Between Waves and Diffusion: Paradoxical Entropy Production in an Exceptional Regime

• Main Authors: Karl Heinz Hoffmann, Kathrin Kulmus, Christopher Essex, Janett Prehl
• Format: Article
• Language: English
• Published: MDPI AG 2018-11-01
• Series: Entropy
• Subjects: entropy production paradox; entropy; fractional diffusion
• Online Access: https://www.mdpi.com/1099-4300/20/11/881

The entropy production rate is a well established measure for the extent of irreversibility in a process. For irreversible processes, one thus usually expects that the entropy production rate approaches zero in the reversible limit. Fractional diffusion equations provide a fascinating testbed for that intuition in that they build a bridge connecting the fully irreversible diffusion equation with the fully reversible wave equation by a one-parameter family of processes. The entropy production paradox describes the very non-intuitive <i>increase</i> of the entropy production rate as that bridge is passed from irreversible diffusion to reversible waves. This paradox has been established for time- and space-fractional diffusion equations on one-dimensional continuous space and for the Shannon, Tsallis and Renyi entropies. After a brief review of the known results, we generalize it to time-fractional diffusion on a finite chain of points described by a fractional master equation.