Talk:Grand canonical ensemble

Dirac brackets

Not a physics expert, but is that formula missing a bracket? Also, which Hamiltonian are we talking about here? The Classical one or quantum one? Any help appreciated. Soo 17:39, 17 August 2005 (UTC)

These (partial brac-kets) are called Dirac brackets. Basically a bra or a ket denotes a quantum mechanical state. And the combination, in 'bra'|'ket' order represents the overlap between two states. Statistical mecahnics aims at summing over all nonoverlapping states. Density matrix is often described as a sum over a series of 'ket'-'bra's. However I personally feel it is quite a convoluted way to define grand canonical ensemble. (It's more like g.c.e. provides a meaning of a mixed state (or an ensemble) described by a density matrix.) It will be a much longer story if I explain Dirac brackets in details, sorry. Please consult some quantum mechanics books. Czhangrice 21:51, 23 May 2007 (UTC)

re merge tag

i suggest no merge. although no one has found the time to expand this stub, the grand canonical ensemble, like the microcanonical ensemble and canonical ensemble, certainly deserves its own page. Mct mht 02:01, 24 August 2006 (UTC)

I agree with the above comment. --HappyCamper 05:00, 25 August 2006 (UTC)

Similar agreement. - Mostly anonymous student: 128.146.34.189 08:32, 16 November 2006 (UTC)

Partition sum

My physics book gives the partion sum as

$\sum _{N}\sum _{i}e^{-\beta (E_{i}-\mu N)}$

where $N$ is the number of particles of the system and runs from 0 to infinity. 82.135.75.113 20:14, 6 April 2007 (UTC)

Your textbook is right. The mentioned formula is wrong.

This problem is fixed. Czhangrice 21:32, 23 May 2007 (UTC)

Ornstein-Zernike

The link seems to be of no relevance for me. As far as I see - I am not a specialist in it - it deals with some rather special questions concerning th canonical(!?) ensemble. In any case the referenced articles does not improve the understanding of thermodynamical potentials at all.

Grand canonical ensemble is roughly a bag of canonical ensembles. In g.c.e, system exchanges particles and energy with the environment (or other systems in the ensemble); In c.e, system only exchanges energy with the environment. I hope that I have made clear upon these points. I didn't see too much relevance of the reference either. Czhangrice 21:37, 23 May 2007 (UTC)

Logarithm of grand canonical partition function

Does anyone know if there is an official name to the logarithm of the grand canonical partition function? I ask because the log of other partition functions have specific names. Notably, the log of the canonical partition function is (up to a factor of temperature) the free energy while the log of the density of states(essentially the partition function of the microcanonical ensemble) is the entropy. Moreover, one could imagine defining a statistical ensemble at fixed pressure and temperature. The log of the associated partition function would then be the enthalpy. So does anyone have any idea about this? Joshua Davis 21:19, 5 July 2007 (UTC)

Yes, take a look at characteristic state function. So, for this case (grand canonical ensemble), it's related to pressure-volume work. --HappyCamper 23:46, 5 July 2007 (UTC)

Thanks. I'm afraid this isn't quite what I had in mind, though. In particular, one doesn't define the pressure-volume work as the log of the grand canonical partition function. Rather, for extensive systems there is the identity, $E=TS+PV+\mu N\,\;$ (I may have gotten some signs wrong and I'm assuming only one chemical species for simplicity). So if you do the appropriate Legendre(or Laplace) transforms to get to grand canonical ensemble, then you end up with $\log \Xi \sim PV\,\;$. But for non-extensive systems(black holes, for example), this doesn't hold, I think. Similarly, the Gibbs free energy for extensive systems satisfies $G=\mu N\,\;$, but one doesn't define it this way. Joshua Davis 04:22, 6 July 2007 (UTC)

Lack of exposition of fundamentals of the subject

This article appears to be afflicted with the common textbook malady of starting to talk about something before making clear what the thing is. The concept of a thermodynamic ensemble is abstract and somewhat challenging to grasp. Despite the difficulty, there is no point discussing the subject at all without conveying the idea of what these ensembles are and why they were invented (I write "invented" rather than "discovered" because ensembles are strictly human ideas, definitely not anything which exists out in the world). I'm going to try adding some brief clarification about the basic concepts. I will be as accurate as I can, but if I make mistakes, I hope someone will correct them. Dratman (talk) 19:43, 10 September 2009 (UTC)

