Open research questions in Statistical Mechanics and Entropy

What the literature leaves open

• The lack of a constructive derivation of thermodynamic extensivity from microscopic conditions on the pair potential. The need for a framework to understand the emergence of extensivity in complex systems.

  Entropy additivity from exponential decay of correlations: a coarse-grained operator approach · 2026

• The paper does not claim to derive gravity or identify Fisher geometry with spacetime geometry. The construction is limited to a five-channel Fisher simplex with two binary rapidity fibers over a ternary context base.

  Bounded Simplex Fields and Unbounded Response Charts: A Sigma-Model Toy Test of Partition Pullbacks, Fisher Kinetics, and Rapidity Blowup · 2026 · DOI

• The paper suggests that future research should focus on the interpretation of partition variables in physical systems. The paper demonstrates the use of bounded asymmetry coordinate and unbounded response charts.

  Bounded Simplex Fields and Unbounded Response Charts: A Sigma-Model Toy Test of Partition Pullbacks, Fisher Kinetics, and Rapidity Blowup · 2026 · DOI

• The convergence properties of the IPFP are not well understood. There is a need for exponential convergence guarantees for the IPFP.

  Exponential Convergence of General Iterative Proportional Fitting Procedures · 2026 · DOI

• Future research could focus on studying the properties of other stochastic processes. Future research could focus on developing new methods for analyzing and modeling complex systems. Future research could focus on applying the results to real-world problems.

  Entropies of Cox–Ingersoll–Ross and Bessel processes as functions of time and of related parameters · 2026 · DOI

• There is a lack of understanding of the long-time asymptotic behavior of entropy measures associated with stochastic processes. There is a need to derive sufficient conditions for the existence of these entropy measures. There is a need to analyze the limiting behavior of these entropy measures as the noncentrality parameter approaches zero.

  Entropies of Cox–Ingersoll–Ross and Bessel processes as functions of time and of related parameters · 2026 · DOI

• In AI for Science, physics-informed losses are increasingly used to train learned compressors for scientific data, but their rate-distortion implications remain poorly understood.

  A Geometric Lens on Physics-Aligned Data Compression · 2026

• The risk of tautology when applying the extraction protocol to a single known trajectory. The need to distinguish two distinct uses of the protocol. The challenge of identifying genuine, falsifiable results rather than relabeling.

  Two-Face Series: Effective Potentials on the Binary Partition Manifold · 2026 · DOI

• The microscopic origin of black-hole entropy is still not well understood. The neutral Schwarzschild case is harder to analyze than the supersymmetric case.

  Schwarzschild Entropy from Quantum-Singularity Cohomology · 2026 · DOI

• Classical physics has limitations in explaining information transfer and the behavior of matter at the quantum level. The need for a new framework that can explain the relationship between energy, mass, information, and entropy.

  (2).Hamzah Quadruple Equivalence Principle (Energy-Mass-Information-Entropy). \(E=\frac{m\cdot c^{2}}{\mathcal{S}_{\text{mat}}+\epsilon _{\text{floor}}}\) :\(E=\frac{\mathcal{I}\cdot \Omega _{H}^{2}}{\mathcal{S}_{\text{mat}}+\epsilon _{\text{floor}}}\) \(\mathcal{I}=m\cdot \frac{c^{2}}{\Omega _{H}^{2}}\implies m\cdot c^{2}=\mathcal{I}\cdot \Omega _{H}^{2}\) · 2026 · DOI

• Future research should focus on further developing the HIP framework and its implications for our understanding of the universe. The paper suggests exploring the practical applications of HIP in the fields of quantum computing and information theory.

  (10).Hamzah Quadruple Equivalence Principle (Energy-Mass-Information-Entropy). \(E=\frac{m\cdot c^{2}}{\mathcal{S}_{\text{mat}}+\epsilon _{\text{floor}}}\) :\(E=\frac{\mathcal{I}\cdot \Omega _{H}^{2}}{\mathcal{S}_{\text{mat}}+\epsilon _{\text{floor}}}\) \(\mathcal{I}=m\cdot \frac{c^{2}}{\Omega _{H}^{2}}\implies m\cdot c^{2}=\mathcal{I}\cdot \Omega _{H}^{2}\) · 2026 · DOI

• The paper identifies the need for a new perspective on the relationship between energy, mass, information, and entropy. The paper identifies the potential of the Hamzah Quadruple Equivalence Principle in explaining various phenomena.

