Lambert function and a new non-extensive form of entropy

We propose a new way of defining entropy of a system, whichgives a general form that is non-extensive like Tsallis entropy,but is linearly dependent on component entropies, like Renyientropy, which is extensive. This entropy has a conceptuallynovel but simple origin and is mathematically easy to defineby a very simple expression involving a derivative. It leadsto a probability distribution function involving the Lambertfunction resulting from optimizing the entropy, which has hithertonever appeared in this context, and is somewhat more complexthan the Shannon or Boltzmann form, but is nevertheless mathematicallyquite tractable. We have compared it numerically with the Tsallisand Shannon entropies. We have also considered constraints onthe energy spectra imposed by the properties of the Lambertfunction, which are absent in the Shannon form. It may turnout to be a more appropriate candidate in a physical situationwhere the probability distribution does not suit any of thepreviously defined forms, especially when the probability densityfunction sought is expected to be stiffer than that resultingfrom maximizing the other entropies. We consider the problemof defining free energy and other thermodynamic functions whenthe entropy is given as a general function of the probabilitydistribution, including that for non-extensive forms. We thenfind that the free energy, which is central to the determinationof all other quantities of interest in a thermodynamic context,can be obtained uniquely, at least numerically, even when itis the root of a transcendental equation. In particular, weexamine the cases of the Tsallis form and the new form proposedby us. We compare the free energy, the internal energy and thespecific heat of a simple system of two energy states for eachof these forms and find significant departures for some quantities,while some others are less sensitive to the parametrization.