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.