Recursive entropy in thermodynamics: establishing the statistical-physics basis of the zentropy approach

TL;DR

The paper addresses the underutilization of entropy's recursive property in thermodynamics by formalizing a rigorous multiscale framework (zentropy) that enables principled coarse-graining. It derives the recursive entropy form $S(K,X_1,\dots,X_n) = -\sum_k p_k \ln p_k + \sum_k p_k S_k$ and redefines the Helmholtz energy and partition function in terms of configuration quantities $F_k$ and $Z$, with $F_k = E_k - T S_k$ and $Z = \sum_k e^{-\beta F_k}$, recovering standard entropy when configurations are fully constrained. It clarifies the separation between configurational entropy $S_{\text{conf}}$ and intra-configurational entropy $S_k$, enabling exact multiscale descriptions at chosen levels and principled coarse-graining. The framework provides a rigorous bridge between microscopic and macroscopic behavior and is applicable to electronic, vibrational, and magnetic degrees of freedom in first-principles settings, supporting emergent phenomena analysis and computational efficiency.

Abstract

The recursive property of entropy is well known in the field of information theory; however, the concept is rarely used in the field of thermodynamics, despite being the field where the concept of entropy originated. This work shows that the equation for entropy used in the zentropy approach, which is an exact multiscale approach to thermodynamics, is a statement of the recursive property of entropy. Further, we clarify the meaning of entropy as the uncertainty arising from unconstrained degrees of freedom and separate configurational and intra-configurational contributions. Building on this, we derive the Helmholtz energy and the partition function, as used in zentropy. The resulting framework is exact for a chosen level of description and enables principled coarse-graining, thereby reducing computational complexity while preserving thermodynamic consistency. These results position zentropy as a rigorous bridge between microscopic and macroscopic behavior, facilitating quantitative predictions and the study of emergent phenomena.

Figures (1)

• Figure 1: Probability tree diagrams illustrating (a) the ungrouped and (b) grouped scenarios.

Theorems & Definitions (10)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1: Chain rule: two random variables
• proof
• Theorem 2: Chain rule: $n$ random variables
• proof
• Definition 4
• Theorem 3: Recursive property
• proof