Talk:Introduction to entropy/Archive 3 Source: en.wikipedia.org/wiki/Talk:Introduction_to_entropy/Archive_3

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There is at least one objection above to leading off with the statistical-mechanical description. There is also at least one insistence that motion be mentioned in the first paragraph. If this article is to be about entropy in general — including the popular concept and information entropy — then it's inappropriate to lead off with a purely thermodynamic account. The statistical-mechanical account applies to both thermodynamics and information, and it explains how the physical concept led to the popular concept. That it "explains but does not define" thermodynamic entropy is a subtlety that seriously gums up the mission of getting entropy across to the English major or the financial analyst. I urge certain editors, for the nth time, to get over it in deference to that mission. If they absolutely cannot, perhaps the distinction can be addressed in the body of the article. -Jordgette [talk] 15:22, 5 December 2020 (UTC)

The words “subtlety that seriously gums up the mission” and “in deference to that mission” remind me that some time ago, Editor Jordgette put his cards on the table thus: “Ironically your argument supports starting this article by describing entropy as a measure of disorder, which is far and away the most intuitive way to describe what it is.” I don't know if he still thinks or intuits so. Such a respectable source as Edwin Thompson Jaynes had an alternative view when he wrote “Glib, unqualified statements to the effect that "entropy measures randomness" are in my opinion totally meaningless, and present a serious barrier to any real understanding of these problems.” I guess intuition and understanding are different.

Assuming that the article is to be re-named to become about entropy in general, it isn't evident to me precisely whither leads Editor Jordgette's just above collection of opinions. In particular, it doesn't mention the most general conception of entropy, that of the mathematical theory of dynamical systems, just a little while ago relegated to the archive. The comments, such as this one, of our IP-only mathematician friend, 67.198.37.16, would return to relevance if the article were re-named to become about entropy in general.Chjoaygame (talk) 00:35, 6 December 2020 (UTC)

Perhaps the 'spread' interpretation (not definition) has not caught on, but it has entered the mainstream. The 'disorder' doctrine has long dominated, but has led to the present situation, in which this article has been a source of complaint for over a decade. Does that remind me of some fable about doing the same thing and expecting a different result? The 'disorder' story may appeal to many, but it is another question as to whether it offers useful understanding; Jaynes and others think not. Amongst those who have taken the spread interpretation seriously, some have found it helpful for teaching, which has a priority in this article. For me, the 'disorder' interpretation is strained or baffling, while I find the 'spread' interpretation natural and enlightening. Is Wikipedia physics a stronghold for the kind of Groupthink that is characterized by active suppression of alternative viewpoints, or does Wikipedia policy favour a neutral point of view?

The beginning of a list of texts using the energy dispersal interpretation

Atkins, P. W., de Paula J. Atkins' Physical Chemistry, 2006, W.H. Freeman and Company, 8th edition, ISBN 9780716787594

″We appear to have found the signpost of spontaneous change: we look for the direction of change that leads to dispersal of the total energy of the isolated system. This principle accounts for the direction of change of the bouncing ball, because its energy is spread out as thermal motion of the atoms of the floor.″

Bell, J., et al., 2005. Chemistry: A General Chemistry Project of the American Chemical Society, 1st ed. W. H. Freeman, 820pp, ISBN 0-7167-3126-6. I would need a trip to the library to get a copy of this.

Brown, T. L., H. E. LeMay, B. E. Bursten, C.J. Murphy, P. Woodward, M.E. Stoltzfus 2017. Chemistry: The Central Science, 10th ed. Prentice Hall, 1248pp, ISBN 9780134414232

"entropy A measure of the tendency for energy to spread or disperse, thereby reducing its ability to accomplish work. In a general sense, it reflects the degree of randomness or disorder associated with the particles that carry the energy." Editor PAR rightly observes that energy spreads not only in ordinary space, but more accurately also in phase space, but that does impugn use of the spreading idea when it may help. Brown et al. admit the compatibility of the 'disorder' and the 'dispersal' interpretations. So could we.

Ebbing, D.D., and S. D. Gammon, 2017. General Chemistry, 11th ed. Centage Learning 1190pp, ISBN 9781305580343

"Entropy (S) a thermodynamic quantity that is a measure of how dispersed the energy of a system is among the different possible ways that a system can contain energy."

Petrucci, Herring, Madura, Bissonnette 2011 General Chemistry: Principles and Modern Applications, 10th edition, 1426 pages, Pearson Canada ISBN 9780132064521

"Entropy, S, is a thermodynamic property related to the number of energy levels among which the energy of a system is spread. The greater the number of energy levels for a given total energy, the greater the entropy."

Perhaps I may seem lazy, but I became bored with this exercise.Chjoaygame (talk) 07:41, 6 December 2020 (UTC)

Just a comment. The fact that Atkins' Physical Chemistry uses the energy dispersal interpretation is important in the UK and a few other countries included Australia, where I am, because it is a text book that is very widely used in chemistry departments. Perhaps chemists are now happier with this interpretation than are physicists. However I have been retired for 17 years now. --Bduke (talk) 08:11, 6 December 2020 (UTC)

The question isn't whether or not spread is a scientifically valid interpretation of energy. Clearly it is (although I'll note again that this interpretation can be explained in light of the statistical mechanic explanation, but not vice versa). The question is whether this interpretation is a useful way to explain the popular interpretation of entropy, which is "disorder," whether we like it or not; my argument has been that it is not. The necessity of this argument stems from the apparent consensus that this article should provide a useful introduction to the scientific idea of entropy to someone who only has some hazy notion of the popular conception. It would be helpful if you could engage with these arguments in some way. DrPippy (talk) 15:30, 6 December 2020 (UTC)

• Replying to Editor DrPippy. Thank you for your comment, Editor DrPippy.

You have asserted that "it is not". I have twice asked you how has the 'spread' interpretation worked in your teaching practice, but you have not replied. I don't see your assertion as amounting to an argument.

The 'popular conception', as you observe, is indeed "hazy". To shift from it into some scientific understanding is our task. In the literature written by physicists who have tackled the matter explicitly, the discussion usually concludes that the 'disorder' account is scientifically almost meaningless. It is true that the 'disorder' account is recited in many textbooks, but few actually try to do more than recite it, or to show explicitly how it has scientific meaning. The 'disorder' account works at a handwaving or loosely metaphorical level, but scarcely more. The notion of 'spread' has obvious physical meaning that is not merely handwaving. For example, Planck says of his thermodynamic systems that they are "chiefly ... homogeneous". 'Evenly spread' would pass for an ordinary language rendition of 'homogeneous'. The spread in entropic motion is not perfectly even, beyond equipartition, and that is why I propose the words 'diversity of motion'. The motions of microscopic bodies obey the laws of physics; to refer to such obedience simply as 'disorder' doesn't quite cover it. As is evident in what I have written above, I am not urging expungement of the 'disorder' metaphor. I am, however, putting the case for the more physical 'spread' language as well.Chjoaygame (talk) 16:46, 6 December 2020 (UTC)

• DrPippy is of course correct, and I would (once again) urge the two minority-opinion editors on this page to devote their energy to other entropy articles, where the mission to connect the popular concept with the physical concept won't continue to be disrupted.

