Legendre transformation, semigrand

Semigrand ensembles—as the term implies—lie between the canonical and the full grand-canonical ensembles. They are formulated by Legendre transformation of some, rather than all, of the chemical terms. Thus we derive an osmotic semigrand ensemble by the most straightforward variant we transform some terms ( = m), and leave others (j = m + 1,..., c) ... 

Legendre transformation with respect to the last term yields a reactive semigrand ensemble... 

It is necessary to know the form of the partition functions to construct transition probabilities that properly sample the ensemble. For instructional purposes we record here the (semiclassical) partition functions that correspond to the osmotic and isomolar semigrand ensembles described above. In both ensembles one must average over moles of the Legendre-transformed species. Thus the osmotic semigrand ensemble partition function is... 

For stability, we then have 8F > 0, which taken at lowest order is thus a constraint on the second derivatives. To find the constraints one commonly carries out a series of linear transformations on the independent variables, which reduces the quadratic to a sum of squares in the transformed variables. From the resulting relations, by elimination of variables and Legendre transformation, the canonical, all the various semigrand canonical, and grand canonical relations can be derived (Valdeavella, Perkyns, and Pettitt 1994). This yields equations that are expressible in terms of the compressibility, the partial molar volumes, and derivatives of the chemical potential, which are directly calculable from the cofactors of a density weighting of the matrix of zeroth moments of the distribution. 

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