Talk:Second law of thermodynamics/Archive 8

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bracketed comment in the lead

Editor Klaus Schmidt-Rohr has made some changes to the lead. I am not here commenting on them.

But there is a thing that I am commenting on, with some passion. Please would he very kindly remove the bracketed comment <!--If, as in the old version, it was not specified that there is no heat exchange, this statement would predict incorrectly that the entropy of a hot body in outer space would increase, while in reality the body would cool by radiation and its entropy would decrease spontaneously.--> As a matter of protocol, I am very much opposed to the insertion of comments in that form, no matter how wise and meritorious might be the sentiments they express. I will not bore you with reasons. Still, very much opposed.Chjoaygame (talk) 00:00, 12 February 2016 (UTC)

Please accept my thanks for your kind response.Chjoaygame (talk) 04:52, 12 February 2016 (UTC)

Some reasons: The correct use of such "invisible" comments is explained at MOS:COMMENT. The examples of appropriate comments given there are only technical reminders to editors: if you change X, remember to do Y. Importantly, the examples do not include actual discussion of the article subject, as this one did. Dirac66 (talk) 16:00, 12 February 2016 (UTC)

reversion to older version

Editor Waleswatcher has here practically reverted to an older version of the article. His edit summary reads "The lede is supposed to be at most four paragraphs, define the subject at the very beginning, concisely summarize the article, and in general, not be bloated and rambling."

I disagree with his reversion. I suppose this may be talked over on this talk page.Chjoaygame (talk) 08:53, 3 February 2016 (UTC)

Perhaps I can start to deal with a few issues now.Chjoaygame (talk) 09:54, 3 February 2016 (UTC)

"The lede is supposed to be at most four paragraphs,"

This can easily be fixed in the previous version, by telescoping some paragraphs, or some other simple remedy.Chjoaygame (talk) 09:54, 3 February 2016 (UTC)

"define the subject at the very beginning,"

This a formal objection. Just what weight should be placed on it may be debatable. In Wikipedia, we are told that no rule, especially a form one, is unbreakable, with certain exceptions, such as for example copyright and BLP. The former version has elected to give a preamble before stating the law. There are, I think, good reasons for this, specific to this article. Most problems with the law arise from loose statements of it. We should try to forestall such problems by giving a tight statement. It is not easy to give a tight statement with the conclusion before the preamble which sets out the range of applicability. For me, this most important. A strong reason I have against the new reversion is that it does not even nearly adequately indicate the scope of the law.Chjoaygame (talk) 09:54, 3 February 2016 (UTC)

"concisely summarize the article"

I am not sure exactly what is involved in this criticism.Chjoaygame (talk) 09:54, 3 February 2016 (UTC)

"and in general, not be bloated and rambling."

It is a matter of opinion, as to whether the former version was bloated and rambling. I think it wasn't. We have repeated requests for a more intuitive account of the law. They should not be dealt with by giving slippery or loose or potted versions.

An ordinary language non-technical account of the meaning of the law may be open to debate. The wording of Guggenheim is I think very good.

In the opinion of Denbigh,

Perhaps one of the most useful verbalisms is 'spread', as used by Guggenheim;^† an increase of entropy corresponds to a 'spreading' of the system over a larger number of possible quantum states. This interpretation is often more appropriate than the one in terms of mixing when the irreversible process in question is concerned less with configurational factors than with a change into a state where there is a greater density of energy levels.

^† Guggenheim, Research, 2 (1949), 450.

A relevant quote from Guggenheim is

To the question what in one word does entropy really mean, the author would have no hesitation in replying 'Accessibility' or 'Spread'. When this picture of entropy is adopted, all mystery concerning the increasing property of entropy vanishes. The question whether, how and to what extent the entropy of a system can decrease finds an immediate answer.

I think Guggenheim is the most reliable source on this topic.Chjoaygame (talk) 09:54, 3 February 2016 (UTC)

response

I am here in response to a ping. This article has been on my watchlist but I have not been following it closely. My own viewpoint is as follows. MOS:LEAD does indeed state as good practice having at most four paragraphs. I think this can be done by consolidating paragraphs as suggested above. The other point about starting the article with the definition also seems reasonable to me. Perhaps the definition of thermodynamics in general can be moved to the second paragraph without loss of coherence? I think we should start with the version before Waleswatcher's edit and prune as required, rather than the earlier, older version. Perhaps one could make a draft on the talkpage and address the objections above, before transfering it to article space. Kingsindian ♝ ♚ 05:18, 4 February 2016 (UTC)

Thank you for that comment. Ok. Here is a draft.

The second law of thermodynamics expresses the irreversibility of transfers of matter and energy that actually occur between bodies of matter and radiation. Each participating body is initially in its own state of internal thermodynamic equilibrium. The bodies are initially separated from one another by walls that obstruct the passage of matter and energy between them. The transfers are initiated by a thermodynamic operation: some external agency makes one or more of the walls less obstructive. This establishes new equilibrium states in the bodies.

The second law of thermodynamics then states that for a process to occur, the sum of the entropies of the participating bodies must increase. In an idealized limiting case, that of a reversible process, this sum remains unchanged. If, instead of making the walls less obstructive, the thermodynamic operation makes them more obstructive, there is no effect on an established thermodynamic equilibrium.

The transfers invariably bring about spread, dispersal, or dissipation of matter or energy, or both, amongst the bodies. They occur because more kinds of transfer through the walls have become possible. Irreversibility in thermodynamic processes is a consequence of the asymmetric character of thermodynamic operations, and not of any internally irreversible microscopic properties of the bodies. Thermodynamic operations are macroscopic external interventions imposed on the participating bodies, not derived from their internal properties.

The second law is an empirical finding that has been accepted as an axiom of thermodynamic theory. Of course, thermodynamics relies on presuppositions, for example the existence of perfect thermodynamic equilibrium, that are not exactly fulfilled by nature. Statistical thermodynamics, classical or quantum, explains the microscopic origin of the law. The second law has been expressed in many ways. Its first formulation is credited to the French scientist Sadi Carnot in 1824 (see Timeline of thermodynamics).