DrPippy, I appreciate your proposal for the introductory paragraphs (dated 2 December[1]); perhaps that is the direction we want to go in. However, entropy is a physical concept first and a popular concept second, derived from the physical concept. So I think we have to lead with the physical concept. The question is how to do that without making the first few sentences highly unpalatable to the lay reader. -Jordgette [talk] 16:25, 6 December 2020 (UTC)

Sounds like the discussion is getting bogged down in generalities again. Chjoaygame continues his "wall of words" replies in which he appeals to "authorities" such as Edwin Thompson Jaynes, Edward A. Guggenheim, Peter Atkins, and college textbook authors. WP:TLDR territory. I don't see these as too relevant to this article, at least not the introductory sections. These scientists were writing for other scientists, physicists and chemists, or students of physics and chemistry at the university level. We are writing for readers at a lower level, who have only taken high school or middle school science or maybe no science at all. That means tailoring explanations to their level of understanding, as I think Jordgette and DrPippy are trying to do. We don't need to be limited to one approach, or form of words. Chjoaygame, I think if you want to appeal to textbooks for wording, the appropriate ones are high school or middle school textbooks.

I would say a more productive use of time would be to start editing the article WP:BRD. One or more editors could write a new introduction. Then arguments could be about specific wording, or whether their approach should be abandoned. --Chetvorno^TALK 18:41, 6 December 2020 (UTC)

It is one thing to snip with the scissors, another to choose a good cloth.Chjoaygame (talk) 18:54, 6 December 2020 (UTC)

As per previous discussions, a distinction MUST be made between thermodynamic entropy (S, a state variable of a thermodynamic system), information entropy (Log[W], a measure of the "spread" of some probability density function), and the statistical mechanics EXPLANATION of thermodynamic entropy, (S=k Log[W], which links the two by assuming that matter is composed of particles, and that every microstate available to a system in a given macrostate is equally probable. The statement that "Entropy is a numerical quantity that measures the number of different ways that the constituent parts of a system can be arranged to get the same overall arrangement" is not good. The "Entropy" referred to here is the information entropy as applied to a system of particles assuming each available microstate is equally probable. It is not thermodynamic entropy, but is related to thermodynamic entropy via Boltzmann's statistical mechanical theory (S=k Log[W]). Thermodynamic entropy is defined by the second law, and the laws of thermodynamics have NOTHING to do with any statistical mechanical concepts and should not be interpreted as such. PAR (talk) 18:19, 11 December 2020 (UTC)

Quoting: "assuming each available microstate is equally probable." The physical problem is to so formulate or specify the microstates that every available one has one and the same probability.Chjoaygame (talk) 10:12, 13 December 2020 (UTC)

This distinction is not rocket science, it's not some esoteric knowledge understandable only to those who have studied these subjects for years. A twelve year old can understand the difference between learning to drive a car (analogous to thermodynamics), and understanding the inner workings of a car which explains why it drives the way it does (analogous to statistical mechanics). PAR (talk) 18:19, 11 December 2020 (UTC)

I agree with PAR that the thermodynamic definition of entropy in terms of state variables should be mentioned in the introduction. Maybe something like: "In thermodynamics entropy is defined as the fraction of internal energy that is unavailable for doing work divided by the temperature". I don't agree with PAR that the thermodynamic concept of entropy has nothing to do with the statistical mechanics concept, but explaining the relationship between the two is probably too involved for the introduction and should be kept in the article body. --Chetvorno^TALK 18:44, 11 December 2020 (UTC)

I didn't mean to imply that the thermodynamic concept of entropy has nothing to do with the statistical mechanics concept. They are most definitely related, by Boltzmann's S=k Log[W] equation. My point was that thermodynamic entropy does not refer to, nor does it need to refer to, any statistical mechanical concept, in order to be completely defined. Boltzmann's equation is not an identity, it is a theory, one that is exceedingly successful in explaining thermodynamic entropy, but it provides two equivalent definitions of thermodynamic entropy in theory only, not in fact. The first and second laws of thermodynamics are a collection of facts organized by the principles of conservation of energy and non-decreasing thermodynamic entropy. They make no reference to any statistical mechanical concept. Statistical mechanics is an extremely successful theory which explains these facts. If any statistical mechanical theory is ever at odds with thermodynamics, then it is simply wrong, and not vice versa. PAR (talk) 19:23, 11 December 2020 (UTC)

Then it looks like we're back to square one on the intro. I was hoping that we could find a way to introduce entropy to people who have no idea what thermodynamics, or internal energy, or work are — but apparently that's simply impossible. I've done what I can, and I'm really not interested in litigating this any further. I'll leave the animation here for review upon its completion, and interested editors can use it or not use it in the article as they please. -Jordgette [talk] 19:26, 11 December 2020 (UTC)

I don't think people need to understand thermodynamics, internal energy, or work in order to understand irreversibility, and that's what is in the introduction. I feel like I am banging my head against a wall, it's so simple. Thermodynamic entropy is about irreversibility, and you can explain irreversibility to a twelve year old, by offering examples. Popular attempts to describe entropy like disorder, spreading, lost information, are all useful but ultimately do not hit the mark. We can use them to give a feel for entropy, but to declare them to be the final word is to lie.

I'm the one who objects to lying in the name of simplification when we don't need to. You are the one who has held people's feet to the fire demanding simplicity, and rightly so. You are the one who made me understand that these popular (mis)conceptions of entropy must be addressed and addressed quickly in the introduction, in order to let people who have these (mis)conceptions know they are on the right page. In what way are we not threading that needle? I've tried, and I wish you would not give up on this. I know, this discussion gets tiring after a while, listening to me and Chjoaygame and Chetvorno and Dr. Pippy and BDuke sounding like we argue about how many angels can dance on the head of a pin, but the bottom line is that we are trying to thread that needle. Please, just understand what the introduction is saying, and point out your objections and explain the flaws. People learn entropy as a mushed up mess of thermodynamics, statistical mechanics, and flawed popular takes on entropy, and then simply won't let go. I will let go if I am wrong, but the only way that happens is a reasoned discussion/argument. PAR (talk) 21:01, 11 December 2020 (UTC)

The reason the discussion gets "tiring" is that you are talking in generalities rather than proposing concrete improvements to the article. What specific sentences do you object to in Jordgette's version above and what would you change them to? --Chetvorno^TALK 02:09, 12 December 2020 (UTC)

