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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)
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)
My physics book gives the partion sum as
where is the number of particles of the system and runs from 0 to infinity. 82.135.75.113 20:14, 6 April 2007 (UTC)
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)
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)
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)
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)
1) At the top of the article we read the exponential expression
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
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 , using the grand partition function rather than the grand potential.
While the relation 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)
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)
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)
From the article:
Minimum grand potential: For given mechanical parameters (fixed V) and given values of T, µ1, …, µs, the ensemble average <E + kT log P − µ1N1 − … µsNs> 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 − … µsNs> 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
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)
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