Chjoaygame (talk) 05:35, 4 February 2016 (UTC)

My comments: MOS:LEAD states that the first sentence, or at least the first paragraph should define the subject. From what I can see, here, it is actually the second paragraph which defines the topic. It is indeed possible that one needs the preliminaries to define the topic, but I wonder if this can be addressed somehow. One could use a bit of jargon in the first sentence/paragraph, and then elaborate it in subsequent paragraphs. Kingsindian ♝ ♚ 06:13, 4 February 2016 (UTC)

References

Another try:

The second law of thermodynamics states that when a thermodynamically defined process of transfers of matter and energy occurs, the sum of the entropies of the participating bodies must increase. In an idealized limiting case, that of a reversible process, this sum remains unchanged.

For a thermodynamically defined process of transfers between bodies of matter and radiation, each participating body is initially in its own state of internal thermodynamic equilibrium. The bodies are initially separated from one another by walls that obstruct the passage of matter and energy between them. The transfers are initiated by a thermodynamic operation: some external agency makes one or more of the walls less obstructive. This establishes new equilibrium states in the bodies. If, instead of making the walls less obstructive, the thermodynamic operation makes them more obstructive, there is no effect on an established thermodynamic equilibrium.

The law expresses the irreversibility of the process. The transfers invariably bring about spread, dispersal, or dissipation of matter or energy, or both, amongst the bodies. They occur because more kinds of transfer through the walls have become possible. Irreversibility in thermodynamic processes is a consequence of the asymmetric character of thermodynamic operations, and not of any internally irreversible microscopic properties of the bodies. Thermodynamic operations are macroscopic external interventions imposed on the participating bodies, not derived from their internal properties.

The second law is an empirical finding that has been accepted as an axiom of thermodynamic theory. Of course, thermodynamics relies on presuppositions, for example the existence of perfect thermodynamic equilibrium, that are not exactly fulfilled by nature. Statistical thermodynamics, classical or quantum, explains the microscopic origin of the law. The second law has been expressed in many ways. Its first formulation is credited to the French scientist Sadi Carnot in 1824 (see Timeline of thermodynamics).

Chjoaygame (talk) 11:04, 4 February 2016 (UTC)

References

The latest try is better than the version I edited. But it is still freighted with too much jargon, and more fundamentally, it's distorted by Chjoaygame's insistence that the article describe the most strict and limited possible interpretation of "the second law of thermodynamics". For instance, the phrase "thermodynamically defined process" in the first sentence is not needed (and in fact is pretty close to gibberish). The second law is a law of physics. It applies to everything, always, without exception, not just to "thermodynamically defined processes", whatever those are. Yes, strict formulations of it refer only to systems that begin and end is exact equilibrium. But such formulations are never applicable in practice, and yet the second law is used constantly and correctly by physicists (and others) and applied to a wide variety of systems that are never in equilibrium (since in reality, nothing ever is). It is simply wrong to write a wikipedia article about this that ignores that. Waleswatcher (talk) 14:54, 6 February 2016 (UTC)

Thank you, Editor Waleswatcher, for this comment of yours. It expresses clearly your view, which has informed your editing of this article for some time. You are right that I prefer a strict statement. You add that you feel that the one I prefer is the "most limited possible interpretation." You disagree with Wikipedia preferring the strict statement.

You propose that Wikipedia ought to give, under the title 'second law of thermodynamics', a statement that you consider to be of a "law of physics". I can agree that thermodynamics is part of physics, and that therefore any law of thermodynamics is a law of physics. But I also think that its being a law of physics does not stop it from being a law of thermodynamics, the subject of the present article. You wish to state a law that covers systems not in thermodynamic equilibrium. I would say that such laws, along the present lines, are not yet clearly formulated. I think you have not offered a clear formulation.

I think we have a difference of opinion here. My concern is that loose or slippery statements are prone to open the way for nonsense that Wikipedia ought not sanction. I could rehearse at length the reasons for the strict form of statement, but I guess that for the present that would overburden this discussion.Chjoaygame (talk) 15:44, 6 February 2016 (UTC)

There are plenty of clear formulations, and they are in plenty of reliable sources, including textbooks on thermal and statistical physics. Some of them used to be in the article and related articles. The problem with an inordinately strict (and rather old-fashioned) interpretation is that it doesn't correspond to how the law is actually applied in the vast majority of contexts, and it actually obscures the extremely important and general physical fact that the second law expresses. It should certainly be in the article, but not in the lede or only briefly - because defining things precisely requires too much background explanation, as is apparent from your attempted rewrites, and that's not what the lede is for. Waleswatcher (talk) 18:08, 6 February 2016 (UTC)

This discussion suggests the following question: what statements about the second law are not exactly true if the systems are NOT in thermodynamic equilibrium? I would think that at least some parts of the theory are still true for the real world. For example, I strongly doubt that one can exceed the Carnot efficiency just by designing a machine which does not follow a strictly reversible path. This seems a good example because the Carnot limit to efficiency is the original historical form of the second law. Dirac66 (talk) 20:17, 6 February 2016 (UTC)

Editor Dirac66 asks a good question. Further, one may ask, how should the law be stated in the lead?

I think there are two main ways of stating the law. An example of the first is in terms of cyclically running heat enines, or the like. The second way is in terms of increase of entropy. Some of these are given in the body of the article. Currently, the statement in the lead is in terms of increase of entropy. It may be asked whether one or both, or which one, should be in the lead.

The lead could try to make statements that are reliable, well-defined, and true, in terms of entropy increase. Thus it would deal with a well-defined physical quantity, entropy of a system in its state of internal thermodynamic equilibrium.

Alternatively, a statement in terms of entropy increase may be loose, slippery, and vague, without attention to whether entropy is well defined. There are endless such ideas to be found in the literature. These include those referred to by Editor Waleswatcher in his sentence "It applies to everything, always, without exception, not just to "thermodynamically defined processes", whatever those are." Their reliability is, I suppose, a matter of opinion. They are unfalsifiable because they are vague. They are not endorsed by the best material in the literature that is focused on thermodynamics for its own sake, the kind of thermodynamics that Einstein thought would not be overthrown. Loose, slippery, and vague statements are hard to judge as to exact truth or falsehood. One might charitably say that they fall in a neutral ground, neither exactly true nor false. I would call them unreliable. More narrowly thinking, one might call them meaningless. They are not exactly true, to answer the question proposed by Editor Dirac. This leads to his subsequent suggestion, that perhaps the lead should state the law in terms of cyclically running heat engines.Chjoaygame (talk) 22:24, 6 February 2016 (UTC)

