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

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.

💡 Research Summary

In this paper the authors present a minimal yet powerful numerical model that demonstrates the breakdown of local equilibrium thermodynamics during a first‑order phase transition under steady heat conduction. The model is a one‑dimensional Hamiltonian Potts system in which a fractional‑order spatial derivative is introduced to mimic the low‑wavenumber density of states of the standard two‑dimensional Potts model. By choosing a fractional exponent (approximately one‑half) the operator D acts non‑locally, converting the one‑dimensional lattice into an effective medium with the same low‑frequency spectral properties as a two‑dimensional system. This “dimensional reduction” enables a genuine first‑order order‑disorder transition to occur in a one‑dimensional setting, something that is impossible with ordinary nearest‑neighbour interactions.

The Hamiltonian contains (n‑1) scalar fields q_i (with n=5 to guarantee a first‑order transition) and their conjugate momenta p_i. The potential energy is constructed from quadratic terms Q_j that enforce the fields to settle into one of n equally spaced minima, which correspond to the vertices of a regular simplex in (n‑1) dimensions. Periodic boundary conditions are imposed, and the fractional derivative D is defined via its Fourier representation Dq(k)=|k|^{α}q(k) with α≈½. This choice yields a constant density of states at low k, reproducing the two‑dimensional behavior.

Numerical simulations are performed on a lattice of size L=256 with spacing Δx=0.25. Heat baths at temperatures T_1 and T_2 (chosen such that T_1 < T_c < T_2) are attached to the left and right ends, establishing a constant heat flux J through the system. Both Langevin dynamics (to generate equilibrium initial conditions) and pure Hamiltonian dynamics (to evolve the non‑equilibrium steady state) are employed. The local temperature is defined through the kinetic energy density, and spatial averages are taken over long time windows to obtain stationary profiles.

The central observation is that a stationary interface separating ordered and disordered domains forms at a position X, and the temperature at this interface, θ = T(X), systematically deviates from the equilibrium transition temperature T_c. This deviation is not a finite‑size artifact; it persists for large L and fine discretization, and its magnitude matches the quantitative prediction of the recently proposed global thermodynamics framework. In that theory the interface temperature is given by

θ_pred = T_c + |J| (1/κ_o – 1/κ_d) X (L – X) / (2 L),

where κ_o and κ_d are the thermal conductivities of the ordered and disordered phases, respectively. The simulations confirm that when κ_o < κ_d the interface is hotter than T_c (super‑heated ordered region), whereas κ_o > κ_d yields a cooler interface (super‑cooled disordered region). Measured values of κ_o, κ_d, J, and X produce θ that agrees with θ_pred within statistical error, providing strong evidence that the global temperature concept correctly captures the thermodynamics of the non‑equilibrium steady state.

The authors also discuss the physical origin of the local equilibrium breakdown. A strong heat current creates steep temperature gradients, which in turn cause the local kinetic energy distribution to deviate from the Maxwell‑Boltzmann form that underlies the usual definition of temperature. Consequently, the local free‑energy landscape is altered: the interface region experiences a shifted effective free‑energy minimum, allowing a metastable phase (either ordered or disordered) to be stabilized by the heat flow. This mechanism is independent of dimensionality, as the same behavior was previously observed in two‑dimensional simulations of the standard Hamiltonian Potts model; the present one‑dimensional fractional‑derivative model reproduces it with far lower computational cost.

Overall, the paper demonstrates three key points: (1) a fractional spatial derivative can endow a one‑dimensional Hamiltonian system with the spectral properties necessary for a first‑order transition; (2) under steady heat conduction the interface temperature deviates from the equilibrium transition temperature, directly violating the local equilibrium assumption; and (3) the deviation is quantitatively described by global thermodynamics, confirming the validity of that framework for non‑equilibrium steady states. The work provides a clean, computationally efficient platform for future studies of non‑equilibrium phase coexistence, metastable state control, and the fundamental limits of thermodynamic descriptions beyond the local equilibrium paradigm.