Universal splitting of phase transitions and performance optimization in driven collective systems

Abstract

Spontaneous symmetry breaking is a hallmark of equilibrium systems, typically characterized by a single critical point separating ordered and disordered phases. Recently, a novel class of non-equilibrium phase transitions was uncovered [Phys. Rev. Res. {\bf 7}, L032049 (2025)], showing that the combined effects of simultaneous contact with thermal baths at different temperatures and external driving forces can split the conventional order-disorder transition into two distinct critical points, determined by which ordered state initially dominates. We show the robustness of this phenomenon by extending a minimal interacting-spin model from the idealized case of simultaneous bath coupling to a finite-time coupling protocol. In particular, we introduce two protocols in which the system interacts with a single bath at a time: a stochastic protocol, where the system randomly switches between the baths at different temperatures, and a deterministic protocol where the coupling alternates periodically. Our analysis reveals two key results: (i) the splitting of phase transitions persists across all coupling schemes -- simultaneous, stochastic, and deterministic -- and (ii) different optimizations of power and efficiency in a collectively operating heat engine reveal that both the stochastic and deterministic protocols exhibit superior global performance at intermediate switching rates and periods when compared to simultaneous coupling. The global trade-off between power and efficiency is described by an expression solely depending on the temperatures of thermal reservoirs as the efficiency approaches to the ideal limit.

Figures (6)

• Figure 1: Schematics and dynamics of the collective engine. The system is composed of an ensemble of all-to-all interacting units, each one described by the single-spin variable $s\in \{0,\pm1\}$ (triangle vertices). Left and right panels show the set of transitions $s' \rightarrow s$ (black arrows) which are favored by the biased driving $F$ according to whether the system is placed in contact with the cold ($\nu=1$; blue) or hot ($\nu=2$; red) thermal reservoirs, at inverse temperatures $\beta_\nu$. Stochastic switching between thermal reservoirs, given the system is at the state $s$, occurs with rate $d$ (black, double arrows); deterministic switching occurs periodically with period $\tau$ (outer arrows; colored according to their contact with the respective reservoir).
• Figure 2: Order parameter $m$ and critical points $\epsilon_c^{A,B}$ as a function of the stochastic switching rate $d$ (a, b) and the deterministic cycle time $\tau$ (c, d). Panels a and c show $m(\epsilon)$ for stochastic and deterministic switching, respectively, illustrating the splitting into phases $A$ and $B$ at distinct critical points $\epsilon_c^{A,B}$ for different values of $d$ and $\tau$. Panels b and d display the corresponding critical points $\epsilon_c^{A}$ (black symbols) and $\epsilon_c^{B}$ (red symbols) as functions of $d^{-1}$ and $\tau$. In both cases, the results converge to the simultaneous-contact limit as $d^{-1}\to0,\,\tau \to 0$. Insets in panels b and d highlight the behavior of the critical points near this limit. Parameters are $F=2,\,\beta_2=1$ and $\beta_1=2$.
• Figure 3: Critical scaling of the order parameter and entropy production near the critical point $\epsilon^{A}_c$. Panels (a) and (c) show the order parameter $m$ versus $\epsilon_c^{A}-\epsilon$; panels (b) and (d) display the reduced entropy production $\sigma_c-\sigma^A$ as a function of $\epsilon_c^{A}-\epsilon$. The top panels (a, b) correspond to the stochastic switching case for several values of the switching rate $d$, whereas the bottom panels (c d) correspond to the deterministic case for several cycle times $\tau$. Black solid lines are guides to the eye indicating power-law behavior, with slope $\beta=1$ for the order parameter and $2\beta$ for the entropy production.
• Figure 4: Power–efficiency Pareto fronts (rainbow–gradient lines) for stochastic switching. (a)$\epsilon,\,d$, and $F$ are varied simultaneously; the color gradient indicates how the optimal non-conservative drive $F$ evolves along the front. The dashed black line corresponds to $\mathcal{P}\sim(1-\eta/\eta_C)^{\alpha}$ with $\alpha=3$. (b) Pareto fronts at fixed $F=2$, obtained by varying only $\epsilon$ and $d$. Here the gradient encodes the optimal $d$, demonstrating that finite-time switching yields globally superior performance---both in power and in efficiency---relative to simultaneous contact. In both panels, the black curves show that the simultaneous-contact limit $d\to\infty$ is entirely dominated by finite-time operation: operating the engine at (optimal) finite time always outperforms the simultaneous-contact limit. Colored lines in the suboptimal regions (shaded gray) represent, respectively, trade-offs at fixed $F$ (panel a) and fixed $d$ (panel b). Colored points mark where these fixed-parameter trade-offs become tangent to the corresponding Pareto fronts. Parameters are $\beta_2=1$ and $\beta_1=2$.
• Figure 5: Power–efficiency Pareto fronts (rainbow–gradient lines) for deterministic switching. (a)$\epsilon,\,\tau$, and $F$ are varied simultaneously; the color gradient indicates how the optimal non-conservative drive $F$ evolves along the front. The dashed black line corresponds to $\overline{\mathcal{P}}\sim(1-\eta/\eta_C)^{\alpha}$ with $\alpha=3$.(b) Pareto fronts at fixed $F=2$, obtained by varying only $\epsilon$ and $\tau$. Here the gradient encodes the optimal $\tau$, demonstrating that finite-time switching yields globally superior performance---both in power and in efficiency---relative to simultaneous contact. In both panels, the black curves show that the simultaneous-contact limit $\tau\to0$ is entirely dominated by finite-time operation: operating the engine at (optimal) finite time always outperforms the simultaneous-contact limit. Colored lines in the suboptimal regions (shaded gray) represent, respectively, trade-offs at fixed $F$ (panel a) and fixed $\tau$ (panel b). Colored points mark where these fixed-parameter trade-offs become tangent to the corresponding Pareto fronts. Parameters are $\beta_2=1$ and $\beta_1=2$.