Yes, I think this point should be added to the intro whose last paragraph now reads The second law has been expressed in many ways. Its first formulation is credited to the French scientist Sadi Carnot in 1824 (see Timeline of thermodynamics). We could add a sentence such asCarnot showed that there is an upper limit to the efficiency of conversion of heat to work in a cyclic heat engine operating between two given temperatures. The interested reader can then find more details in the section on Carnot's principle and the linked article on the Carnot (heat) engine. Dirac66 (talk) 00:08, 7 February 2016 (UTC)

I agree with the foregoing proposal by Editor Dirac66.Chjoaygame (talk) 03:45, 7 February 2016 (UTC)

Thanks. Done. Dirac66 (talk) 12:46, 7 February 2016 (UTC)

Stimulated by the just-mentioned edit, I would like to add a comment. That there exist heat pumps or refrigeration engines is a good illustration of why the concept of "tampering with things" is important. One can feel somehow intuitively easy enough about heat flowing down and driving a heat engine, as Carnot thought. But for heat to flow up? Intuitively, that obviously requires tampering with things.Chjoaygame (talk) 00:20, 12 February 2016 (UTC)

Here is another try:

The second law of thermodynamics states that for a thermodynamically defined process to actually occur, the sum of the entropies of the participating bodies must increase. In an idealized limiting case, that of a reversible process, this sum remains unchanged.

A thermodynamically defined process consists of transfers of matter and energy between bodies of matter and radiation, each participating body being initially in its own state of internal thermodynamic equilibrium. The bodies are initially separated from one another by walls that obstruct the passage of matter and energy between them. The transfers are initiated by a thermodynamic operation: some external agency intervenes to make one or more of the walls less obstructive. This establishes new equilibrium states in the bodies. If, instead of making the walls less obstructive, the thermodynamic operation makes them more obstructive, no transfers are occasioned, and there is no effect on an established thermodynamic equilibrium.

The law expresses the irreversibility of the process. The transfers invariably bring about spread, dispersal, or dissipation of matter or energy, or both, amongst the bodies. They occur because more kinds of transfer through the walls have become possible. Irreversibility in thermodynamic processes is a consequence of the asymmetric character of thermodynamic operations, and not of any internally irreversible microscopic properties of the bodies.

The second law is an empirical finding that has been accepted as an axiom of thermodynamic theory. When its presuppositions may be only approximately fulfilled, often enough, the law can give a very useful approximation to the observed facts. Statistical thermodynamics, classical or quantum, explains the microscopic origin of the law. The second law has been expressed in many ways. Its first formulation is credited to the French scientist Sadi Carnot in 1824 (see Timeline of thermodynamics). Carnot showed that there is an upper limit to the efficiency of conversion of heat to work in a cyclic heat engine operating between two given temperatures.

References

Chjoaygame (talk) 00:59, 12 February 2016 (UTC)

criticism of the current lead

The first two paragraphs of the current version of lead are unsatisfactory. They read as follows:

The second law of thermodynamics states that in every real process the sum of the entropies of all participating bodies is increased. In the idealized limiting case of a reversible process, this sum remains unchanged. The increase in entropy accounts for the irreversibility of natural processes, and the asymmetry between future and past.

While often applied to more general processes, the law technically pertains to an event in which bodies initially in thermodynamic equilibrium are put into contact and allowed to come to a new equilibrium. This equilibration process involves the spread, dispersal, or dissipation of matter or energy and results in an increase of entropy.

The first sentence reads "The second law of thermodynamics states that in every real process the sum of the entropies of all participating bodies is increased." This is faulty because it implies that entropy is defined for any system at all, including systems not in their own states of internal thermodynamic equilibrium. This is false. No matter how one may appeal to rhetorical needs for the lead, a grossly misleading statement such as this is unacceptable. The second paragraph reminds one of "but I didn't inhale." The use of the word "technical" is deliberately intended to mislead. It suggests that the limitation to equilibrium states is insubstantial, and merely scholastic. It suggests a kind of double-think in physics. Not good. The pretendedly correcting sentence (that starts "While often applied to more general processes, the law technically pertains to an event in which bodies initially in thermodynamic equilibrium") reinforces rather than corrects the initial wrong impression given by the first sentence, that the law applies to every kind of process. The error is important because, amongst other reasons, it hides the conceptual structure of thermodynamics. The fundamental presupposition of thermodynamics is that there exist systems in their own states of internal thermodynamic equilibrium, a proposition so deeply presupposed that it is often hardly articulated as a presupposition. It says that any system left isolated by walls for an infinitely long time may be regarded as in thermodynamic equilibrium. It does not cover unwalled systems such as parts of galaxies, which may suffer adventures such as encounters with black holes, with the opposite of spreading. This presupposition is not part of the second law, but is logically prior to it. The Carnot cycle is discussed in terms of fictive reversible processes, which require and presuppose states of thermodynamic equilibrium.

The third sentence of the faulty version reads: "The increase in entropy accounts for the irreversibility of natural processes, and the asymmetry between future and past." The first clause of this sentence is faulty. The dynamics of a natural process is reversible. This is exemplified in Poincaré recurrence, a phenomenon of equilibrium on a vast time scale. Unmentioned is the unnaturalness of thermodynamic operations that accounts for irreversibility. A thermodynamic operation is normally applied in ignorance of the timing of the possible Poincaré-recurrent gross aberrations, which belong to the natural dynamics of the system. Looking back in time from a recent intervention one sees a "timed relaxation" to near homogeneity. Looking forward from it, one has practically no idea when the Poincaré recurrence will occur. The sentence's second clause is faulty. Best current opinion is that the arrow of time is not derived from the second law. The article doesn't say that it is. It was lazy editing that threw out the three references for the word "spread". Unmentioned are many kinds of decrease of obstructiveness of walls beyond 'putting in contact', such as for example allowing radiative exchange.

It is fair to critically examine the reasons or arguments that are offered above for the new version of the lead. They rely on unstated faulty assumptions.

For example, there is nothing modern about loose statements of the second law. They go right back to Clausius, whose famous statement is called sibylline by Clifford Truesdell. "The entropy of the universe tends to a maximum." This statement was grandiose and pollyanna, and therefore appealing to the heart. But it was not physics. It has not been made into physics by advances since Clausius.

It seems to be assumed that there is something about the lead that justifies its being unreliable. I think such an assumption is not suitable here. Yes, perhaps for The Huffington Post, but not for Wikipedia.

