Generalized entropies from first principles

We present a derivation of power law canonical distributions from first principle statistical mechanics, including the exponential distribution as a It is presented a derivation of power law canonical distributions from first principle statistical mechanics, including the exponential distribution as a particular case. It is shown that these distributions arise naturally, and that the heat capacity of the heat bath is the condition that determines its type. As a consequence, it is given a physical interpretation for the parameter $q$ of the generalized entropy.