• The first paragraph: "Entropy is a concept in physics, specifically the field of thermodynamics, that has entered popular usage and is also employed in information theory. Entropy is a numerical quantity that measures the number of different ways that the constituent parts of a system can be arranged to get the same overall arrangement." This is so wrong in a number of ways. There is thermodynamic entropy and there is information entropy. They are two different things, yet the distinction is destroyed in this sentence by referring to "entropy" as if they were the same thing. Thermodynamic entropy is "a concept in physics, specifically the field of thermodynamics". Thermodynamic entropy is defined by a measurement of thermodynamic variables on a real system. It is not "employed" in information theory. Information entropy is a measure of the "spread" of a hypothetical probability distribution. Information entropy, combined with the atomistic assumption and the assumption of equal probability of microstates, yields the statistical mechanical entropy - "a numerical quantity that measures the number of different ways that the constituent parts of a system can be arranged to get the same overall arrangement." If we assume that matter is composed of particles, we can define a microstate and a macrostate and if the probability of every microstate is assumed to be equal (=1/W), then we can define a statistical mechanical entropy k Log[W] and propose that it is equal to the thermodynamic entropy. These need not be esoteric concepts. Thermodynamic entropy reveals itself to our senses qualitatively and quantitatively as irreversibility. Easy to understand, given a few examples. The more difficult part is the statistical mechanical explanation of thermodynamic entropy. We have to introduce microstates and macrostates in as simple a way as possible, and show how the microstate is constantly changing, and propose that each microstate is equally probable. Now we can show by simple examples how the macrostate is almost always the equilibrium state, which presents itself to us as an unvarying macrostate. This will hopefully demonstrate the core conclusions of statistical mechanics.

• The second paragraph: A further description of information entropy applied to statistical mechanics. As long as it doesn't imply that this is a representation of thermodynamic entropy, or that entropy is completely explained by the spreading process, that's fine.

• the third paragraph - a conceptual link between thermodynamic entropy and stat mech entropy is introduced. That's fine, but it's out of order.

• the fourth paragraph - Finally, thermodynamic entropy is introduced. This should come first.

• The fifth paragraph introduces statistical mechanics - a subject covered by most of the above paragraphs. Statistical mechanics should be introduced after thermodynamics, and then described. Then there is the statement: "Later, Claude Shannon applied the concept of entropy to information, such that the entropy of a message transmitted along a wire can be calculated." This statement is totally false. Shannon, working on information theory, developed a measure of (missing) information, which he called "information entropy". It is a measure of the "spread" of a hypothetical probability distribution function. He applied that to the practical problem of message transmission. In retrospect, this same information entropy was applied by Boltzmann in his development of statistical mechanics. It is just the "Log(W)" in his famous equation.

In contrast, I believe the present introduction avoids these problems and roughly attempts to present things in a simple, logical order. Please list your objections to the present introduction. PAR (talk) 06:22, 12 December 2020 (UTC)

I'm fundamentally at odds with PAR's position at this. The laws of thermodynamics emerge from statistical mechanics, not the other way around. The reason that entropy increases is fundamentally due to the statistical properties of large systems of particles; if thermodynamics didn't have the Second Law, statistical mechanics would demand it. If I'm understanding PAR's point about the relationship between thermo and stat mech (e.g., "If any statistical mechanical theory is ever at odds with thermodynamics, then it is simply wrong, and not vice versa."), it simply amounts to an argument that theories have to conform to observations. This is not in dispute, but doesn't really provide any useful guidance for why the thermodynamic view of entropy should take precedence over the statistically mechanical view. The fact of the matter is that stat mech entropy explains thermo entropy; the reverse is not true. Nor does thermo entropy provide a particularly clear explanation of why the second law implies irreversibility, tendency to disorder, etc.; the stat mech version does.

Thermodynamics says that there's some quantity defined by dS=dQ/T, and that dS>=0 for any spontaneous process in an isolated system, etc. This is sort of like observing that planetary orbits are elliptical. This observation is explained by Newton's law of gravity, which also explains a number of other phenomena. Arguing that we should start introducing the concept of entropy with the narrower thermodynamic definition strikes me as similar to arguing that we should start an introduction to gravity by talking about the shape of planetary orbits rather than with Newton's law.

I also think that the stat mech version is potentially easier to understand. I'm not sure that there's a way to frame entropy as a measurement of heat that's unavailable to do work (or whatever version of this you prefer) that isn't going to feel pretty abstract to the uninitiated; at least, I haven't heard one that really works for me. If we don't mind sacrificing some degree of technical rigor (and I think that's okay in this context), we could simply say that entropy is a measurement of the probability that a system will find itself in some particular state/arrangement/configuration, and thus the second law is simply a result of a system in a relatively improbably state moving to increasingly probably states as the deck is slowly shuffled, so to speak. In fact, I think I would prefer something along these lines even to the "number of arrangements" wording that Jordgette and I have been working with. (This definition is technically wrong, but it's wrong in sort of the same way that Newtonian gravity is wrong compared to GR, and you definitely wouldn't want to start off a layman's introduction to gravity by talking about geodesics in curved spacetime and all that.)

TL;DR version: stat mech provides an explanation of the concept of (physical) entropy which is more explanatory, more intuitive, and more fundamental than thermodynamics does, so we should lead with that. I would prefer to explain the entropy in terms of probability instead of multiplicity, but I feel less strongly about that.

We seem to be at a bit of an impasse: I think Jordgette and I are more or less on the same page, and possibly Bduke as well; PAR and Chjoaygame are not on that page, but their objections seem to some from different directions. Is it time to pursue some sort of dispute resolution process (RfC, etc.)? DrPippy (talk) 14:26, 12 December 2020 (UTC)

Excellent - Thank you for a clear explanation of your point of view that recognizes the difference and the relationships between thermodynamic entropy and statistical mechanical entropy. I like the analogy with planetary motion. I think (maybe) this list a set of things we can agree on:

• Thermodynamics is a phenomenological theory, a condensation of experimental, macroscopic facts, which are constrained by the phenomenological laws of thermodynamics. It defines thermodynamic entropy. This situation is similar to the theory of planetary motion at the time of Kepler - a series of measurements which showed that planetary orbits were elliptical, and constrained by Keplers three phenomenological laws of planetary motion.

• Statistical mechanics is a theory which, using information theory and the idea that matter is particulate, and that microstates corresponding to a particular macrostate all have equal probability, and that energy and particle number is conserved, is able to explain and clarify thermodynamic entropy, and the laws of thermodynamics. Statistical mechanics defines a statistical mechanical entropy (k Log(W)) and equates it to the thermodynamic entropy (S). This situation is similar to Newton's law of gravitation, which explains and clarifies Keplers laws.

• Information theory is a mathematical abstraction which can be applied to many different situations involving probability distributions. Information entropy is also a mathematical abstraction. It is somewhat similar to the integral and differential calculus that Newton invented to deal with his theory of gravitation, but which can be applied to a much broader set of problems.

If you find these statements agreeable, then the thing we disagee on is which theory, thermodynamics or statistical mechanics, comes a priori and which comes a posteriori, both conceptually and from a pedagogical (teaching) "introduction to entropy" point of view. Good, that we can argue about. My point of view is that thermodynamic entropy comes first (a priori), then statistical mechanics (a posteriori), both historically, conceptually and pedagogically. You disagree, on the last two and I can't "prove" you to be wrong. Have I described the situation accurately? If so, this is a lot of progress, and our point of contention is clarified.

