Abstract: Using the concept of quantum wave probability, combined with the identity principle, we can derive the Boltzmann distribution, Fermi distribution, and Bose distribution. Different distributions correspond to different conditions. The Boltzmann condition corresponds to the Boltzmann distribution. The Fermi condition corresponds to the Fermi distribution. The Bose condition corresponds to the Bose distribution. This demonstrates that the foundation of these three statistical distributions is quantum wave probability, all originating from quantum mechanics. The Boltzmann distribution is also an independent quantum distribution and is not simply a sparse limit of the Fermi or Bose distributions. The essence of the Boltzmann distribution is a uniform distribution. The Fermi and Bose distributions are deviations from the uniform distribution. Boltzmann entropy based on the quantum wave probability can resolve the Gibbs paradox. Using this new approach, we can also derive the results of eigenstate thermalization. The equilibrium state is the state in which all eigenstates have the same temperatures. We need to rethink the fundamentals of statistical mechanics.

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This paper advances Coherence Thermodynamics for understanding systems composed purely of information and coherence. It derives five laws of coherence thermodynamics and applies them to two case studies. Three canonical modes of coherent informational systems are developed: Standing State, Computation Crucible, and Holographic Projection. Each mode has its own dynamics and natural units, with thermodynamic coherence defined as the reciprocal of the entropy–temperature product. Within this theory, reasoning is proposed to emerge as an ordered, work‑performing process that locally resists entropy and generates coherent structure across universal features.

Abstract: We present a fully controlled thermodynamic study of the two-dimensional dipolar $Q$-state clock model on small square lattices with free boundaries, combining exhaustive state enumeration with noise-free evaluation of canonical observables. We resolve the complete energy spectra and degeneracies $\{E_n,c_n\}$ for the Ising case ($Q=2$) on lattices of size $L=3,4,5$, and for clock symmetries $Q=4,6,8$ on a $3\times3$ lattice, tracking how the competition between exchange and long-range dipolar interactions reorganizes the low-energy manifold as the ratio $\alpha = D/J$ is varied. Beyond a finite-size characterization, we identify several qualitatively new thermodynamic signatures induced solely by dipolar anisotropy. First, we demonstrate that ground-state level crossings generated by long-range interactions appear as exact zeros of the specific heat in the limit $C(T \rightarrow 0,\alpha)$, establishing an unambiguous correspondence between microscopic spectral rearrangements and macroscopic caloric response. Second, we show that the shape of the associated Schottky-like anomalies encodes detailed information about the degeneracy structure of the competing low-energy states: odd lattices ($L = 3,5$) display strongly asymmetric peaks due to unbalanced multiplicities, whereas the even lattice ($L = 4$) exhibits three critical values of $\alpha$ accompanied by nearly symmetric anomalies, reflecting paired degeneracies and revealing lattice parity as a key organizing principle. Third, we uncover a symmetry-driven crossover with increasing $Q$: while the $Q=2$ and $Q=4$ models retain sharp dipolar-induced critical points and pronounced low-temperature structure, for $Q \ge 6$ the energy landscape becomes sufficiently smooth to suppress ground-state crossings altogether, yielding purely thermal specific-heat maxima. Altogether, our results provide a unified, size- and symmetry-resolved picture of how long-range anisotropy, lattice parity, and discrete rotational symmetry shape the thermodynamics of mesoscopic magnetic systems. We show that dipolar interactions alone are sufficient to generate nontrivial critical-like caloric behavior in clusters as small as $3\times3$, establishing exact finite-size benchmarks directly relevant for van der Waals nanomagnets, artificial spin-ice arrays, and dipolar-coupled nanomagnetic structures.

Abstract: As remarked by Boltzmann, the Second Law of Thermodynamics is notable for the fact that it is readily proved using elementary statistical arguments, but becomes harder and harder to verify the more precise the microscopic description of a system. In this article, we investigate one particular realization of the 2nd Law, namely Joule heating in a wire under electrical bias. We analyze the production of entropy in an exactly solvable model of a quantum wire wherein the conserved flow of entropy under unitary quantum evolution is taken into account using an exact formula for the entropy current of a system of independent quantum particles. In this exact microscopic description of the quantum dynamics, the entropy production due to Joule heating does not arise automatically. Instead, we show that the expected entropy production is realized in the limit of a large number of local measurements by a series of floating thermoelectric probes along the length of the wire, which inject entropy into the system as a result of the information obtained via their continuous measurements of the system. The decoherence resulting from inelastic processes introduced by the local measurements is essential to the phenomenon of entropy production due to Joule heating, and would be expected to arise due to inelastic scattering in real systems of interacting particles.