I completely agree. I am a well-educated layman (beginning a PhD in Applied Mathematics), and I found this article lacked a general explanation to ground me in the subject matter. Furthermore, it lacks links to basic concepts such as Partition function (statistical mechanics). I am not qualified to edit this article, but I hope that somebody will. Chrislaing (talk) 01:14, 21 June 2011 (UTC)

Another significant problem of the article is that the definition of the grand-canonical partition function includes the fugacity, which is given as a function of the chemical potential. The chemical potential however is a thermodynamic quantity, which can not be derived from stat mech basic principles, but is rather recovered after linking stat mech to thermodynamics.The fugacity should be introduced as the one particle partition function summing over all its degrees of freedom. — Preceding unsigned comment added by 193.175.8.66 (talk) 12:18, 16 December 2011 (UTC)

Parasitic surface

Dear Nanite (talk)!

The problem is not that in nature there is no system, surrounded by a closed surface. The problem is that nobody knows that GCE is a centaur --- open system for bulk properties and closed for the surface. This led, in particular, to the fact that the problem of adsorption has not been resolved for 70 years. If you do not like the name of the subkey that you can change it, but to remove such important information is unacceptable.Luksaz (talk) 07:16, 30 December 2014 (UTC)

(Note: the article mentioned below has been deleted - Wikipedia:Articles for deletion/Open statistical ensemble. Latrissium (talk) 20:24, 27 May 2020 (UTC))

(for readers' sake, see also related discussion at Talk:Open statistical ensemble)

I have to admit I don't fully understand what you're saying Luksaz. Do you have some examples to illustrate your point here? What is a strong example of a physical system where I cannot apply GCE but the OSE treats correctly? You mention adsorption, but I thought the GCE-based Langmuir adsorption model worked well for this problem? --Nanite (talk) 13:23, 31 December 2014 (UTC)

Langmuir adsorption model is not the fundamental model and how you can easily see, does not start from GCE. Attempts to address this problem from the viewpoint of statistical physics (within the GCE) was carried out with about the middle of last century. This took part American, Polish, Russian, Japanese and others researchers. But the problem was solved within the OSE in 2011 alone. Even Henry adsorption constant could not count under the GCE (roughly a little). For example, see Henry adsorption constant and references therein. Luksaz (talk) 14:25, 31 December 2014 (UTC)

Definition of grand potential and grand partition function

1) At the top of the article we read the exponential expression

$P=e^{\frac {\Omega +\mu N-E}{kT}},$

for the probability and then:

The number Ω is known as the grand potential ... many important ensemble averages can be directly calculated from the function Ω(µ, V, T).

Shouldn't Ω be defined in some way? Is it given arbitrarily or is it computed from something? How can we tell that it should be a function of µ, V, T ?

So following the link from grand potential, one reads the definition Φ = U - T S - μ N for a Φ that is called the grand potential. Is this definition we are supposed to use here? If it is the definition we want, then what are S and N? Or is this Φ definition is only valid for large "thermodynamic" systems with a non-fluctuating total U, S, N ?

Much later in the article, the relation

$\Omega =-kT\ln {\Big (}\sum _{\rm {microstates}}e^{\frac {\mu N-E}{kT}}{\Big )}$

is given, and it's observed that this follows immediately from the sum of the probabilities being 1. Then this is the definition of Ω that should be used at the top of the present article, isn't it?

2) Note on formulation An alternative formulation for the same concept writes the probability as $\textstyle P={\frac {1}{\mathcal {Z}}}e^{(\mu N-E)/(kT)}$, using the grand partition function $\textstyle {\mathcal {Z}}=e^{-\Omega /(kT)}$ rather than the grand potential.

While the relation $\textstyle {\mathcal {Z}}=e^{-\Omega /(kT)}$ is a true fact about Z, it's not the definition of Z. If you try to use it as a starting point for the theory instead of the usual sum, you go in circles. That makes the "alternate formulation" via Z misleading; it's not self-contained.