PS - I found a very interesting web page that I keep referring back to: [2]. Although it roughly supports my present point of view, your above explanation causes me to question my present point of view, so I take this article as informative rather than definitive. See especially Einstein's quote on constructive theories versus theories of principle. PAR (talk) 00:16, 13 December 2020 (UTC)

Quoting from that article: "It is well accepted that protein and lipid self-assembly is a direct consequence of the second law of thermodynamics." It doesn't violate the second law, but I think it presumes a lot to say "direct consequence".

Here, Editor PAR is referring to Einstein's inversion of Boltzmann's formula. Einstein also allowed fluctuations of 'entropy' at times. We may also remember that Einstein thought that classical thermodynamics, within its scope of applicability, will never be overthrown.Chjoaygame (talk) 01:39, 13 December 2020 (UTC)

That is an interesting article! I'll need to go back and look at it more closely. I'd nitpick a little bit regarding the extent to which statistical mechanics borrows from information theory (which I unfortunately don't know much about). As I understand it, though, information theory has some version of "entropy" which has a similar mathematical form to stat-mech entropy, but there's no version of the second law for info theory. Physical entropy, though, is important (as I understand it) precisely because of the second law, which explains a lot of things ranging from the direction of heat flow to the ultimate fate of the universe. So I'm a little leery of trying to draw interpretive connections between physical and informational entropy, because it's not immediately obvious what those connections should be, or even if they exist beyond the fact that they look mathematically similar.

With all that said, I think there's probably a way to start with the thermodynamic version of entropy and then move from there to the stat-mech version, and certainly that's what happened historically. I'm very happy to wait and see what y'all come up with and then offer critiques and suggestions; this is probably fairer and more constructive than me focusing on potential conceptual problems with the approach. Regardless of how it turns out, it's been a fun and interesting conversation! DrPippy (talk) 15:29, 17 December 2020 (UTC)

Good to have your comment. Glad to hear you've found it fun.

Agreed that statistical mechanics doesn't borrow from information theory. As I see it, information theory provides a fresh way of reading or interpreting or expressing the ideas of statistical mechanics. As you say, this is largely because they use the same mathematical formulas. The link is in combinatorics.

Disagree that the second law explains things. For me, it just describes them, using language derived from the distinction between heat and work. Strongly reject the widely made proposal that the second law has much useful to say about the fate of the universe.Chjoaygame (talk) 16:05, 17 December 2020 (UTC)

@Dr. Pippy - Again, thank you for putting a fine point on the nature of our disagreements. Because of that, and the article I mentioned, I have been practicing the "statistical mechanics is a priori" approach. I still ultimately don't like it, but it took some blinders off. Also, statistical mechanics most certainly does "borrow" from information theory:

A quick take on info entropy (H) in statistical mechanics: In information theory, for the general discrete case:

${\displaystyle H=-\sum _{i=1}^{W}p_{i}{\textrm {Log}}(p_{i})}$

where p_i is the probability of the i-th outcome, whatever you choose that to be: Dice roll, message, microstate, whatever. There are a total of W outcomes so that:

${\displaystyle \sum _{i=1}^{W}p_{i}=1}$

As applied in statistical mechanics, each microstate (outcome) is assumed to be to be equally probable, so that, by the second equation, p_i = 1/W. Plug that into the first equation and you get the information entropy H = Log(W), which is the same Log(W) in Boltzmann's equation S = k Log(W). PAR (talk) 16:39, 17 December 2020 (UTC)

Boltzmann's equation is such a beautiful thing - it unites the information entropy, Log(W), Boltzmann's constant k, and the theoretical statistical mechanical entropy k Log(W) which it equates to the thermodynamic entropy S. PAR (talk) 17:13, 17 December 2020 (UTC)

Since we are going into detail, acccording to Cercignani,^[1] Boltzmann knew and published the formula,

${\displaystyle H=-\sum _{i=1}^{W}p_{i}{\textrm {Log}}(p_{i})}$,

though that fact is not widely celebrated. (Very tied up right now, please give me till tomorrow to find page number.)Chjoaygame (talk) 19:37, 17 December 2020 (UTC)

Quoting from Cercignani, p. 8, quoting Boltzmann: "... erroneous to believe that the mechanical theory of heat is therefore afflicted with some uncertainty because the principles of probability theory are used. ... It is only doubly imperative to handle the conclusions with the greatest strictness."

Quoting from Cercignani, p. 8, commenting on Boltzmann: "But he also seemed to think that he had obtained a result which, except for these fluctuations, followed from the equations of mechanics without exception."

I don't recall us mentioning the origin of the 'disorder' doctrine. Perhaps this quote from Cercignani, p. 18, may help, though I think it isn't enough to settle the matter: "In 1877 he published his paper “Probabilistic foundations of heat theory”, in which he formulated what Einstein later called the Boltzmann principle; the interpretation of the concept of entropy as a mathematically well-defined measure of what one can call the "disorder" of atoms, which had already appeared in his work of 1872, is here extended and becomes a general statement."

Coming to the present point. On page 18, Cercignani writes "In the same year [1877] he also wrote a fundamental paper, generally unknown to the majority of physicists, who by reading only second-hand reports are led to the erroneous belief that Boltzmann dealt only with ideal gases; this paper clearly indicates that he considered mutually interacting molecules as well, with non-negligible potential energy, and thus, as we shall see in Chapter 7, it is he and not Josiah Willard Gibbs (1839-1903) who should be considered as the founder of equilibrium statistical mechanics and of the method of ensembles."

On page 55, Cercignani quotes Boltzmann: "The assumption that the gas-molecules are aggregates of material points, in the sense of Boscovich, does not agree with facts." Boltzmann knew, from spectroscopy, that atoms must be intricately complex objects.

On page 64, Cercignani writes: "Thermodynamics ... can be regarded as a limitation of our ability to act on the mechanics of the minutest particles of a body ..." I think this is a wise statement. My reason is that it does not appeal to probability. Maxwell's demon has the ability that we lack.

On page 82, Cercignani writes about Carnot: "Essentially he saw that there was something that was conserved in reversible processes; this was not heat or caloric, however, but what was later called entropy." This might help us talk about the relation between (ir)reversibility and entropy, a matter that Editor Chetvorno has raised. The characteristic of entropy is that it increases as a result of a thermodynamic process. The pure mode of entropy generation is in heat transfer. In contrast, ideally pure work transfer generates no entropy. (At the risk of being accused of some crime, I may say that work is like heat transfer from a body that is at infinite temperature: ${\displaystyle \delta Q/T=0\,}$; such transfer can heat any body in the surroundings. To extract all the internal energy of a body as work, we need a heat reservoir at zero temperature.)

