Efficiency at maximum power and efficiency fluctuations in a linear Brownian heat-engine model

We investigate the stochastic thermodynamics of a two-particle Langevin system. Each particle is in contact with a heat bath at different temperatures $T_1$ and $T_2 \left(<T_1\right)$, respectively. Particles are trapped by a harmonic potential and driven by a linear external force. The system can act as an autonomous heat engine performing work against the external driving force. Linearity of the system enables us to examine thermodynamic properties of the engine analytically. We find that the efficiency of the engine at maximum power ${\eta }_{MP}$ is given by ${\eta }_{MP}=1-\sqrt{T_2/T_1}$. This universal form has been known as a characteristic of endoreversible heat engines. Our result extends the universal behavior of ${\eta }_{MP}$ to nonendoreversible engines. We also obtain the large deviation function of the probability distribution for the stochastic efficiency in the overdamped limit. The large deviation function takes the minimum value at macroscopic efficiency $\eta =\overline{\eta }$ and increases monotonically until it reaches plateaus when $\eta \le {\eta }_L$ and $\eta \ge {\eta }_R$ with model-dependent parameters ${\eta }_R$ and ${\eta }_L$.

Figure 1

Function diagram of the linear engine model. The dashed lines are the boundary of the stable region.

Figure 2

Density and contour plots for the engine power $w$.

Figure 3

Linear heat engine model in the $(s_1,s_2)$ plane. The model acts as a heat engine in the shaded area satisfying $(1-{\eta }_C)s_2<s_1<s_2$. The dotted curve represents a characteristic of the engine. The maximum power is achieved when the curve is tangential to a straight line of slope $1-{\eta }_C$.

Figure 4

In the $\mathbf{\lambda }=({\lambda }_W,{\lambda }_Q)$ plane, we draw schematically the boundary $C$ of the ${\chi }_{\mathsf{J}}=1$ domain. The dotted straight line of slope $-\overline{\eta }$ passing through the origin corresponds to the line $\mathrm{\Lambda }=0$, while the dashed straight line of slope $-\overline{\eta }$ corresponds to the line $\mathrm{\Lambda }={\mathrm{\Lambda }}_m$, where $\mathrm{\Lambda }={\lambda }_Q+\overline{\eta }{\lambda }_W$. The intersections between the line $\mathrm{\Lambda }=0 \left({\mathrm{\Lambda }}_m\right)$ and the curve $C$ are marked with closed circles and labeled as $a$ and $c (b$ and $d)$. The gray arrows indicate that the cumulant generating function $\varphi$ in (23) is minimum along the line $\mathrm{\Lambda }={\mathrm{\Lambda }}_m$ and increases as one moves away from it. To a given value of $\eta ,L\left(\eta \right)$ is obtained from the minimum value of $\mu ({\lambda }_Q,{\lambda }_W)$ along the segment of the straight line $l_{\eta }$ passing through the origin $O$ with slope $-\eta$ inside the boundary $C$. When $\eta =\overline{\eta }$, the line $l_{\eta }$ coincides with the line $\mathrm{\Lambda }=0$ where $\mu =0$. Hence, $L\left(\overline{\eta }\right)=0$. (i) When ${\eta }_L\le \eta <\overline{\eta }$, the right intersection point $e$ (open circle) of $l_{\eta }$ and $C$ lies on a segment between $a$ and $b$. Thus, the LDF is determined by the $\mathrm{\Lambda }$ value at $e,L\left(\eta \right)=-H\left({\mathrm{\Lambda }}_e\right)$. The left intersection point is irrelevant since it is farther from the line $\mathrm{\Lambda }={\mathrm{\Lambda }}_m$ than $e$. The territory ${\eta }_L$ is determined by the condition that the point $e$ coincides with $b$. (ii) When $\eta <{\eta }_L$, the line $l_{\eta }$ intersects with the line $\mathrm{\Lambda }={\mathrm{\Lambda }}_m$ within $C$ at point $f$ (open circle). Hence, $L\left(\eta \right)=-{\mu }_m$. (iii) When $\overline{\eta }<\eta \le {\eta }_R$, the left intersection point $g$ (open circle) lies on a segment between $c$ and $d$. Thus, the LDF is given by $L\left(\eta \right)=-H\left({\mathrm{\Lambda }}_g\right)$. The territory ${\eta }_R$ is determined by the condition that $g$ coincides with $d$. (iv) When $\eta >{\eta }_R$, the line $l_{\eta }$ intersects with the line $\mathrm{\Lambda }={\mathrm{\Lambda }}_m$ at point $h$ (open circle) and $L\left(\eta \right)=-{\mu }_m$.

Figure 5

The LDF for efficiency with the set of parameters $K=1,T_1=2,T_2=1,\epsilon =1/2$, and $\delta =3/8$. The solid line represents the analytic result obtained from Eq. (24), while closed circles represent the numerical results from the fixed initial condition. The numerical results are obtained by extrapolating $-\frac{1}{t}ln{P}_t\left(\eta \right)$ at $t=512$ (dashed line), 1024 (dotted line), and 2048 (dash-dotted line). Details for the simulations are explained in the text. We also present the numerical results (open circles) that are obtained from the steady-state initial condition.