178.38.76.171 (talk) 00:06, 13 April 2015 (UTC)

Good points. Perhaps it's best to say something along the lines of "Ω is a normalizing factor that ensures the probabilities add up to one (then include the equation, sum_microstates P = 1). But oh, by the way, it turns out that $\Omega =\langle E\rangle -\langle N_{1}\rangle \mu _{1}\ldots -\langle N_{s}\rangle \mu _{s}-ST$ (see properties section), a quantity known as the grand potential.". I think a similar change is necessary for canonical ensemble and its Helmholtz free energy.

Regarding 2), I mean that Z and Ω are just two alternative mathematical conventions or notations for describing the exact same concept. I know most physicists have a preference of one over the other but I figured I would go with Ω since it was first, and at least thermodynamic averages are written more conveniently in terms of Ω. That said if one uses variables $\beta =1/(kT)$ and $\alpha =-\mu /(kT)$ then it is much more elegant to use either Z or ln(Z) (aka free entropy). Unfortunately it seems that is unconventional.

PS: To be honest I am not even sure that "grand potential" is a widely recognized name for this average. In the books of Landau&Lifshitz and Balescu it's just called the thermodynamic potential or free energy associated with µ,V,T. Some books like Huang or Kittel don't even mention it, using entirely the grand partition function (a.k.a. Gibbs sum or grand sum). Kubo's 1968 Thermodynamics does say "grand potential" with the immediately following disclaimer: "This function has no particular name widely accepted. In recent literature of statistical thermodynamics it is sometimes called this way because it is related to the grand canonical partition function." Maybe the grand potential article needs a tweak. --Nanite (talk) 18:37, 17 April 2015 (UTC)

Some material in "Applicability" section seems mistaken or unclear

The grand canonical ensemble is the ensemble that describes the possible states of an isolated system that is in thermal and chemical equilibrium with a reservoir

How can a system be isolated and at the same time be in thermal and chemical equilibrium with something outside itself?

the derivation proceeds along lines analogous to the heat bath derivation of the normal canonical ensemble, and can be found in Reif

We really need the derivation to understand the role of μ and Ω, which are quite mysterious here !

The condition that the system is isolated is necessary in order to ensure it has well-defined thermodynamic quantities and evolution.

1) To have well-defined thermodynamic properties, you don't need the system to be isolated !! It just needs to have time to get into equilibrium within itself and with its neighbors.

2) We don't need or want a well-defined evolution, however. Thermodynamics (at this level, anyway) is about equilibrium situations. The transitions between them are left somewhat mysterious, except for starting points and end results. (I realize the authors know this; it's the reader who can be misled.)

In practice, however, it is desirable to apply the grand canonical ensemble to describe systems that are in direct contact with the reservoir, since it is that contact that ensures the equilibrium.

This makes it sound like contact with the reservoir is an unpleasant practical imperfection of a grand canonical ensemble. Actually, it's a defining condition, isn't it? (Of course, once equilibrium is reached, you could intermittently interrupt contact with the reservoir and no-one would notice.)

By the way, what is the difference between "direct contact" and "contact"?

The use of the grand canonical ensemble in these cases is usually justified either 1) by assuming that the contact is weak, or 2) by incorporating a part of the reservoir connection into the system under analysis, so that the connection's influence on the region of interest is correctly modelled.

The words "usually justified" bother me. It seems that somewhere during this paragraph we have moved from idealized definitional situations, which serve to explain the concept -- to bona-fide practical situations, where there's a real concern about making the model fit. But the transition wasn't clearly announced.

become equivalent in some aspects

This makes me uneasy, of course. But maybe it's just hard to put these into words. And it's valuable to say.

As a result, the grand canonical ensemble can be highly inaccurate when applied to small systems of fixed particle number, such as atomic nuclei.

The emphasis is misleading. The "can" suggests that it's a habit of practitioners. But it wouldn't be "highly inaccurate"; it would be a mistake, wouldn't it?

178.38.76.171 (talk) 22:11, 12 April 2015 (UTC)

Thanks for the input! I agree with your feeling. I wrote most of the article as it is, including the section in question, but the wording has been bothering me for a while now. I suppose I just haven't found the right words. My effort was to boil down what is found in Gibbs' book.