On page 83, Cercignani writes: "The first attempts at explaining the Second Law on the grounds of kinetic theory are due to Rankine [22, 23]." Rankine (I forget the exact date, but about 1849 or 1850) used a quantity that he called "the thermodynamic function", later called 'entropy' by Clausius.

On pages 83–84, Cercignani writes: "Boltzmann himself makes his first appearance in the field with a paper [25] [1866] in which he tries to prove the Second Law starting from purely mechanical theorems, under the rather restrictive assumption that the molecular motions are periodic, with period ${\displaystyle \tau }$, and the awkward remark, which might perhaps be justified, that “if the orbits do not close after a finite time, one may think that they do in an infinite one”. Essentially, Boltzmann remarks that temperature may be thought of as the time average of kinetic energy, while heat can be equated to the average increase in kinetic energy; if we compute the unspecified period from one of the relations and substitute the result into the other, it turns out that the heat divided by the temperature is an exact differential. This part of the paper appears to be a rather primitive justification of the first part of the Second Law; as for the second part, Boltzmann's argument belongs more to pure thermodynamics than to statistical mechanics and leads to the conclusion that entropy must increase in an irreversible process." I guess that ${\displaystyle \tau }$ is the Poincaré recurrence time.

My automatic tldr alarm is ringing loudly and I will stop for now.Chjoaygame (talk) 16:04, 18 December 2020 (UTC)

Here is a comment from the point of view of Jaynes's 'mind projection fallacy' argument.

The particles can't read the observer's mind. So the physical question is not 'when and how much will the particles decide to move randomly so as to violate the laws of motion?' No, the physical question is 'at what stage of the Poincaré recurrence cycle will the observer choose to observe the body in its state of thermodynamic equilibrium?' That tells us the physical source of the 'subjectivity' of the probabilistic view of entropy.Chjoaygame (talk) 18:57, 18 December 2020 (UTC)

For the microscopic view of a body of matter and radiation in its own state of internal thermodynamic equilibrium, the physical entropy belongs to the trajectory of the constituent particle set through its phase space. Physical entropy measures how the trajectory in the chosen body extends in or partly fills phase space. A trajectory may be characterized by the Lie group of the law of motion that generates it, along with its Poincaré recurrence time. Mathematical entropy is labeled by the Lie group of the law of motion that generates the trajectory. It does not change in time. The observer subjectively or randomly chooses when to observe or sample the trajectory. Statistical mechanics is a mathematical algorithm to find probabilities for the various states which the trajectory will have reached at the respective various sampling times, as well as to calculate the time-invariant entropy of the trajectory.Chjoaygame (talk) 23:24, 18 December 2020 (UTC)

Our task here is to find good ways for the article to introduce ideas about entropy. On page 81, Cercignani offers a thought: "In any case there remained the important unsolved problem of deducing the Second Law of Thermodynamics. As is well known from elementary physics, this principle is often subdivided into two parts, according to whether we consider just reversible processes or irreversible processes as well." Carnot considered the most efficient imaginable heat engine, that conserved entropy, thought of by Carnot as the precious "caloric", not to be wasted, running on the quasi-equilibrium and infeasible Carnot cycle. Later, physicists considered other processes, feasible and producing entropy, wasting energy as heat.

Cercignani remarks on page 84 that Maxwell knew the reason for the second law: “"Hence, if the heat consists in the motion of its parts, the separate parts which move must be so small that we cannot in any way lay hold of them to stop them" [19]. In other words, the Second Law expresses a limitation on the possibility of acting on those tiny objects, atoms, with our usual macroscopic tools.” The reason for the second law is our incompetence and caprice, not nature's imputed gambling. Many of us would prefer to exonerate our incompetence and caprice, and instead convict nature of gambling.

On page 86, Cercignani points to the mathematical trick used in ergodic theory: "As remarked by M.J. Klein [6], Boltzmann interprets Maxwell's distribution function in two different ways, which he seems to consider as a priori equivalent: the first way is based on the fraction of a sufficiently long time interval, during which the velocity of a specific molecule has values within a certain volume element in velocity space, whereas the second way (quoted in a footnote to paper [3]) is based on the fraction of molecules which, at a given instant, have a velocity in the said volume element." I find the first way more physical, the second more statistical. Cercignani continues: "It seems clear that Boltzmann did not at that time feel any need to analyse the equivalence, implicitly assumed, between these two meanings, which are so different."

On page 89, Cercignani tells us that Boltzmann knew that his derivation for ideal gases did not extend to more general substances, and that his H-function is not in general a thermodynamic entropy: "Boltzmann however warned his readers against the illusion of an easy extension of his calculations to the case of more complicated interaction laws." The illusion remains seductive today.

On page 99, Cercignani quotes for us how Boltzmann dealt with the famous objection raised by his good friend Loschmidt: "One therefore cannot prove that, whatever may be the positions and velocities of the spheres at the beginning, the distribution must become uniform after a long time; rather one can only prove that infinitely many more initial states will lead to a uniform one after a definite length of time than to a non-uniform one." Boltzmann assumes that the observation time is prescribed and that the initial microscopic instantaneous state is randomly chosen. Alternatively, randomness can be created from a given initial microscopic instantaneous by the physicist's capricious choice of when to make the final observation. In contrast, physical entropy measures extent of physical diversity of lawful motion of microscopic constituents.

Also on page 99, Cercignani tells us his ideas on Boltzmann's word "order": “We remark that the use of terms such as "ordered states" or "uniformity" may be confusing, since there are various levels of description at which one can consider "order".”

On page 146, Cercignani quotes Poincaré on the same topic: “Thus the notions of molar geordnet (ordered at a molar level) or molekular geordnet (ordered at a molecular level) do not seem, in my opinion, to have been defined with sufficient precision.”

Why would the novice not suffer the confusion felt by Cercignani, Poincaré, Jaynes, Grandy, and others?

On page 100, Cercignani writes: "It is only when we pass to a reduced description, based on the one-particle distribution function, that we lump many states into a single state and we can talk about highly probable (disordered) states; these are states into which, in the reduced description, an extremely large number of microscopic states are lumped together." It takes some intellectual effort to see to the relevance of disorder for physics. Should we demand that from the novice?

On page 104, Cercignani quotes Grad (1961) on the problem of multiple different meanings that people find in the one word 'entropy': "On the other hand, much of the confusion in the subject is traceable to the ostensibly unifying belief (possibly theological in origin!) that there is only one entropy. Although the necessity of dealing with distinct entropies has become conventional in some areas, in others there is an extraordinary reluctance to do so." I think that Cercignani himself at times fails on this.

On page 123, Cercignani tells of Boltzmann's use of the above ${\displaystyle \Sigma \,p_{i}\log p_{i}}$ formula.Chjoaygame (talk) 06:50, 19 December 2020 (UTC)

I did not know that Boltzmann came up with Shannon's entropy before Shannon did. I'm not sure how the rest of the exposition relates to the previous statement.PAR (talk) 07:34, 19 December 2020 (UTC)

According to Cercignani, it's not too widely celebrated. It was new to me when I read Cercignani. We are familiar with Boltzmann's use of the formula, but we didn't know that he used it in considering the canonical ensemble under the name 'holode', before Gibbs.

