Universal theory of efficiency fluctuations

Using the fluctuation theorem supplemented with geometric arguments, we derive universal features of the (long-time) efficiency fluctuations for thermal and isothermal machines operating under steady or periodic driving, close or far from equilibrium. In particular, the probabilities for observing the reversible efficiency and the least likely efficiency are identical to those of the same machine working under the time-reversed driving. For time-symmetric drivings, this reversible and the least probable efficiency coincide.

Figure 1

Typical contour lines of the LDF $I({\sigma }_1,{\sigma }_2)$. The point C corresponds to the most probable value $I(\langle {\sigma }_1\rangle ,\langle {\sigma }_2\rangle )=0$. A straight line through the origin with slope ${\eta }_D$ touches the contour line, whose $I$ value equals $J\left({\eta }_D\right)$ (idem for point B sharing the same $J$ value). The maximum of $J\left(\eta \right)$ corresponds to $I$ value of the contour crossing the origin $J\left({\eta }^{*}\right)=I(0,0)$ (blue long dashed line), while $J\left(\infty \right)$ to that of the contour touching the ${\sigma }_2$ axis in A (red dash-dotted line).

Figure 2

Typical shape of the efficiency LDF $J\left(\eta \right)$. For steady-state machines or machines with time-symmetric driving cycles, the shape is the same and the maximum is at the reversible efficiency ${\eta }^{*}={\overline{\eta }}_{\mathrm{rev}}=1$. The horizontal asymptote corresponds to point A of Fig. 1.

Figure 3

Typical shape of the efficiency LDF $J\left(\eta \right)$ and $\widehat{J}\left(\eta \right)$. The two curves cross at $\eta ={\overline{\eta }}_{\mathrm{rev}}=1$ and take the same maximum value at their least likely efficiency ${\eta }^{*}$ and ${\widehat{\eta }}^{*}$.

Figure 4

(Top) External driving $h\left(t\right)$ following a piecewise constant protocol of period $\tau$. The driving takes the value $h^{\pm }=h_0\pm a$ during ${\alpha }^{\pm }\tau$, with ${\alpha }^{-}+{\alpha }^{+}=1$. (Bottom) Time evolution of ${\Gamma }_{\nu }\left(t\right)$ indicating the coupling with cold reservoir $\nu =c$ for the blue solid line and with the hot reservoir $\nu =h$ for the red dashed line. At driving $h^{\pm }$, the hot reservoir coupling is ${\Gamma }_{\pm }$ and the cold reservoir coupling is ${\Gamma }_{\mp }$.

Figure 5

Diagram representing the various operating modes of our thermal machine subjected to the driving cycle depicted in Fig. 4, as a function of the inverse temperature difference ${\beta }_{-}^{ch}={\beta }_c-{\beta }_h$ and the amplitude $a$ of the driving ($h^{\pm }=h_0\pm a$). We used: sum of inverse temperatures ${\beta }_{+}^{ch}={\beta }_c+{\beta }_h=2$, period $\tau =1$, cyclic ratio ${\alpha }^{-}=0.3$, bare field $h_0=1$, and coupling constants ${\Gamma }_{-}=0.25$ and ${\Gamma }_{+}=4$. These parameters are also used for all figures using this model (excepted Fig. 10). Black dots correspond to the values of $a$ and ${\beta }_{-}^{ch}$ used in Figs. 6, 7, and 8. I: ${\beta }_{-}^{ch}=0.05$, $a=0.05$; II: ${\beta }_{-}^{ch}=0.5$, $a=0.5$; III: ${\beta }_{-}^{ch}=0.1$, $a=0.02$; IV: ${\beta }_{-}^{ch}=0.8$, $a=0.2$.

Figure 6

LDF of work and heat for the three operating modes of the machine: heat engine (a,d), heat pump (b,e), and refrigerator (c,f), far from (top row) or close to equilibrium (bottom row). The parameter values correspond to the black dots in Fig. 5. The solid black contour lines corresponding to $I({\sigma }_1,{\sigma }_2)$ and the orange dashed lines corresponding to $I(-{\sigma }_1,-{\sigma }_2)-{\sigma }_1-{\sigma }_2$ perfectly coincide, illustrating the fluctuation theorem for symmetric drivings under time-reversal. The yellow dotted line is the straight line of slope ${\overline{\eta }}_{\mathrm{rev}}$ crossing the origin.

Figure 7

Efficiency LDF for the three main types of machines working close to equilibrium (CE). Lines are for $J\left(\eta \right)$ obtain from Eq. (26) and Table 2 and symbols come from the close-to-equilibrium prediction of Eq. (31). The chosen parameters correspond to point I of Fig. 5 for the heat pump and the refrigerator and to point III of the same figure for the heat engine. The indicated type of machine corresponds to the average behavior.

Figure 8

Efficiency LDF for the three main types of machines working far from equilibrium. The chosen parameters correspond to point II of Fig. 5 for the heat pump and the refrigerator and to point IV of the same figure for the heat engine. The indicated type of machine corresponds to the average behavior.

Figure 9

(Top) External driving following a piecewise constant protocol of period $\tau$. The driving takes three different values $h_j$ with $j=1,2,3$ during three time intervals ${\alpha }_j\tau$, with ${\alpha }_1+{\alpha }_2+{\alpha }_3=1$. (Bottom) Time evolution of ${\Gamma }_{\nu }\left(t\right)$ indicating the coupling with the cold reservoir $\nu =c$ for the blue solid line and the hot reservoir $\nu =h$ for the red dashed line. Note that the reverse driving cycle is defined by $\widehat{h}\left(t\right)=h(\tau -t)$ and ${\widehat{\Gamma }}_{\nu }\left(t\right)={\Gamma }_{\nu }(\tau -t)$.

Figure 10

Efficiency LDF for a refrigerator (average behavior) working far from equilibrium (squares) or close to equilibrium (circles) with the asymmetric driving cycle under time reversal of Fig. 9. The empty (full) symbols are for the direct (reverse) driving. The kinetics is described by Fermi rates and the common parameters to all curves are ${\alpha }_1=0.6$, ${\alpha }_2={\alpha }_3=0.2$, $\tau =1$, ${\Gamma }_1=0.1$, ${\Gamma }_2=1$, ${\Gamma }_3=10$, and ${\beta }_h=1$. For far-from-equilibrium case, $h_1=0.5$, $h_2=1.5$, $h_3=2$, and ${\beta }_c=1.5$. For close-to-equilibrium case, $h_1=1.45$, $h_2=1.5$, $h_3=1.55$, and ${\beta }_c=1.05$.