Universal features of nonequilibrium Ising models in contact with two thermal reservoirs

Abstract

We derive generic properties of nonequilibrium phase transitions in all-to-all Ising models placed in contact with two thermal reservoirs, in which parameters (temperatures, interactions and field parameters) assume arbitrary values depending on the contact with each thermal bath. The presence of different kinds of external parameters leads to remarkably different sort of phase transitions. While continuous, discontinuous and even tricritical points are presented when external parameters are symmetric (e.g. the case of energetic barriers or different couplings between the system and thermal baths), the tricriticality is absent when external parameters are antisymmetric (e.g. the case of magnetic fields or biased drivings) implying that solely critical or discontinuous are possible. In such latter case, the probability distribution acquires the Boltzmann-Gibbs like form, irrespectively the model parameters when the switching between thermal reservoirs is sufficiently fast. Our work sheds light about the differences between equilibrium and nonequilibrium ingredients and theirs consequences upon phase transitions.

Figures (5)

• Figure 1: Schematics of a model composed of 4 spins with an arbitrary connectivity. The system is connected to a hot or a cold thermal bath, with reciprocal temperatures $\beta_2$ and $\beta_1$, respectively, that stochastically exchange with each other with rate $\kappa$.
• Figure 2: The behavior of Ising model for antisymmetric external parameters ($\delta_1=\delta_2=0$) and different $\lambda$'s ($\lambda_1=\lambda_2=\lambda=0,\pm1$). Top and bottom panels depict the total magnetization ${\overline m}$ for $\theta_1=\theta_2=1$ and $\theta_1=-\theta_2=1$, respectively. Vertical dashed lines denote the corresponding transition points. Inset: Log-log plot of $\overline{m}$ versus $\epsilon-\epsilon_c$, consistent with scaling $\overline{m}\sim(\epsilon-\epsilon_c)^{1/2}$. Parameters: $\beta_1=6,\beta_2=1\text{ and }\gamma=1$
• Figure 3: Global magnetization $\overline{m}$ versus $\epsilon$ for the symmetric external parameters ($\theta_1=\theta_2=0$) and different $\lambda_\nu$'s. Vertical dashed lines denote the corresponding transition points. The orange ($\lambda_1=16.2$, $\lambda_2=3$), light blue ($\lambda_1=0.5$, $\lambda_2=-0.69$) and dark blue ($\lambda_1=0.5$, $\lambda_2=1.31$) curves correspond to the second-order, first-order and tricritical transitions, respectively. Inset: Log-log plot of $\overline{m}$ versus $\epsilon-\epsilon_c$, whose slopes are consistent with $1/2$ and $1/4$, respectively. Parameters: $\beta_1=6,\beta_2=1,\gamma=1$ and $\delta_1=\delta_2=1$.
• Figure 4: The entropy production $\langle\dot{\sigma}\rangle$ versus $\epsilon$ for the antisymmetric (a) ($\delta_\nu=0,\theta_\nu=1$ and $\lambda=0,\pm1$) and symmetric (b) ($\delta_\nu=1,\theta_\nu=0$) parameters in the fast switching limit. In $(b)$, we set $(\lambda_1,\lambda_2)$ as $(0,0)$, $(0.5,-0.69)$ and $(0.5,1.31)$ for the second-order, first-order and tricritical transitions, respectively. Parameters: $\beta_1=6,\beta_2=1,\gamma=1$.
• Figure 5: Depiction of phase transitions for finite switchings between thermal baths. $(a)$ shows transition points $\epsilon_c$ versus $\kappa^{-1}$ for the absence ($\circ$)(from Eq.\ref{['ec_nodriving_kappa']}), symmetric (triangles), respectively. Panels $(b)$ and $(c)$ show tricritical and critical lines $\lambda_1\times\lambda_2$ for different $\kappa$'s, respectively, for $\delta_1=\delta_2=1$ and $\lambda_1=\lambda_2=0.1$ (symmetric) and $\theta_1=\theta_2=1$ and $\lambda_1=-\beta_2\lambda_2/\beta_1$ with $\lambda_2=0.1$ (antisymmetric). They approach to Eqs. (\ref{['tric_condition']}) and \ref{['cond_Anti']} as $\kappa\rightarrow\infty$. Parameters: $\gamma=1$, $\beta_1=6$, $\beta_2=1$.