Abstract
We study the class of self-similar probability density functions with finite mean and variance, which maximize Rényi’s entropy. The investigation is restricted in the Schwartz space S(Rd) and in the space of l-differentiable compactly supported functions Clc (Rd). Interestingly, the solutions of this optimization problem do not coincide with the solutions of the usual porous medium equation with a Dirac point source, as occurs in the optimization of Shannon’s entropy. We also study the concavity of the entropy power in Rd with respect to time using two different methods. The first one takes advantage of the solutions determined earlier, while the second one is based on a setting that could be used for Riemannian manifolds.
Highlights
The last two decades have witnessed an enormous growing interest in using information concepts in diverse fields of science
The critical point is a local maximum of the functional
We have proven that the pdfs that maximize Rényi’s entropy under the conditions of finite variance and of zero or non-zero mean are given by a one-parameter family of functions, which belong to S(Rd ) for α < 1 and to Ccl (Rd ) for α > 1
Summary
The last two decades have witnessed an enormous growing interest in using information concepts in diverse fields of science. To prove its global nature, we use the concept of the relative Rényi entropy and examine its positivity at the critical point. This procedure can be generalized on Rd straightforwardly. The difference is in the diffusion coefficient, which depends on the shape and the size of a molecule, and on the order α of the Rényi entropy and the dimension of the space. The solutions of the first problem guarantee concavity on the condition that the second time derivative of Rényi’s entropy fulfils Inequality (49).
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