  (8).Hamzah Quadruple Equivalence Principle (Energy-Mass-Information-Entropy). \(E=\frac{m\cdot c^{2}}{\mathcal{S}_{\text{mat}}+\epsilon _{\text{floor}}}\) :\(E=\frac{\mathcal{I}\cdot \Omega _{H}^{2}}{\mathcal{S}_{\text{mat}}+\epsilon _{\text{floor}}}\) \(\mathcal{I}=m\cdot \frac{c^{2}}{\Omega _{H}^{2}}\implies m\cdot c^{2}=\mathcal{I}\cdot \Omega _{H}^{2}\) · 2026 · DOI

• The lack of understanding of the informational nature of the universe.

  (9).Hamzah Quadruple Equivalence Principle (Energy-Mass-Information-Entropy). \(E=\frac{m\cdot c^{2}}{\mathcal{S}_{\text{mat}}+\epsilon _{\text{floor}}}\) :\(E=\frac{\mathcal{I}\cdot \Omega _{H}^{2}}{\mathcal{S}_{\text{mat}}+\epsilon _{\text{floor}}}\) \(\mathcal{I}=m\cdot \frac{c^{2}}{\Omega _{H}^{2}}\implies m\cdot c^{2}=\mathcal{I}\cdot \Omega _{H}^{2}\) · 2026 · DOI

• The paper suggests future research directions, including the development of new technologies based on the theory of Hamzah Information Physics. The paper suggests further experimentation to prove the existence of the HamzahXCell.

  (5).Hamzah Quadruple Equivalence Principle (Energy-Mass-Information-Entropy). \(E=\frac{m\cdot c^{2}}{\mathcal{S}_{\text{mat}}+\epsilon _{\text{floor}}}\) :\(E=\frac{\mathcal{I}\cdot \Omega _{H}^{2}}{\mathcal{S}_{\text{mat}}+\epsilon _{\text{floor}}}\) \(\mathcal{I}=m\cdot \frac{c^{2}}{\Omega _{H}^{2}}\implies m\cdot c^{2}=\mathcal{I}\cdot \Omega _{H}^{2}\) · 2026 · DOI

• The method is not limited to the ground states, but can also be applied in dynamical cases. The method can also be applied in higher dimensional systems. The computation of the Renyi entropy is not limited to the second order, but can be extended to the general m-th order.

  A universal approach to Renyi entropy of multiple disjoint intervals · 2026 · DOI

• The localized characterization of the quantum Tsallis relative entropy is an unsolved problem. The existing quantum Tsallis relative entropy does not have localized characteristics.

  Quantum Tsallis Relative Entropy with Localization Characteristics · 2026 · DOI

• Classical physics has reached stalemates and unphysical infinities. The material world requires an extrasystem software root to render and debug its divergences.

  (1).Hamzah Quadruple Equivalence Principle (Energy-Mass-Information-Entropy). \(E=\frac{m\cdot c^{2}}{\mathcal{S}_{\text{mat}}+\epsilon _{\text{floor}}}\) :\(E=\frac{\mathcal{I}\cdot \Omega _{H}^{2}}{\mathcal{S}_{\text{mat}}+\epsilon _{\text{floor}}}\) \(\mathcal{I}=m\cdot \frac{c^{2}}{\Omega _{H}^{2}}\implies m\cdot c^{2}=\mathcal{I}\cdot \Omega _{H}^{2}\) · 2026 · DOI

• Development of a complete dynamical theory based on the constraint-based formalism. Application of the framework to specific physical theories and systems.

  Constraint Operators as a Unified Structural Framework in Physics. · 2026 · DOI

• To apply the approach in dynamical cases and higher dimensional systems. To extend the method to the general m-th order. To study entanglement entropy in various systems using the approach.

  A universal approach to Renyi entropy of multiple disjoint intervals · 2026 · DOI

• Further study of the implications of the results for thermodynamic systems with long-range interactions. Development of new thermodynamic models and simulations based on the approach. Exploration of the relationship between the coarse-graining operator approach and other methods for deriving entropy additivity.

  Entropy additivity from exponential decay of correlations: a coarse-grained operator approach · 2026

• To build a library of extracted numerator classes from solved fields across domains. To search for repeated normalized potential shapes across independent systems. To fit V on part of a family and predict held-out trajectories.

  Two-Face Series: Effective Potentials on the Binary Partition Manifold · 2026 · DOI

• Further study of the properties of quantum-singularity cohomology and its relation to black-hole entropy. Application of the results to various astrophysical and cosmological contexts.

  Schwarzschild Entropy from Quantum-Singularity Cohomology · 2026 · DOI

• Experimental verification of the Hamzah Quadruple Equivalence Principle. Further development of the principle and its applications. Discussion of the implications of the principle for our understanding of the universe.

