Virial Expansion – A Brief Introduction

The existence of interactions among the particles of a system is a common situation in statistical physics, for instance in real gases or weak liquid solutions. On the one hand, these interactions limit the range of applicability of the ideal equation of state. On the other hand, they give rise to a whole new class of exciting phenomena, such as thermodynamic instability and metastability, phase transition, phase separation, and self-assembled micro-structuring. The virial expansion is the simplest and most general theory addressing these effects, and constitutes the foundation for more advanced and specific models, such as Ornstein-Zernike Integral Schemes, Path Integral Statistical Physics, and Phase Field Theories. In these notes, we introduce the virial expansion using elementary mathematical methods. The reader needs to be only familiar with the contents of firstand second-year university courses. A basic understanding of classical mechanics and thermodynamics is recommended, including the equation of state of the ideal gas. In the first section, we briefly review the equation of state of the ideal gas. We discuss the problems encountered when trying to extend that result to systems of interacting particles, and elucidate the general strategy of the virial expansion. Then, we introduce the second virial coefficient and derive the corresponding second-order virial equation of state in two different ways, using the virial theorem and the cluster expansion. Finally, we present an example based on the square well potential, and connect the virial expansion to the familiar van der Waals equation of state for real gases, and thus connect microscopic interaction parameters to macroscopic equation of state, one of the key points in condensed matter. Starting point: equation of state Matter can be found in different thermodynamic phases. For instance, H2O in a closed box is a liquid under normal conditions, but can be converted