Exact Mapping of Nonequilibrium to Equilibrium Phase Transitions for Systems in Contact with Two Thermal Baths

Abstract

We show that a large class of nonequilibrium many-body systems in contact with two thermal baths admit an exact mapping onto equivalent equilibrium systems. This mapping provides direct access to nonequilibrium phase transition points from known equilibrium results, irrespective of the model, interaction topology, or distance from equilibrium. We verify the universality of this correspondence using paradigmatic models (Ising, Potts, and Blume-Capel), and highlight distinctive features in entropy production close to critical and tricritical points. Our findings connect equilibrium and nonequilibrium statistical mechanics, with implications for microscopic thermal machines and stochastic thermodynamics.

Figures (5)

• Figure 1: (a) An illustration of a setup for the class of systems considered here, containing $N=4$ units with interaction parameters (represented by colored edges) whose values may depend on the contact with either of two baths at temperatures $T_1$ and $T_2$. Center and right: simulation results for the $L\times L$ square-lattice Ising model, with various system sizes $L$, under both field combinations (see main text). (b) Main plot shows the order-parameter variance $\chi$ versus $T_2$, while insets show, in log scale, the size-dependence of the order parameter $\langle |m|\rangle$ and its variance $\chi$ at criticality. Dashed lines in the insets have slopes consistent with Ising critical exponents. (c) Main plot shows the entropy production $\langle {\dot \sigma}\rangle$ per site versus $T_2$ for $L=30$, while inset shows no evidence of a singular behavior for $\langle {\dot \sigma}\rangle$ at criticality. Parameters: $\mathcal{J}^{(1)}=\mathcal{J}^{(2)}=1$, $T_1=2$; for field combination (II), $h^{(1)}=-1/2$.
• Figure 2: Simulation results for the nonequilibrium square-lattice $q$-state Potts model. The left (right) panel shows results for $q=3$ ($q=6$), for which the order-disorder transition is continuous (discontinuous). Left panel: rescaled variance $\chi^*=\chi L^{-\gamma/\nu_\perp}$ (main panel) and rescaled order parameter $m^*=\langle m\rangle L^{\beta/\nu_\perp}$ (inset) versus rescaled distance to criticality $y=(T_{2}-T_{2c})L^{1/\nu_\perp}$, with $\beta=1/9$, $\gamma=13/9$, and $\nu_{\perp}=5/6$. Right panel: $\chi^*=\chi L^{-2}$ (main panel) and $\langle\dot{\sigma}\rangle$ (inset) versus $y=(T_{2}-T_{2c})L^{-2}$ and $T_2$, respectively. The critical bath temperature $T_{2c}$ is given by $T_{2c}=\mathcal{J}^{(2)}/\left[2\ln\left(1+\sqrt{q}\right)-\beta_1\mathcal{J}^{(1)}\right]$ . Parameters: $\mathcal{J}^{(1)}=1$, $\mathcal{J}^{(2)}=2$ and $\beta_1=1$.
• Figure 3: Results for the $L\times L$ square-lattice BC model with $\beta_1=2$, $\beta_2=1$ and $\Delta^{(2)}=0$. (a) Points are estimates for the location of either discontinuous or tricritical transitions. In the discontinuous case, the points follow the (dashed) line $\frac{1}{2}\beta_1\mathcal{J}^{(1)} + \frac{1}{2}\beta_2\mathcal{J}^{(2)} = 2$, with $\frac{1}{2}\beta_1\Delta^{(1)}+\frac{1}{2}\beta_2\Delta^{(2)}=3.984$, while in the tricritical case the (dotted) line corresponds to $\frac{1}{2}\beta_1\mathcal{J}^{(1)} + \frac{1}{2}\beta_2\mathcal{J}^{(2)} = 1.642$, with $\frac{1}{2}\beta_1\Delta^{(1)}+\frac{1}{2}\beta_2\Delta^{(2)}=3.233$. Both lines nicely follow the predictions of Eq. \ref{['eqe']}, according to the equilibrium data in Refs. Silva2006PhysRevE.92.022134. (b) Behavior of $\langle|m|\rangle$ (main panel) and of $\langle\dot\sigma\rangle$ (upper inset) as functions of $\mathcal{J}^{(1)}$, for various values of $L$ and the particular choice $\left(\mathcal{J}^{(2)},\Delta^{(1)}\right)=(1,3.984)$ along the discontinuous line in (a). The lower inset shows $\langle|m|\rangle$ as a function of the rescaled distance to the transition, measured by $y=\left(\mathcal{J}^{(1)}-\mathcal{J}_c^{(1)}\right)L^2$, in which $\mathcal{J}_c^{(1)}$ is the value of $\mathcal{J}_c^{(1)}$ signaled by the vertical dotted line in the main panel, marking the discontinuous transition for the informed parameters according to Eq. \ref{['eqe']}. (c) Behavior of the Binder cumulant $U_4$ as a function of $\mathcal{J}^{(1)}$, for various values of $L$ and the particular choice $\left(\mathcal{J}^{(2)},\Delta^{(1)}\right)=(1,3.233)$ along the tricritical line in (a). The crossing of the various curves occurs at a value of $\mathcal{J}^{(1)}$, signaled by the vertical dotted line, compatible with Eq. \ref{['eqe']}. (d) Finite-size dependence, at the same tricritical point as in (c), of $\chi$ and of the derivative of the entropy production, $\sigma^*\equiv d\langle {\dot \sigma} \rangle/d \mathcal{J}^{(1)}$. The corresponding slopes of the log-log plots yield estimates of the tricritical exponents $\gamma_t/\nu_\perp=1.82(1)$, and $\zeta_t/\nu_\perp=1.59(1)$. Inset: rescaled order-parameter $m^*=\langle|m|\rangle L^{\beta_t/\nu_\perp}$ versus $y=\left(\mathcal{J}^{(1)}-\mathcal{J}_c^{(1)}\right)L^{1/\nu_\perp}$ for the equilibrium exponent $\beta_t/\nu_\perp=3/40$. The estimate for $\gamma_t/\nu_\perp$ is also in excellent agreement with the equilibrium equilibrium value $37/20$. The value of $\zeta/\nu_\perp$ is strongly dependent on the choice of parameters.
• Figure 4: Comparison between steady-state order-parameter probability distributions $p^{\mathrm{st}}(m)\equiv p^\mathrm{st}\left((1+m)N/2,(1-m)N/2,N\right)$ obtained from Eq. (\ref{['eq:pstexp']}) (solid lines) and Eq. (\ref{['alltoall']}) (symbols). Left panel corresponds to $h^{(1)}=1/4$, $h^{(2)}=-1/2$, while right panel corresponds to $h^{(1)}=1/8$, $h^{(2)}=-1/7$. Black curves: $\mathcal{J}^{(1)}=\mathcal{J}^{(2)}=1/4$. Red curves: $\mathcal{J}^{(1)}=\mathcal{J}^{(2)}=1/2$. Green curves: $\mathcal{J}^{(1)}=\mathcal{J}^{(2)}=1/10$. In all cases $N=100$.
• Figure 5: For the same parameters as in Fig. \ref{['fig1']} (main text), we plot (left) the entropy production $\langle {\dot \sigma}\rangle$ for different alternation rates $d$ (with $L=30$) and (right) the critical point $T_{2c}$ versus $d^{-1}$. The red symbol denotes the exact value for $d\rightarrow \infty$, which is equivalent to the simultaneous contact model. The straight line highlights the linear dependence on $d^{-1}$ for large $d$.