  The postulates of thermodynamics were originally formulated for macroscopic
systems. They lead to the definition of the entropy, which, for a homogeneous
system, is a homogeneous function of order one in the extensive variables and
is maximized at equilibrium. We say that the macroscopic systems are extensive
and so it is also the entropy. For a mesoscopic system, by definition, the size
and the contacts with other systems influence its thermodynamic properties and
therefore, if we define an entropy, this cannot be a homogeneous of order one
function in the extensive variables. So, mesoscopic systems and their entropies
are non-extensive. While for macroscopic systems and homogeneous entropies the
equilibrium conditions are clearly defined, it is not so clear how the
non-extensive entropies should be applied for the calculation of equilibrium
properties of mesoscopic systems--for example it is not clear what is the role
played by the boundaries and the contacts between the subsystems. We propose
here a general definition of the entropy in the equilibrium state, which is
applicable to both, macroscopic and mesoscopic systems. This definition still
leaves an apparent ambiguity in the definition of the entropy of a mesoscopic
system, but this we recognize as the signature of the anthropomorphic character
of the entropy (see Jaynes, Am. J. Phys. 33, 391, 1965).
  To exemplify our approach, we analyze four formulas for the entropy (two for
extensive and two for non-extensive entropies) and calculate the equilibrium
(canonical) distribution of probabilities by two methods for each. We show that
these methods, although widely used, are not equivalent and one of them is a
consequence of our definition of the entropy of a compound system.
