Foundations, Volume 5, Issue 3 (September 2025) – 7 articles

The often used analytical representation of the Maxwell–Boltzmann classical speed distribution function (F) for elastic, indivisible particles assumes an infinite limit for the speed. Consequently, volume and the number of particles (n) extend to infinity: Both infinities contradict assumptions underlying this non-relativistic formulation. Finite average kinetic energy and temperature (T) result from normalization of F removing n: However, total energy (i.e., heat of the collection) remains infinite because n is infinite. This problem persists in recent adaptations. To better address real (finite) systems, wherein T depends on heat, we generalize this one-parameter distribution (F, cast in energy) by proposing a two-parameter gamma distribution function (F*) in energy which reduces to F at large n. Its expectation value of kT (k = Boltzmann’s constant) replicates F, whereas the shape factor depends on n and affects the averages, as expected for finite systems. We validate F* via a first-principle, molecular dynamics numerical model of energy and momentum conserving collisions for 26, 182, and 728 particles in three-dimensional physical space. Dimensionless calculations provide generally applicable results; a total of 10⁷ collisions suffice to represent an equilibrated collection. Our numerical results show that individual momentum conserving collisions in three-dimensions provide symmetrical speed distributions in all Cartesian directions. Thus, momentum and energy conserving collisions are the physical cause for equipartitioning of energy: Validity of this theorem for other systems depends on their specific motions. Our numerical results set upper limits on kinetic energy of individual particles; restrict the n particles to some finite volume; and lead to a formula in terms of n for conserving total energy when utilizing F* for convenience. Implications of our findings on matter under extreme conditions are briefly discussed. Full article

Knots are ubiquitous in polymers and biological macromolecules such as DNA and proteins, yet their behavior and functionality are still not sufficiently explored. Here we investigate the impact of Poiseuille flow on simple knots in flexible polymers placed in a quasi-rectangular micro-channel by systematically varying the flow strength for different chain lengths. Hydrodynamic interactions are accounted for by means of Multi-Particle Collision Dynamics (MPCD). We find that initially loosely localized knots in polymer coils typically tighten under shear to several segments beyond a certain body force threshold. At higher shear rates, intermittent transition from chain stretching to tumbling is observed which correlates with strong fluctuations in the knot size. Somewhat unexpectedly, our results indicate that the influence of channel width on tightening steadily increases with growing width even at equal mean shear rate $\overline{\dot{\gamma }}$. Full article

The aim of this work was to construct explicit expressions for the summation of Dirichlet Beta functions with odd arguments and Zeta functions with even arguments. In the established literature, this is typically done using Fourier series expansions or Bernoulli numbers and polynomials. Here, instead, we achieve our goal by employing tools from probability: specifically, we introduce a generalisation of a technique based on multiple integrals and the algebra of random variables. This also allows us to increase the number of nested integrals and Cauchy random variables involved. Another key contribution is that, by generalising the exponent of Cauchy random variables, we obtain an original proof of the reflection formulae for the Digamma and Trigamma functions. These probabilistic proofs crucially utilise the Mellin transform to compute the integrals needed to determine probability density functions. It is noteworthy that, while understanding the presented topic requires knowledge of the rules for calculating multiple integrals (Fubini’s Theorem) and the algebra of continuous random variables, these are concepts commonly acquired by second-year university students in STEM disciplines. Our study thus offers new perspectives on how the mathematical functions considered relate and shows the significant role of probabilistic methods in promoting comprehension of this research area, in a way accessible to a broad and non-specialist audience. Full article

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

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

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 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