where the value of the Lagrange multiplier [lambda] is given by [lambda] = 1/T as a consequence of the Maxwell relation TdS - dU PdV = 0.

But when the dispersion relation [epsilon] (p, T) nontrivially depends on the temperature, the standard expressions of the thermodynamic potentials--derived in statistical mechanics--for a gas of weakly interacting constituents do not satisfy the thermodynamic relations (or Maxwell relations).

It is shown that the accurate determination of the quasi-particles vacuum energy completes the construction of the thermodynamic potentials satisfying the Maxwell relations. It is verified that, in agreement with the Landau principle, the value of the gap corresponds to a stationary point of the grand potential [OMEGA](T, V, [mu]).

By means of a Bogoliubov transformation acting on the creation and annihilation operators of the electron fluctuations around the Fermi sphere, the full expression of [OMEGA](T, V, [mu]) satisfying the Maxwell relations is obtained.

This means that the Maxwell relations associated with [OMEGA] are satisfied.

Consequently, when constructing the thermodynamic relations by means of the first derivatives of the potentials, [DELTA] effectively behaves like a constant term and does not alter the Maxwell relations. Thus, because of the validity of the gap equation, the quasi-particles description of the systems, which is given--in the low temperature limit--by the grand potential (50), is perfectly consistent with the validity of the standard thermodynamic relations.

His topics include the first three laws of thermodynamics, the structure of thermodynamic theories, thermodynamic potentials and

Maxwell relations, the van der Waals equation, the approach to absolute zero, and some mathematical aspects of thermodynamics.

of Amsterdam) start from the basic principles of modern theory and progress to an experimental analysis of one of the most important consequences of thermodynamics, namely

Maxwell relations. As they progress, they provide a comprehensive review of glassy phenomena and present the physically realistic approach of exactly solvable models, including two-temperature thermodynamics and previously paradoxical problems.