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A while back I was looking into the concept of entropy and trying to understand it.

I am relatively familiar with the statistical mechanical and classical definitions, but I'm not that great at physics, so I'm not sure how firm my grasp is. However, I've been thinking of this sort of information-based interpretation.

equation of entropy as lost work

My interpretation would basically say that the entropy term (T * S) of an energy equation is any energy that is not explicitly accounted for in another way. So the total energy of a system is divided into two categories: 1) information, which is explicitly accounted for as kinetic or potential or internal energy, or 2) entropy, which is "unlabeled" energy.

In this equation, entropy is interpreted as "lost work." My thought is that we can consider "information" to be the energy accounted for in a set of equations that are attributed to explicitly identified entities in the system.

In a heat engine system, the two reservoirs as well as whatever is having work done on it are identified, and they have terms associated with them which account for the energy that is contained within or is a part of them. The energy which is "lost" is precisely the energy which is not flowing into a body which is explicitly referenced in the equation. If it were explicitly referenced, we would consider it to be reversible work done on that body.

In a chemical reaction, it seems the "labeled" elements are the reactants, the products, and the overall system. The heat is considered as work done on the system, and the internal energies of each reactant and product are considered. Any other energy is considered irreversibly lost.

In statistical mechanics, it's a little tougher. Each possible state is labeled, so there is no possibility of something being totally "unlabeled." But the system is only in those labeled states probabilistically, so the labels themselves are uncertain. Knowing which state a system is in allows us to extract energy via something like Maxwell's demon, however if we only know the state probabilistically, then Maxwell's demon has a maximum amount of efficiency. So in this case the extent to which particles are "labeled" is continuous based on how much information is in the probability distribution of its states. However the entropy ends up measuring not how "labeled" the system is but precisely how "unlabeled" it is. A system with maximum entropy will have equal probability of all states, and so the distinction between the different labels will be useless. A system in which any state is equally probably cannot have any energy extracted via Maxwell's demon. So in this case what is being calculated is essentially the accuracy of the labels.

Is this interpretation accepted or valid at all? Is it close to being valid? Is it just dead wrong for obvious reasons that I'm not seeing?

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  • $\begingroup$ Related if not a Duplicate. $\endgroup$
    – Kurt G.
    Aug 25, 2022 at 9:52
  • $\begingroup$ It's related, yes, thanks. I don't think it's a duplicate because to me there's no clear connection between shannon entropy and the choice of the entities being explicitly referenced in the equations. $\endgroup$
    – scl
    Aug 25, 2022 at 10:16
  • $\begingroup$ "My interpretation would basically say that entropy is any energy" "In this equation, entropy is interpreted as 'lost work.'" Unit mismatches (J vs. J/K) immediately preclude both interpretations. That way lies deep confusion. Entropy is a measure of what one can't know about a system, i.e., a measure of how many microstates could provide the macrostate one observes. $\endgroup$ Aug 25, 2022 at 18:07
  • $\begingroup$ I edited my question to correct this mistake. I should have said "the entropy term (T * S) of an energy equation"... not just "entropy" "Entropy is a measure of what one can't know about a system" this seems to match well with what I'm saying... in a classical system this would be anything that doesn't have a variable in the equation representing it, correct? $\endgroup$
    – scl
    Aug 26, 2022 at 0:28
  • $\begingroup$ @Chemomechanics. Perhaps too flippant, but what you said made me think of imgur.com/a/UkoReYi $\endgroup$
    – RC_23
    Aug 26, 2022 at 3:20

2 Answers 2

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https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.129.011602

Entropy is a complicated matter which does not only depend on microstates as in the case for systems like the thermodynamics of a black hole, which only depends on the area. Here in a recent study on hydrodynamics, it is found that there are far from equilibrium states that approaches some constancy despite the driving force is largely nonentropic.

Still this interpretation is interesting... I wonder how many energy loss in physical process are due to unaccounted transformations...

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  • $\begingroup$ "there are far from equilibrium states that approaches some constancy despite the driving force is largely nonentropic" ... wouldn't the equilibrium state being a small subset of available microstates indicate low entropy from a statistical mechanics point of view? $\endgroup$
    – scl
    Aug 25, 2022 at 10:04
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The statistical interpretation of entropy has to do with the number of microscopic possibilities under a macroscopic observation. Here is what this means.

Suppose we know about the system only this: It has $N$ particles, they are contained in a box with volume $V$, and their total energy is $E$. Knowing only this we are asked to guess the precise position and momentum of every particle. Of course we can only guess because the three pieces of information we have, $(E,V,N)$, are not enough to specify the $6N$ position and momentum vectors of the particles. Suppose there are $\Omega(E,V,N)$ possible microstates the system might be in. Statistical mechanics gives this recipe: assume they are all equally probable. Then entropy is $$S = k \ln \Omega(E,V,N)$$ Entropy in this case is associated with our ignorance about the detailed microstate. It is through ignorance that entropy is related to information

What Maxwell's demon can do, but humans cannot, is "see" actual microstates, which is to say, the demon is privy to the complete information about the state of $N$ particles. We humans, on the other hand, can only access incomplete information, for example $(E,V,N)$, or some other combination of a small number of macroscopic variables. The demon can see a hot or cold particle and open or close the gate as needed, and extract useful work out of the motion of particles. We, on the other hand cannot see the speed of particles. We open and close the gate at random, and so we are stuck with the average: for every hot particle we let into the trap, we let one out, so no net work. This is the price for our ignorance.

The lost work of classical mechanics is energy that becomes randomly dispersed in such a way that to extract it we need microscopic knowledge that we don't have. There is a qualitative real-life analog of this. Suppose I write down your phone number, then I tear the piece of paper and throw it in the trash. The information still exists, just not in the form of a piece of paper, rather in a large number of tiny pieces, each at some different spot in the trash can. A demon who has access to the pieces can still call you. To me this information is lost.

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