Cyclic active refrigerators

TL;DR

The paper develops a formal framework for cyclic active refrigerators within stochastic thermodynamics by incorporating an excess entropy term $\mathcal{I}$ that captures activity-driven dissipation. It shows that the conventional Carnot bound on refrigerators can be surpassed by naive metrics like $\mathrm{COP}_{ps}$, while a properly defined COP that includes $\mathcal{I}$ remains bounded by the Carnot value $\mathrm{COP}_c$. Through an exactly solvable 1D model and a more realistic 2D model centered on active baths, the authors demonstrate three distinct operation regimes: enhanced refrigeration, Maxwell-demon-like heat flow without work, and a hybrid engine-refrigerator that performs both tasks. These results highlight the fundamental role of activity in reshaping thermodynamic performance and point to experimental realizations in colloidal systems immersed in active baths as well as extensions to more complex multi-degree-of-freedom cycles.

Abstract

Thermodynamic cycles are idealized processes that can convert heat into work or produce heat flow against a temperature gradient with the input of work. They remain an active area of research in modern stochastic thermodynamics. In particular, cyclic active heat engines have been shown to display a rich phenomenology, such as ``violations'' of the Carnot bound on efficiency and an improved performance in comparison to their passive counterparts. We introduce the concept of cyclic active refrigerators using a previously derived second law for cyclic active systems. We show that for cyclic active refrigerators, a naive definition of the coefficient of performance can exceed the bound set by the standard second law for passive refrigerators. We also show that cyclic active systems can behave like a Maxwell's demon, with heat flowing from the cold to the hot reservoir without any work input. Beyond this phase, cyclic active systems can enter a hybrid phase, functioning as both a heat engine and a refrigerator simultaneously. Our results are obtained with two models that involve active Brownian particles, a simpler one that allows for analytical results and a more realistic one that is analyzed through numerical simulations.

Figures (5)

• Figure 1: Depiction of the protocol for a refrigerator with four steps. A colloidal particle subjected to a harmonic potential is represented by the equation $\kappa x^2/2$. The energy change is represented by a change in the stiffness of the harmonic potential $\kappa$. The circles with arrows represent active particles and are a pictorial representation of the fact that the heat engine is active.
• Figure 2: Performance of an active refrigerator. The number in the lines separating two regions indicate the values of the $\textrm{COP}$ or the parameter $\xi$ for the hybrid phase. (a) Refrigerator phase for $\kappa= 5$ and $F=5$, where $\textrm{COP}_{ps}>\textrm{COP}_c$ for $\Delta \kappa$ less than the value indicated by the red dotted line. (b) Maxwell's demon phase with $\Delta \kappa= 0$ and $F=5$. (c) Hybrid phase for $\kappa=5$ and $F=5$. The inverse temperatures are set to $\beta_c=2$ and $\beta_h=1$ in all three figures.
• Figure 3: Phase Diagram for the exactly solvable model. The parameters are set to $\kappa= 5$, $F=5$, $\beta_c= 2$, and $\beta_h=1$.
• Figure 4: Performance of of an active refrigerator. (a) Refrigerator phase with $\textrm{COP}_{ps}>\textrm{COP}_c$. The parameters are set to $\kappa_1= 416.5$, $\kappa_2=83.3$, $u_1= u$, and $u_2= 15 u$. (b) Realization of Maxwell's demon with positive $-Q_c$ and no work input. The parameters are set to $\kappa_1 = \kappa_2 = 83.3$, $u_1 = u$, and $u_2=10u$.(c) Realization of hybrid phase. The parameters are set to $\kappa_1= 141.6$, $\kappa_2=83.3$, $u_1=u$, and $u_2 = 15u$.
• Figure 5: Phase Diagram for the second model in the plane $(\Delta\kappa= \kappa_2-\kappa_1,\Delta u=u_2-u_1)$. Parameters are set in the following way: we fixed the sums $\kappa_1+\kappa_2= 333.2$ and $u_1+u_2=100$.