Foundations

6 pages, 198 KiB

Open AccessOpinion

Relation Between Diffusion Equations and Boundary Conditions in Bounded Systems

by Fabio Sattin and Dominique Franck Escande

Foundations 2025, 5(3), 26; https://doi.org/10.3390/foundations5030026 (registering DOI) - 31 Jul 2025

Viewed by 62

Differential equations need boundary conditions (BCs) for their solution. It is widely acknowledged that differential equations and BCs are representative of independent physical processes, and no correlations between them are required. Two recent studies by Hilhorst, Chung et al. argue instead that, in the specific case of diffusion equations (DEs) in bounded systems, BCs are uniquely constrained by the form of transport coefficients. In this paper, we revisit how DEs emerge as fluid limits out of a picture of stochastic transport. We point out their limits of validity and argue that, in most physical systems, BCs and DEs are actually uncorrelated by virtue of the failure of diffusive approximation near the system’s boundaries. When, instead, the diffusive approximation holds everywhere, we show that the correct chain of reasoning goes in the direction opposite to that conjectured by Hilhorst and Chung: it is the choice of the BCs that determines the form of the DE in the surroundings of the boundary. Full article

(This article belongs to the Section Physical Sciences)

13 pages, 900 KiB

Open AccessHypothesis

Beyond Classical Multipoles: The Magnetic Metapole as an Extended Field Source

by Angelo De Santis and Roberto Dini

Foundations 2025, 5(3), 25; https://doi.org/10.3390/foundations5030025 - 14 Jul 2025

Viewed by 189

Abstract

We introduce the concept of the magnetic metapole—a theoretical extension of classical multipole theory involving a fractional j pole count (related to the harmonic degree n as j = 2ⁿ). Defined by a scalar potential with colatitudinal dependence and no radial variation, the metapole yields a magnetic field that decays as 1/r and is oriented along spherical surfaces. Unlike classical multipoles, the metapole cannot be described as a point source; rather, it corresponds to an extended or filamentary magnetic distribution as derived from Maxwell’s equations. We demonstrate that pairs of oppositely oriented metapoles (up/down) can, at large distances, produce magnetic fields resembling those of classical monopoles. A regularized formulation of the potential resolves singularities for the potential and the field. When applied in a bounded region, it yields finite field energy, enabling practical modeling applications. We propose that the metapole can serve as a conceptual and computational framework for representing large-scale magnetic field structures particularly where standard dipole-based models fall short. This construct may have utility in both geophysical and astrophysical contexts, and it provides a new tool for equivalent source modeling and magnetic field decomposition. Full article

(This article belongs to the Section Physical Sciences)

► Show Figures

Figure 1

19 pages, 342 KiB

Open AccessArticle

Fisher Information in Helmholtz–Boltzmann Thermodynamics of Mechanical Systems

by Marco Favretti

Foundations 2025, 5(3), 24; https://doi.org/10.3390/foundations5030024 - 4 Jul 2025

Viewed by 257

Abstract

In this paper, we review Helmholtz–Boltzmann thermodynamics for mechanical systems depending on parameters, and we compute the Fisher information matrix for the associated probability density. The divergence of Fisher information has been used as a signal for the existence of phase transitions in finite systems even in the absence of a thermodynamic limit. We investigate through examples if qualitative changes in the dynamic of mechanical systems described by Helmholtz–Boltzmann thermodynamic formalism can be detected using Fisher information. Full article

(This article belongs to the Section Physical Sciences)

► Show Figures

Figure 1

22 pages, 323 KiB

Open AccessArticle

Mathematical Formalism and Physical Models for Generative Artificial Intelligence

by Zeqian Chen

Foundations 2025, 5(3), 23; https://doi.org/10.3390/foundations5030023 - 24 Jun 2025

Viewed by 332

Abstract

This paper presents a mathematical formalism for generative artificial intelligence (GAI). Our starting point is an observation that a “histories” approach to physical systems agrees with the compositional nature of deep neural networks. Mathematically, we define a GAI system as a family of sequential joint probabilities associated with input texts and temporal sequences of tokens (as physical event histories). From a physical perspective on modern chips, we then construct physical models realizing GAI systems as open quantum systems. Finally, as an illustration, we construct physical models realizing large language models based on a transformer architecture as open quantum systems in the Fock space over the Hilbert space of tokens. Our physical models underlie the transformer architecture for large language models. Full article

(This article belongs to the Section Physical Sciences)

Journal Menu

Journal Browser

Foundations, Volume 5, Issue 3 (September 2025) – 4 articles

Further Information

Guidelines

MDPI Initiatives

Follow MDPI