Thermodynamic Framing Fundamentals

• Thermodynamic framing is the rigorous structuring of thermodynamic concepts, precisely defining quantities and laws across classical, quantum, and economic systems.
• It employs compositional, geometric, variational, and information-theoretic methods to reconstruct and unify traditional and modern thermodynamic principles.
• The framework resolves ambiguities in work, heat, and entropy, extending thermodynamic reasoning to nonequilibrium, inferential, and cross-disciplinary applications.

Thermodynamic framing is the explicit, often mathematically rigorous, choice or construction of a conceptual and formal structure within which thermodynamic concepts, quantities, and laws are precisely defined, interrelated, and applied. Historically rooted in macroscopic heat theory, contemporary thermodynamic framing encompasses a diverse array of abstract, geometric, statistical, operational, and information-theoretic approaches. These frameworks serve to unify, generalize, or reconstruct thermodynamic reasoning across equilibrium and nonequilibrium, classical and quantum, microscopic and macroscopic, and even economic and inferential domains. Recent developments emphasize compositional, geometric, variational, and information-based paradigms, providing formal clarity, extensibility, and cross-disciplinary reach.

1. Abstract and Categorical Foundations

Modern thermodynamic framing often begins from the abstraction of state spaces and entropy functions:

• Compositional Thermostatics defines a thermostatic system as a convex space of states $X$ together with a concave entropy function $S : X \to \overline{\mathbb{R}}$. This abstraction supports composition via entropy maximization subject to convex constraints, captured operadically as algebras of convex relations. The result is a universal, category-theoretic framework that subsumes classical, statistical, quantum, and generalized probabilistic thermodynamics. Classical ensembles, density matrices, and even heat baths are treated as particular algebras, with the Maxwell, Slutsky, and Legendre relations appearing as categorical consequences (Baez et al., 2021).
• Contact and Symplectic Geometries furnish a geometric framing: the thermodynamic phase space is realized as a contact manifold (for equilibrium) and its symplectization (for global structural analysis and dynamics), with thermodynamic potentials, conjugate variables, and state equations encoded as homogeneous Lagrangian submanifolds. Non-equilibrium processes are constructed as Hamiltonian flows constrained by the first and second laws, and port-thermodynamic systems allow for modular, power-preserving interconnections akin to network systems theory (Schaft et al., 2018, Baldiotti et al., 2016).

2. Axiomatic and Operational Reconstructions

Fundamental thermodynamic laws can be reconstructed or reframed axiomatically and operationally:

• Constructor Theory of Thermodynamics eliminates reliance on ensembles or coarse-graining by formulating the zeroth, first, and second laws as scale-independent statements about the possibility and impossibility of physical tasks (“constructors” for transformations). Adiabatic accessibility, work, and heat are precisely defined via the existence of transformation tasks and the side-effects on work or heat media. This “task-based” ontology clarifies the distinction between work and heat at all scales and uncovers an information-theoretic underpinning for the first law: work media are, in this setting, information media supporting distinguishable states, and conservation/interoperability of work is formalized as the compositionality of information carriers under additive conserved charges (Marletto, 2016).
• Fundamental Equation–First Framing dispenses with “heat” and “work” as primitive constructs, instead deriving all process and irreversibility statements from the fundamental thermodynamic equation (e.g., $dU = TdS - PdV + \sum \mu_n dN_n$) and associated state balances. Different definitions of "heat" and "work" are seen as convenient but not fundamental; all thermodynamic structure flows from a differential-geometric narrative centered on the state function $U$ and entropy $S$, with process invariants encoding the symmetry between system and environment (Anacleto, 2 Jul 2025).

3. Information-Theoretic, Statistical, and Variational Framings

A major thrust in contemporary thermodynamic framing is the recasting or deriving of thermodynamic structure from statistical and information-theoretic primitives:

• Thermodynamics Reconstructed from Information Theory starts from chosen observables (coarse-grainings) and a reference measure, yielding "information volume" for each observational cell. An augmented entropy functional, incorporating both Shannon and log-volume terms, is maximized under constraints using a minimum-relative-entropy (KL divergence) principle, from which temperature, chemical potential, and information-theoretic "pressure" arise as Lagrange multipliers. The resulting formalism admits a full Legendre structure, a first-law-like differential system, and a nonequilibrium extension wherein entropy production is rendered as path-space KL divergence, with heat as the component of dissipation not accounted for by the system-entropy change. Hidden dissipation and information flow are captured as projection-induced KL gaps in coarse-grained dynamics (Tsuruyama, 31 Dec 2025).
• Thermoinformational State Construction provides a constructive route from empirical data to thermodynamic structure. First, a generative energy function is inferred by fitting a Boltzmann-type model to observed data, inducing a data-driven energy axis. An inverse maximum-entropy problem is then solved, yielding an optimized (possibly non-Shannon) entropy functional matched to the empirical statistics. Macroscopic variables, including internal energy, entropy, and a thermoinformational temperature $T^{-1} = dS/dU$, are thereby assigned in a system-specific, data-consistent manner. An explicit H-theorem ensures the Lyapunov property of entropy under gradient ascent dynamics, and the framework extends naturally to non-equilibrium, multimodal distributions far beyond classical MaxEnt surrogates (Domenikos et al., 27 Apr 2026).
• Unified Nonequilibrium Statistical Framing utilizes the effective number of accessible states $N_{\text{eff}} = \exp(S)$ as a geometric-mean measure of entropy, defines thermodynamic distance via KL divergence, and obtains the equilibrium manifold as the entropy-maximizing, stationary solution of a constrained Markov process or more general dynamical generator. The total entropy production is decomposed into adiabatic (“housekeeping”) and nonadiabatic parts, the latter providing a Lyapunov functional for the approach to steady state. The dynamic structure admits fluctuation symmetries (Gallavotti–Cohen), thermodynamic uncertainty relations, and speed bounds on current fluctuations, all unified by variational and information-geometric principles (Taye, 10 Sep 2025).

