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We present the full thermodynamics of a fluid confined by an arbitraryexternal potential based on the virial expansion of the grand potential. Thefluid may be classical or quantum and it is assumed that interatomicinteractions are pairwise additive. We indicate how the appropriate"generalized" volume and pressure variables, that replace the usual volume andhydrostatic pressure, emerge for a given confining potential in thethermodynamic limit. A discussion of the physical meaning and of themeasurement of these variables is presented. We emphasize that this treatmentyields the correct equation of state of the fluid and we give its virialexpansion. We propose an experiment to measure the heat capacity, so that withthis quantity and the equation of state, the complete thermodynamics of thesystem may be extracted. As a corollary, we find that the so-called {\it localdensity approximation} for these systems follows in the thermodynamic limit,although we also point out that it cannot be used indiscriminately for alllocal variables. Along the text we discuss the relevance of these findings inthe description of the currently confined ultracold gases.

Thermodynamics and the virial expansion for trapped fluids in arbitrary external potentials