  (2).Hamzah Quadruple Equivalence Principle (Energy-Mass-Information-Entropy). \(E=\frac{m\cdot c^{2}}{\mathcal{S}_{\text{mat}}+\epsilon _{\text{floor}}}\) :\(E=\frac{\mathcal{I}\cdot \Omega _{H}^{2}}{\mathcal{S}_{\text{mat}}+\epsilon _{\text{floor}}}\) \(\mathcal{I}=m\cdot \frac{c^{2}}{\Omega _{H}^{2}}\implies m\cdot c^{2}=\mathcal{I}\cdot \Omega _{H}^{2}\) · 2026 · DOI

• Experimental verification of the model. Development of a more realistic model that takes into account imperfect vacuums. Exploration of the potential applications of the model.

  (3).Hamzah Quadruple Equivalence Principle (Energy-Mass-Information-Entropy). \(E=\frac{m\cdot c^{2}}{\mathcal{S}_{\text{mat}}+\epsilon _{\text{floor}}}\) :\(E=\frac{\mathcal{I}\cdot \Omega _{H}^{2}}{\mathcal{S}_{\text{mat}}+\epsilon _{\text{floor}}}\) \(\mathcal{I}=m\cdot \frac{c^{2}}{\Omega _{H}^{2}}\implies m\cdot c^{2}=\mathcal{I}\cdot \Omega _{H}^{2}\) · 2026 · DOI

• Further analysis of the computational drag and its implications. Experimental verification of the predictions made by Hamza Information Physics (HIP).

  (9).Hamzah Quadruple Equivalence Principle (Energy-Mass-Information-Entropy). \(E=\frac{m\cdot c^{2}}{\mathcal{S}_{\text{mat}}+\epsilon _{\text{floor}}}\) :\(E=\frac{\mathcal{I}\cdot \Omega _{H}^{2}}{\mathcal{S}_{\text{mat}}+\epsilon _{\text{floor}}}\) \(\mathcal{I}=m\cdot \frac{c^{2}}{\Omega _{H}^{2}}\implies m\cdot c^{2}=\mathcal{I}\cdot \Omega _{H}^{2}\) · 2026 · DOI

Most-cited papers in Statistical Mechanics and Entropy

• Extropy: Complementary Dual of Entropy · Statistical Science · 2015 · 294 citations
• An algebra of observables for de Sitter space · Journal of High Energy Physics · 2023 · 255 citations
• Jeffreys' prior is asymptotically least favorable under entropy risk · Journal of Statistical Planning and Inference · 1994 · 228 citations
• On cumulative entropies · Journal of Statistical Planning and Inference · 2009 · 226 citations
• Spatial Entropy · Geographical Analysis · 1974 · 221 citations
• Statistical Continuum Theories · American Journal of Physics · 1968 · 204 citations
• Introduction to special section on Origins and Properties of Kappa Distributions: Statistical Background and Properties of Kappa Distributions in Space Plasmas · Journal of Geophysical Research Space Physics · 2015 · 202 citations
• On the dynamic cumulative residual entropy · Journal of Statistical Planning and Inference · 2006 · 196 citations
• The Meaning of Kurtosis: Darlington Reexamined · The American Statistician · 1986 · 189 citations
• Toward a Redefinition of Density · Environment and Behavior · 1975 · 186 citations

Most recent work

• How Occam's razor guides human decision-making · bioRxiv · 2026
• Gauge invariance and hyperforce correlation theory for equilibrium fluid mixtures · The Journal of Chemical Physics · 2026
• rKAN: Rational Kolmogorov-Arnold networks · Neural Networks · 2026
• Non-Hermitian Green’s function theory with N -body interactions: The coupled-cluster similarity transformation · Physical Review Research · 2026
• New modified cosmology from a new generalized entropy · Physics Letters B · 2026
• Nonlinear asymptotes of dual information dimension of complex networks · Chaos Solitons & Fractals · 2026
• Approximating optimal μ-point distributions over a generalized sphere for Riesz s-energy using a gradient descent algorithm · Information Sciences · 2026
• First excursion probability sensitivity in stochastic linear dynamics by means of multidomain line sampling · Reliability Engineering & System Safety · 2026
• Generalized informational functionals and new monotone measures of statistical complexity · Communications in Nonlinear Science and Numerical Simulation · 2026
• Symbolic regression and differentiable fits in beyond the standard model physics · Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences · 2026

Related topics in Physics and Astronomy