Time Evolution of Relative Entropies for Anomalous Diffusion

Abstract

The entropy production paradox for anomalous diffusion processes describes a phenomenon where one-parameter families of dynamical equations, falling between the diffusion and wave equations, have entropy production rates (Shannon, Tsallis or Renyi) that increase toward the wave equation limit unexpectedly. Moreover, also surprisingly, the entropy does not order the bridging regime between diffusion and waves at all. However, it has been found that relative entropies, with an appropriately chosen reference distribution, do. Relative entropies, thus, provide a physically sensible way of setting which process is "nearer" to pure diffusion than another, placing pure wave propagation, desirably, "furthest" from pure diffusion. We examine here the time behavior of the relative entropies under the evolution dynamics of the underlying one-parameter family of dynamical equations based on space-fractional derivatives.

Generalized Boundary Conditions for the Time-Fractional Advection Diffusion Equation

“…Fractional calculus also plays a significant part in studies of entropy. It should be emphasized that entropy is also used in the analysis of anomalous diffusion processes and fractional diffusion equations [8][9][10][11][12][13][14][15][16][17][18]. The entropy production rate for fractional diffusion processes was calculated in [8][9][10].…”

Bounding Extremal Degrees of Edge-Independent Random Graphs Using Relative Entropy

“…Take ε = 3p(ln n)/(c 2 (1 − p)n) in the inequality (15). It is direct to check that ε → 0 and ε/p → 0 as n → ∞ under our assumptions.…”

“…It measures the difference between two probability distributions; see e.g., [11][12][13][14][15][16][17] for various applications of relative entropy on physical, chemical and engineering sciences.…”

Approximate Analytical Solutions of Time Fractional Whitham–Broer–Kaup Equations by a Residual Power Series Method

“…For example, entropies based on fractional calculus could be used more widely than traditional Shannon entropy [6]. Due to its wide application, fractional entropy has become a hot research field [7]. Another example is fractional differential equations, which are powerful for modeling various phenomena [8].…”