It is claimed that ""thermodynamically defined processes", whatever those are" is "gibberish". This claim is cavalier, careless, and inaccurate. Thermodynamically defined processes are processes defined in thermodynamics, not physical processes in general. However much one might wish for a general rule that covers all physical processes along these lines, such is not provided by thermodynamics. If it is thought that some more general law is known and can be precisely stated, for physical processes beyond the scope of thermodynamics, then perhaps an article on that new law could be written. But such a putative new law should not be slipped unannounced into the lead of the article on the second law of thermodynamics by dint of a loose statement. The law of conservation of energy is far more general than the first law of thermodynamics, and has its own article, as is proper. If it should be stated and be valid, the claimed more general "law of physics" would likewise deserve its own article.

There are many usages of the word 'entropy' that have arisen over the century and a half since its invention by Clausius. Shannon's usage arose because von Neumann suggested that no one would know exactly what the word should mean; this is not a good basis for an expression of a physical law. Boltzmann knew that his H-function was not a thermodynamic entropy; that's why he didn't call it so. Thermodynamic entropy specifically describes thermodynamic equilibrium, and it is misleading to suggest otherwise, as do many loose or slippery proposed statements of the law. Phil Attard has admirably tried to do what I think is promising: define new concepts that extend the virtues of entropy from its valid base in thermodynamic equilibrium; this is advanced but unfinished research, not material for a Wikipedia lead. At present, extended versions of the second law, and extended usage of the word 'entropy', are not well enough supported to be regarded as Wikipedia-reliable.Chjoaygame (talk) 16:41, 7 February 2016 (UTC)

Since I wrote the above paragraphs, the first paragraph of the lead has been made more elaborate, and some other smaller edits have been made. I do see these changes as notable improvements.Chjoaygame (talk) 21:53, 12 February 2016 (UTC)

Edit war re Entropy and disorder

There seems to be an edit war today over whether or not to include a mention of the idea that entropy is a measure of molecular disorder, and therefore that the second law is about increasing disorder. The controversial sentence is Entropy is commonly understood as a measure of molecular disorder within a macroscopic system.. I agree that this is not entirely satisfactorily, but perhaps we can find a solution based on an improved text.

If I understand correctly, the idea of entropy as disorder goes back to Clausius and has been included in many leading textbooks for a long time. More recently however, it has become recognized as simplistic and poorly defined, and has started to disappear from many textbooks. This trend is documented in detail at Entropy Sites — A Guide.

So what should be included about disorder in this article? I think that this idea has been so widespread over the past century in science textbooks and articles that Wikipedia cannot just ignore it. Readers look to Wikipedia to explain points they read about elsewhere, such as, yes, the relation between entropy and disorder. On the other hand, the text should also explain that this particular idea is losing credibility.

Perhaps the intro can include something like: Entropy has often been described (refs) as a measure of molecular disorder within a macroscopic system. However many recent authors (refs) instead describe entropy as a measure of energy dispersal within such a system, so that the second law requires the increase of energy dispersal. Then further down we can include a more detailed section, perhaps labelled Increase of molecular disorder or of energy dispersal?

I am willing to make changes to these suggestions, and I do not insist on any of the exact words written above. However please, everyone, calm down and try to find a compromise solution which recognizes that there is some merit in the opposing position. Dirac66 (talk) 02:13, 13 February 2016 (UTC)

Good to have sober thoughts here, thank you for them.

I undid the edit because I thought it was in excessive detail for its place in the article, as I wrote in my edit summary. It seems that Editor DMacks also thinks the detail/place reason was sufficient.

Editor Dirac66 makes explicit a further important aspect of this matter: that the 'disorder' view of entropy is not discussed in the article. He proposes to put in some info about this.

The two issues are not entirely separate, and are not entirely separated from the general status of the lead.

As a further matter, there has been some activity on my talk page which I think properly belongs here, not there. Here it is, shifted from there, with a re-format of the section header:

edits

hello. It's obvious that you think you own some of these articles, but you don't. Entropy is commonly understood as a measure of molecular disorder within a macroscopic system. It was not "undue detail"...but valid elaboration and from other WP articles themselves... No valid reason to rudely undo...because of "I don't like". Against WP policy....restored valid modification. You gave no actual rationale for your disrespectful reverting. Do it again, and you will be reverted again. Or go to Talk. Don't edit war.. Thanks. 71.183.133.173 (talk) 22:19, 12 February 2016 (UTC)

@71.183.133.173:, don't forget WP:BRD since you are interested in behavioral standards. Different editors have different feelings about how much detail goes where...being "valid" doesn't make it indisputably appropriate. For the record, I agree that at least some of your insertions are excessive detail for the context. DMacks (talk) 22:30, 12 February 2016 (UTC)\

Hello DMacks. Frankly speaking, you can "agree" with a wrong revert all you want, but "I don't like" is not a WP valid reason for removing accurate, valid, good-faith, and/or sourced information and elabs...as it's just your OPINION that it's "unnecessary" or "excessive". You "agreeing" uptightly about things like that don't have solid WP backing or recommendation. "I don't like" (which is really what it is, behind all verbiage) is not an argument. This is a wiki. No one person owns any article. Regards .... 71.183.133.173 (talk) 22:34, 12 February 2016 (UTC)

Do it's your opinion that it does belong. I never asserted bad-faith (in fact I noted the contrary) or that it is invalie or inaccurate. If you don't wish to collaborate and discuss, respect others' thoughts as you wish yours to be, and recognize that you might not be right about what should be in an article, then maybe wikipedia isn't for you. DMacks (talk) 22:36, 12 February 2016 (UTC)

In accord with the advice of Editor Dirac66 I will think this over.Chjoaygame (talk) 08:19, 13 February 2016 (UTC)

Specifically in response to the comments of IP user 71.183.133.173, I would observe that Wikipedia articles, as cited by him, are not reliable sources, and his hint of other reliable sources is not supported by details. Wikipedia is a reporter of reliable sources, not of what is "commonly understood", to quote the edit in question. If a source nowadays puts the 'disorder' doctrine as currently valid, I would say that, in that respect, it is not reliable.

As for edit in general: In recent times, till now, the article contained no mention of the 'disorder' doctrine for entropy. The topic of the article is not primarily entropy, nor the history of intuitive takes on entropy; it is the second law.

