Realizing an "unstable state" that cannot exist in equilibrium through heat flow

Hitomi Endo (Master's student, second year) and Professor Michikazu Kobayashi (Faculty of Engineering, Science and Technology) at Kochi University of Technology have analyzed phase transition phenomena in a "non-equilibrium steady state*1)" where heat continuously flows, using a "one-dimensional computational model" they developed. They revealed that the temperature at the boundary (interface) where different states coexist deviates from the conventional "transition temperature in equilibrium" understanding.

This achievement strongly supports the universality of the new thermodynamic theory "Global Thermodynamics*3)" proposed in recent years, and it indicates the possibility of stably controlling states that are normally unstable and quickly disappear, by utilizing heat flow. Furthermore, the introduction of a model (fractional calculus*4)) that extends high school calculus into an analysis that typically requires large-scale computation, significantly reducing the computational cost for high-precision simulations, was highly praised.

This research was published on June 2, 2026, in the prestigious international academic journal "Physical Review E," published by the American Physical Society.

[Key Research Findings]

Discovered that the "common sense of phase transition" changes when there is heat flow.

Showed the possibility of stably realizing "unstable states" that cannot exist in equilibrium.

Supports the universality of the new non-equilibrium thermodynamics theory, "Global Thermodynamics."

Solved the challenge of large-scale simulations with the mathematical method "fractional calculus*4)."

[Research Content and Results]

Substances exist in different states such as solid, liquid, and gas depending on temperature, and the temperature at the boundary is called the "phase transition temperature." For example, in an equilibrium state, water coexists as ice and water at 0°C and as water and water vapor at 100°C. Such phase transitions*2) are understood by equilibrium thermodynamics*5), but many aspects of phase transitions in non-equilibrium states where heat continuously flows remain unclear.

In this study, we investigated the temperature of the interface separating two phases in a non-equilibrium steady state*1) with heat flow using numerical simulations. As a result, we revealed that the interface temperature significantly deviates from the phase transition temperature in the equilibrium state. This means that metastable states, which are unstable and cannot exist for long in equilibrium, are stabilized by heat flow.

Furthermore, we confirmed that these results are in good agreement with the predictions of the thermodynamic theory for non-equilibrium steady states, "Global Thermodynamics*3)," proposed in recent years. This research demonstrates that the local equilibrium hypothesis, widely assumed in non-equilibrium states, breaks down at the phase coexistence interface, and it is an important result supporting the effectiveness of global thermodynamics.

Another significant feature of this research is the introduction of a mathematical method called "fractional calculus*4)," an extension of calculus learned in high school, to efficiently analyze phase transition phenomena, which typically require very large-scale computations and can be difficult to reproduce or verify. This method has succeeded in significantly reducing the computational cost required for high-precision simulations of non-equilibrium phase coexistence states.

[Future Prospects]

The phenomenon confirmed in this study may be a universal phenomenon independent of specific models or spatial dimensions. In the future, experimental verification will be pursued for phenomena such as liquid-gas transitions*6), solid-liquid transitions*7), and nematic transitions in liquid crystals*8). Theoretical clarification of the relationship with the statistical mechanical description of non-equilibrium steady states is also expected.

Furthermore, the numerical analysis method using fractional calculus employed in this study is expected to have a wide range of applications as a technique that will significantly advance research on non-equilibrium phase transitions, which have been difficult due to high computational costs.

[Glossary]

*1) Non-equilibrium steady state: Refers to a state where, despite the continuous inflow and outflow of energy and matter from the outside, the overall state remains unchanged and stable. While "equilibrium state" represents "static stability" with complete cessation of movement, non-equilibrium steady state is characterized by "stability while continuously moving." An example is the constant temperature distribution of a metal rod where heat continuously flows from a hot region to a cold region. Many natural and life phenomena, such as cellular activity, engine combustion, and traffic flow, exist in this state, and it is one of the most important themes awaiting elucidation in modern physics.

*2) Phase transition: The phenomenon of a substance changing into different states such as solid, liquid, or gas.

*3) Global Thermodynamics: A new theoretical framework that describes the behavior of energy and matter by viewing the entire system (global) as a single entity, rather than accumulating changes in parts of the system (local). It is an attempt to clarify "universal laws" that permeate the system as a whole, even with intense energy flow (non-equilibrium) within it. It enables the consistent explanation of complex phenomena, previously studied individually, with a single overarching rule. It is also expected to guide next-generation energy management technologies and nanotechnology.

*4) Fractional calculus: A generalized mathematical concept of differentiation learned in high school. While ordinary differentiation uses only nearby information, fractional calculus can incorporate the influence of distant locations. In this study, this property was utilized to reproduce phase transition phenomena, which normally require large-scale simulations, with low-cost simulations.

*5) Equilibrium thermodynamics: A branch of physics that deals with the states of matter and energy changes. It is the study for elucidating the properties of substances in a "stable state where energy exchange with the surroundings has stopped," such as water in a glass that has reached the same temperature as its surroundings and is in a state of equilibrium without evaporation or condensation. It developed in the 19th century with the aim of improving the efficiency of steam engines and forms the basis of current physics, chemistry, and materials engineering. On the other hand, it is also used as an important standard for understanding "non-equilibrium" phenomena, such as life activities and meteorological phenomena, where energy continuously flows and changes.

*6) Liquid-gas transition: The phenomenon of a substance changing from a "liquid" state to a "gas" state, or vice versa. For example, when water is heated, it boils and turns into steam at 100°C, and a key characteristic is that the density and properties of the substance change drastically at a certain point rather than continuously (phase transition).

*7) Solid-liquid transition: The phenomenon of a substance changing from a "solid" state to a "liquid" state (melting), or from a "liquid" state to a "solid" state (solidification). The everyday phenomenon of ice melting into water is an example.

*8) Nematic transition: The phenomenon of a substance changing from a "liquid (isotropic phase)" where molecules point in random directions to a "liquid crystal (nematic phase)" where the molecular orientation is aligned in a specific direction. By controlling this "degree of alignment" with heat or electricity, the way light passes through can be altered, which is the fundamental principle of liquid crystal displays. This phenomenon is at the core of all devices that manipulate light, not only in liquid crystal televisions but also in smartphone screens, and next-generation optical switches and sensors.

[Research Funding]

This research was supported by Grants-in-Aid for Scientific Research (KAKENHI) from the Japan Society for the Promotion of Science (Grant Numbers: 23K22492, 24K00593, 26K07020).

[Publication Information]

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

Authors: Hitomi Endo and Michikazu Kobayashi

Journal: Physical Review E 113, 064109 (2026)

Publication Date: June 2, 2026

DOI: 10.1103/rz4l-krv6