Thus, the last two sections should be cropped out ( actually, the last section should maybe be merged into Grand canonical ensemble since the material doesn't seem to be there ), and the links to this article put in the proper context.
22.
As a result, the only consistent way to include " fully distinguishable " particles in the grand canonical ensemble is to consider every possible distinguishable type of those particles, and to track each and every possible type with a separate particle counter and separate chemical potential.
23.
In the grand canonical ensemble a further logical inconsistency occurs : the number of distinguishable permutations depends not only on how many particles are in the system, but also on how many particles are in the reservoir ( since the system may exchange particles with a reservoir ).
24.
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 ( the derivation proceeds along lines analogous to the heat bath derivation of the normal canonical ensemble, and can be found in Reif ).
25.
Physicists call the result of applying the Gibbs algorithm the Gibbs distribution for the given constraints, most notably Gibbs's grand canonical ensemble for open systems when the average energy and the average number of particles are given . ( See also " partition function " ).
26.
In a 2012 study Pigolotti et al . conducted a thorough study of the unique bid auction in the grand canonical ensemble, finding a theoretical expression for the Nash equilibrium distribution and showing that real-world players play according to this distribution when the number of players in the auction is low.
27.
In subsequent works Baeurle et al . ( Baeurle 2002, Baeurle 2002a ) applied the concept of tadpole renormalization, which originates from quantum field theory and leads to the " Gaussian equivalent representation " of the partition function integral, in conjunction with advanced MC techniques in the grand canonical ensemble.
28.
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.
29.
:general comments : derivations for the microcanonical ensemble and canonical ensemble are already given, IMO in a much better fashion, are given in their respective pages . grand canonical ensemble is a stub, but at least everything is correct and one can just add to it . now the article in question apparently proposes to give separate derivations for these ensembles, and is clearly confused, IMO . along with the mistakes i cited, i thought the overall awkwardness and the redundancy warranted deletion . if someone thinks my judgement of the article is overly harsh and think it's salvagable, that's fine with me . as long as the information on there is correct.
