The natural and Dyson orbitals are studied for small helium drops comprising 5 to 20 helium atoms interacting via a soft two-body gaussian potential. The wave functions of these drops have been obtained in the hyperspherical cluster model (HCM) which provides a correct description of the single-particle behaviour at large separations from the system. The natural orbitals are obtained from diagonalization of the nonlocal one-body density matrix, while Dyson orbitals are constructed by direct overlap of the wave functions of two drops differing by one boson. This overlap converges with increasing basis of the HCM. The shapes and occupancies of the natural orbitals as well as their link to Dyson overlaps and evolution with increasing number of atoms are discussed. Both natural and Dyson orbitals can be used to represent the density of the system. However, the natural orbitals representation is demonstrated to be superior. With increasing boson numbers the difference between Dyson and natural orbitals becomes less prominent and it is expected to disappear in infinitely large systems of identical bosons.
The construction of Dyson spheres, megastructures designed to capture the total radiative output of stars, can be one of the most compelling techno-signature scenarios for advanced extraterrestrial civilizations. By considering equilibrium temperatures, we investigate the luminosities and fluxes of Dyson spheres built around two promising classes of host stars: white dwarfs and red M-dwarfs. Using radiative balance arguments and representative stellar parameters, we compute the temperature-radius relationship for full energy interception and place these hypothetical structures on the Hertzsprung-Russell (H-R) diagram to assess their observational signatures. Our results show that Dyson spheres around white dwarfs produce cooler and fainter blackbody emissions, peaking in the near- to mid-infrared, while those around M-dwarfs radiate more strongly but at longer wavelengths. In both cases, the equilibrium temperature decreases as R_ D^-1/2, while the total luminosity and observed bolometric flux remain fixed by the stellar output. These findings highlight the astrophysical suitability of low-luminosity stars as Dyson sphere hosts and provide practical constraints for future techno-si
The goal of this short paper is to investigate the regularity of the solutions of the Dyson equation. In the work of Bertucci and al. [3, 4, 5], a new notion of solutions for the Dyson equation has been introduced using the viscosity solutions theory and they proved a regularization of solutions in L^{$\infty$} . According to the work of Biane [6] we should expect a regularization in C^{1/3} of solutions (and not better). In the spirit of the works of Bertucci and al., we shall prove using PDE methods that for almost all time t \ge 0 the solution is as expected in C^{1/3} . Our approach allows us to extend this result in addition to a drift term in the Dyson equation for which the semi explicit solutions is not necessarily known. We shall also give an application of this result, proving that the solutions of the periodic Dyson equation converge in long-time toward the uniform distribution on the circle in L^{$\infty$} norm which was an open question in [5].
The recent works \cite{EFMV2024} and \cite{JOP2023} have studied the spectral properties of the Dyson model in the absence of an external field. This paper is a continuation of \cite{EFMV2024} and aims to bridge the gap in the literature by investigating the Dyson model in a field.\\ In this paper, we prove that, for high temperatures or strong magnetic fields, there exists a non-negative, integrable (with respect to the unique half-line Gibbs measure) eigenfunction of the transfer operator for the Dyson model if $α\in(\frac 3 2,2]$. However, unlike in the zero-magnetic-field case, this eigenfunction is not continuous.
In this paper, we study the unitary Dyson Brownian motion through a partial differential equation approach recently introduced for the real Dyson case. The main difference with the real Dyson case is that the spectrum is now on the circle and not on the real line, which leads to particular attention to comparison principles. First we recall why the system of particles which are the eigenvalues of unitary Dyson Brownian motion is well posed thanks to a containment function. Then we proved that the primitive of the limit spectral measure of the unitary Dyson Brownian motion is the unique solution to a viscosity equation obtained by primitive the Dyson equation on the circle. Finally, we study some properties of solutions of Dyson's equation on the circle. We prove a L $\infty$ regularization. We also look at the long time behaviour in law of a solution through a study of the so-called free entropy of the system. We conclude by discussing the uniform convergence towards the uniform measure on the circle of a solution of the Dyson equation.
