The additive generalization of the Boltzmann entropy
  There exists only one generalization of the classical Boltzmann-Gibbs-Shannon entropy functional to a one-parametric family of additive entropy functionals. We find analytical solution to the corresponding extension of the classical ensembles, and discuss in some detail the example of the deformation of the uncorrelated state. 
Alexander N. Gorban, Iliya V. Karlin and Hans Christian Ottinger