I learnt of the objections to the old doctrine of 'disorder' by reading a section on it at pp. 55–58 in Grandy, W.T., Jr (2008), Entropy and the Time Evolution of Macroscopic Systems, Oxford University Press, Oxford, ISBN 978-0-19-954617-6. Another monograph that considers the matter is Denbigh, K.G., Denbigh, J.S. (1985), Entropy in Relation to Incomplete Knowledge, Cambridge University Press, Cambridge UK, ISBN 0-521-25677-1, on pages 43–44. They conclude on page 44: "The inference to be drawn from these examples is that entropy cannot be interpreted in a reliable and comprehensive manner as increase of disorder." They refer to the 'disorder' doctrine as "the prevailing mythology". These two in my opinion are reliable sources. They are not mentioned in the review article by Lambert cited above as Entropy Sites — A Guide. They are thus independent reliable sources that concur with the view of that article, summarized by Editor Dirac66 thus: "it has become recognized as simplistic and poorly defined, and has started to disappear from many textbooks".

From this I conclude that the now obsolete doctrine of 'disorder' is hardly notable in the present article on the Second law of thermodynamics. Since it is at present not mentioned in the body of the article, and since I think it is hardly notable in this article, and since there is no end of attempts to get ideas into prominence in Wikipedia by trying to attach them to the second law, and since space is at a premium in the lead, I think that the new insertion has no just place in the lead. If Editor Dirac66 very kindly writes a section or some such within the body of the article, I suppose he will make it clear that the 'disorder' doctrine is of historic rather than current value. I think it will still not merit a place in the lead, as not sufficiently notable. At present, I do not agree with the suggestion that the lead should make a place for the 'disorder' doctrine by such words as he proposes: "Entropy has often been described (refs) as a measure of molecular disorder within a macroscopic system. However ...."Chjoaygame (talk) 17:11, 13 February 2016 (UTC)Chjoaygame (talk) 18:20, 13 February 2016 (UTC)

Two points: I think Guggenheim's word 'spread' is better than 'dispersal', though I have no objection to 'dispersal'. I think that not only energy, but also matter, are spread.Chjoaygame (talk) 18:26, 13 February 2016 (UTC)

Entropy as disorder is somewhat useful, but ultimately misleading. It should be mentioned in the article only because the idea is so pervasive, but should not be offered as an explanation. Ultimately, the static idea of lack of information and the kinetic idea of irreversibility are the only intuitive concepts that give a good intuitive idea of entropy that I can think of. The idea of disorder, when you inspect it closely, falls apart. If you keep refining the definition of disorder to plug the logical holes, you wind up with the definition of entropy, which gives you no more insight than you had at the beginning. "Order" is in the mind of the beholder. If you have what most people would call a messy (high entropy) desk, but you know where everything is and that it belongs there, then it's a very ordered desk. Science is not only about gathering information, but sharing it as well. Once you share your information about that messy desk, the person who thought it was messy will have to agree its very ordered. So now where are we? The "messy" desk has not changed, but the concept of what constitutes a mess has, and that has to do with a lack of information being eliminated in the second person. No, the concept of entropy as disorder is a mess. PAR (talk) 19:59, 14 February 2016 (UTC)

It is great to hear from Editor PAR. It may be of interest to add an idea from Tisza, L., Quay, P.M. (1963), 'The statistical thermodynamics of equilibrium', Annals of Physics, 26: 48–90, on pp. 65, 88. They suggest that, more reasonable than the name that von Neumann suggested to Shannon, 'entropy', "because no one will know what it means", the name 'dispersal of the distribution' should be given to the quantity −Σ_i f_i ln f_i. Accepting this suggestion of Tisza & Quay, one will find it intuitively appealing and reasonable to say that the entropy is the maximum possible value that can be attained by the dispersal. I would say that 'spread' would be even better than 'dispersal' for this purpose. If this were to become customary terminology, one would be spared the horrible pain of hearing that the "entropy increases as the system settles to equilibrium". Almost always, the spread increases towards its maximum. 'Spread' can also be a verb. Thus matter and energy nearly always spread. If you will watch for a Poincaré recurrence time, you will see, very nearly, recurrence of the initial lack of spread, and, if you then play the tape backwards, exact 'return' to the initial state, because the laws of motion are time reversible.Chjoaygame (talk) 23:30, 14 February 2016 (UTC)

Thanks to Editor Chjoaygame for adding a brief note to the history section re molecular disorder, which I think is probably sufficient for this article. As a source I have added a link to Lambert's site which directs to much other information. I do not have access to the books by Grandy and by Denbigh, but you may wish to add them as well. (I do remember owning Denbigh's excellent book and reading it about 30 years ago, but I no longer have it.) Dirac66 (talk) 02:35, 15 February 2016 (UTC)

where's the spread ?

Hmm. I don't like "spread". If you have a container with only hydrogen and oxygen in it, let's say at 300 degrees C, in it and you wait for the reaction to reach equilibrium, there will be hydrogen, oxygen and water vapor. The entropy will have increased. Where's the "spread"?

I also don't like "dispersal of the distribution" as a name for the Shannon entropy of a distribution. A uniform distribution is a function that is constant (1/w) over a width of w, centered at, say x=m. It has an entropy of log(w). If you have two such distributions of width w at half the height (1/2w), thats a good distribution, as long as they don't overlap. This "double uniform" distribution has an entropy of log(2w). The two distributions could be very close to each other, or separated by a million miles, it doesn't matter, the entropy is log(2w) in both cases. If entropy is dispersal, then both possibilities have the same "dispersal", which seems counter to the notion of dispersal. Variance is a measure of dispersal, not entropy. The variance of the double uniform distribution increases as the distance between them increases. Entropy is lack of information. If you want to know which of the two boxes a particle is in, you only have to ask one question. Is it this one? If yes you know its this box, if its no, its the other box. How far apart the boxes are is irrelevant. You are tying down the particle position to within width w. log(2w)-log(w)=log(2) = 1 bit (one question). PAR (talk) 04:51, 15 February 2016 (UTC)

It's great to have this criticism from Editor PAR. These are important questions for this article.

We are judging between three stories (1) 'disorder', (2) information, and (3) spreading.

I will not here further consider the 'disorder' story.

I think the information and spreading stories are the same with different labeling.

