Talk:Microcanonical ensemble

reason for revert

reason for a recent revert:

some edits are of dubious nature:

1. the symbol Z is used before ever defining ( it is in fact the partition function, in this case, multiplicity. and that's explained in the current version.)

2. there is no need to used generalized function here. one can just simply take that symbol, in this case, to mean the density function is 1 on the level set of interest and zero elsewhere.

3. the phase space measure is the Lebesgue measure. if you'd like to add something, don't remove what's correct.

4. Liouville's thm states that that measure is invariant under the flow. again, welcome to add the Poisson bracket, don't remove what's correct.

5. it's already obvious from the discussion that $\;\Omega$ is the multiplicity. Mct mht 14:56, 23 May 2006 (UTC)

Further,

6. there's no need to embed the constant energy surface of interest into the whole phase space then call it a submanifold.

7. ergodicity consideration are much more natural in the measure-theoretic context. Mct mht 16:17, 23 May 2006 (UTC)

Can we at least keep some of the matrix formalism which the previous posts alluded to? --HappyCamper 17:03, 23 May 2006 (UTC)

if you're referring to the density operator of a MCE, it's already in the article. Mct mht 17:07, 23 May 2006 (UTC)

Ah yes, I see it now. Still rather technical of an article though. I think later I might add some stuff when I get a chance. --HappyCamper 17:13, 23 May 2006 (UTC)

suggestion for addition to article

an ensemble of quantum mechcanical harmonic oscillators (or some other system that lends itself easily to both treatments) can be added as a good example comparing the semi-classical approach and the density matrix formulation. Mct mht 19:03, 23 May 2006 (UTC)

reply to "reason for revert"

I think my (Top.Squark) revision is correct and should be restored. Explanation:

1. Z is defined when first used: "Z is a normalization constant" I.e. Z is such a constant that the integral of rho over the phase-space is 1. You are right that it is the same as $\;\Omega$, which should be made explicit.

2. Rho cannot be "1 on the level set of interest and zero elsewhere" because:

a. Integral of rho would be _zero_ rather than 1 as suitable for a probability distribution since it's supported on a set of _zero measure_.

b. The expectation value of an obserable A is integral of A rho and for your function it would be _always zero_.

This is precisely the kind of thing generalized functions were invented for!

3. The phase space (Liouville) measure_is not_ the measure induced by the microcanonical ensemble since the later is supported on a level submanifold of H and is a probability measure while the former is neither.

4. Liouville theorem states the _Liouville measure_ is invariant, not the microcanical ensemble measure.

5. It was not stated that $\;\Omega$ is the multiplicity earlier (in my revision).

6. What do you mean? The constant energy surface _is_ embedded in the phase space.

7. What it has to do with my changes?

Top.Squark 10:18, 24 May 2006 (UTC)

the above statements are wrong and/or don't make sense.

1. it already is explicit.

2. a. "supported on a set of zero measure"? as in the support of ρ has Lebesgue measure 0? then the ρ is essentially the zero measure?! clearly nonsense. how exactly does one get from the discussion on phase space measure to the zero measure condition you brought up? in where is any set of zero measure mentioned or seems relevant? again, pure nonsense.

b. no it won't be. read it again.

Further, there's seems to be a misunderstanding of what a generalized function is.

3. that statement doesn't really make sense. for a good explanation of what's meant by Liouville measure, see a recent discussion in Talk:Dynamical system. there, the user XaosBits gave a good explanation.

4. you're merely contesting the jargon. for the microcanonical ensemble, the microcanonical ensemble measure is clearly the Liouville measure.

5. that's right, it wasn't stated in you edits. that's why it's not clear.

6. your revision refers to the constant energy surface as a submanifold.

7. because ergodicity is clearly relevant here, and therefore, not only is it unnecessary to refer to generalized functions, it is also awkward.

4. and 6. are minor points, so is 1, i wouldn't object a change in mere language. the others, however, seem to indicate confusion, which should be in no way reflected by the article. Mct mht 16:28, 24 May 2006 (UTC)

Reply to Mct mht

1. In my revision it wasn't explicit and I agreed it should be explicit. This is the part where I'm agreeing with you.

2. "he support of ρ has Lebesgue measure 0? then the ρ is essentially the zero measure?! clearly nonsense." Right, nonsense!! What you describe is nonsense _because_ it leads to rho being the zero measure, which it shouldn't be! The correct function is rho = Omega^-1 delta(H-E), _not_ "1 on the level set of interest and zero elsewhere".

