Scale-Invariant Work and State Functions: Connecting Kullback-Leibler and Bregman Divergences to Chemical Conversions and Fundamental Equations of State
- 1Department of Chemical and Process Engineering, Karlsruhe Institute of Technology, Kaiserstr. 12, Karlsruhe, Germany, 76131
- 2Institute of Beam Physics and Technology, Department of Physics and Mathematics, Hermann-von-Helmholtz Platz 1, Eggenstein-Leopoldshafen, Germany, 76344
Abstract
Thermodynamic work function vectors formalize the energetic principles governing scale-invariant process trajectories and equilibria via relative entropy representations along a directional path trajectory e p . While macroscopic processes are inherently irreversible, chemical transformations originate from microscopic reversibility along a positive time coordinate, where the non-dimensionalized work function vector maps directly to the statistical Kullback-Leibler divergence D KL. This divergence determines chemical conversion and equilibrium, altering the intensive properties that drive macroscopic boundary gradients. By transforming this statistical description into a geometric representation over Hessian manifolds, the formulation systematically maps non-ideal fluid phases and localized structural configurations, extending explicitly to long-range gravitational fields. This mapping establishes that the non-dimensionalized thermodynamic state space of the non-flat system is directly corrected by the asymmetric Bregman divergence D Ψ formulated with respect to a physically fixed ideal reference baseline. The resulting fundamental equation of state resolves fluid phase coexistence directly via the global convex envelope of the work function landscape. Through this formulation, the developed work and state functions unify the microscopic phase space and the macroscopic state space into a single, consistent conceptual framework mediated by the non-dimensionalized entropy vector s . To track non-equilibrium trajectories within long-range gradients, a secondary geometrical operator acts as a kinetic functional, yielding a dual-metric framework. This structure defines a total time-dependent state transformation vector l state(τ ) that isolates path-dependent transport kinetics from the static landscape. The transport dynamics enforce a monotonic reduction in the transient distance metric until the trajectory collapses onto the stationary identity defined via the Bregman divergence, validating the work function vector as a scale-invariant process property anchored across all spatial magnitudes. Consequently, an energy-time duality is established, reconciling microscopic reversibility along the forward trajectory with macroscopic thermodynamic irreversibility.
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BibTeX
@misc{grohmann2026scaleinvariant,
title = {{Scale-Invariant Work and State Functions: Connecting Kullback-Leibler and Bregman Divergences to Chemical Conversions and Fundamental Equations of State}},
author = {Steffen Grohmann},
howpublished = {ChemRxiv},
year = {2026},
month = aug,
doi = {10.26434/chemrxiv.15007507/v2},
url = {https://chemrxiv.org/doi/full/10.26434/chemrxiv.15007507/v2},
note = {CC BY},
}
RIS
TY - UNPB
TI - Scale-Invariant Work and State Functions: Connecting Kullback-Leibler and Bregman Divergences to Chemical Conversions and Fundamental Equations of State
AU - Steffen Grohmann
PY - 2026
DA - 2026/08/31
PB - ChemRxiv
DO - 10.26434/chemrxiv.15007507/v2
UR - https://chemrxiv.org/doi/full/10.26434/chemrxiv.15007507/v2
AB - Thermodynamic work function vectors formalize the energetic principles governing scale-invariant process trajectories and equilibria via relative entropy representations along a directional path trajectory e p . While macroscopic processes are inherently irreversible, chemical transformations originate from microscopic reversibility along a positive time coordinate, where the non-dimensionalized work function vector maps directly to the statistical Kullback-Leibler divergence D KL. This divergence determines chemical conversion and equilibrium, altering the intensive properties that drive macroscopic boundary gradients. By transforming this statistical description into a geometric representation over Hessian manifolds, the formulation systematically maps non-ideal fluid phases and localized structural configurations, extending explicitly to long-range gravitational fields. This mapping establishes that the non-dimensionalized thermodynamic state space of the non-flat system is directly corrected by the asymmetric Bregman divergence D Ψ formulated with respect to a physically fixed ideal reference baseline. The resulting fundamental equation of state resolves fluid phase coexistence directly via the global convex envelope of the work function landscape. Through this formulation, the developed work and state functions unify the microscopic phase space and the macroscopic state space into a single, consistent conceptual framework mediated by the non-dimensionalized entropy vector s . To track non-equilibrium trajectories within long-range gradients, a secondary geometrical operator acts as a kinetic functional, yielding a dual-metric framework. This structure defines a total time-dependent state transformation vector l state(τ ) that isolates path-dependent transport kinetics from the static landscape. The transport dynamics enforce a monotonic reduction in the transient distance metric until the trajectory collapses onto the stationary identity defined via the Bregman divergence, validating the work function vector as a scale-invariant process property anchored across all spatial magnitudes. Consequently, an energy-time duality is established, reconciling microscopic reversibility along the forward trajectory with macroscopic thermodynamic irreversibility.
ER -