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Michael K. -H. Kiessling

Publications and source records attributed to Michael K. -H. Kiessling.

At least 55 records · Page 3Linked to original sources
arXiv2012

The Hartree limit of Born's ensemble for the ground state of a bosonic atom or ion

The non-relativistic bosonic ground state is studied for quantum N-body systems with Coulomb interactions, modeling atoms or ions made of N "bosonic point electrons" bound to an atomic point nucleus of Z "electron" charges, treated in Born--Oppenheimer approximation. It is shown that the (negative) ground state energy E(Z,N) yields the monotonically growing function (E(l N,N) over N cubed). By adapting an argument of Hogreve, it is shown that its limit as N to infinity for l > l* is governed by Hartree theory, with the rescaled bosonic ground state wave function factoring into an infinite product of identical one-body wave functions determined by the Hartree equation. The proof resembles the construction of the thermodynamic mean-field limit of the classical ensembles with thermodynamically unstable interactions, except that here the ensemble is Born's, with the absolute square of the ground state wave function as ensemble probability density function, with the Fisher information functional in the variational principle for Born's ensemble playing the role of the negative of the Gibbs entropy functional in the free-energy variational principle for the classical petit-canonical configurational ensemble.

math-ph↗
arXiv2012

Order and Chaos in some Trigonometric Series: Curious Adventures of a Statistical Mechanic

This paper tells the story how a MAPLE-assisted quest for an interesting undergraduate problem in trigonometric series led some "amateurs" to the discovery that the one-parameter family of deterministic trigonometric series $\pzcS_p: t\mapsto \sum_{n\in\Nset}\sin(n^{-{p}}t)$, $p>1$, exhibits both order and apparent chaos, and how this has prompted some professionals to offer their expert insights. It is proved that $\pzcS_p(t) = α_p\rm{sign}(t)|t|^{1/{p}}+O(|t|^{1/{(p+1)}})\;\forall\;t\in\Rset$, with explicitly computed constant $α_p$. Experts' commentaries are reproduced stating the fluctuations of $\pzcS_p(t) - α_p{\rm{sign}}(t)|t|^{1/{p}}$ are presumably not Gaussian. Inspired by a central limit type theorem of Marc Kac, a well-motivated conjecture is formulated to the effect that the fluctuations of the $\lceil t^{1/(p+1)}\rceil$-th partial sum of $\pzcS_p(t)$, when properly scaled, do converge in distribution to a standard Gaussian when $t\to\infty$, though --- provided that $p$ is chosen so that the frequencies $\{n^{-p}\}_{n\in\Nset}$ are rationally linear independent; no conjecture has been forthcoming for rationally dependent $\{n^{-p}\}_{n\in\Nset}$. Moreover, following other experts' tip-offs, the interesting relationship of the asymptotics of $\pzcS_p(t)$ to properties of the Riemann $ζ$ function is exhibited using the Mellin transform.

math-ph↗
arXiv2011

Some uniqueness results for stationary solutions to the Maxwell-Born-Infeld field equations and their physical consequences

Uniqueness results are established for time-independent finite-energy electromagnetic fields which solve the nonlinear Maxwell--Born--Infeld equations in boundary-free space under the condition that either the charge or current density vanishes. In addition, it is also shown that the simpler Maxwell--Born equations admit at most a unique stationary finite-energy electromagnetic field solution, without the above condition. In these theories of electromagnetism, the following physical consequences emerge: source-free field solitons moving at speeds less than the vacuum speed of light $c$ do not exits; any purely electrostatic (resp. magnetostatic) field is the unique stationary electromagnetic field for the same current-density-free (resp. charge-density-free) sources. Our results put to rest some interesting speculations in the recent physics literature.

math-ph↗
arXiv2011

Convergent perturbative power series solution of the stationary Maxwell--Born--Infeld field equations with regular sources

The stationary Maxwell-Born-Infeld field equations of electromagnetism with integrable regular sources in a Hoelder space are solved using a perturbation series expansion in powers of Born's electromagnetic constant. The convergence of the power series for the fields is proved with the help of Banach algebra arguments and complex analysis. The finite radius of convergence depends on the norm of both, the Coulombfield generated by the charge density and the Amp`ere field generated by the current density. No symmetry is assumed.

