Shannon?s inequality for the R?nyi entropy and an application to the uncertainty principle

摘要

We consider Shannon's inequality for the Renyi entropy. The Renyi entropy corresponds to a nonlinear diffusion equation, for example, the porous-medium equation. Shannon's inequality implies the positivity of the relative Lyapunov functional for a nonlinear Fokker-Planck type equation. We derive Shannon's inequality for the Renyi entropy and give the other proof of this positivity. Moreover, by combining the Shannon and the logarithmic Sobolev inequalities for the Renyi entropy, we obtain the Heisenberg type uncertainty principle.(c) 2022 Elsevier Inc. All rights reserved.