In his 1856 Adams Prize essay, James Clark Maxwell demonstrated that Saturn's rings cannot be comprised of a uniform rigid body. This is a consequence of the two-body gravitational interaction between a ring and planet resulting in instability. Similarly, it is also known that a so-called Dyson sphere encompassing a single star would be unstable due to Newton's shell theorem. A surprising finding is reported here that both a ring and a sphere (shell) can be stable in the restricted three-body problem. First, if two primary masses are considered in orbit about their common centre of mass, a large, uniform, infinitesimal ring enclosing the smaller of the masses can in principle be stable under certain conditions. Similarly, a Dyson sphere can, be stable, if the sphere encloses the smaller of the two primary masses, again under certain conditions. These findings extend Maxwell's results on the dynamics of rings and have an interesting bearing on so-called Ringworlds and Dyson spheres from fiction. Moreover, the existence of passively stable orbits for such large-scale structures may have implications for so-called techno-signatures in search for extra-terrestrial intelligence studies.
In this paper, we introduce a new stochastic process of $N$ interacting particles on the line that evolve via Dyson Brownian motion (DBM) with Dyson's index $β> 0$ and undergo simultaneous resetting to their initial positions at a constant rate $r$. We call this process the resetting Dyson Brownian motion (RDBM) -- in short the $β$-RDBM. For $β= 1,2,4$, the positions of the particles in the RDBM can be interpreted as the eigenvalues of a random matrix ensemble where the entries of an $N x N$ Gaussian matrix evolve as simultaneously resetting Brownian motions (with rate $r$) in the presence or absence of a harmonic trap. For $r=0$ and in the presence of a harmonic trap, this system reaches an equilibrium Gibbs-Boltzmann state of the so called Dyson log-gas. However, the stochastic resetting drives the system at long time to a nonequilibrium stationary state (NESS). We compute exactly the joint distribution of the positions of the particles in this NESS for all $β>0$ and calculate several observables for large $N$: the average density profile of the gas, the extreme value statistics, the spacing between two consecutive particles and the full counting statistics. We show that a
We use the well established duality of topological gravity to a double scaled matrix model with the Airy spectral curve to define what we refer to as topological gravity with arbitrary Dyson index $\upbeta$ ($\upbeta$ topological gravity). On the matrix model side this is an interpolation in the Dyson index between the Wigner-Dyson universality classes, on the gravity side it can be thought of as interpolating between orientable and unorientable manifolds in the gravitational path integral, opening up the possibility to study moduli space volumes of manifolds ``in between''. Using the perturbative loop equations we study correlation functions of this theory and prove several structural properties, having clear implications for the generalised moduli space volumes. Additionally we give a geometric interpretation of these properties using the generalisation to arbitrary Dyson index of the recently found Mirzakhani-like recursion for unorientable surfaces. Using these properties, we investigate whether $\upbeta$-topological gravity is quantum chaotic in the sense of the Bohigas-Giannoni-Schmit conjecture. Along the way we answer this question for the symplectic Wigner-Dyson class, not
In 1942 Mann solved a famous problem, the $α+β$ conjecture, about the lower bound of the Shnirel'man density of sums of sets of positive integers. In 1945, Dyson generalized Mann's theorem and obtained a lower bound for the Shnirel'man density of rank $r$ sumsets. His proof introduced the Dyson transform, an important tool in additive number theory. This paper explains the background of Dyson's work, gives Dyson's proof of his theorem, and includes several applications of the Dyson transform, such as Kneser's inequality for sums of finite subsets of an arbitrary additive abelian group.
We study large fluctuations of the current in a Dyson gas, a 1D system of particles interacting through a logarithmic potential and subjected to random noise. We adapt the macroscopic fluctuation theory to the Dyson gas and derive two coupled partial differential equations describing the evolution of the density and momentum. These equations are nonlinear and non-local, and the `boundary' conditions are mixed: some at the initial time and others at the final time. If the initial condition can fluctuate (annealed setting), this boundary-value problem is tractable. We compute the cumulant generating function encoding all the cumulants of the current.