Abstract: We develop an informational extension of spacetime thermodynamics in which local entropy production is coupled to spacetime curvature within an effective covariant framework. Spacetime is modeled as a continuum limit of finite-capacity information registers, giving rise to a coarse-grained entropy field whose gradients define an informational flux. Within a nonminimally coupled scalar–tensor formulation, the resulting field equations imply that the local divergence of this flux is sourced by the Ricci scalar, establishing a direct relation between curvature and entropy production. The corresponding integral form links cumulative entropy generation to the integrated spacetime curvature over a causal region. In stationary limits, the framework reproduces the Bekenstein–Hawking entropy of horizons, while in homogeneous expanding cosmologies it yields monotonic entropy growth consistent with the observed arrow of time. The construction remains compatible with unitarity at the microscopic level and with holographic entropy bounds in the stationary limit. Numerical solutions in flat FLRW backgrounds are used as consistency checks of the coupled evolution equations and confirm the expected curvature–entropy behavior across cosmological epochs. Overall, the results provide a thermodynamically consistent interpretation of curvature as a geometric source of irreversible information flow, without modifying the underlying gravitational field equations.

Abstract: Generative diffusion models have emerged as a powerful class of models in machine learning, yet a unified theoretical understanding of their operation is still developing. This paper provides an integrated perspective on generative diffusion by connecting the information-theoretic, dynamical, and thermodynamic aspects. We demonstrate that the rate of conditional entropy production during generation (i.e. the generative bandwidth) is directly governed by the expected divergence of the score function's vector field. This divergence, in turn, is linked to the branching of trajectories and generative bifurcations, which we characterize as symmetry-breaking phase transitions in the energy landscape. Beyond ensemble averages, we demonstrate that symmetry-breaking decisions are revealed by peaks in the variance of pathwise conditional entropy, capturing heterogeneity in how individual trajectories resolve uncertainty. Together, these results establish generative diffusion as a process of controlled, noise-induced symmetry breaking, in which the score function acts as a dynamic nonlinear filter that regulates both the rate and variability of information flow from noise to data.

Abstract: The Kelvin formulation of the second law of thermodynamics permits the following generalization: If the efficiency of a heat engine approaches unity, then the rejected work vanishes. This generalization allows deriving the behavior of a Carnot cycle near absolute zero of temperature. Also, the unattainability of absolute zero can be shown. In turn, these results allow deriving the behavior of the entropy near absolute zero, as has already been shown previously. The point of view is the phenomenological, macroscopic, and non-statistical one of classical thermodynamics.

Abstract: Self-organizing open systems, sustained by continuous fluxes between sources and sinks, convert stochastic motion into structured efficiency, yet a first-principles explanation of this transformation remains elusive. We derive the time-dependent Average Action Efficiency (AAE)—defined as events per total action—from a stochastic–dissipative least-action principle formulated within the Onsager–Machlup and Maximum Caliber path-ensemble frameworks. The resulting Lyapunov-type identity links the monotonic rise of AAE to the variance of action and to the rate of noise reduction, delineating growth, saturation, and decay regimes. Self-organization emerges from a reciprocal feedback between dynamics and structure: the stochastic dynamics concentrates trajectories around low-action paths, while the resulting structure, through the evolving feedback precision parameter β(t), modulates subsequent dynamics. This self-reinforcing coupling drives a monotonic increase of the dimensionless Average Action Efficiency α_t =η/⟨ I⟩_t , providing a quantitative measure of organizational growth. In the deterministic limit, the theory recovers Hamilton’s Principle. The increase of AAE corresponds to a decrease in path entropy, yielding an information-theoretic complement to the Maximum Entropy Production and Prigogine–Onsager variational formalisms. The framework applies to open, stochastic, feedback-driven systems that satisfy explicit regularity conditions. In Part II, agent-based ant-foraging simulations confirm sigmoidal AAE growth, plateau formation, and robustness under perturbations. Because empirical AAE requires only event counts and integrated action, it offers a lightweight metric and design rule for feedback-controlled self-organization across physical, chemical, biological, and active-matter systems.

