Thermodynamics Reconstructed from Information Theory:An Axiomatic Framework via Information-Volume Constraints and Path-Space KL Divergence
Tatsuaki Tsuruyama
TL;DR
This work reframes thermodynamics as an information-theoretic theory built from two primitives: an observational map $M$ and a reference measure $\tau$, which together define an information volume and lead to a Legendre structure where $T$, $\mu$, and $P$ are dual variables to energy, particle number, and information volume via $\frac{1}{T}=\beta$, $-\frac{\mu}{T}=\alpha$, and $\frac{P}{T}=\pi$. Nonequilibrium dissipation is captured by the path-space KL divergence $\Sigma_{0,T}=D_{\mathrm{KL}}(\mathbb{P}\|\mathbb{P}^{\mathrm R})$, with a gauge-invariant total dissipation that can be decomposed into observable and hidden contributions through projection-induced gaps, and observational entropy $S_{M,\tau}$ serving as the system entropy. Heat is reconstructed as an information-theoretic representation, $Q = T(\Sigma_{0,T}-\Delta S_{\mathrm{sys}})$, which reduces to conventional cross-entropy forms in isothermal settings and relates to Landauer-type arguments, all without assuming local detailed balance or explicit bath models. By unifying static Min-KL inference with dynamic path-KL dissipation and embedding correlation free energy through mutual information, the framework provides a model-independent, gauge-aware foundation for both equilibrium and nonequilibrium thermodynamics, with potential applications to biochemical and complex systems.
Abstract
We develop an axiomatic reconstruction of thermodynamics based entirely on two primitive components: a description of what aspects of a system are observed and a reference measure that encodes the underlying descriptive convention. These ingredients define an "information volume" for each observational cell. By incorporating the logarithm of this volume as an additional constraint in a minimum-relative-entropy inference scheme, temperature, chemical potential, and pressure arise as conjugate variables of a single information-theoretic functional. This leads to a Legendre-type structure and a first-law-like relation in which pressure corresponds to information volume rather than geometric volume. For nonequilibrium dynamics, entropy production is characterized through the relative-entropy asymmetry between forward and time-reversed stochastic evolutions. A decomposition using observational entropy then separates total dissipation into system and environment contributions. Heat is defined as the part of dissipation not accounted for by the system-entropy change, yielding a representation that does not rely on local detailed balance or a specific bath model. We further show that the difference between joint and partially observed dissipation equals the average of conditional relative entropies, providing a unified interpretation of hidden dissipation and information-flow terms as projection-induced gaps.