Mathematical Foundations of Thermodynamics (Springer 1984).
  – It gives examples of the non-uniqueness of entropy, and studies the structure of the set of entropy-functions.

• Samohýl, Pekar: The Thermodynamics of Linear Fluids and Fluid Mixtures (Springer 2014).
  – I warmly recommend chapter 2 of this book for an overview of all points above with simple examples.

• Truesdell (ed.): Rational Thermodynamics (Springer 1984).

• Day: An objection to using entropy as a primitive concept in continuum thermodynamics (1977) https://doi.org/10.1007/BF01180089