I think that the only proper way to talk about the formulas such as −Σ_i f_i ln f_i is in terms of such things as Kullback–Leibler divergence, described, for example in Kullback, S. (1959), Information Theory and Statistics, Wiley, New York. I have to admit that name of the topic is 'information theory', but I claim that the fundamental concept is the so-called 'divergence'. Divergence tells how far apart things are, which measures their spread. And one-way spread: the divergence is 'directional' (= sensed). Therefore I welcome Tisza & Quay's update, 'dispersal', on Shannon's patched up term 'entropy'. The information story may be labeled by the topic name 'information', but I claim the core concept is measured by formulas such as −Σ_i f_i ln f_i, which following Tisza & Quay I claim are better called 'dispersal' than 'entropy'. In support, I cite the Denbighs' above-quoted 1985 book. In their concluding section, on page 117, they write "Notions about 'loss of information' can sometimes be intuitively useful. But they can also, like the comparable concept of 'disorder', give rise to mistakes." On the other hand they are in favour of Guggenheim's word 'spread', though admittedly in a slightly different context.

To tackle Editor PAR's container of hydrogen and oxygen question. It's a good test case. I submit that talk of "wait" is out of order for classical thermodynamics. Time scales are in question. For the second law, the initial conditions are required to be in equilibrium as constrained by the walls. If the two gases are stopped from interacting by my patent anti-catalyst, one could say they are in equilibrium, regarding my patent anti-catalyst as a wall. Remove my patent anti-catalyst, and a new equilibrium is thereby established. From the initial condition of mixed gases, much potential energy will eventually, over an intermediate time scale, be released as kinetic energy, and steam will appear. On the time scale of equilibrium (the Poincaré recurrence time), the initial reverse-fluctuated state will be nearly approximated. It was the removal of the wall that increased the entropy, not the relaxation from the anti-fluctuation (Pippard). The potential energy has spread into new degrees of freedom, in new, previously unoccupied, kinetic energy states of new previously non-existent molecules. Without my patent anti-catalyst being initially present, the system did not start in equilibrium and the entropy does not increase, because it is initially undefined; no increase of entropy and Editor PAR sees no spread; agreement. Maybe he doesn't buy my patent anti-catalyst. He says the mixture is initially in metastable equilibrium. I say he will tamper (Guggenheim) with the system to make it explode, somewhere somehow locally removing a potential energy barrier. The potential energy has spread into new degrees of freedom, in new, previously unoccupied, kinetic energy states of new previously non-existent molecules. I accept that this is a bit of a stretch. But I see it as superior to the disorder story, and I don't see it as truly different from the information story, as noted above. I concede that 'spread into new degrees of freedom' is not exactly orthodox classical thermodynamics!!

I wish to explicitly say that I don't like the wording "Entropy is lack of information." I accept that it is current doctrine, but I think it too abstract. I prefer to be more physical, and more intuitive, replacing it with 'Entropy is maximal spread', with no change of meaning.Chjoaygame (talk) 11:50, 15 February 2016 (UTC)

Hi Chjoaygame. Yes, I agree, you have to be careful about formulas like −Σ_i f_i ln f_i, the argument of the logarithm must be dimensionless. Kullback and Liebler's divergence was a very perceptive solution to this problem. It also renders my "double uniform distribution" argument suspect, because log(w) is the logarithm of a distance. The whole argument can be recovered another way by dividing all distances by a some "standard distance" like the classical electron radius, or the distance between two marks on a platinum bar in Paris, or whatever. So the argument stands. Chjoaygame, you say "Divergence tells how far apart things are, which measures their spread." I point out in the double uniform case how the information entropy is not a measure of how far apart the two peaks are. Please, I beg you to understand this example, otherwise everything else I say will be meaningless. The distance between the two boxes has no effect on the entropy. Below is an ascii sketch of the distribution:

         ------        ------
         |    |        |    |   
    -----     ---------      ---------> x

The width of each box is w, the height of each box is 1/2w. The area under each box is 1/2, so its a good probability distribution. The entropy is log(2w), or log(2w/λ) if you prefer, where λ is some standard length. The whole point is that the entropy of the distribution does not depend on the distance between the boxes. This little example clearly shows that entropy is NOT a measure of how far apart things are.

Regarding the container of hydrogen and oxygen, yes, perhaps I should have said that the mixture was ignited so that equilibrium was reached on a reasonable time scale. Or maybe chosen two different gases which react more quickly. But this is beside the point.

• 1 The concept of "wait" is essential. Yes, thermodynamics deals with equilibrium states and which equilibrium state results from another. If you have two gases separated by a wall, and then you remove the wall, and speak of the equilibrium state that results, you have to "wait" for that equilibrium state to develop. Without the concept of "wait" all such thought-experiments must be discarded, and that is not acceptable.

• 2 In the two reacting gases case, there is no "spread" of anything in time or space, and so "spread" is not a valid characterization of an increase in entropy as it applies to time and space. Yes, you can say the potential energy has "spread" into new degrees of freedom, but this is just repairing a logical flaw in the "spread" argument in order to keep the idea of "spread". Even then, it is still misleading. When the energy "spreads" in this abstract "degree of freedom" space, you are dealing with two probability distributions, one in which the microstates are fewer in number, and a second set of microstates of which there more. You can define other probability distributions from these basic distributions, like describing the equilibrium distribution of the number of particles in each energy level. This gives you the Maxwell-Boltzmann distribution (or Bose-Einstein, or whatever). Whenever you have a probability distribution with distribution function f, you have an information entropy −Σ_i f_i ln f_i, and so, yes, and in the before and after case, you see a "spread" in this "degree of freedom" space. But the only reason you see a spread is due to the special nature of the probability function. When you restrict yourself to nice, smooth, single-peak probability distributions, like the Maxwell-Boltzmann distribution, higher entropy correlates well with the variance, variance being the true measure of "spread". What you are really saying when you say "higher entropy means a spread of energy into new degrees of freedom" is that "higher variance means a spread of energy into new degrees of freedom, and I am restricting myself to distributions for which entropy and variance are rather correlated, and so higher entropy correlates with a spread of energy into the new degrees of freedom". I refuse to make the mistake of saying it is the increase in entropy which creates the spread. As illustrated by the double-uniform example, entropy has nothing to do with spread, although in some special cases entropy may correlate well with spread (variance).

I don't see the idea of "entropy is lack of information" as too abstract. When applied to a gas in a container, for example, the thermodynamic entropy of the gas is just Boltzmann's constant times the information entropy ln(W), as Boltzmann's famous equation says. W is the number of microstates, and the log(W) is −Σ_i f_i ln f_i where the sum is over all W microstates, and each f=1/W (i.e. each microstate has an equal probability).

