Two coupled, driven Ising spin systems working as an engine

Miniaturized heat engines constitute a fascinating field of current research. Many theoretical and experimental studies are being conducted that involve colloidal particles in harmonic traps as well as bacterial baths acting like thermal baths. These systems are micron-sized and are subjected to large thermal fluctuations. Hence, for these systems average thermodynamic quantities, such as work done, heat exchanged, and efficiency, lose meaning unless otherwise supported by their full probability distributions. Earlier studies on microengines are concerned with applying Carnot or Stirling engine protocols to miniaturized systems, where system undergoes typical two isothermal and two adiabatic changes. Unlike these models we study a prototype system of two classical Ising spins driven by time-dependent, phase-different, external magnetic fields. These spins are simultaneously in contact with two heat reservoirs at different temperatures for the full duration of the driving protocol. Performance of the model as an engine or a refrigerator depends only on a single parameter, namely the phase between two external drivings. We study this system in terms of fluctuations in efficiency and coefficient of performance (COP). We find full distributions of these quantities numerically and study the tails of these distributions. We also study reliability of the engine. We find the fluctuations dominate mean values of efficiency and COP, and their probability distributions are broad with power law tails.

Figure 1

Cartoon of the model discussed in the text.

Figure 2

(a) Engine mode of operation. $\langle {\dot{q}}_L\rangle >0, \langle {\dot{q}}_R\rangle <0$ and $\langle \dot{w}\rangle <0$ for certain values of the phase $\varphi$. Parameter values are $h_0=0.25, \tau =190, T_L=1.0, T_R=0.1$. At $\varphi =0.7\pi$, maximum work is extracted from the system (inset). In all the results discussed further these parameters are considered to be optimal for engine mode of operation. (b) Pump-refrigerator mode of operation. Parameter values are $h_0=0.25, \tau =225, T_L=0.5, T_R=0.5$. See, for example, at $\varphi =0.7\pi$, we have $\langle {\dot{q}}_R\rangle >0, \langle \dot{w}\rangle >0$, and $\langle {\dot{q}}_L\rangle <0$ implying heat is taken from the right bath, work is done on the system, and heat is dissipated into the left bath. In all the results discussed further, these parameters are considered to be optimal for pump-refrigerator mode of operation. Zero line is just a guide for the eyes. See Ref. [25] for details.

Figure 3

(a) Shows the phase diagram as a function of the phase difference $\varphi$ for engine mode of operation. Black solid circles indicate Heater 2 mode and open black circles indicate engine mode. (b) Phase diagram for the refrigerator mode. Solid red circles heater 1 $(\mathrm{for}0\le \varphi <\pi /2$ and $3\pi /2<\varphi \le 2\pi )$ and solid green circles refrigerator $(\mathrm{for}\pi /2\le \varphi <\pi )$, solid black circles heater 2 mode $($ for $\pi \le \varphi <3\pi /2)$. Different modes are also indicated in the figure.

Figure 4

(a) Phase diagram for Engine mode of operation. Here $T_R=0.1, \varphi =0.7\pi , h_0=0.25$, and $J=1.0$. Black solid circles represent engine operation while black open circles heater 2 operation. These modes are also indicated in the figure. (b) Distribution $P\left(\epsilon \right)$ of efficiency $\epsilon$ in the engine mode of operation, parameters used are $T_L=1.0, T_R=0.1, \varphi =0.7\pi , h_0=0.25$, and $\tau =190$. Inset shows the tail part of the distribution for $\tau =10$. Tail of the distribution can be fitted to a power law $a{\epsilon }^{-\alpha }$ with exponent close to 2 (solid black line). (c) Phase diagram for refrigerator mode of operation. Parameters are $T_R=0.5, \varphi =0.7\pi , h_0=0.25$. Red portion (middle portion) shows heater 1 operation, while Black portion represents heater 2 operation (upper right $\mathrm{part})$, and green refrigerator operation $($ bottom right part $)$ is indicated in the figure. For refrigerator mode of operation one requires the temperature difference between $T_L$ and $T_R$ to be small. (d) Distribution $P\left(\eta \right)$ of coefficient of performance $\eta$ in the refrigerator mode of operation. Parameters used are $T_L=0.5, T_R=0.5, \varphi =0.7\pi , h_0=0.25$, and $\tau =225$. Inset shows the tail part of the distribution. Tail of the distribution can again be fitted to a power law $b{\eta }^{-\alpha }$ with exponent close to 2 (solid black line).

Figure 5

Plot of $\langle {\dot{q}}_L\rangle , \langle {\dot{q}}_R\rangle , \langle \dot{w}\rangle$ as a function of time period of external driving $\tau$ in different modes of operation. Here $h_0=0.25, \varphi =0.7\pi$. (a) Engine mode, $T_L=1.0, T_R=0.1$. (b) Refrigerator mode, $T_L=0.5, T_R=0.5$. (c) Efficiency $\overline{\epsilon }$ as a function of time period of external driving $\tau$ in engine mode, $T_L=1.0, T_R=0.1$. Zero line is just a guide for the eyes.

Figure 6

Plot of Power generated namely $\langle \dot{w}\rangle /\tau$ as a function of time period of external driving $\tau$. For two sets of parameters $h_0=0.25, \varphi =0.7\pi , T_L=1.0, T_R=0.1$ and $T_L=0.5, T_R=0.5$.