Violation of local equilibrium thermodynamics in one-dimensional Hamiltonian-Potts model

TL;DR

The paper addresses non-equilibrium first-order transitions and the breakdown of local equilibrium under steady heat conduction. It introduces a minimal one-dimensional Hamiltonian-Potts model with a fractional spatial derivative to mimic the low-k density of states of a 2D system and realize phase coexistence with a stationary interface. The study finds that the interface temperature deviates from the equilibrium transition temperature Tc and that this deviation quantitatively agrees with global thermodynamics predictions, indicating metastable states can be stabilized by a steady heat current. This supports the universality of non-equilibrium thermodynamic breakdown across dimensions and provides a controlled model for testing the limits of local-equilibrium descriptions.

Abstract

We investigate non-equilibrium phase coexistence associated with a first-order phase transition by numerically studying a one-dimensional Hamiltonian-Potts model with fractional spatial derivatives. The fractional derivative is introduced so as to reproduce the low-wavenumber density of states of the standard two-dimensional model, allowing phase coexistence to occur in a minimal one-dimensional setting under steady heat conduction. By imposing a constant heat flux through boundary heat baths, we observe stable coexistence of ordered and disordered phases separated by a stationary interface. We find that the temperature at the interface systematically deviates from the equilibrium transition temperature, demonstrating a clear violation of the local equilibrium description. This deviation indicates that equilibrium metastable states can be stabilized and controlled by a steady heat current. Furthermore, the interface temperature obtained in our simulations is in quantitative agreement with the prediction of global thermodynamics for non-equilibrium steady states. These results confirm that the breakdown of local equilibrium and the stabilization of metastable states are intrinsic features of non-equilibrium first-order phase transitions, independent of spatial dimensionality. Our study thus provides a minimal and controlled numerical model for exploring the fundamental limits of thermodynamic descriptions in non-equilibrium steady states.

Figures (3)

• Figure 1: Equilibrium properties of one-dimensional Hamiltonian-Potts model. The ensemble-averaged energy density $\langle h \rangle$ is shown as a function of the temperature $T$ for simulations based on the Langevin equation \ref{['eq:Langevin_eq']}. The orange and green curves correspond to protocols in which the temperature is gradually increased from $T = 0$ and decreased from $T = 2$, respectively, revealing a clear hysteresis characteristic of a first-order phase transition. The blue curve shows the average temperature $\langle T \rangle$ as a function of the energy density $h$ obtained from the Hamiltonian dynamics \ref{['eq:Hamilton']}, where a plateau indicates the equilibrium transition temperature.
• Figure 2: Spatial profiles of the local temperature $\langle T(x) \rangle$ in the non-equilibrium steady states with $T_1 = 0.6$ and $T_2 = 0.7$ (panel (a)), and $T_1 = 0.35$ and $T_2 = 0.7$ (panel (b)). The data in the central region $2\Delta x \leq x \leq L_x - 2\Delta x$ are fitted by the piecewise linear function $T_{\rm fit}(x)$ defined in Eq. \ref{['T_fit']}. The equilibrium transition temperature $T_{\rm c} \simeq 0.634$ is indicated for reference. The estimated interface temperature are $\theta \simeq 0.645 \pm 6.92 \times 10^{-5}$ in panel (a) and $\theta \simeq 0.656 \pm 7.50 \times 10^{-4}$ in panel (b), both exceeding $T_{\rm c}$.
• Figure 3: Dependence of the interface temperature $\theta$, the predicted interface temperature $\theta_{\rm pred}$ from global thermodynamics [Eq. \ref{['prediction3']}], and the interface position $X$ on the bath temperature $T_1$. The temperature of the hot bath is fixed to $T_2 = 0.7$. Error bars represent statistical uncertainties obtained from fitting $\langle T(x) \rangle$ with $T_{\rm fit}(x)$ in Eq. \ref{['T_fit']}. The horizontal red line indicates the equilibrium transition temperature $T_{\rm c}$, and the shaded region represents its uncertainty.