How much information about a thermodynamic process is contained in ΔS?

[u/Unique-Rice9999]

Thermodynamics can tell us, for example, what happened to the entropy along the path, but ΔS is not the path itself. Two different chains of transitions may well have the same initial and final states and the same change in a certain measure, but different internal structures.

So the “entropy table” may well turn out to be one column in a much larger table.

And we started with heat, because it is one of the phenomena that we have known since the very beginning of our existence. Then we found ways to obtain and control it, and after some time eventually formalized the second law, for example as dS_total ≥ 0 for an isolated system. But this is only one aspect for describing the transition, not the transition itself.

ΔS may characterize one aspect of a transition, while a more detailed description could characterize the realized structure of the process itself: the sequence and organization of interactions along the path.

[u/7ieben_]

That's basically a long way of saying: state quantitys are path independed, whilst process quantitys are not. Mathematically this implies state quantitys being an total differential, whilst process quantitys are not.

[u/Unique-Rice9999 → u/7ieben_]

yes, I used this distinction as a condition for my question.
since ΔS is path-independent, different realized paths can map to the same value of ΔS. what I am interested in is the information that is lost under that mapping: what additional descriptors would distinguish the internal structure of those paths?
so I am treating entropy as one scale/coordinate of description and asking what a larger description of the transition itself would require.

f:γ ↦ ΔS

γ₁ ≠ γ₂, but f(γ₁) = f(γ₂)

what distinguishes γ₁ from γ₂

edit: and does not mean that σ_γ₁(x, t) = σ_γ₂(x, t)

[u/7ieben_ → u/Unique-Rice9999]

What do you mean by gamma here? We are missing a definition.

The whole idea is, that two identical states are indistuingishable and as such contain the exact same information.

If you want information like what the process was or what the initial state was you came from you must look at the process quantitys and their change (not the state quantitys).

[u/Unique-Rice9999 → u/7ieben_]

this is about another comment in the thread that suggested using Boltzmann's Entropy Formula and I add this description to keep it all together in one place

[u/7ieben_ → u/Unique-Rice9999]

I'm still not quite certain about your question.

You've basically converted the integral balance into an local one, but the statements made remain the same: entropy itselfe is a state quantity and as such neither itselfe nor its change tells you anything about the process (besides what state you started at, obviously).

For analysing the actual process we must use process quantitys, most generally these are work and heat.

[u/Unique-Rice9999 → u/7ieben_]

thank you — I agree that entropy is a state function and that heat and work are relevant process quantities. I should clarify what I mean by the “table” or “scale.”

a table can list states and their properties, but constructing a path through those states also requires rules governing the possible transitions. where spatial effects matter, this includes position and spatial distributions, as well as the environment and the conditions of interaction.

given a sequence,

x₀ → x₁ → … → xₙ,

the corresponding entropy values,

S(x₀) → S(x₁) → … → S(xₙ),

provide one profile of that sequence. Neither this profile nor the total heat and work uniquely specifies the full process.

my question is therefore: how do we move from a static description of states and their properties to a description that lets us construct the possible paths under specified environmental and boundary conditions?

I understand that thermodynamics already accounts for the surroundings. I am trying to make the distinction between listing properties and specifying the dynamics explicit.

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[u/7ieben_ → u/Unique-Rice9999]

That's what we use Balance Equations for.

What you call static is also called global or integral balance and what you are interested in, I suspect, is called local or differential balance (local here includes time). Those are differential equations which are parametized by the boundarys conditions.

Alternativly, as I'm not 100 % sure, I suspect you are asking about the Least Action Principle. The principle comes from mechanics, those is mathematically generalized and robust. Basically stating: of all possible curves, the one minimizing the Action is realized.

Otherwise please provide a specific example.

[u/Unique-Rice9999 → u/7ieben_]

yes this is a possible part of the way, thank you !thanks

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[u/Playful-Painting-527]

∆S was primarily invented as an engineering tool to built more efficient heat engines. In thermodynamics, the states of a system are of interest, not the precise way of how a system gets from one state to the next on a microscopic level. If you want to know more details you have to study the individual components more cloesly or do a CFD simulation of the entire circuit. The insights gained from understanding the exact paths S takes may be of academic interest but I doubt they will help us create better heat engines.

[u/Unique-Rice9999 → u/Playful-Painting-527]

my question is slightly different, is there a useful intermediate description between a state-function change such as ΔS and a complete microscopic or CFD description?

for example, if two processes connect the same initial and final states and have the same ΔS, they may still differ in the sequence and organization of intermediate interactions. I am interested in whether that structure of the realized transition can itself be characterized without specifying every microscopic trajectory.

so the question is not really whether entropy should contain this information, but what additional quantities or frameworks are used when the structure of the path itself matters.

[u/Playful-Painting-527 → u/Unique-Rice9999]

You can calculate Entropy from the microstate using Boltzmann's Entropy Formula. For this to work you need to know the state of every particle in a fluid. In other words: You can't know the path entropy took without resolving every particle in a fluid.

The only discipline to my knowledge where the path of entropy actually matters, is turbulence theory. Here we do precisely that: Solve the Navier Stokes equations using DNS to obtain a space and time resolved measure of entropy.

[u/Unique-Rice9999 → u/Playful-Painting-527]

so, if I understand correctly, the same total entropy change ΔS can correspond to different internal spatiotemporal structures of the process, which can be distinguished by quantities such as the local entropy production σ(x,t) and the resolved dynamical fields?

[u/Ch3cks-Out]

It is unclear what are you asking, specifically. State functions like entropy, by definition, do not provide information about the system along the path between states.

[u/Unique-Rice9999 → u/Ch3cks-Out]

thank you. I should clarify what I mean by the “table” or “scale.”

a table can list states and their properties, but constructing a path through those states also requires rules governing the possible transitions. where spatial effects matter, this includes position and spatial distributions, as well as the environment and the conditions of interaction.

given a sequence,

x₀ → x₁ → … → xₙ,

the corresponding entropy values,

S(x₀) → S(x₁) → … → S(xₙ),

provide one profile of that sequence. Neither this profile nor the total heat and work uniquely specifies the full process.

my question is therefore: how do we move from a static description of states and their properties to a description that lets us construct the possible paths under specified environmental and boundary conditions?

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[u/Ch3cks-Out → u/Unique-Rice9999]

constructing a path through those states also requires rules governing the possible transitions

Like I've said, a state function like entropy cannot help you about such "rules" for transitions - it only depends on the end states. On the other hand, if you had specified a particular path with the x_n intermediate states selected, what else would you want to specify the "full" process? There will be many micro-configurations for a large multitude of possible paths, but thermodynamics only deals with various population statistics of them, not each of them individually!

[u/Unique-Rice9999 → u/Ch3cks-Out]

> a state function like entropy cannot help you about such "rules" for transitions - it only depends on the end states.

in other words, I am not asking entropy to determine the path. I am asking what information about the path is lost when the process is represented only through a particular set of observable states

edited
in part of way it may be Balance Equations

[u/More-Damage8294]

Seems like you're asking about the structural categories it confines, here’s my 2 cents.

Total energy: 1
Exergy: 0.5 usable
Anergy: 0.5 unusable;
in the sense of performing duty work that exergy does.

But Anergy can still be utilized in other ways, such as maintaining higher-order structure by recycling it within a set system and diverting usable exergy; so the structure remains equal. Think of energy as a building block for higher-order structural maintenance.

Separate the building concerns:
1 main-band; 1 sub-band.
One is duty-facing, one is not.