Out-of-Equilibrium Clock Model at the Verge of Criticality

We consider an out-of-equilibrium lattice model consisting of 2D discrete rotators, in contact with heat reservoirs at different temperatures. The equilibrium counterpart of such a model, the clock model, exhibits three phases: a low-temperature ordered phase, a quasiliquid phase, and a high-temperature disordered phase, with two corresponding phase transitions. In the out-of-equilibrium model the simultaneous breaking of spatial symmetry and thermal equilibrium gives rise to directed rotation of the spin variables. In this regime the system behaves as a thermal machine converting heat currents into motion. In order to quantify the susceptibility of the machine to the thermodynamic force driving it out of equilibrium, we introduce and study a dynamical response function. We show that the optimal operational regime for such a thermal machine occurs when the out-of-equilibrium disturbance is applied around the critical temperature at the boundary between the first two phases, namely, where the system is mostly susceptible to external thermodynamic forces and exhibits a sharper transition. We thus argue that critical fluctuations in a system of interacting motors can be exploited to enhance the machine overall dynamic and thermodynamic performances.

Figure 1

Circles, squares, lines: equilibrium transition temperatures for the nonchiral ($\phi =0$) and the chiral clock model ($\phi \ne 0$) as given by Eq. (1). Lines: the expressions for $T_1^{\mathrm{eq}}$ (full) and $T_2^{\mathrm{eq}}$ (dotted) are taken from Ref. [17]. Open circles: $T_1^{\mathrm{eq}}$ for $N_s\le 6$, $\phi =0,\mathrm{\Delta }T=0$ [17], the dashed line is a guide to the eye for the results with $N_s\le 6$. Full circles: numerical results for the first transition temperature $T_1^{\mathrm{eq}}$ as a function of $N_s$ ($N_s\le 40$) for the equilibrium model with $\phi =\overline{\phi }$. Squares: numerical results for the second transition temperature $T_2^{\mathrm{eq}}$ as a function of $N_s$, for the equilibrium model with $\phi =\overline{\phi }$. Triangles: temperature $T^{*}$ that maximizes the dynamic susceptibility ${\chi }_{\omega }$ in the out-of-equilibrium clock model with $\phi =\overline{\phi }$, see text and Fig. 3. Inset: enlargement of the small $N_s$ region.

Figure 2

(a) Average angular velocity per spin as a function of the mean lattice temperature $T_m$, for $N_s=16$ and $\mathrm{\Delta }T=4\times {10}^{-4}$. (b) Ratio of the average angular velocity to the temperature gradient as a function of the mean lattice temperature $T_m$, for $N_s=16$ and $L=64$. The vertical dashed lines correspond to the equilibrium transition temperature $T_1^{\mathrm{eq}}$ for the clock model ($\phi =0$, light gray) [17], and the chiral clock model ($\phi \ne 0$, dark gray), see Fig. 1. In both panels we considered trajectories consisting of 1000 $L^2$ MC steps. In (a) the results are obtained by averaging over ($10^6$, $2\times {10}^5$, $5\times {10}^4$, 8000, 8000, 1000) independent trajectories for the different system sizes ($L=4$, 8, 16, 32, 64, 128). The angular velocity is expressed in units of 1/(number of MC steps per spin). In (b) the results are obtained by averaging over ($10^4,8000,200,100,100$) trajectories for increasing $\mathrm{\Delta }T$.

Figure 3

(a) Out-of-equilibrium dynamical susceptibility ${\chi }_{\omega }$ as a function of the mean lattice temperature $T_m$, for a different number of states $N_s$, $L=64$ and $\mathrm{\Delta }T\simeq T_1^{\mathrm{eq}}/100$. We consider 8000 independent trajectories, each consisting of 1000 $L^2$ MC steps, after the system has reached its steady state. For the system with $N_s=6$ we indicate the temperature $T^{*}$ for which ${\chi }_{\omega }$ is maximum. The BKT temperature $T_2^{\mathrm{eq}}$, indicated in panel (a), is independent of the number of states $N_s$, see Fig. 1. Inset: ${\chi }_{\omega }$ as a function of $T_m$ for larger values of $N_s$. (b) Ratio of the average angular velocity to the heat current $z=⟨\omega ⟩/⟨{\dot{Q}}_{+}⟩$, as a function of the mean temperature. Inset: $z$ as a function of $T_m$ for larger values of $N_s$.

Figure 4

Points: maximum value of the dynamical susceptibility ${\chi }_{\omega }^{*}$ as a function of the number of states $N_s$, from the data in Fig. 3, as obtained with MC simulations for a different number of states $N_s$, $L=64$, and $\mathrm{\Delta }T\simeq T_1^{\mathrm{eq}}/100$. The full line is a linear fit to the data, with slope $b=(941.0\pm 1.6)\times {10}^{-6}$. Upper inset: enlargement of the small $N_s$ region. Lower inset: average angular velocity $⟨{\omega }^{*}⟩$ at $T^{*}$ as a function of the number of states.