A generalized function, to your information, is a functional on a function space is context-dependant in general, but is usually taken either as the Shwartz space (smooth functions that decay faster than polynomial in infinity together with all derivatives) or the space of compactly supported smooth functions. In this case the functional is integral over M_E divided by M_E's hyperarea (Omega).

3. In the discussion you mentioned XaosBits correctly says that dp dq (in the simplest case, i.e. a point particle in 1 space dimension). If you intented that dp dq is invariant under the flow according to Liouville's, that's correct and can be integrated into the article. However, your statement "this is also the measure induced by the microcanonical ensemble density function" is incorrect since the later is Omega^-1 delta(H-E) dp dq, not just dp dq.

4. Not at all (see above)

5. It _was_ stated in my revision, but not earlier!! First you claim that "it's _already obvious_ from the discussion that Omega is the multiplicity" and now you say "that's right, it wasn't stated in you edits. that's why _it's not clear_"??

6. The constant energy surface _is_ a submanifold. What's the problem?

7. I don't see any contradiction between ergodicity and generalized function. What are you getting at?

Top.Squark 19:05, 24 May 2006 (UTC)

it seems you're misinterpreting the notation and the disagreement is more minor than i first thought. the notation you gave (\delta ...) means precisely what i said, see, for instance, Reed and Simon. it does not lead to the measure having zero support. if you wanna consider the whole phase space, not just the constant energy surface, and add the delta symbol, that's fine. but that's highly misleading, if you do embed the constant energy surface into the whole phase space, then, in the dimension of the whole phase space, not just the constant energy surface, in the induced measure of that dimension, your energy surface could have measure zero, the very error you said your edit using this notation corrects. for instance, your constant energy surface could be a line and has zero Lebesgue measure in R^2, where the phase space is R^2. it's more accurate to consider the constant energy surface as a manifold in its own right, and used the natural measure on this manifold, rather than embed.

similarly, if you wanna identify the measure with a distribution, that's fine too. althought that's not really necessary either. as for ergodicity, it's rarely, if ever, talked about in terms of distributions. since the subject of article naturally leads to a discussion on ergodicity, the measure-theoretic language shouldn't be abandoned. Mct mht 19:53, 24 May 2006 (UTC)

by the way, Reed and Simon does use the notation $\delta (H-E)dpdq$, but they explain clearly that the measure is induced in the dimension of a constant energy surface of interest, not in the dimension of the whole phase space. so if that symbol is to be put in the article, should be accompanied by similar clarification. Mct mht 20:09, 24 May 2006 (UTC)

led me also say, whatever the disagreements may be, these are informative discussions. so thanks. Mct mht 20:27, 24 May 2006 (UTC)

as long as we're on the issue of the phase space. in the case of the microcanonical ensemble, it perhaps is more appropriate to replace the phase space by the disjoint union, a bundle-like object, of the constant energy surfaces, indexed by the possible energy values. this bundle, if you will, and the induced measure, then automatically has the suitable dimension. the density function is then 1, on the one energy surface of interest, and 0 elsewhere. this avoids the confusion brought about from various the notations one might use. this seems to be particular to the microcanonical ensemble, as it doesn't appear to be an issue for canonical ensemble. Mct mht 03:37, 25 May 2006 (UTC)

arguably, this could be related to the "degeneracy" of the microcanonical ensemble, referred to in the 2nd sentence of the article. Mct mht 03:47, 25 May 2006 (UTC)

comment on recent edit

at first glance the main issue (of the conversation above) is resolved by a recent edit by 62.0.156.172(Top quark?). whoever editted it, can you provided a reference on the way that measure is obtained? i didn't see anything incorrect, just would like a reference. also, the measure theoretic language regarding Liouville's theorem should be added back, as it's not immediately clear one can talk about Poisson's bracket in a weak(distribution) sense. (if you wanna be fully rigorous, as you seem to, better to use the measure you constructed in detail.) Mct mht 16:45, 25 May 2006 (UTC)

It was indeed me who did the edit (just forgot to log in). The measure is obtained from the delta-function expression, i.e. it is defined so that the integral of any function f on the whole phase space against delta(H-E) (using the usual Liouville [= Lebsegue in the R^2n case] measure) is the same as the integral of f on the constant energy hypersurface of this measure. One way to see my expression satisfies this condition is to note that dA is defined in such manner that dq^n dp^n = dA dH. This equation is just the infinitesimal version of the definition I provided. It indeed implies the compatibility condition since

(integral) f delta(H-E) dq^n dp^n = (integral) f delta(H-E) dA dH = (integral on H=E) f dA

the last transition by integrating over H first.

