Stochastic efficiency of an isothermal work-to-work converter engine

We investigate the efficiency of an isothermal Brownian work-to-work converter engine, composed of a Brownian particle coupled to a heat bath at a constant temperature. The system is maintained out of equilibrium by using two external time-dependent stochastic Gaussian forces, where one is called load force and the other is called drive force. Work done by these two forces are stochastic quantities. The efficiency of this small engine is defined as the ratio of stochastic work done against load force to stochastic work done by the drive force. The probability density function as well as large deviation function of the stochastic efficiency are studied analytically and verified by numerical simulations.

Figure 1

Two scenarios are shown here. (a) $g({\lambda }_1,{\lambda }_2)$ is analytic for all $({\lambda }_1,{\lambda }_2)\in {\mathbb{R}}_1$. (b) The light red region ${\mathbb{R}}_2$ represents the area where $g({\lambda }_1,{\lambda }_2)$ is imaginary, whereas it is real in ${\mathbb{R}}_1-{\mathbb{R}}_2$. The red contour (thick dashed) is the singularity line and corresponds to Eq. (51). End points $\left({\lambda }_{1,2}^{\pm }\right)$ are given in Appendix. In both cases, the black solid contour represents the region ${\mathbb{R}}_1$ bounded by $({\lambda }_1\left(\varphi \right),{\lambda }_2\left(\varphi \right)), \varphi \in [-\pi ,\pi ]$. Black dashed lines show the maximum and minimum values of ${\lambda }_1\left(\varphi \right)$ and ${\lambda }_2\left(\varphi \right)$ as given by Eqs. (44) and (45).

Figure 2

Contour (red solid line) separates two regions depending upon if $g({\lambda }_1,{\lambda }_2)$ has singularities or not.

Figure 3

Distribution of stochastic efficiency $P(\eta ,t)$ is plotted against the stochastic efficiency $\eta$ for $\theta =0.1$ and $\alpha =0.5$ at three different times: $t/t_{\gamma }=20,50,100$. Yellow $(t/t_{\gamma }=20)$, red $(t/t_{\gamma }=50)$, and cyan $(t/t_{\gamma }=100)$ dashed lines are plotted for analytical expression given in Eq. (62) while the blue dots are for the numerical simulations at the corresponding times $t/t_{\gamma }$.

Figure 4

Two possibilities are shown. (a) Saddle point $w_2^{*}\left(\eta \right)$ stays in the saddle-point region of $P_t(-\eta W_2,W_2)$. (b) Saddle point $w_2^{*}\left(\eta \right)$ stays in both the saddle- and branch-point regions of $P_t(-\eta W_2,W_2)$. Light blue (S.P. region) $[h(-\eta w_2,w_2)>0]$ and light red (B.P. region) $[h(-\eta w_2,w_2)<0]$ shaded areas are for saddle- and branch-point contributions of $P_t(-\eta W_2,W_2)$, respectively. Black solid line corresponds to $h(-\eta w_2,w_2)=0$. Blue points are given by $({\eta }_{-},w_2^{*}\left({\eta }_{-}\right))$ and $({\eta }_{+},w_2^{*}\left({\eta }_{+}\right))$.

Figure 5

Possibility I: The distribution of efficiency $P(\eta ,t)$ can be computed by only the saddle-point solution given by Eq. (53). Possibility II: The distribution of efficiency $P(\eta ,t)$ requires the branch-point contribution of $P_t(-\eta W_2,W_2)$.

Figure 6

Distribution of stochastic efficiency $P(\eta ,t)$ is plotted against the stochastic efficiency $\eta$ for $\theta =2$ and $\alpha =0.5$ at time $t/t_{\gamma }=10$. Red dashed lines plotted for analytical expression given in Eq. (62), while the blue dots are for the numerical simulation.

Figure 7

Distribution of stochastic efficiency $P(\eta ,t)$ is plotted against the stochastic efficiency $\eta$ for $\theta =3$ and $\alpha =0.5$ at time $t/t_{\gamma }=50$. The red dashed line is plotted for the distribution $P(\eta ,t)$ evaluated numerically as explained in Sec. 5d, while the blue dots correspond to the numerical simulation.

Figure 8

Phase diagram indicates the sign of $C_{1,2}^{\pm }$. Region I: $g({\lambda }_1,{\lambda }_2)$ is analytic. In subregions IIa: $C_{1,2}^{\pm }>0$. In subregion IIb: $C_{1,2}^{+}>0, C_1^{-}<0$, and $C_2^{-}>0$. In subregion IIIa: $C_{1,2}^{+}<0, C_1^{-}<0$, and $C_2^{-}>0$. In subregion IIIb:: $C_1^{+}>0, C_2^{+}<0, C_1^{-}<0$, and $C_2^{-}>0$. In region IV: $C_1^{+}>0, C_2^{+}<0$, and $C_{1,2}^{-}<0$.