State Variables and Constraints in Thermodynamics of Solids and Their Implications

TL;DR

This paper argues that the mean atom positions and internal energy form a complete set of thermodynamic state variables for solids, challenging the conventional view of a limited number of variables.

Contribution

It introduces a framework where all atom positions and internal energy serve as state variables, providing a full description of thermodynamics in solids.

Findings

Mean atom positions and internal energy constitute a complete set of state variables.

An infinite number of atom positions do not conflict with thermodynamics.

Specific heat can be derived from the fundamental relation of equilibrium.

Abstract

There is a common view in thermodynamics that the behavior of a macroscopic system can be described by only a few state variables. Although this is true for many cases, it is unclear whether it is meaningful to ask how many state variables are acceptable. This is indeed a problem when solids are investigated within the framework of thermodynamics, which is scarcely discussed in textbooks. The present study gives an answer to this question: the mean values of all the atom positions of a given solid together with the internal energy constitute a commensurate set of state variables (thermodynamic coordinates, TCs). The argument begins with constructing consistent definitions of equilibrium and TCs. TCs are created by the constraints which characterize the system under consideration. The values of TCs are uniquely determined in equilibrium and the mutual relationships between them constitute the fundamental relation of equilibrium (FRE). Specific heat can be deduced from the FRE. Therefore, the TCs of a solid must be to give a full expression of the specific heat in the entire range of temperature, from which the above conclusion is deduced. Contrary to the conventional view, an infinite number of the atom positions and their microscopic characters do not conflict with the principles of thermodynamics. The most important requirement for TCs to meet is the uniqueness of their values in equilibrium against random motions of the constituent particles. This conclusion is compatible with the principle of information theory that the information needed to determine the probability distribution of states is the expectation values of statistical variables. A few working examples of TCs in solids are given.