Regarding the Poisson bracket, it does make sense to use it as one can derivate generalized functions and multiply them by smooth ones. Alternatively it is indeed possible to prove the time-invariance from the definition of dA. Top.Squark 12:53, 27 May 2006 (UTC)

Compromise

Is there a way to incorporate both teaching methodologies into the article? --HappyCamper 19:31, 24 May 2006 (UTC)

seems that the differences are mainly linguistic, as long as everything is explained clearly, that's fine withe me. Mct mht 20:16, 24 May 2006 (UTC)

p and q

Can we add a brief comment about the p's and q's in the article? --HappyCamper 20:35, 24 May 2006 (UTC)

target audience

This is one of several articles in this area that can easily loose a large part of its target audience. Can undergraduate physicists and chemists read the first 3 paragraphs and get a clear idea of the ensemble, and browse the remaining paragraphs without getting jargon overload? Bob aka Linuxlad 20:51, 24 May 2006 (UTC)

New version

I submitted a new revision which hopefully combines both point of view.

The old revision refers to the constant energy hypersurface as the "phase space" (which is confusing) and works w.r.t. the natural measure there without stating so explicitely (even using notation which suggests the contrary, e.g. dp dq which only correctly describes the measure on the _full phase space_). That "local" point of view is the most natural in the context of ergodic theory, but is less favorable in some other contexts, for instance when one wishes to compare the microcanonical ensemble to other ensembles used in thermodynamics (e.g. the canonical ensemble).

The new verision uses accurate notation (using the "global" phase space as a starting point) but does not de-emphasize the local point of view like my previous revision. I added a paragraph explaining how the natural measure on the constant energy hypersurface is defined in a rigorous manner (it of course follows from the Dirac delta-function expression for the density on the full phase space).

Also, I noticed there have been complaints about the level of technicality of the article. I personally do not think technical jargon should or can be avoided in a satisfactory presentation. If one would integrate the detailed explanation of every technical term (e.g. the Dirac delta-function, the Liouville measure etc.) into the article it would become oversized and unfocused. On the hand, avoiding the use of such notions altogether would make the article unrigorous, lower its information content and render it completely unsatisfactory for the more sophisticated reader. The reader unfamiliar with one of these notions should address the separate Wikipedia article on the subject (especially given most of those are linked to this one).

Top.Squark 16:36, 25 May 2006 (UTC)

overall i think it's a good edit, see comment above. Mct mht 16:56, 25 May 2006 (UTC)

I suggest you look at some of the expositions of these matters on university sites, with a target audience of eg postgraduate astronomy students or chemists. It can, and should be done without the explicit use of Liouville measure etc. Bob aka Linuxlad 18:37, 25 May 2006 (UTC)

I think the Liouville stuff will stay -- it just means that this article needs more development. Most articles on Wikipedia do not mature until well over a year or two of continuous editing. Well, unless there's an active push for it, if we're lucky, a few months. --HappyCamper 19:49, 25 May 2006 (UTC)

I'm quite happy with Liouville's theorem being in there (It's pretty fundamental to the way 3rd year physicists were taught it years ago - and it clarifies that the phase space units dp.dq occur as the appropriate weightings in Boltzman etc), but we didn't need the Liouville or Lebesgue measures, and even now I'm not too sure what a sub-manifold is... Bob aka Linuxlad

per Top.Squark's comment, as far as technical sophistication goes, this article is just fine. these are essentially rigorous versions of presentations found in graduate (possibly undergraduate as well) physics texts, see, for example, Statistical Mechanics by Pathria. if one would like, one can maybe add a note saying, for example, in case the reader is not familiar with the mathematical machinery used, some notions can be interpreted naively and article can still be understood. also, the title "deeper relations..." is rather misleading. it's also misleading to say "There are many different approaches to deriving the ensemble properties..." Mct mht 20:56, 25 May 2006 (UTC)

by the way, the description of the ensemble in the current version that's essentially given by Top.Squark is excellent and very informative, IMHO, and hopefully will editted only if the editor's sure the resulting change will be an improvement. Mct mht 21:03, 25 May 2006 (UTC)

"Deeper" for want of a better word...

