Talk:Grand potential

Validity Considerations
I can't find many indications of when this is valid, but it appears to be valid for open systems (where particles can flow in and out of an arbitary fixed volume), which are attatched to a heat reservoir. The differential formula may also require particles to be quasi-static, and possibly a monatomic ideal gas. --H2g2bob 15:06, 28 May 2006 (UTC)

Needs an expert
I did my best, but I'm not an expert in this. This really needs someone with better knowledge on this, to clean it up and make it more rigorous, especially with the valididty constraints (my best guesses are above). Are you an expert? Please help out! --H2g2bob 15:19, 28 May 2006 (UTC)

GCE and diversions
My background is gravity and high energy physics so black holes are a very important set of non-extensive systems for me. I'm aware of Tsallis and have flipped through a couple of his papers. I haven't been too excited by the work, but I may not have seen the good stuff. With regards to the GCE, have you heard the phrase "Landau potential" used? I seem to remember picking that up from a textbook somewhere but I'm not sure. Joshua Davis 17:25, 6 July 2007 (UTC)


 * Found the reference on "Landau potential" that I was thinking of. The book States of Matter by David Goodstein defines it on page 19 as $$ \Omega = F- \mu N \,\;$$ where F is the Helmholtz free energy. For homogeneous systems, one obtains $$ \Omega = -PV \,\;$$. I don't know if this nomenclature is in widespread use. Joshua Davis 17:06, 9 July 2007 (UTC)


 * Yes, they seem the same to me. Joshua Davis 21:18, 9 July 2007 (UTC)

Thermodynamic potentials
Hi Sadi - have you heard of the term "Landau potential" used for grand potential or vice versa? --HappyCamper 20:42, 9 July 2007 (UTC)


 * I read a bit about the Landau potential in Joon Chang Lee's 2002 book Thermal Physics - Entropy and Free Energies (ch. 5); he defines the Landau potential as:



\Omega \ \stackrel{\mathrm{def}}{=}\ F - \mu N = U - T S - \mu N $$

He uses this as the potential for open systems, at fixed volume, with a variable number of molecules. If this is exactly the same as the grand potential, I'm not for sure (off the top of my head); some authors use different stipulations, e.g. some define systems in an external field using "E", whereas those not in an external field use "U". Although, Pierre Perrot’s A to Z of Thermodynamics (encyclopedia) defines the grand potential as:



\Omega \ \stackrel{\mathrm{def}}{=}\ F - \Sigma n_i \mu_i $$

which is basically the same, so they could be the synonyms, but I would have to read more into the history of each term to confirm. I hope this helps. --Sadi Carnot 23:25, 9 July 2007 (UTC)


 * The latter equation was also used by STIG LJUNGGREN, JAN CHRISTER ERIKSSON and PETER A. KRALCHEVSKY; “Minimization of the free energy of arbitrarily curved interfaces”;  Journal of Colloid and Interface Science;  1997/7/15;  191 (2): pp. 424-441. for grand potential, as distinct from the un-summed version in the article.
 * —DIV (128.250.80.15 (talk) 07:48, 28 March 2008 (UTC))

Summation over μN
The internal energy definition uses a summation over μN. This article does not. Can someone knowledgeable check if this difference is correct? David Hollman (Talk) 10:12, 31 August 2010 (UTC)

Outline
This equation at the beginning $$ \Phi_{G} = - k_{B} T \ln(\Xi) = - k_{B} T Z_{1} e^{\beta \mu} $$. I do not understand the last part. Where does it come from?

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