Critical time scale of coarse-graining entropy production

We study coarse-grained entropy production in an asymmetric random walk system on a periodic one-dimensional lattice. In coarse-grained systems, the original dynamics are unavoidably destroyed, but the coarse-grained entropy production is not hidden below the critical time-scale separation. The hidden entropy production is rapidly increasing near the critical time-scale separation.

Figure 1

For fixed $\lambda =2$, the critical time-scale separation and the exponent are (a) ${\tau }_c^{\prime }=0.515$ and (b) $\alpha =0.51$, respectively. The plots in (a) are rapidly increasing near ${\tau }_c^{\prime }\simeq 0.515$ for $\lambda =2$. The inset in (a) shows a log-log plot but does not conclusively confirm whether the plot follows a power law or not.

Figure 2

For fixed $D=1/4$, entropy production rate for (a) fixed $\lambda =2$ and (b) fixed $N=40$.

Figure 3

The coarse-grained entropy productions are zero at every periodic time scale, i.e., $\dot{S}\left(n{\tau }_p\right)=0$ for $n\in \{1,2,3,\cdots \}$. In the inset, the $y$ axis is log-scaled.

Figure 4

In (a), the entropy production drops drastically as ${\tau }^{\prime }$ crosses 0.515 for fixed $\lambda =2$ and large $N$. In (b), the critical time-scale separation is approaching 1 as $\lambda$ increases for fixed $N=80$.

Figure 5

(a) Critical time-scale separation ${\tau }_c^{\prime }\mathrm{s}$ for $N⋙1$. It follows a power law, $({\tau }_c^{\prime }-0.5)\sim {(\lambda -1)}^{1.67}$ for $\lambda <5$ (inset). (b) Exponents $\alpha$ of the power law ${\dot{S}}_{\mathrm{hidden}}^{\prime }\sim N^{-\alpha }$ for $N⋙1$.