Well, the title explains it all.. --HappyCamper 21:09, 25 May 2006 (UTC)

The natural measure on the constant energy hypersurface

There is a slight technicality about the natural measure on the constant energy hypersurface. The Liouville measure on the full phase is just the Lebesgue measure on R^2n (given the phase space is R^2n). However, the induced measure on the constant energy hypersurface is not hyperarea in the sense of Euclidean geometry. For example, given a 1-dimension particle with phase space R^2, and Hamiltonian H = p^2 / 2m + V(q), the measure on the p^2 / 2m + V(q) = E curve is not just arc-length. The correct measure of an arc is the time for the system to transgress it. For instance, if two plateaus (V = const) with different height are present, the measure on each will be m/sqrt(2m(E-V)) dq, hence normalized differently (due to different V). In the harmonic oscillator example, the measure will equal (up to a constant) arc-length only for a special choice of coordinates (sqrt(k/2)q, sqrt(1/2m)p) and for most systems no natural choice of coordinates will do the trick. Therefore I removed a paragraph stating dA is "just" the natural measure up to a constant: the definition of "natural measure" is precisely the definition of dA I wrote, nothing less.

Top.Squark 12:01, 27 May 2006 (UTC)

Can we add a bit more material relating to Gibbs work? --HappyCamper 14:26, 27 May 2006 (UTC)

section from recent edit

An anonymous editor introduced the following paragraph:

Distinguishing Microstates from Macrostates

To understand the meaning of the microcanonical ensemble it is instructive to understand the distinction between a microstate and a macrostate:

The term macrostate indicates the overall thermodynamic state in a system. In an isolated system, the macrostate is defined by the type of substance considered, the amount of substance, $N$, its internal energy, $U$, and its volume, $V$. A macrostate thus refers to the bulk properties of a given system.

A microstate on the other hand is the most accurate and precise (without violating Heisenberg's uncertainty principle) description of a physical system. If a system is characterised by a certain macrostate $(N,U,V)$, there is an enormous number of possible microstates. For example, if you know the position and the momentum of every particle in your system, you know its microstate. It therefore becomes apparent that the microstate of a physical system that contains a reasonable number of constituents (eg particles or strings), say $N=10^{23}$, changes at a rate of about $10^{23}$ times every second... Clearly then some kind of averaging procedure is needed and this is where the microcanonical ensemble comes into play. The microcanonical ensemble, $\Omega$, equals the total number of microstates of a given system. For example, for the whole observable universe

$\Omega \sim e^{10^{86}}$.

As one would then expect, quantum mechanics comes to our rescue because it is (at least conceptionally) far easier to count a discrete number of states than it is to count a continuum of states.

while i don't dispute its veracity (for the most part), may i suggest considering relocating it to microstate (statistical mechanics). if it's insisted that the section be in the article, perhaps retool it a bit (the last three sentences in particular)? Mct mht 17:39, 10 April 2007 (UTC)

In the Entropy Section

I think this text needs to be rewritten: 'Notice that, for the microcanonical ensemble, Ω plays the role of the partition function in the canonical and grand canonical ensembles.'Paranoidhuman (talk) 21:59, 11 March 2009 (UTC)

I have already

haned out my homwework of SM. Thanks god.

Tomorrow,QM h.w. remains only one problem in it. —Preceding unsigned comment added by 140.123.72.3 (talk) 05:17, 27 October 2009 (UTC)

volume entropy vs. surface entropy

Gibbs discussed two possible analogues for "entropy" and "temperature", from the microcanonical ensemble. One was based on the measure of phase space "surface area" at energy E, and the other based on the phase space "volume" at energy E. The surface area definition (used in this article) is certainly very popular, but I've seen other places use volume entropy. They both work fine for macroscopic objects so seems to be a matter of taste, which one to use. However they are different for microscopic objects and lately, with nanoscience and all that, the distinction seems to be a real issue.

If you're wondering why in the world someone might use volume entropy or what the difference is, this article might be helpful. Disclaimer, it is in favour of volume entropy. Personally, I'm not sure I buy their arguments and I think I may have a bias towards the simplicity of surface entropy (and I sort of like the idea of negative temperatures). For a more balanced view, Gibbs' Elementary Principles in Statistical Mechanics is a harder read but (typical of Gibbs' thoroughness and near-prescience) touches on relevant pro's and con's for small systems. From page 179 "The choice between the variables [volume or surface entropy] will be determined partly by the relative importance which is attached to average and probable values.", and:

Volume Entropy

[pro]: average values are important and they lend themselves better to analytical transformations.