The rest of the exposition is because I read Cercignani more closely to find the reference and on the way found lots of more or less relevant bits that might be handy for us.Chjoaygame (talk) 09:39, 19 December 2020 (UTC)

• I don't think the relabeling of the section from "Introductory descriptions of Entropy" to "Versions of Entropy" is a good idea. I think there should be two subsections - basically thermodynamic and statistical mechanical approximate takes on entropy. The Statistical mechanical would include what is now "Microstate dispersal" (with corrections, see below), disorder, and missing information. The "Microstate dispersal" section is in the realm of statistical mechanics. PAR (talk) 10:49, 14 December 2020 (UTC)

As you please.Chjoaygame (talk) 12:39, 14 December 2020 (UTC)

• In the "Microstate dispersal" section, the statement "ordinary language interpretation of entropy as 'dispersal of microstates throughout their accessible range'." is wrong. Microstates do not disperse, and Guggenheims original statement "If instead of entropy one reads number of accessible states, or spread, the physical significance becomes clear." does not justify the rewording. I also think that a phrase that contains "... that characterizes the fluid, crystalline oscillatory, phononic, molecular, atomic, subatomic, and photonic structure and composition of bodies of matter and radiation." is not appropriate for an introductory article. PAR (talk) 10:49, 14 December 2020 (UTC)

I have tried to address this.Chjoaygame (talk) 18:25, 14 December 2020 (UTC)

• In the introduction, I think removing the paragraph which introduces and explains equilibrium and reversibility and replacing it with a paragraph that uses those as-yet-undefined terms is wrong. I have re-inserted the lost paragraph before that. PAR (talk) 10:49, 14 December 2020 (UTC)

Defined thermodynamic equilibrium, with a view.Chjoaygame (talk) 18:43, 14 December 2020 (UTC)

As I read you, the re-inserted paragraph is

Entropy does not increase indefinitely. As time goes on, the entropy grows closer and closer to its maximum possible value.^[1] For a system which is at its maximum entropy, the entropy becomes constant and the system is said to be in thermodynamic equilibrium. In some cases, the entropy of a process changes very little. For example, when two billiard balls collide, the changes in entropy are very small and so if a movie of the collision were run backwards, it would not appear to be impossible. Such cases are referred to as almost "reversible". Perfect reversibility is impossible, but it is a useful concept in theoretical thermodynamics.

footnote

I don't like that paragraph because it would let the novice forget that a thermodynamic process is defined by the values of its initial and final state variables, and that, given them, it is allowed to take any path including those far from thermodynamic equilibrium.Chjoaygame (talk) 23:01, 14 December 2020 (UTC)

I agree - I will change it. PAR (talk) 05:43, 15 December 2020 (UTC)

Great.Chjoaygame (talk) 06:12, 15 December 2020 (UTC)

Thank you for that. To save possible re-edits, I will talk here.

As I read it now, the relevant paragraph reads

Entropy does not increase indefinitely. A body of matter and radiation eventually will reach an unchanging state, with no detectable flows, and is then said to be in a state of thermodynamic equilibrium. Thermodynamic entropy has a definite value for such a body and is at its maximum value. When bodies of matter or radiation, initially in their own states of internal thermodynamic equilibrium, are brought together so as to intimately interact and reach a new joint equilibrium, then their total entropy increases. For example, a glass of warm water with an ice cube in it will have a lower entropy than that same system some time later when the ice has melted leaving a glass of cool water. Such processes are irreversible: An ice cube in a glass of warm water will not spontaneously form from a glass of cool water. Some processes in nature are almost reversible. For example, the orbiting of the planets around the sun may be thought of as practically reversible: A movie of the planets orbiting the sun which is run in reverse would not appear to be impossible.

The sentence "Entropy does not increase indefinitely" is problematic. It seems to be a consequence or residue from the previous paragraph, which is itself more problematic, or unacceptable. The problem is here: "which says that in an isolated system (a system not connected to any other system) which is undergoing change, entropy increases over time.^[1]"

note

The technical term 'system' is not ideal for an introductory lead, because it is a technical term. '... an isolated system ... which is undergoing change' is not a thermodynamic system in its own state of internal thermodynamic equilibrium and so does not have a defined thermodynamic entropy. Thermodynamic entropy increases process by process. "coffee can be unmixed" calls for a bit of imagination. Part of the problem is that the mixing was not a spontaneous thermodynamic process; it was driven by "you", an animate agency. Once lit, wood burns by itself. Perhaps it may be convenient to omit the coffee and cream?

Instead of "Irreversibility is described by an important law of nature known as the second law of thermodynamics, which says that in an isolated system (a system not connected to any other system) which is undergoing change, entropy increases over time.[2]", I would suggest instead 'Irreversibility is described by a law of nature known as the second law of thermodynamics, which says that when bodies of matter and radiation interact intimately, their total entropy increases.'

If this is accepted, then the sentence "Entropy does not increase indefinitely" can well be omitted.

If this is accepted, then the second statement of the second law can be omitted.

Adressing some details:

"An isolated body of matter and radiation eventually will reach an unchanging state ..."

"Thermodynamic entropy has a definite value for such a body. and is at its maximum value" The words "and is at its maximum value" are problematic. They don't say with respect to which constraints is the maximum. The novice can scarcely do better than to guess. I propose to omit them.

"... a glass of warm water with an ice cube in it will have a lower entropy than that same system some time later when the ice has melted leaving a glass of cool water." A problem is that "a glass of warm water with an ice cube in it" is not a thermodynamic system in its own state of internal thermodynamic equilibrium and so has no defined entropy. A thermodynamic process needs a thermodynamic operation to start. I suggest 'For example, when an ice cube is put into a glass of warm water, and allowed to melt, leaving a glass of cool water, then the total entropy increases.'

"Such processes are irreversible: ..." I would write 'Such a process is irreversible: ...'

"... would not appear to be impossible." I would write '... would appear to be possible.'

Assembling this, here is a proposal.

The word 'entropy' has entered popular usage to refer a lack of order or predictability, or of a gradual decline into disorder.^[1] Physical interpretations of entropy refer to spread or dispersal of energy and matter, and to extent and diversity of microscopic motion.

In thermodynamics, entropy is a numerical quantity that shows that many physical processes can go in only one direction in time. For example, once lit, a piece of wood burns by itself, and doesn't "unburn". Again, when an ice cube is put into a glass of warm water, and allowed to melt, it leaves a glass of cool water; the ice cube will not re-form, leaving itself in warm water. Such one-way processes are said to be irreversible. Some processes in nature may for many purposes be regarded as reversible, for example, the orbiting of the planets around the sun. If you reversed a movie of burning wood, you would see things that you know are impossible in the real world. On the other hand, run in reverse, a movie of the planets orbiting the sun would appear to be possible.

