Nonequilibrium equalities in absolutely irreversible processes

We generalize nonequilibrium integral equalities to situations involving absolutely irreversible processes for which the forward-path probability vanishes and the entropy production diverges, rendering conventional integral fluctuation theorems inapplicable. We identify the mathematical origins of absolute irreversibility as the singularity of probability measure. We demonstrate the validity of the obtained equalities for several models.

Figure 1

(a) Forward protocol of free expansion. Initially, a single-particle gas is confined in the left box at temperature $T$. Then, the wall is removed and the gas expands to the entire box. (b) Backward protocol of free expansion. Initially, the particle is in equilibrium with respect to the entire box. Then, the wall is inserted. The particle is in the left or right box. The backward path ending in the right box (indicated by the blue arrow) has no corresponding forward path. This is a singular path with negatively divergent entropy production.

Figure 2

Schematic illustration of the Lebesgue decomposition. The abscissa represents coordinates of phase space and the ordinate shows the probability density. Vertical lines represent $\delta$-function-like localization. Here, ${\mathcal{M}}^{\mathrm{r}}$ (solid curve) is decomposed into two parts for a given $\mathcal{M}$ (dashed curve). In the absolutely continuous part, the probability ratio of ${\mathcal{M}}_{\mathrm{AC}}^{\mathrm{r}}$ to $\mathcal{M}$ is well defined. In the singular part, the ratio of ${\mathcal{M}}_{\mathrm{S}}^{\mathrm{r}}$ to $\mathcal{M}$ is divergent.

Figure 3

Schematic illustration of the stronger version of the Lebesgue decomposition. Here, ${\mathcal{M}}^{\mathrm{r}}$ (solid curve) is decomposed into three parts for a given $\mathcal{M}$ (dashed curve). In the absolutely continuous part (bottom left), the probability ratio of ${\mathcal{M}}_{\mathrm{ac}}^{\mathrm{r}}$ to $\mathcal{M}$ is well defined. In the singular continuous part (bottom middle), the ratio of ${\mathcal{M}}_{\mathrm{sc}}^{\mathrm{r}}$ to $\mathcal{M}$ is divergent because the probability density of $\mathcal{M}$ vanishes. In the discrete part (bottom right), the ratio of ${\mathcal{M}}_{\mathrm{d}}^{\mathrm{r}}$ to $\mathcal{M}$ is divergent because the probability density of ${\mathcal{M}}^{\mathrm{d}}$ diverges whereas $\mathcal{M}$ is finite. Note that $\mathcal{M}$ involves no $\delta$-function-like localization, which is the assumption for the stronger Lebesgue decomposition and needed to guarantee the uniqueness of the decomposition.

Figure 4

(a) Schematic illustration of an overdamped Langevin system consisting of $n$ harmonic potential wells on a one-dimensional ring. (b) Probability density of work performed on the system for several values of $n$. The triangles indicate the points of $W=k_{B}Tlnn$. (c) Averaged values of $\langle exp\left[-\beta W\right]\rangle$ at each $n$. Superimposed is a fitted $1/n$ curve. (d) Averaged values of $\langle \beta W\rangle$. The line represents the minimum dissipation given by $k_{B}Tlnn$. The parameters are chosen as follows: diffusion constant $D={10}^{-13}{\mathrm{m}}^2/\mathrm{s}$; temperature $T=300\mathrm{K}$; duration of the process $\tau =10\sec$; half width of a single potential $a={10}^{-6}\mathrm{m}$; the initial stiffness of potential $K$ is chosen so as to satisfy $K{a}^2/2=5k_{B}T$. The nonequilibrium process is repeated ${10}^6$ times for each $n$.

Figure 5

(a) An overdamped Langevin system with a trap. (b) Probability density of the total entropy production for several values of $\tau$. The triangles indicate the points where $\Delta s^{\mathrm{tot}}=-ln(1-p_{\mathrm{trap}}^{†})$. (c) Averaged values of $\langle exp[-\Delta s^{\mathrm{tot}}]\rangle$ versus the trapping probability $p_{\mathrm{trap}}^{†}$. Superimposed is a fitted $1-p_{\mathrm{trap}}^{†}$ line. The parameters $D$ and $T$ are the same as in Fig. 1. The distance between the trap and the center of the potential is $x_c={10}^{-6}\mathrm{m}$. The initial stiffness $K$ is set so as to satisfy $K{a}^2/2=10k_{B}T$. The duration of the process $\tau$ is varied between $1\sec$ and $100\sec$. The process is repeated ${10}^6$ times for each $\tau$.

Figure 6

(a) A measurement and the subsequent trivial feedback control. Initially, a particle is in equilibrium with the entire box. Then, we perform an error-free position-measurement to determine whether the position $X$ is the left $L$ or right $R$. After this measurement, we do nothing. (b) The time-reversed protocol. The particle is in equilibrium and we do nothing. At the time of the measurement, the particle is probabilistically in either the left or right side. For the outcome $X=L$, the case in which the particle is in the right side is singular (blue arrows), and similarly for the outcome $X=R$.

Figure 7

Two-particle Szilard engine. (a) Forward protocol. Initially, two particles are in thermal equilibrium. A hard wall is inserted in the middle and a position measurement is performed to find the number $n$ of particles in the right box (second row). Based on the measurement outcome, the wall is isothermally shifted to such a position that the extracted work is maximal (third row). Finally, the wall is removed. (b) Time-reversed protocol. A wall is inserted (third row) and then moved in accordance with the value of the outcome $n$ (second row). The case of $n=1$ contains two singular paths, which are absent in the forward protocol as indicated by the blue arrows.

Figure 8

Virtual process corresponding to free expansion. Initially, the particle is in the global equilibrium state; the particle is in the left box with probability $l$ and in the right one with probability $1-l$. Then, the wall is removed.