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

Abstract

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 increase 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.

Keywords: entropy; entropy production paradox; fractional diffusion.

Figure 1

The entropy $S(\gamma ,t)$ is shown over $\gamma$ and t—for small times, a monotonic increasing entropy decreasing $\gamma$. At times $t>t_c$, a maximum S at $\gamma >1$ is found for each t, as indicated by the red dots.

Figure 2

The entropy $S(\gamma ,{\tau }_{\gamma }\left(\widehat{x}\right))$ is shown over $\gamma$ and $\widehat{x}$. We observe a monotonic decreasing function of entropy going from $\gamma =2$ to 1. This is emphasized by the red dots, indicating the maximum of S, which is always given for $\gamma =1$.

Figure 3

The entropy $S(\alpha ,t)$ is shown over $\alpha$ and t. For small times $t<t_c$, a monotonic increasing entropy for reaching $\alpha =2$, i.e., the reversible limit, is observed. At larger times, a maximum of S at $\alpha <2$ is found for each t, as indicated by the red dots.

Figure 4

The entropy $S(\alpha ,{\tau }_{\alpha }(-\widehat{x}))$ is shown over $\alpha$ and $\widehat{x}$. We observe a monotonic increasing function of entropy going from $\alpha =1$ to 2. This is emphasized by the red dots, indicating the maximum of S, which is always given for $\alpha =2$.

Figure 5

Dynamics on a linear chain of finite length $m=4$ with only nearest neighbor connections.

Figure 6

The entropy production rate is given for the short time regime $t\in [0.26,0.83]$. Here, it behaves paradoxically, as the entropy production rate for $\gamma =1.5$ is larger than for the $\gamma =1.3$.

Figure 7

The entropy production rate is given for the intermediate time regime $t\in [100,800]$. The entropy production rates are crossing each other several times, i.e., the entropy production shows alternating normal and paradoxical behavior.

Figure 8

The entropy production rate is given for the long time regime t$\in [$2000, 20,000]. Here, it shows regular ordering with a higher production rate for smaller $\gamma$.

Figure 9

The entropy production rate (EPR) is shown as a function of time for two different values of $\gamma$ and $\eta$.

Figure 10

Comparison of the entropy production rate for different $\eta$. The influence of the initial probability change on the smaller value of $\gamma$ persists longer.