4. Non-equilibrium, Local, and Kinetic Framings

Advanced thermodynamic framing must account for spatial, nonlocal, and rate-dependent features:

• Global Thermodynamics for Heat Conduction defines a global temperature (a weighted average over spatially inhomogeneous profiles) for systems with stationary heat currents and extends the fundamental relation—e.g., $dF = -S\,dT_g - p\,dV$—to global variables, accurate up to $O(\Delta T^2)$. Variational principles and generalized Maxwell constructions are derived, permitting non-equilibrium phase coexistence analysis, and predicting interface temperature shifts inaccessible to local equilibrium theory (Nakagawa et al., 2019).
• Thermokinetics extends the state space to include vectorial and orientational variables (distribution moments, angles), making the total macroscopic energy $\Theta$ a function not only of extensive variables but also of their spatial and orientational distributions. The fundamental differential

$S : X \to \overline{\mathbb{R}}$0

incorporates generalized potentials, forces, and torques, enabling the treatment of heterogeneous, finite-rate, and dissipative processes without restricting to quasi-static or reversible idealizations. Classical thermostatics is the spatially homogeneous limit (Etkin, 2014).

5. Quantum, Relativistic, and Coherently Driven Systems

Quantum and relativistic generalizations necessitate refined thermodynamic framing:

• Quantum Reference Frame and Thermodynamics realize exact or approximate unitary operations on a system while strictly enforcing conservation laws by introducing explicit quantum reference frames ("batteries") that absorb back-action in non-commuting observables (e.g., energy, angular momentum). This permits explicit micro-physical realization of the first and second laws, the construction of thermodynamic tasks with compatible battery degrees of freedom, and reversibly saturating quantum resource inequalities up to $S : X \to \overline{\mathbb{R}}$1 (Popescu et al., 2018).
• Relativistic Dissipative Fluid Frames clarify that the choice of flow-frame (Eckart: particle, Landau-Lifshitz: energy, Jüttner: thermometer) is not a matter of convention but has physical content with respect to the stability and causality of the resulting non-equilibrium theory. Thermodynamic framing here determines the nature of irreversible fluxes and their coupling to dissipative processes; only the Jüttner (thermometer) frame yields symmetric hyperbolicity and robust stability under general constitutive laws (Ván et al., 2013).
• Coherently Driven Quantum Systems require thermodynamic frameworks that respect the explicit accessibility of output modes. When output light is treated as an accessible battery, entropy production splits into coherent (work) and noisy (heat) parts, yielding second law inequalities strictly tighter than conventional treatments. The output field’s noise necessarily exceeds the input's, forbidding net coherence extraction from purely coherent inputs without added entropy sources (Schrauwen et al., 13 May 2025).

6. Applications, Extensions, and Cross-disciplinary Framings

Thermodynamic framing is not confined to physical systems:

• Thermal Macroeconomics applies the axiomatic structure of thermodynamics to aggregate economic phenomena, identifying entropy with aggregate utility, temperature with the inverse marginal utility of money, and deriving all classical macroeconomic laws (pricing, Slutsky symmetry, Le Chatelier–Samuelson principle) from extensive, convexity, and uniqueness principles analogous to thermodynamic systems. Equilibrium, irreversibility, and market adjustments map directly onto entropy maximization and canonical flow/force structures (Chater et al., 2024).
• Thermodynamic Structure of Inference reinterprets statistical estimation as a thermodynamic process, with sample size and estimator variance forming a state manifold. Shannon entropy plays the role of "uncertainty," an integrating factor organizes a first-law-type balance, and information-theoretic efficiency is naturally capped by irreducible representation noise, producing analogues of the laws of thermodynamics within the theory of inferential physics (Wong, 26 Feb 2026).

7. Synthesis and Theoretical Significance

The contemporary landscape of thermodynamic framing replaces historical reliance on ambiguous, macroscopic constructs with a network of rigorous, compositional, and highly generalized mathematical formalisms. These yield:

• Universality: Covering quantum, classical, macroscopic, microscopic, equilibrium, nonequilibrium, information-theoretic, and economic settings.
• Structural clarity: All major constructs (work, heat, entropy, irreversibility, information) arise as precisely defined relations, objects, or invariants under specified framings.
• Extensibility: Formalisms permit the natural integration of, e.g., constraints, control, feedback, and system composition, without ad hoc amendments.
• Resolution of controversies: Apparent paradoxes and ambiguities in the definition or role of work, heat, and entropy are resolved by explicit choices of framing and invariance structure.
• Cross-disciplinary reach: The same categorical, geometric, or information-theoretic tools apply far beyond traditional thermodynamics, unifying concepts across diverse domains.

The technical apparatus of thermodynamic framing outlined in these research threads establishes a high-precision, cross-cutting foundation for modern thermodynamics and its applications (Baez et al., 2021, Baldiotti et al., 2016, Tsuruyama, 31 Dec 2025, Domenikos et al., 27 Apr 2026, Taye, 10 Sep 2025, Ván et al., 2013, 2402.22605, Schrauwen et al., 13 May 2025, Chater et al., 2024).