So the entropy of the gas is equal to Boltzmann's constant (converted to entropy per bit) times the number of yes/no questions you would have to ask about the macro state, in order to determine the microstate. This seems very non-abstract to me. PAR (talk) 14:11, 15 February 2016 (UTC)

That is all very well, but it's not to the point. I want you to place an order for $ 1000000 worth of my patent anti-catalyst. That's the point! Also I have very nice clock that is calibrated in units of the Poincaré recurrence time that I can offer you at a very cheap price!! To be more serious, I am very keen on the idea of talking about such formulas in terms of the Kullback–Leibler divergence. I think doing so forces one to put one's cards on the table, which I think is a number-one good idea. It's not a measure of how far apart the peaks lie along the axis, but it is a measure of how far apart the distributions are in some sort of conceptual space, relative to some reference distribution. I am rather attached to the saying that the entropy is the greatest possible value of the spread!Chjoaygame (talk) 14:40, 15 February 2016 (UTC)

I did explicitly address that point: "It's not a measure of how far apart the peaks lie along the axis, but it is a measure of how far apart the distributions are in some sort of conceptual space, relative to some reference distribution." There I am conceding that the spread is not always, or perhaps not even most often, in ordinary physical space. In that sense, I am conceding that the spread is decidedly in an abstract space. Yet I objected to the information story because it was too abstract. What's sauce for the goose is sauce for the gander. I can only reply that in some easy cases, the spread story is easy to see as in ordinary physical space. In contrast, for the information story, it's never as concrete-seeming as that. Information is a way of estimating spread. I made some jokes because to a degree this is a matter of taste and habit of thought, personal preferences, and I felt that outright intellectual disagreement would not express my position. I am happy to concede, as I did above, that the spread isn't essentially about distance in ordinary physical space. The variance is a measure of spread, but not, I think, relative to a reference distribution, and not in an abstract space of distributions. The spread, as a re-reading of Shannon's entropy, is new to me, since I just read it in the cited Tisza & Quay paper that I was alerted to by reading the Denbigh book, for this discussion. But it answers for me a long-felt discomfort with the Shannon account of things. I have long been unhappy with the use of the word by Shannon. Very unhappy. Tisza & Quay soothed that itch instantly. I was prepared for it by finding the Guggenheim spread story in K.G. Denbigh's Principles of Chemical Equilibrium. I had to go to an out-of-town library stack to get a copy of Guggenheim's 1949 paper in Research. I think it a very valuable paper. It also makes me very unhappy to see people read Boltzmann's H-function as an "entropy". I have thought of it for myself as a measure of distance from a homogeneous but random distribution, whatever that might be, nothing to do with entropy. The present conversation seems to cement that for me. Again, I did reply explicitly first-up to your two-peaked distribution argument.

For my own reasons, I will not here reply to your very reasonable point about the wait for equilibrium.Chjoaygame (talk) 16:38, 15 February 2016 (UTC)

Well, ok then. I can't afford the anti-catylist, but please put the Poincare clock on my tab and send it along. As far as "entropy" is concerned, to me its only a word, its the concepts that I find most interesting. PAR (talk) 19:20, 15 February 2016 (UTC)

It's great that you are around. I have just now sent the Poincaré clock by teleporting by Bell's theorem. I imagine it arrived at your place about two hours ago. On a different note, I want to develop a good habit in the use of the conceptual frame of the Kullback–Leibler divergence. The conceptual frame is important.Chjoaygame (talk) 19:35, 15 February 2016 (UTC)

Perhaps a quote from Tisza & Quay would be useful: "We consider now three typical situations in which the dispersal of the d.f. assumes the character of entropy, information, and uncertainty, respectively. However, these cases are by no means exhaustive. In particular, the concept of information has many ramifications which are outside the scope of the present discussion." The paper is reprinted in Tisza 1966 Generalized Thermodynamics.Chjoaygame (talk) 22:32, 15 February 2016 (UTC)

The key is not that the divergence is a complication derived from the simple case −Σ_i f_i ln f_i. It is that the simple case −Σ_i f_i ln f_i is a special case of the natural one −Σ_i f_i ln (f_i/g_i). The directed divergence is a measure of relation between two distributions. The Shannon case is when one of them is considered so basic that it is not even mentioned. In the land of the blind, a man with one eye is king. The directed divergence is for men with two eyes, who can see in three dimensions. The Shannon case is that of the man with one eye in the land of the blind. A directed distance is essentially a function of two variables. A norm is a distance from the zero, which is the same as a distance to the zero. But a directed distance is different between the 'to' case and the 'from' case. The Shannon story gives little hint of that. The result of the directed nature of the Kullback–Leibler divergence is that one can compare two distributions, but in general not more than two at once. There is only a trace of it in the Tisza & Quay paper; they don't seem to notice it, as far as I can see.Chjoaygame (talk) 10:04, 16 February 2016 (UTC)

Yes, the word "divergence" refers to the divergence between the two functions, not a "spread" of either. Just reiterating that the entropy of a probability distribution in absolutely no way measures its "spread". I agree, the continuum Shannon entropy cannot be used without understanding the source of what people call its "problems" (negative values, etc.). Just look at the Shannon entropy of the uniform distribution - its log(w), and if you are talking about a probability distribution in space, w is a dimensioned variable, and logs of dimensioned variables are very bad. If w is smaller than 1, the entropy is negative. KLD deals with this problem by introducing another distribution Q along with P, and having P/Q in the logarithm. But we can deal with it another way, by multiplying P by a "standard" length xo, rendering (P xo) dimensionless. This doesn't require the introduction of a second distribution, and it clarifies (as does KLD) the meaning of entropy. If we cannot deal with the "problems" of Shannon entropy, either by introducing a standard or using KLD, then to that extent, we don't know what we are doing, really. PAR (talk) 10:46, 16 February 2016 (UTC)

Thanks for these points. I will think it over some more.Chjoaygame (talk) 11:14, 16 February 2016 (UTC)

a revanchist?

I have looked at Callen's second edition. On page 380, he writes: "In fact, the conceptual framework of "information theory", erected by Claude Shannon¹ in the late 1940s, provides a basis for interpretation of the entropy in terms of Shannon's measure of 'disorder'."

