Cyclic Heat Engine with the Ising model: role of interactions and criticality

Abstract

Heat engines that convert thermal energy into work are a cornerstone of classical thermodynamics and remain an active area of contemporary research. Notable examples include microscopic heat engines, trade-off relations between power and efficiency, and the attainability of Carnot efficiency at finite power. We propose a cyclic heat engine based on the Ising model, in which the thermodynamic cycle involves variations of both temperature and magnetic field. We analyze the one-dimensional and mean-field Ising models, which allow for simple analytical results and provide new insight into the role of interactions in cyclic heat engines. In particular, we show that interactions can enhance both power and efficiency. Moreover, a system that does not operate as an engine in the absence of interactions can become an engine upon tuning the interaction strength. The mean-field model enables us to investigate the relevance of the phase transition for the performance of this Ising heat engine. Owing to the emergence of spontaneous magnetization, the mean-field model can still operate as an engine even when one of the magnetic fields is set to zero. Remarkably, when the work is maximized, we find that the optimal parameters are numerically consistent with this regime, in which one magnetic field vanishes and the cycle explores the phase transition. We also consider an alternative cycle for the mean-field model, obtained by varying the interaction strength while keeping both temperatures below the critical temperature and setting the magnetic field to zero throughout the cycle. The power and efficiency of this cycle are analyzed as well. Finally, while our analytical results are valid for the limit of large period we use numerical simulations for finite periods and show that the power decreases monotonically with the period.

Figures (7)

• Figure 1: Cyclic protocol for the Ising model. The thermodynamic parameters are the inverse temperature $\beta$ and the external magnetic field $H$. The changes in temperature are assumed to be instantaneous. Effectively the system stays in the first part of the cycle for half of the period and in the third part of the cycle for the other half of the period, with the second and fourth part having a negligible duration.
• Figure 2: Contour plots for the 1D model in the $H_2 \times J$ plane. (a) Work $W$. (b) Efficiency $\eta$. The remaining parameters are set to $\beta_h=0.1$ and $H_1=1$.
• Figure 3: Efficiency at maximum power for the 1D model. (a) Efficiency at maximum power $\eta^*$ as a function of $\beta_h$. (b) Ratio $\theta$ as a function of $\beta_h$.
• Figure 4: Contour plots for the MF model in the $H_2\times J$ plane. (a) Work $W$. (b) Efficiency $\eta$. The other parameters are set to $\beta_h=0.1$ and $H_1=1$.
• Figure 5: Efficiency at maximum power for the MF model. (a) Efficiency at maximum power $\eta^*$ as a function of $\beta_h$. (b) Ratio $\theta$ as a function of $\beta_h$.