What I am trying to allude to is the following:

1. To have definite mechanical evolution, the system's Hamiltonian should have a well defined form, which technically excludes any interactions with other dynamic systems. This also excludes contact with a reservoir. I know it's quite common in textbooks to "prove" the GCE from the MCE using contact with a reservoir, however in a rigorous derivation it is impossible to do this in general unless the total Hamiltonian can be decoupled into system Hamiltonian and reservoir Hamiltonian. But if such a decoupling is possible, then they are not in contact!
2. If we bend the rules a bit we can say that weak contact is allowed, for example, maybe the system and reservoir only interact through a relatively tiny shared interface. But is contact necessary? If the system is already in GCE then the contact is not even needed. We only need contact when we want to describe the approach to equilibrium, i.e., nonequilibrium statistical mechanics, but is a huge topic which is outside the scope of this article. Still, it might be worth mentioning how real life systems arrive at the GCE, but setting aside fundamental questions about the approach to equilibrium.
3. When I say equilibrium, I mean that the ensembles of the two systems would not change as a result of being brought into weak contact. However they do not actually need to be in contact. (By contact I mean they are allowed to weakly interact and exchange particles. E.g., if the two systems are bottles of gases, we could imagine opening up a tiny hole between them.) Just to be clear this definition would also mean that two canonical ensembles are in thermal "equilibrium" if their temperatures are equal, regardless of the distance between them, even if one is at Alpha Centauri and one is here. However I would say a system and reservoir remain in equilibrium even if the contact is interrupted.
4. As you say, "you could intermittently interrupt contact with the reservoir and no-one would notice" --- this is what I am trying to get at. Moreover, suppose you removed contact and never restored it... your system would continue to remain described by the GCE forever, even though it is mechanically isolated. The initial uncertainties about energy and particle number would not spontaneously disappear.

Well, I hope you can see what I am getting at. You're right that the article text needs to change... --Nanite (talk) 17:42, 17 April 2015 (UTC)

Sections "Properties of the grand ensemble" and "Grand potential, ensemble averages, and exact differentials"

I split the previous section into two sections, at a logical place, in the hopes of stimulating someone to expand them.

The two sections are full of gold, but very condensed. I have to say I appreciate the summary list format, even though it's deprecated. Still, these sections need more prose -- to explain the connecting logic to a newcomer !!

178.38.76.171 (talk) 00:21, 13 April 2015 (UTC)

Minimum grand potential

From the article:

Minimum grand potential: For given mechanical parameters (fixed V) and given values of T, µ₁, …, µ_s, the ensemble average <E + kT log P − µ₁N₁ − … µ_sN_s> is the lowest possible of any ensemble.

My comment:

From the definition of the grand potential Ω that was given (or not given!) earlier in the text, it is difficult to recognize the quantity <E + kT log P − µ₁N₁ − … µ_sN_s> as being the grand potential Ω of an ensemble that is competing with the grand canonical ensemble. The problem is that Ω was never defined for an arbitrary ensemble, indeed was never defined at all (!). It was a number (of mysterious origin) that appeared in the definition of the grand canonical ensemble (only).

So even though this minimizing property is really important in the theory, in this section it looks either circular, or not a property of Ω.

Of course, if one defines Ω as a function of the microstate ω by

Ω(ω) = E(ω) + kT log P(ω) − µ₁N₁(ω) − … µ_sN_s(ω),

then the assertion makes sense and <Ω> is minimized (I didn't actually check this). However, if one doesn't say this formal stuff, the reader is lost.

If this is indeed what's intended, shouldn't the description be Minimum average grand potential?

178.38.76.171 (talk) 01:17, 13 April 2015 (UTC)

If I had to define grand potential for a general ensemble it would be not as the microscopic quantity but as the thermodynamic quantity, already averaged: Φ ≡ <E + kT log P − µ₁N₁ − … µ_sN_s>. I say this because I feel it ought to be a thermodynamic potential and it includes entropy, which is an already-averaged quantity (we don't speak of −k log P being entropy). In that case what is written at the moment would be true.

In any case no reference I can find actually suggests such a generalized definition of this potential for arbitrary ensembles, so I'll omit my generalization and stick to the facts: the quantity <E> - TS − µ₁<N₁> − … µ_s<N_s>, whatever you call it, is minimized. (Actually, it is maximized if T is negative... but is that worth mentioning?)

--Nanite (talk) 18:36, 17 April 2015 (UTC)

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