On the connection between linear combination of entropies and linear combination of extremizing distributions

We analyze the distribution that extremizes a linear combination of the Boltzmann–Gibbs entropy and the nonadditive q-entropy. We show that this distribution can be expressed in terms of a Lambert function. Both the entropic functional and the extremizing distribution can be associated with a nonlinear Fokker–Planck equation obtained from a master equation with nonlinear transition rates. Also, we evaluate the entropy extremized by a linear combination of a Gaussian distribution (which extremizes the Boltzmann–Gibbs entropy) and a q-Gaussian distribution (which extremizes the q-entropy). We give its explicit expression for q=0, and discuss the other cases numerically. The entropy that we obtain can be expressed, for q=0, in terms of Lambert functions, and exhibits a discontinuity in the second derivative for all values of q<1. The entire discussion is closely related to recent results for type-II superconductors and for the statistics of the standard map.