An isolated body of matter and radiation, in an unchanging state, with no detectable flows, is said to be in a state of thermodynamic equilibrium. Thermodynamic entropy has a definite value for such a body. Irreversibility is very precisely described in terms of entropy by a law of nature known as the second law of thermodynamics, which says that when bodies of matter and radiation interact intimately, their total entropy increases.

While the second law, and thermodynamics in general, is accurate in its predictions of intimate interactions of bodies of matter and radiation, scientists are not content with simply knowing how things behave, but want to know also WHY they do so. The question of why entropy increases in thermodynamic processes was tackled over time by several physicists, regarding bodies of matter and radiation as composed of extremely small particles, such as atoms. A masterly answer was given in 1877 by Ludwig Boltzmann, in terms of a theory that is now known as statistical mechanics. It explains thermodynamics in terms of the diverse microscopical motions of the particles, analysed statistically. The theory explains not only thermodynamics, but also a host of other phenomena which are outside the scope of thermodynamics.

Your thoughts?Chjoaygame (talk) 05:08, 17 December 2020 (UTC)

I see what you are saying. I am thinking of the statistical meechanical entropy, which IS defined for nonequilibrium states. Any state, disequilibrium or not, has a macrostate, and an associated set of microstates, and therefore the statistical mechanical entropy k Log(W) has meaning, even though the thermodynamic entropy S does not. I think you are saying that since we have not yet introduced the stat mech entropy, we should not be talking in stat mech terms. I looked at the Second law of thermodynamics page, and the use of the words "increase" and "decrease" are all over the place. Strictly speaking, they should never be used to describe a thermodynamic entropy change. Only "larger than" or "equal" or "smaller than" should be used. I wonder if this is a case where rigor is a hinderance to the newcomer rather than a help. I have been accused of being too rigorous, but I have my limits. What do you think? signature added: PAR, posted 20:45, 17 December 2020.

Quoting: I looked at the Second law of thermodynamics page, and the use of the words "increase" and "decrease" are all over the place. Strictly speaking, they should never be used to describe a thermodynamic entropy change. Only "larger than" or "equal" or "smaller than" should be used. I wonder if this is a case where rigor is a hinderance to the newcomer rather than a help. I have been accused of being too rigorous, but I have my limits. What do you think?

I am not too worried about this. I guess that increases and decreases can be stepwise or gradual and continuous.

More concerning to me is Editor Chetvorno's comment:

Quoting: But the title of the article is not Irreversibility, it is Entropy, the introduction gives readers no idea of what that is. "...entropy refers to spread of energy or matter, or to extent and diversity of microscopic motion." is not going to mean anything to readers below college level.

Three points here. 'Spread of energy or matter' is incomprehensible. The topic of the article is 'entropy', not 'irreversibility'. The lead does not give an applicable definition of 'thermodynamic entropy'.

The usual Wikipedia custom is that the lead is a summary. I am distinguishing between the lead and a possible section, in the body of the article, entitled 'Introduction'. I see the lead as a kind of guidepost to the body of the article, not containing any more explanatory material or technical detail than is necessary. This view of mine seems a bit at odds with some other views on this page, that seem to want the lead to explain and define in detail. I think when Editor Chetvorno writes "introduction" he is referring to what I call the lead. His three points call for close attention.

Quoting: Any state, disequilibrium or not, has a macrostate, and an associated set of microstates, ...

It depends on what you mean by 'macrostate'. For the general non-equilibrium body of matter and radiation, in general, the thermodynamic state variables, such as temperature, pressure, and in some cases internal energy and suchlike, are undefined. That they are defined in special cases is what makes thermodynamics. The general non-equilibrium state has non-zero flows of matter and energy, and such flows are permitted to be turbulent: hard to define a macrostate then. Boltzmann's H-function is based on a very special non-equilibrium state, and it may be very hard indeed to define precisely a corresponding function, with a suitable sample space or ensemble, for other non-equilibrium states. People look at Boltzmann's H-function, see that it looks like Shannon's function, say to themselves 'Shannon's function has been labeled 'entropy', therefore Boltzmann's H-function is an entropy, therefore I can say that it defines physical non-equilibrium entropy'. As far as I know, the only serious attempt to define a true thermodynamic entropy for non-equilibrium is Phil Attard's hierarchy of multi-time 'entropies'. He wrote to me that he didn't know how to measure them. Futuristic stuff. Off the table for us, I think. In the meantime, I think that it is just a card-trick or word game to speak of 'non-equilibrium entropy' in general as if it had physical statistical mechanical meaning.

Quoting: I think you are saying that since we have not yet introduced the stat mech entropy, we should not be talking in stat mech terms.

I didn't have that thought in mind. Not concerned about it yet.Chjoaygame (talk) 03:42, 18 December 2020 (UTC)

You said: "I am not too worried about this. I guess that increases and decreases can be stepwise or gradual and continuous." But you said you had a problem with the statement "Entropy does not increase indefinitely". PAR (talk) 07:36, 19 December 2020 (UTC)

The problem I see with the sentence "Entropy does not increase indefinitely" is that it relies for its logic on the paragraph previous to it, and that paragraph is faulty. Chjoaygame (talk) 09:26, 19 December 2020 (UTC)

I am concerned that you are referring to a paragraph that has been removed. Could you re-read the previous paragraph and tell me if you have objections to it and, if so, what they are.PAR (talk) 20:32, 19 December 2020 (UTC)

You quoted Chetvorno: "But the title of the article is not Irreversibility, it is Entropy, the introduction gives readers no idea of what that is." This is not true, as seen by the statement "In thermodynamics, entropy is a numerical quantity that shows that many physical processes can go in only one direction in time.". The only way thermodynamic entropy presents itself to our senses and measuring instruments is through its change. Thermodynamic entropy cannot be experimentally measured, only its change. PAR (talk) 07:36, 19 December 2020 (UTC)

I don't mean that Editor Chetvorno's sentence is fully compelling; "no idea" is an exaggeration. I just mean that we should attend to his concern that 'entropy' has not been fully and precisely defined at that point. An attribute of thermodynamic entropy has been declared, but not a fully precise definition. He is perhaps concerned that other attributes of entropy still remain to be declared. It's not immediately obvious how to respond to his concern, but I think some attention is needed. Chjoaygame (talk) 09:26, 19 December 2020 (UTC)

I think a definition of thermodynamic entropy is too much for an introduction section and should go in the explanation section.PAR (talk) 20:36, 19 December 2020 (UTC)

I tend to agree, but I think we need to attend to Editor Chetvorno's concern. I like to distinguish between a lead and a section in the body of the article entitled 'Introduction'. Chjoaygame (talk) 20:49, 19 December 2020 (UTC)