He then more or less demolishes the 'disorder' metaphor. Nevertheless, he proceeds to use the word as a routine, defining it in terms of Shannon's "entropy". I did not find him using the term 'dispersal'. Nor the word 'spread'.Chjoaygame (talk) 14:10, 16 February 2016 (UTC)

Now checking Bailyn 1994. He offers a part of a chapter, with the title 'Part C. Information and Entropy'. It starts : "The association of entropy with information can be traced all the way back to Boltzmann (1894), ..." In section 5.12 he cites Shannon & Weaver (1949): "Information is a measure of one's freedom of choice when one selects a message." He comments: "This definition of information is therefore not all that intuitive." He concludes this section: "In fact, −Σ_i p_i log₂ p_i was called entropy by Shannon, a usage of a technical term somewhat out of context, and not universally welcomed (32 [K. Denbigh and J. Denbigh, Entropy in Relation to Incomplete Knowledge, (Cambridge University Press, 1985), Chap. 5.]) His Section 5.15 is headed 'Order and Disorder'. I may sketch a summary of it, as saying that order and disorder often increase together. He doesn't, so far as I can see, talk about spread or dispersal, but Clausius' term 'disgregation' has three index entries. He writes on page 114: "Disgregation meant "the degree in which the molecules of a body are separated from each other."" 'Disgregation' was apparently Clausius' first stab at giving a name to a state variable pretty close to entropy, the name he invented 11 years later, in 1865. Bailyn offers a section headed 'Entropy and disgregation'. Clausius defined disgregation also for non-equilibrium, with the virtual tacit assumption of local thermodynamic equilibrium. Disgregation was configurational entropy, as against kinetic entropy.Chjoaygame (talk) 16:19, 16 February 2016 (UTC)

Wikipedia actually has an article on disgregation as a thermodynamic term. If I understand the (only) reference in that article, Clausius (who was German) communicated and then published his work in 1862 first in German, and then in English and French. So presumably he first some used German word, which was then translated (by him?) as disgregation.

I would think a more modern translation would be disaggregation. Ideally we could dig up his original German term and check the translation with someone who knows German well. Dirac66 (talk) 20:46, 16 February 2016 (UTC)

Bailyn gives 1862 for Clausius' first use of 'disgregation'. I get the feeling that's the traditional physics translation. It's not convenient right now for me to hunt the German original and its translations; later perhaps. I guess the word was invented by him for the purpose, to judge from his invention of 'entropy'. I am not keen on putting that word in this article. I just mentioned it because Bailyn does. I would also say that disgregation is pretty close to dispersal.Chjoaygame (talk) 21:42, 16 February 2016 (UTC)

I agree with not putting the word in this article, given that it is not really standard English. Dirac66 (talk) 23:59, 16 February 2016 (UTC)

I agree. Not only is it non-standard, when associated with entropy, it implies some kind of dispersion, which is false.

Another way to see that entropy is not about dispersion - take a piece of paper, the bottom edge being the x axis. Draw a curve on it, and take scissors and cut along the curve, discard the top part. Take the scissors, cut it into many narrow strips of width Δx, then put them in a bag and shake them up. Now ask someone to calculate the area under the curve. No problem. The area of a strip is practically its length f(x) times its width Δx if Δx is small. Measure the area of each strip, add them up, and thats the area under the curve. I can create a new function by pasting the strips back together in any random way I want. Even leave gaps, huge gaps, but use every strip. The area under that jagged curve will be identical to the area under the original curve. The thing is, each strip has its own entropy as well. Its length is f(x), its area is f(x)Δx and its entropy is f(x) log(f(x)) Δx. So calculate the entropy of each strip, add them up and thats the entropy of the original curve. Again, paste them back in any random way you want. The entropy of that jagged curve will be identical to the entropy of the original curve. The entropy of each strip is a measure of the amount of information it can carry, and the entropy of the curve is a measure of the amount of information the curve can carry. (after normalizing the area, dealing with negative entropies, etc. etc.). PAR (talk) 02:10, 17 February 2016 (UTC)

The foregoing on the face of it is labeled an argument that entropy is not about dispersion. On closer examination it appears to be an argument that information is not about dispersion. The labeling as one about entropy arises from the naming of information as entropy. Take away the tie between information and entropy, and the argument is not about entropy. In this light it does not touch on the proposed link between dispersal and entropy as defined in thermodynamics. If the argument is valid (a question I do not tackle), it may consequently suggest that it would be better not use the name information for entropy as defined in thermodynamics.

Tisza and Quay write "... the dispersal of the distribution. Of course, this term is to be distinguished from the dispersion of a random variable, a synonym for its variance or relative quadratic moment. Thus in a uniform distribution over g degenerate states the dispersion of the energy vanishes whereas the dispersal of the distribution is ln g."Chjoaygame (talk) 08:21, 20 February 2016 (UTC)

Strictly speaking, its an argument against using Q(f)=−Σ_i f_i ln f_i as a measure of "dispersion" of the probability distribution f. I have used Q(f) to avoid any implications that it has anything other than a pure mathematical meaning. Gibbs used Q(f) as a measure of the thermodynamic entropy S divided by kB, Boltzmann's constant, when f is the probability distribution of microstates. This reaffirmed Boltzmann's famous equation S=kB ln (W), in which each microstate had equal a priori probability. (f=1/W so Q(f) = ln(W)). The fact that Shannon also used it to describe a lack of information in his theory, and called it entropy, is irrelevant. In other words, its not about entropy and information, its about entropy and Q(f). I'm not promoting, at this point, the idea that thermodynamic entropy is about information, only that entropy as "dispersion of energy", in physical space or abstract probability space, is simply not right. Sometimes it "looks" right, but it's not.

So yes, if you throw out the connection between thermodynamic entropy and Q(f), my argument is pointless. But to throw out that connection, you throw out the work of Gibbs, Boltzmann, all of statistical mechanics, basically. PAR (talk) 12:43, 20 February 2016 (UTC)

Thank you for these challenging and valuable points. To be precise (as per Tisza & Quay above), the topic is 'dispersal' not "dispersion".

I am not the inventor of the 'dispersal' view: as far as I have learnt, Guggenheim was the culprit. He thought he was basing his view on the statistical mechanics that your comment says it would "throw out". I don't think I can do better than Guggenheim in presenting his reasons. As I mentioned above, the journal Research is in only some recherché libraries. I don't know if there is such a one near you. Tisza, who also advocates 'dispersal', does not see himself as throwing out the usual statistical mechanics. His article is reprinted in his 1966 Generalized Thermodynamics, to which on a past occasion you have had access.Chjoaygame (talk) 14:41, 20 February 2016 (UTC)