Recursive entropy in thermodynamics: the partition function under coarse-graining and the basis of the zentropy approach

Abstract

The recursive property of entropy is well known in information theory; however, the concept is underappreciated in thermodynamics, the field in which entropy originated. The zentropy approach is built on this idea, and it has emerged as a useful framework for describing thermodynamic systems across multiple scales, yet its statistical-physics foundation has not been fully articulated. In this work, we derive the partition function under coarse-graining taking the recursive form of entropy as primary. Using probability tree diagrams, we first show that the recursive property separates entropy into a configurational term and an intra-configurational term, and that it reproduces the Gibbs entropy whenever each microstate belongs to only one coarse-grained configuration. Maximizing entropy in the recursive form yields a partition function in which each configuration's Helmholtz energy replaces a microstate energy, recovering Rushbrooke's 1940 result for temperature-dependent energy levels while maintaining a clear distinction between the physical and statistical aspects of statistical mechanics. We argue that the recursive form should be regarded as the primary expression for entropy: the Gibbs form is its special case in which every configuration is a microstate, and recursivity is itself one of the axioms that uniquely determine the Gibbs entropy. Two representative applications, magnetic materials and liquids, illustrate how the framework enables physically meaningful coarse-graining and captures emergent behavior arising from probability redistribution among configurations. Together, these results place the zentropy approach on a maximum-entropy foundation and provide an intuitive understanding of the coarse-graining of systems.