Regarding my statement "Any state, disequilibrium or not, has a macrostate, and an associated set of microstates, .". Yes, I have to amend that statement. It holds for cases where, e.g. the pressure and temperature for each type of particle are defined functions of space and time (local thermodynamic equilibrium). Now entropy density can be defined and total entropy can be said to decrease, assuming we can talk about the rates of energy and entropy tranfer between two equilibrated systems in contact and almost in equilibrium with each other. But the question is, how does this apply to an introductory article? Can we use the words "Entropy does not increase indefinitely" as an introductory statement, knowing that it is flawed? I think the verbal gymnastics we have to go thru to keep things rigorous but simple are almost overwhelming in this case. PAR (talk) 07:36, 19 December 2020 (UTC)

Local thermodynamic equilibrium is a relatively new concept. I think it was introduced by Milne to deal with radiation. I am not sure of its exact history. It was used by others such as Onsager, Prigogine, Gyarmati, and De Groot & Mazur. It's an approximation, reliant on strong assumptions. I think that Prigogine 1947 has conceptual problems. Yes, I agree that it is not for the lead of this articleChjoaygame (talk) 09:26, 19 December 2020 (UTC)

I thought it might be useful to take stock of where we are in terms of points of agreement or disagreement. So here's what I see as three major questions where we need to achieve consensus, together with what I've seen as the likely answers. (I trust y'all will add anything that I've missed here!) Not trying to break any new ground here; just hoping to provide an organizational framework where it might be easier to keep track of the discussion (which I'm having a bit of trouble following, tbh). Hopefully this is something we can !vote on, and it'll make it clearer where things stand. DrPippy (talk) 14:58, 12 December 2020 (UTC)

It may be kept in mind that Wikipedia articles are subject to perpetual re-editing by individual editors.Chjoaygame (talk) 20:33, 19 December 2020 (UTC)

Thats the whole point of achieving a consensus - If six editors can compromise their way to a consensus, that will be six editors who will object to somebody who just read their first pop-entropy book and is determined to "set things straight". PAR (talk) 20:44, 19 December 2020 (UTC)

Some individual editors are not like that. And a consensus is not necessarily needed to knock off such a straight-setting.Chjoaygame (talk) 20:58, 19 December 2020 (UTC)

Option 1: No math at all
Option 2: No math in the introduction, some basic math in the body of the article
Option 3: Basic math throughout, including introduction
Option 4: The article should incorporate an advanced mathematical treatment of entropy; introduction might or might not have some math, but we should end up in the deep end.

• Option 1 on this for me; I think this would hang together better as a purely conceptual explanation of the idea of entropy in various contexts. I'd be okay with option 2 as well. I think any math in the intro is a bad idea. DrPippy (talk) 14:58, 12 December 2020 (UTC)
• Option 2, but keep it at a very simple level in the body.PAR (talk) 00:27, 13 December 2020 (UTC)
• Option 2, agree with PAR --Chetvorno^TALK 20:06, 17 December 2020 (UTC)

Option 1: Thermodynamic
Option 2: Stat mech
Option 3: Informational
Option 4: Popular (disorder, etc.)
Other?: I'm sure I've left some out...

• Options 1, 2, 4 for sure, no opinion on 3. Again, I think the approach of trying to explain the popular view of entropy in terms of the scientific view is exactly the right one; we just need to figure out which version of the scientific view is the right one. DrPippy (talk) 14:58, 12 December 2020 (UTC)
• Options 1,2,3,4, low emphasis on the nuts and bolts of information entropy.PAR (talk) 00:27, 13 December 2020 (UTC)
• Options 1,2,3,4 Since this article's title is Introduction to entropy, we have to define all the types of entropy in the introduction, but I agree Shannon entropy should just be briefly defined WP:SUMMARYSTYLE. --Chetvorno^TALK 20:20, 17 December 2020 (UTC)
• Comment. 'Other' might include or point to general abstract mathematical entropies, that might pretend or intend to 'really capture' the 'underlying concept' of 'entropy'.Chjoaygame (talk) 20:39, 19 December 2020 (UTC)

Option 1: Thermodynamic
Option 2: Stat mech
Option 3: Informational
Option 4: Popular (disorder, etc.)
Option 5: Other/none of the above/all of the above

• Option 2 is my strong preference, for the reasons given above. The short version is that the stat mech picture of entropy (1) explains the popular view; (2) explains the thermodynamic view; (3) explains many other related concepts (diffusion, irreversibility, heat flow, others); and, most importantly (4) does all of these in an intuitive way which is accessible to a popular audience. DrPippy (talk) 14:58, 12 December 2020 (UTC)
• Option 1 is the best way to lead a new reader into an understanding of thermodynamic entropy. Pose the problem (thermodynamic entropy) then the solution (statistical mechanics). Vice versa is backwards. PAR (talk) 00:27, 13 December 2020 (UTC)
• Boltzmann thought that the physics of thermal motion must reflect Newton's laws (we agree, modulo technicalities). Phase space expresses or relies on Newtonian physics. The mathematical problem, expressed in terms of phase space, posed here by Newton's physical laws, taken literally, is formidable, practically overwhelming for Boltzmann and for some generations after him. Based on phase space, probabilistic thinking is a mathematical artifice, relying on ergodic theory, to get around the mathematical difficulties. Some physicists say that 'mathematics is the language of physics'. Thermodynamics is about physical facts. One approach wants ordinary language conceptions, explanation, theory, and mathematical reasoning, to be placed in the article ahead of physical facts. Editor PAR wants to put the physical facts first.

'Entropy' is a word that expresses several more or less distinct concepts that belong to several respective distinct frames of thinking.Chjoaygame (talk) 01:19, 13 December 2020 (UTC)

• I agree with user PAR that we should start with thermodynamic entropy. The reason is that it is more clearly related (in many cases) to familiar macroscopic experience. Most readers know that hot water mixes with cold water to give lukewarm water, or that gases under pressure expand, etc. etc. Thermodynamic entropy gives a simple way of predicting the direction of a process, as introduced by Carnot with purely macroscopic reasoning. Statistical reasoning explains why thermodynamic entropy works, but it requires a fairly detailed molecular picture to understand properly, so in an Introduction article it should be presented after thermodynamic entropy and only briefly. Dirac66 (talk) 02:06, 13 December 2020 (UTC)

• Option 2: The problem with Option 1 is that it gives no understandable definition of entropy. The existing introduction explains thermodynamic entropy is related to irreversibility. I like the approach and I think some of it should be saved in the new introduction. But the title of the article is not Irreversibility, it is Entropy, and the introduction gives readers no idea of what that is. "...entropy refers to spread of energy or matter, or to extent and diversity of microscopic motion." is not going to mean anything to readers below college level. The thermodynamic definition, the change in thermal energy per unit temperature, similarly doesn't give readers any insight into why this should be related to irreversibility (although I think it probably should be in the introduction). Therefore we have to give some kind of statistical mechanics def up front. That, and the diffusion animation, will explain the relation to irreversibility, and also the popular meaning of entropy as disorder. --Chetvorno^TALK 20:06, 17 December 2020 (UTC)