How to find a complete understanding the 2nd law of Thermodynamics in terms of forms?

Question

I have two straightforward question, and below I introduce more context to interpret them:

1. What is, or is there, an order relation for forms that one can use to make sense of the 2nd law of thermodynamics for processes (reversible or not)? Or is the 2nd law fundamentally given in an integral way?
2. Is there a sense in which a form $dS$ can be "path-dependent", to accommodate the distinction between reversible and irreversible statements of the 2nd law (with saturation or not of the inequality)?

I'm trying to understand the laws of thermodynamics from a differential forms formalism, and in doing so I stumbled upon concepts I could not make sense of very well. I'll start with the second law of thermodynamics for reversible/quasi-static processes (which at every instant are at equilibrium) as in Quantum Thermodynamics by Mahler et al. (eq. 3.8 p. 26): $$ dS = \delta Q/T , $$ where $d$ is the exterior derivative and $\delta$ gives an infinitesimal difference. At this point I take this statement as an equality of forms, so $\delta Q$ is not a number, but a 1-form.

In extending this to general thermodynamic process (irreversible, no longer quasi-static, involving non-equilibrium states), Mahler et al. write $$ \delta S \geq \delta Q / T . $$ Here I understand as statement of numbers, with both $\delta$ meaning a very small difference. But with the previous considerations, this should be an inequality on forms, where $\delta S$ is a 1-form, albeit perhaps no longer exact. More importantly, they are related through an inequality, which motivates the 1st question.

My first guess is yes, and it is given by the order relation on the real number line given by integrating these forms. Would that be enough to define such an order relation? I would imagine there could be some caveats to this (e.g. it's just a pre-order...). I can see how that would make sense with Mahler's internal logic, where we'd make the substitution $\delta S \to \int dS$ (whilst maintaining the meaning of $\delta$ in the $\delta Q$ notation).

Another version of this question could be thought of when looking at the book on Mechanical Foundations of Thermodynamics by Campisi, where the 2nd law is stated as $$ dS \geq \delta Q / T , $$ although here the author seems to admit from the outset on using a definition of an order relation of forms.

The second question actually stems from the statement, already given, that the second law depends on the nature of the process.

If so, it even seems this form would even no longer be exact whilst also being path-dependent, in such a way that we would prefer to write it as $\delta S(\gamma_\text{gen})$, and reduce it to $dS = \delta S(\gamma_\text{rev})$, for a path $\gamma$. As far as I'm understanding, this is also a different statement from the path-dependence on the integral of an inexact form.

Could this path-dependence of forms be understood in terms of a coarse-graining from the microstate space to the macrostate space? Viewing the state spaces as a manifold, an irreversible path, by accessing microstates, would require a notion where many 1-forms, each defined at a point on the macrostate manifold, are associated to a path. I could then imagine that, given an underlying theory for the microstates such as quantum mechanics, this path-dependence should come into light. Is there a path already laid out in this direction?

Answer 1

The proper way to deal with Physics, particularly Thermodynamics, is to use Mathematics as a language to say something about the world and not to ask how to make the world fit a particular mathematical formalism.

In thermodynamics, some differential forms appear quite naturally. However, this does not imply that every thermodynamic quantity can be expressed as a differential.

This introduction was motivated by the presence in the original question of a few statements indicating a not full appreciation of the previous sentences. In particular, in all the formulas containing a $\delta Q$, it is essential to realize that the $\delta$ symbol has nothing to do with differences or differentials.

It was a significant achievement by the founders of Thermodynamics to understand that heat ($Q$) is quantity referring to a process and not to a state. There is nothing like "the heat contained in a system." Instead, we can speak about the heat transferred from or to the system. As such, it is not something depending only on the thermodynamic quantities of the system. Even worse, in some cases (non-equilibrium processes), it doesn't depend on them at all.

Differentials enter into play only for the particular case of reversible processes. Therefore, concentrating on differentials means that we exclude an essential part of Thermodynamics from our understanding due to the irreversible nature of every real-world process.

The constructive part of the present answer is that, once one has realized that some functions of the state can be defined through special processes, one can deal with them, and their differentials, to say something about the effect of any process (equilibrium or not equilibrium) connecting equilibrium states. In some cases (connected to the 2nd principle) the conncection has the form of an inequality.

A final note is about the past part where microstates and coarse graining have been introduced. From the thermopdynamical point of view, there is no microstate. States in thermodynamics are aways macrostates. There is no space for microstates. Therefore no space for any concept of coarse-graining.

Answer 2

For your first question, I guess you could, but it would be a mere translation of the inequalities of integrals. For example, when people write $dS\geq \delta Q/T$, they mean $\int dS (=\Delta S)\geq \int \delta Q/T$ along all paths. Mathematically, you can prove that this defines a partial order on the $1-$forms, but it is a bit pedantic and thinking in terms of integrals is more transparent and physically relevant.

For your second question, by construction you want to build $S$ as a state function. This is why $\Delta S$ between two states is defined by $\int \delta Q/T$ along a reversible path linking these two (with implicit assumption that it exists). This is consistent thanks to Clausius inequality which states $\oint \delta Q/T=0$ for a reversible cycle. With this definition, the above inequality can be derived easily. I know that people sometimes introduce the notion of "created entropy" which is defined as $\delta S_c = dS-\delta Q/T$ which captures this path dependent $1-$form, with which the second law becomes $\delta S_c \geq 0$. This is the closest notion to what you hinted as a path dependent entropy.

Finally, for your final question on the microscopic origin of the 2nd principle, the coarse graining approach is the usual way to interpret it, as was proposed by Jaynes. Careful though, the coarse graining is done on macrostates. In QM, there is also the notion of partial trace and measurement which gives rise to entropy increase. In this case the order in which you coarse grain/measure will influence the end result.

Hope this help and tell me if you find some mistakes.