Upper bound for the average entropy production based on stochastic entropy extrema

The second law of thermodynamics, which asserts the non-negativity of the average total entropy production of a combined system and its environment, is a direct consequence of applying Jensen's inequality to a fluctuation relation. It is also possible, through this inequality, to determine an upper bound of the average total entropy production based on the entropies along the most extreme stochastic trajectories. In this work, we construct an upper bound inequality of the average of a convex function over a domain whose average is known. When applied to the various fluctuation relations, the upper bounds of the average total entropy production are established. Finally, by employing the result of Neri, Roldán, and Jülicher [Phys. Rev. X 7, 011019 (2017)], we are able to show that the average total entropy production is bounded only by the total entropy production supremum, and vice versa, for a general nonequilibrium stationary system.

Figure 1

By definition, a convex function $f\left(x\right)$ (yellow curve) must lie below a secant line (blue line) connecting any two points on the curve. Denote these points by $(m,f\left(m\right))$ and $(M,f\left(M\right))$. Let $\overline{x}$ be the average of random variable $x$. Inequality (4) states that the average $\overline{f\left(x\right)}$ must be bounded from above by a point on the secant line at $\overline{x}$. This average is also bounded from below according to $f\left(\overline{x}\right)\le \overline{f\left(x\right)}$ by the direct application of Jensen's inequality. The green line indicates the region of possible values of $\overline{f\left(x\right)}$.

Figure 2

The surface normal at $\mathbf{v}-\mathbf{x}$ on the surface of $S^{*}$ is given by $\mathbf{u}\left(\mathbf{v}\right)-\mathbf{x}$ such that $\left[\mathbf{u}\right(\mathbf{v})-\mathbf{x}]·(\mathbf{v}-\mathbf{x})=n$. The volume of the shaded hyperpyramid with infinitesimal base area $da\left(\mathbf{v}\right)$ is given by $(1/n)|\mathbf{v}-\mathbf{x}|cos\left(\theta \right)da\left(\mathbf{v}\right)=da\left(\mathbf{v}\right)/\left|\mathbf{u}\right(\mathbf{v})-\mathbf{x}|$, which gives rise to the geometrical interpretation of $V$ and $d\lambda (\mathbf{v},\mathbf{x})$ in (8) as the hypervolume of $S^{*}$ and the fractional volume of a hyperpyramid with base area $da\left(\mathbf{v}\right)$, respectively.

Figure 3

Possible outcomes of $\mathbf{x}$, whose average is $\overline{\mathbf{x}}$, occur inside region $S$ shown in green. Its polar dual $S^{*}$ (in yellow) lies within the boundaries formed from edges $e_1$ to $e_6$ (dual to vertices ${\mathbf{u}}_1$ to ${\mathbf{u}}_6$) and a part of curve $c_1^{*}$ (dual to $c_1$). Part $c_2$ of $\partial S$ between ${\mathbf{u}}_3$ and ${\mathbf{u}}_4$ is nonconvex and thus has no dual correspondence on $\partial S^{*}$.

Figure 4

Entropy productions along some sampling stochastic trajectories are plotted. The supremum and infimum of the total entropy production at time $t_{\text{f}}$ over all trajectories are, respectively, represented by $s_{\text{sup}}\left(t_{\text{f}}\right)$ and $s_{\text{inf}}\left(t_{\text{f}}\right)$. There is only one pair of such quantities per ensemble. Each trajectory, however, has one entropy production infimum along its trajectory, $s_{\text{inf}}^{\text{s.s.}}\left(t_{\text{f}}\right)$, which is located at the lowest point of the trajectory during time interval $[0,t_{\text{f}}]$. The ensemble average over all $s_{\text{inf}}^{\text{s.s.}}$'s is equal to $-1$ [17].

Figure 5

This plot shows the region of possible values of the average entropy production $\langle s_{\text{tot}}\rangle$ and the trajectorywise total entropy production supremum $s_{\text{sup}}$. The value of $s_{\text{sup}}$ is bounded from below by a curve described in Eq. (19). This bound quickly approaches $(1+e^{}\langle s_{\text{tot}}\rangle )/(e^{}-1)$ (dashed line) for a sufficiently large value of $\langle s_{\text{tot}}\rangle$.

Figure 6

Probability distribution $p\left(s_{\text{tot}}\right)$ of the trajectorywise total entropy production $s_{\text{tot}}$ during the time interval of $0.1\tau$ for the one-dimensional overdamped system of one particle under potential $V\left(x\right)=k_{B}Tln[cos(2\pi x/L)+2]$. In this example, the constant applied force of $f=0.2$ acts on the particle. The histogram is generated from 10 000 independent trajectories. The average entropy production $\langle s_{\text{tot}}\rangle$, the steady-state maximum bound $S_{\text{sup}}^{}\text{s.s.}$ from Eq. (18), and the maximum bound $S_{\text{sup}}$ from Eq. (5) are shown as dashed lines, respectively, from left to right.

Figure 7

The area of the polar dual $S^{*}$ to domain $S$ is divided into seven regions.