The work of Boltzmann on entropy in the years 1865-1905 is really amazing. Beyond important combinatorial aspects, one of the general ideas behind his work is that along certain dynamics, some functional is monotonic, and thus, the long time equilibrium, if it exists, is related to the optimum of the functional over the constraints related to the conservation laws. For the original Boltzmann equation $latex \partial_tf_t=A(f_t)$ which comes from kinetic gases modelling, the entropy is $latex H(f)=-\int\!f(x)\log f(x)dx$, and is maximized by Gaussians under a variance constraint. Here the variance constraint corresponds to the convervation law of the energy. One may call entropies such functionals. Boltzmann was the first to use a partial differential equation to describe the evolution of a probability density function, dozens of years before the rigorous analysis of such concepts in mathematics.