  Complex systems that are characterized by strong correlations and fat-tailed
distribution functions have been argued to be incompatible within the framework
of Boltzmann-Gibbs entropy. As an alternative, so-called generalized entropies
were proposed and intensively studied. Here we show that this incompatibility
is a misconception. For a broad class of processes, Boltzmann entropy the log
multiplicity remains the valid entropy concept, however, for non-i.i.d.,
non-multinomial, and non-ergodic processes, Boltzmann entropy is not of Shannon
form. The correct form of Boltzmann entropy can be shown to be identical with
generalized entropies. We derive this result for all processes that can be
mapped reversibly to adjoint representations where processes are i.i.d.. In
these representations the information production is given by the Shannon
entropy. We proof that over the original sampling space this yields functionals
that are identical to generalized entropies. The problem of constructing
adequate context-sensitive entropy functionals therefore can be translated into
the much simpler problem of finding adjoint representations. The method
provides a comprehensive framework for a statistical physics of strongly
correlated systems and complex processes.
