However, hysteresis has long posed a problem for thermodynamics. In conventional thinking, the state of a material should be described by state variables, such as temperature and volume. But in solids, the same temperature and volume can correspond to different material properties depending on the material's past treatment.

For this reason, hysteresis has traditionally been treated as a nonequilibrium phenomenon, outside the standard framework of thermodynamics.

Prof. Koun Shirai, at the Graduate School of Engineering, The University of Osaka, has now shown that hysteresis in solids can be described thermodynamically by reconsidering what counts as an equilibrium state and what variables are needed to define the state of a solid.

The study, published in International Journal of Thermophysics, argues that the difficulty arises because temperature and volume alone are not enough to specify the state of a solid. Instead, the complete state of a solid must include its atomic configuration—the time-averaged equilibrium positions of all atoms making up the material.

In other words, history dependence does not mean that thermodynamics fails. It means that some necessary state variables have been overlooked.

A common way to describe hysteresis is to say that a material's response, Y, depends on an input, X, in a way that forms a loop. But Shirai points out that this description is incomplete.