Abstract: Entropy appears in physics in many forms—thermal, quantum, informational, gravitational—yet its conceptual foundations remain disparate. We propose a unified definition of entropy grounded in global physical constraints. A constraint set C determines the admissible microstate region Γ(C), and the entropy is defined as S(C) = k_BlnVol[Γ(C)]. This constraint–volume formulation applies uniformly to classical and quantum systems, to internal and external degrees of freedom, and to finite or continuous state spaces, without invoking coarse-graining, ensembles, or subjective information. Local interactions generically weaken global constraints such as coherence, correlations, gradients, and entanglement structure. We prove a structural Second Law: whenever constraints decay under dynamical evolution, C(t +∆t) ⊆ C(t), the entropy must increase. This mechanism explains thermodynamic irreversibility, decoherence, thermalization, and hydrodynamic mixing as manifestations of constraint erosion, while identifying integrable and symmetry-protected systems as the exceptional cases in which constraints persist. The framework clarifies how macroscopic entropy can grow evenwhenmicroscopic dynamics are reversible, and why time itself is not a form of entropy. Classical thermodynamic entropy, quantum von Neumann entropy, and black hole entropy all emerge as special cases of the same structural principle.

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The historical context of Szilard’s thought experiment is considered: Maxwell’s demon, Brownian motion, and naturalization of Maxwell’s demon by Smoluchowski. After that, the discussions of Szilard’s thought experiment in the second half of the 20th century are described: the penetration of information into statistical mechanics, the works of Brillouin, Bennett (Landauer’s principle) and Zurek. The second part of the paper is devoted to the criticism of thermodynamics of information. The critique of Earman and Norton is extended by considering levels of organization. Also, the problem of coordination with respect to information is discussed.

Abstract: In this paper, during the simulation analysis of a Curie phase transition shielding model, it was found that the cumulative mechanical work performed on a moving ferromagnet after one Curie phase transition cycle of the model is non-zero. Further analysis based on Curie phase transition theory and the thermodynamics of magnetic media indicates that after completing one phase transition cycle, the cumulative magnetization work on the Curie phase transition ferromagnet is zero, the internal energy of both the Curie ferromagnet and the entire model is conserved, yet mechanical work remains separately, which contradicts the conservation of energy. This discovery suggests that there are exceptions to the conservation of energy, and it is not absolute.

Abstract: This paper separately analyzes the thermodynamic processes of adiabatic superconducting phase transition in superconductors and adiabatic Curie phase transition in ferromagnets in magnetic fields. Through analysis, it is concluded that for an object undergoing a phase transition cycle, when the accumulated magnetization work is zero, the overall internal energy of the adiabatic phase transition system is conserved. However, in the model, the accumulated mechanical work done on the permanent magnet is not zero, which leads to non-conservation of energy in the model, contradicting the law of conservation of energy. This indicates that the law of conservation of energy also has exceptions and is not absolute.

Abstract: This paper introduces a Thermodynamic Unified Field Theory (UFT) where prime-enforced symmetry constraints emerge from helical recoils in photrino dynamics, unifying phase behaviors and transport phenomena through a covariant fugacity-Hessian equation. By deriving the viscous stress tensor from entropy maximization without pa-rameters, the framework resolves Navier-Stokes limitations (e.g., infinite speeds, non-Fourier transport) and reproduces empirical phase diagrams for substances like he-lium, water, and neon via prime-locked gears. We demonstrate how primes arise from triad indivisibility, leading to rational direction cosines that enforce shell uniformity and curvature floors. Applications to catalysis, superfluidity, and non-equilibrium systems highlight UFT's potential as a parameter-free TOE candidate, with time and gravity as emergent distortions in the flux sea.

Abstract: The problem of coordination for thermodynamic entropy as a physical quantity is expressed in two related questions: 1) What counts as a measurement of entropy? 2) What is entropy? These issues are considered in this paper for thermodynamic properties of pure substances in classical thermodynamics. The conceptual model to define entropy in the second law of thermodynamics cannot be used directly to produce an ideal experiment related to real measurements. Thus, the solution of the problem of coordination for entropy is based on the tight integration of entropy with other thermodynamic properties in the formalism of classical thermodynamics. Therefore, the solution of the problem of coordination for entropy is related to the simultaneous solution of the problem of coordination for other thermodynamic quantities, such as heat capacity, internal energy, enthalpy, and the Gibbs energy.

Abstract: In his paper 'The Impossible Process: Thermodynamic Reversibility', John Norton criticized the concept of reversible processes in classical thermodynamics and suggested that they should be considered approximations. In mathematics, however, the term approximation is related to the approximately equal sign, which is inappropriate for the mathematical formalism of thermodynamics. This paper examines the relationship between the formalism of thermodynamics, conceptual models and the world, and then discusses Norton's proposal from this point of view. In conclusion, a proposal by mathematician Zorich for the reversible process of heat transfer between two bodies with different temperatures is described.