...some more, though I don't understand exactly...

Surface Entropy

[pro]: conceptually simpler.

[pro]: the area of the energy-E surface can be finite even when the volume of phase space might be infinite.

[con]: we have to keep making exceptions for systems of 1 or 2 degrees of freedom.

...some more, though I don't understand exactly...

Worth a mention in the article, I think? --Nanite (talk) 19:41, 26 August 2013 (UTC)

I prefer to look at this from a more fundamental physical point of view. The omega dependence, while inconvenient from a puritan mathematical perspective, is entirely natural from a physics point of view. Entropy expressed in bits is the amount of information needed to specify the exact physical state the system is in, given its macroscopic specification. Obviously, if you make omega twice as small that saves you one bit of information (assuming constant density of states over energy range of omega), which is insignificant for large systems (and vanishes altogether in the thermodynamic limit when considering specific entropy). But that omega dependence which is important for small systems, is an important reminder that entropy is ultimately a "subjective" quantity that expresses the amount of ignorance we have about the exact physical state of the system. Count Iblis (talk) 17:39, 12 November 2013 (UTC)

Thanks for the reply. You probably noticed that I just made a big revision to this article. I've been reading Gibbs carefully since my last talk post (also other books, but regrettably there seems to be no clearer treatment than in Gibbs) and so I've added a section on entropy computation leaning heavily on his results in classical ensemble mechanics. However I'm in want of a similar fundamental treatment on quantum ensemble mechanics. I suspect that von Neumann might be the go-to source, however I've yet to get deeply into his works.

I agree with you that (besides the thermodynamic entropy) the information entropy is really the only thing that deserves the title of entropy, and it is unfortunate that the word seems to be burdened with a lot of subtly different meanings (I can think of five statistical entropies off the top of my head). It seems that the difficulties in the microcanonical ensemble are simply a symptom of an impossible assumption that we can know perfectly the energy of a system. In quantum mechanics that basically amounts to saying you know exactly the state of the system, whereas in classical mechanics it amounts to saying that you can tell me the energy correctly to as many decimal points as are desired. To be honest, I'm surprised that the microcanonical ensemble is still considered by some to be foundational in statistical mechanics, given that other equilibrium ensembles with finite uncertainty in energy are both more realistic and more mathematically manageable.

The section on entropies as I wrote it is rather unsatisfying, I admit. At the moment it looks like a pointless debate over what is the "true entropy" however it would be much more useful if it just focussed on providing the best analogies between the microcanonical ensemble and thermodynamics. My hope is that a learner of physics coming to the article will come away understanding what the microcanonical ensemble is and how it is used in real calculations, but also advised that a better description of thermodynamic equilibrium is found in the canonical ensemble. --Nanite (talk) 23:10, 13 November 2013 (UTC)

Delete section "Perfect gas entropy"?

As it stands I'm not sure the content of this section is appropriate for a wikipedia article, as it reads more like a homework solution than a general overview of the microcanonical ensemble. I would propose deleting this section, but otherwise the section needs a cleanup regardless as it is full of spelling mistakes and formatting problems.

AwkwardWhale (talk) 14:05, 9 December 2020 (UTC)

Symbol confusion

The text has two symbols $\nu (E)$ and $v(E)$ for the what seems like the phase space volume. Are they meant to be distinguished in some way?

Questions regarding ideal gas example

This is my first time here so I apologize if I am doing this wrong. But I have a few questions and I hope that this is the correct place to ask them.

Under the subsection Ideal gas within Examples section:

1. Regarding the Sackur–Tetrode equation, what exactly is the energy E here? (Following the link on this page gives a slightly different, and to me more familiar and explicit, form of it.)
2. Furthermore, is the E in the derivation of chemical potential the ideal gas' internal energy (sometimes denoted by U, like on the Sackur–Tetrode equation page)? Is there a equation for it in some simple case?
3. Finally, I think there might be a missing minus sign in said chem. potential equation. Am I missing something in the derivation?

Thank you! Ivaya007 (talk) 14:47, 17 August 2023 (UTC)

How is $\omega$ defined?

As far as I can tell, $\omega$ is only vaguely defined as the 'energy' spacing. It is mentioned in the expressions for the entropy and is therefore important to at least know the definition. 131.211.54.68 (talk) 10:17, 22 April 2024 (UTC)