A Dyson ring is a hypothetical megastructure, that a very advanced civilization would build around a star to harness more of its energy. Satellite propagation is of high priority in such a vast world where distances could very well be measured in astronomical units. We analyze the ring's parameters and stability and propose a stable helical orbit around the Dyson ring influenced by the gravity of the Dyson ring and the Sun. Taking theoretically explainable values for all parameters, we describe our approach to finding this orbit and present the successful simulation of a satellite's flight in this path.
At the 1987 Ramanujan Centenary meeting Dyson asked for a coherent group-theoretical structure for Ramanujan's mock theta functions analogous to Hecke's theory of modular forms. Many of Ramanujan's mock theta functions can be written in terms of $R(ζ_p,q)$, where $R(z,q)$ is the two-variable generating function of Dyson's rank function and $ζ_p$ is a primitive $p$-th root of unity. In his lost notebook Ramanujan gives the $5$-dissection of $R(ζ_5,q)$. This result is related to Dyson's famous rank conjecture which was proved by Atkin and Swinnerton-Dyer. In 2016 the first author showed that there is an analogous result for the $p$-dissection of $R(ζ_p,q)$ when $p$ is any prime greater than $3$, by extending work of Bringmann and Ono, and Ahlgren and Treneer. It was also shown how the group $Γ_1(p)$ acts on the elements of the $p$-dissection of $R(ζ_p,q)$. We extend this to the group $Γ_0(p)$, thus revealing new and surprising symmetries for Dyson's rank function.
We give series solutions to single insertion place propagator-type systems of Dyson--Schwinger equations using binary tubings of rooted trees. These solutions are combinatorially transparent in the sense that each tubing has a straightforward contribution. The Dyson--Schwinger equations solved here are more general than those previously solved by chord diagram techniques, including systems and non-integer values of the insertion parameter $s$. We remark on interesting combinatorial connections and properties.
The ordering between Wigner--Yanase--Dyson function and logarithmic mean is known. Also bounds for logarithmic mean are known. In this paper, we give two reverse inequalities for Wigner--Yanase--Dyson function and logarithmic mean. We also compare the obtained results with the known bounds of the logarithmic mean. Finally, we give operator inequalities based on the obtained results.
We define the Dyson diffusion process on a curved smooth closed contour in the plane and derive the Fokker-Planck equation for probability density. Its stationary solution is shown to be the Boltzmann weight for the logarithmic gas confined on the contour.
The crank is a partition statistic requested by Dyson in 1944 in order to combinatorially prove a Ramanujan congruence of Euler's partition function $p(n)$. In this paper, we provide connections between Dyson's crank and unimodal compositions. Somewhat unrelated, we give a combinatorial proof of a new truncated Euler pentagonal number theorem due to Xia and Zhao.
Functional epistemology is about ways to access functional objects by using varieties of methods and procedures. Not all such means are equally capable of reproducing these functions in the desired consistency and resolution. Dyson's argument against the perturbative expansion of quantum field theoretic terms, in a radical form (never pursued by Dyson), is an example of epistemology taken as ontology.
The Dyson rank of an integer partition is the difference between its largest part and the number of parts it contains. Using Fine-Dyson symmetry, we give formulas for the number of partitions of n with rank larger than n/2, and we prove identities for counts of partitions with large rank in fixed residue classes.
Dyson's model in infinite dimensions is a system of Brownian particles that interact via a logarithmic potential with an inverse temperature of $ β= 2$. The stochastic process can be represented by the solution to an infinite-dimensional stochastic differential equation. The associated unlabeled dynamics (diffusion process) are given by the Dirichlet form with the sine$ _2$ point process as a reference measure. In a previous study, we proved that Dyson's model in infinite dimensions is irreducible, but left the ergodicity of the unlabeled dynamics as an open problem. In this paper, we prove that the unlabeled dynamics of Dyson's model in infinite dimensions are ergodic.