Abstract: This paper introduces a novel entropy formulation — multiplicative entropy — defined as the product of energy values across all units in a quantized homogeneous invariant network. Unlike traditional statistical entropy, this approach explicitly tracks irreversible energy redistribution pathways, offering an analytic and path-dependent description of entropy growth. Logarithmic transformation recovers classical entropy forms, while preserving temporal directionality and quantum-scale resolution. The model enables precise simulation of thermodynamic processes, supporting the development of Analytic Quantum Thermodynamics as a new framework for understanding entropy-driven dynamics. Crucially, the analytical multiplicative entropy formula proposed here responds to Planck’s long-sought vision of an entropy expression in analytic form, enabling the resolution of entropy to match the high resolution of quantum processes.

Abstract: Serrin’s works provided a new perspective on classical thermodynamics through his statements of the first law and the accumulation function, and of the second law and the accumulation theorem, as well as the subsequent result by Huilgol that the work done in a thermal cycle implies an inequality where the important temperatures of the thermal cycle and an integral similar to that of Clausius appear. Based on these pioneering works, explicit forms of the accumulation function have been derived for the Otto, Diesel, Stirling and Ericsson cycles. In this article, a straightforward alternative derivation is presented to obtain the inequality for the work done in a cycle, following the approaches of Serrin and Huilgol. The derivation of the accumulation function for the ideal air-standard Brayton cycle is provided, where the temperature constraints of the adiabatic compression and expansion processes under which it operates are analyzed. Finally, a practical example is presented to illustrate the application of Serrin’s accumulation function to the ideal air-standard Brayton cycle.

Abstract: Jos Uffink rejected the existence of the arrow of time in classical thermodynamics, and his views turned out to be influential in the philosophy of physics. However, this position omits the connection between classical thermodynamics and continuum mechanics and is based on a simplistic view of equilibrium states. In this article, this issue is discussed with an example of a temperature field; we examine the relationship between the Fourier heat equation and classical thermodynamics in the 19th century. With this example the relationship of the Clausius inequality to time and transport equations of continuum mechanics is clarified. The question is raised whether it makes sense to carry out the axiomatization of classical thermodynamics without continuum mechanics.

Abstract: The material–immaterial divide remains one of the deepest conceptual tensions in both science and philosophy. This paper introduces the Dual Entropy Uncertainty Frame-work (DEUF)—a quantitative, information-based model that unifies material structure and immaterial information under a single uncertainty law, SE × OE ≥ C. Structural Entropy (SE) and Optical Entropy (OE) represent complementary Shannon-type measures of configurational and informational uncertainty. Building upon the Hei-senberg Uncertainty Principle and Shannon Information Theory, DEUF provides a testable framework linking quantum duality, biological evolution, and Platonic ideal-ism. Empirical implications are outlined in terms of compensatory dynamics between structure and information. The framework resolves long-standing philosophical and scientific challenges and suggests that matter and information form a single, entropi-cally balanced continuum. The DEUF formulation opens a new path for studying en-tropy-driven self-organization and the informational basis of natural laws.

Abstract: Quantitative Geometrical Thermodynamics (QGT) exploits the entropic Lagrangian-Hamiltonian canonical equations of state as applied to entities obeying the holographic principle and exhibiting Shannon information, the creation of which measures the (validly defined) “entropic purpose” of the system. QGT provides a physical description for what we might consider the true ‘atoms’ of physical science, and has also recently enabled a number of significant advances: including accounting ab initio for the chirality of DNA and the stability of Buckminsterfullerene; the size of the alpha particle (and other nuclear entities) and the lifetime of the free neutron; and the shape, structure and stability of the Milky Way galaxy. All these entities, ranging in size over more than 38 orders of magnitude, can each be considered to be an ‘atom’; in particular, the size of the alpha is calculated from QGT by assuming that the alpha is a “unitary entity” (that is, than which exists no simpler). The surprising conclusion is that clearly compound entities may also be physically treated as unitary (“uncuttable”) according to a principle of scale relativity, where a characteristic size for such an entity must be specified. Since QGT is entropic, and is therefore described using a logarithmic metric (involving hyperbolic space), it is not surprising that the length scale must be specified in order to account for unitary properties and for an entity to be appropriately considered an ‘atom’. The contribution to physics made by QGT is reviewed in the context of the related work of others.