Entropies for complex systems: generalized-generalized entropies

Many complex systems are characterized by non-Boltzmann distribution functions of their statistical variables. If one wants to -- justified or not -- hold on to the maximum entropy principle for complex statistical systems (non-Boltzmann) we demonstrate how the corresponding entropy has to look like, given the form of the corresponding distribution functions. By two natural assumptions that (i) the maximum entropy principle should hold and that (ii) entropy should describe the correct thermodynamics of a system (which produces non-Boltzmann distributions) the existence of a class of fully consistent entropies can be deduced. Classical Boltzmann-Gibbs entropy is recovered as a special case for the observed distribution being the exponential, Tsallis entropy is the special case for q-exponential observations.