[FRIAM] Entropy RE-redux

[Stephen Guerin]

And a follow up on Frank's prompt which was for equilibrium where the
integral /change of entropt was zero to out of equilibrium systems like
weather.

Entropy (S) in Thermodynamics and Beyond

Definition of a State Variable (Equilibrium Context):
A state variable depends only on the system’s current equilibrium state,
not on the path taken to reach it.
In equilibrium, entropy is one such state variable: its value is uniquely
determined by macroscopic variables like T, P, V.

Entropy in Nonequilibrium Systems
Local Entropy:
  Out of equilibrium, we can often define a local entropy density s(r,t)
based on local variables (if local equilibrium holds).
  S_total = ∫ s(r,t) dV
  but this depends on internal gradients, flows, and structure — no single
global state function S(T,P,V) describes the system.

Entropy Production:
  In nonequilibrium systems, the total entropy changes due to both exchange
and internal production:
  dS = δQ/T + dS_gen
  where dS_gen > 0 for irreversible processes (e.g., viscous dissipation,
heat conduction).

Negentropy and Free Energy:
  Systems far from equilibrium possess negentropy — a measure of their
potential to do work or sustain structure.
  Free energy reflects this potential:
    F = U - T S
  More negentropy → more free energy → greater capacity for work or
structure formation.

Closed Path Integrals:
  In equilibrium: ∮ dS = 0 (S is exact differential)
  Out of equilibrium: ∮ dS = ∮ dS_gen > 0 (irreversibility generates
entropy)

Summary:
  In equilibrium, entropy is a state variable with a unique value for each
state.
  Out of equilibrium, entropy production, local entropies, and negentropy
describe the system’s internal structure and potential to do work.
  The simple equilibrium picture no longer applies.
------------------------------


____________________________________________
CEO Founder, Simtable.com
stephen.guerin at simtable.com

Harvard Visualization Research and Teaching Lab
stephenguerin at fas.harvard.edu

mobile: (505)577-5828

On Mon, Jun 16, 2025, 2:02 PM Frank Wimberly <wimberly3 at gmail.com> wrote:

> You've probably done this. Nick.
>
> Yes, entropy (S) is a state variable in thermodynamics.
> Here's what that means:
>  * Definition of a State Variable: A state variable (or state function) is
> a property of a thermodynamic system that depends only on the current state
> of the system, not on the path or process taken to reach that state. If a
> system is in a particular equilibrium state, its state variables will have
> specific, unique values.
>  * Why Entropy is a State Variable:
>    * Path Independence: The change in entropy between two states is the
> same regardless of the reversible or irreversible path taken to go from the
> initial state to the final state.
>    * Unique Value for a Given State: For any given equilibrium state of a
> system (defined by other state variables like temperature, pressure, and
> volume), there is a unique value of entropy.
>    * Mathematical Property: Mathematically, a quantity is a state variable
> if its differential is an exact differential, meaning that its integral
> over a closed path is zero. This is true for entropy.
>  * Contrast with Path Variables: In contrast, quantities like heat (Q) and
> work (W) are not state variables. The amount of heat transferred or work
> done depends entirely on the specific path followed during a process.
> In summary, entropy is a fundamental property that characterizes the state
> of a thermodynamic system and is independent of its history.
>
>
> ---
> Frank C. Wimberly
> 140 Calle Ojo Feliz,
> <https://www.google.com/maps/search/140+Calle+Ojo+Feliz,++Santa+Fe,+NM+87505?entry=gmail&source=g>
> Santa Fe, NM 87505
> <https://www.google.com/maps/search/140+Calle+Ojo+Feliz,++Santa+Fe,+NM+87505?entry=gmail&source=g>
>
> 505 670-9918
> Santa Fe, NM
>
> On Sun, Jun 15, 2025, 9:27 AM Nicholas Thompson <thompnickson2 at gmail.com>
> wrote:
>
>> Same setup as before. Cylinder with two plungers and a peggable slider at
>> dead center.  Lets fill our compartments with bbs of the same volume. Unpeg
>> the divider.  Now lets tap on the two opposite plungers with a series of
>> blows of the same average magnitude but different but different positive
>> skew in the distribution of magnitudes. Will the divider move?
>> --
>> Nicholas S. Thompson
>> Emeritus Professor of Psychology and Ethology
>> Clark University
>> nthompson at clarku.edu
>> https://wordpress.clarku.edu/nthompson
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