Efficiency Fluctuations of Stochastic Machines Undergoing a Phase Transition

We study the efficiency fluctuations of a stochastic heat engine made of $N$ interacting unicyclic machines and undergoing a phase transition in the macroscopic limit. Depending on $N$ and on the observation time, the machine can explore its whole phase space or not. This affects the engine efficiency that either strongly fluctuates on a large interval of equiprobable efficiencies (ergodic case) or fluctuates close to several most likely values (nonergodic case). We also provide a proof that despite the phase transition, the decay rate of the efficiency distribution at the reversible efficiency remains largest one although other efficiencies can now decay equally fast.

Figure 1

Graph of the discrete state space of the collective machine. Blue edges are for channel 1 and red edges for channel 2.

Figure 2

Stable (black) and unstable (light blue) mean field steady state densities $x^{\mathrm{MF}}$ versus interaction energy $V$. Insets: density LDFs versus $x$ for four values of $V$ indicated by vertical dashed lines. The parameters are $E_a=2$, $E=0$, ${\beta }^{(1)}=10$, $F=0.5$ for panel (a) and $F=0$ for panel (b). In all the Letter we take ${\beta }^{(2)}=1$ to set the energy scales, while the timescale is set by Eq. (1).

Figure 3

Left column, CGFs $\mathrm{\Phi }$ defined in (11) as a function of ${\gamma }^{\mathsf{q}}$ and ${\gamma }^{\mathsf{w}}$ for three different $V$ (a)–(c). Right column, the corresponding efficiency LDF (d)–(f). On the left, the diagonal black dotted line of slope one is there to guide the eye and the blue dashed line is the contour line $\mathrm{\Phi }=0$ enclosing the $\mathrm{\Phi }<0$ region that is relevant to calculate (13). The green solid line between $A$ and $B$ defines the degenerate minimum of the CGF. Its boundaries belong to the dashed gray critical lines separating regions with different dominant stationary solutions $x^{*}$. The slopes ${\eta }_A$ and ${\eta }_B$, of the lines ($OA$) and ($OB$), respectively, give the efficiencies delimiting the higher plateaux of the efficiency LDFs on the right. When $V>V_{\mathrm{cr}}^{\mathrm{MF}}=1.92$, one critical line touches the origin indicating bistability in the MF dynamics. On the right, $J^{**}(\eta )$, $J(\eta )$, and the finite $N$ efficiency LDFs are given, respectively, by the light blue thick solid line, the thin green solid line and the different dashed lines. The solid black (empty blue) squares show the location of the stable (unstable) MF efficiencies. The parameters are those of Fig. 2.