math-ph↗
arXiv2010

Monotonicity of quantum ground state energies: Bosonic atoms and stars

The N-dependence of the non-relativistic bosonic ground state energy is studied for quantum N-body systems with either Coulomb or Newton interactions. The Coulomb systems are "bosonic atoms," with their nucleus fixed, and the Newton systems are "bosonic stars". In either case there exists some third order polynomial in N such that the ratio of the ground state energy to the respective polynomial grows monotonically in N. Some applications of these new monotonicity results are discussed.

math-ph↗
arXiv2010

Hard-Sphere Fluids with Chemical Self-Potentials

Existence, uniqueness and stability of solutions is studied for a set of nonlinear fixed point equations which define self-consistent hydrostatic equilibria of a classical continuum fluid that is confined inside a container and in contact with either a heat and a matter reservoir, or just a heat reservoir. The local thermodynamics is furnished by the statistical mechanics of a system of hard balls, in the approximation of Carnahan-Starling. The fluid's local chemical potential per particle at is the sum of the matter reservoir's contribution and a self contribution which is computed by convoluting the fluid density distribution with a van der Waals, a Yukawa, or a Newton kernel. We prove the existence of a grand canonical phase transition, and a petit canonical phase transition which is embedded in the former.

math-ph↗
arXiv2010

The Vlasov continuum limit for the classical microcanonical ensemble

For classical Hamiltonian N-body systems with mildly regular pair interaction potential it is shown that when N tends to infinity in a fixed bounded domain, with energy E scaling quadratically in N proportional to e, then Boltzmann's ergodic ensemble entropy S(N,E) has the asymptotic expansion S(N,E) = - N log N + s(e) N + o(N); here, the N log N term is combinatorial in origin and independent of the rescaled Hamiltonian while s(e) is the system-specific Boltzmann entropy per particle, i.e. -s(e) is the minimum of Boltzmann's H-function for a perfect gas of "energy" e subjected to a combination of externally and self-generated fields. It is also shown that any limit point of the n-point marginal ensemble measures is a linear convex superposition of n-fold products of the H-function-minimizing one-point functions. The proofs are direct, in the sense that (a) the map E to S(E) is studied rather than its inverse S to E(S); (b) no regularization of the microcanonical measure Dirac(E-H) is invoked, and (c) no detour via the canonical ensemble. The proofs hold irrespective of whether microcanonical and canonical ensembles are equivalent or not.

math-ph↗
arXiv2009

Misleading signposts along the de Broglie-Bohm road to quantum mechanics

Eighty years after de Broglie's, and a little more than half a century after Bohm's seminal papers, the de Broglie--Bohm theory (a.k.a. Bohmian mechanics), which is presumably the simplest theory which explains the orthodox quantum mechanics formalism, has reached an exemplary state of conceptual clarity and mathematical integrity. No other theory of quantum mechanics comes even close. Yet anyone curious enough to walk this road to quantum mechanics is soon being confused by many misleading signposts that have been put up, and not just by its detractors, but unfortunately enough also by some of its proponents. This paper outlines a road map to help navigate ones way.

physics.hist-ph↗
arXiv2009

A note on classical ground state energies

The pair-specific ground state energy of Newtonian N-body systems grows monotonically in N. This furnishes a whole family of simple new tests for minimality of putative ground state energies obtained through computer experiments. Inspection of several publically available lists of such computer-experimentally obtained putative ground state energies has yielded several dozen instances which failed (at least) one of these tests. Although the correct ground state energy is not revealed by this method, it does yield a better upper bound on it than the experimentally found value whenever the latter fails a monotonicity test. The surveyed N-body systems include in particular N point charges with 2- or 3-dimensional Coulomb pair interactions, placed either on the unit 2-sphere or on a 2-torus (a.k.a. Thomson, Fekete, or Riesz problems).

math-ph↗
arXiv2009

On the relativistic Vlasov-Poisson system

The Cauchy problem is revisited for the so-called relativistic Vlasov-Poisson system in the attractive case. Global existence and uniqueness of spherical classical solutions is proved under weaker assumptions than previously used. A new class of blowing up solutions is found when these conditions are violated. A new, non-gravitational physical vindication of the model which (unlike the gravitational one) is not restricted to weak fields, is also given.

math-ph↗
arXiv2008

On Ruelle's construction of the thermodynamic limit for the classical microcanonical entropy

In this note we make a very elementary technical observation to the effect that Ruelle's construction of the thermodynamic limit of the classical entropy density defined with a regularized microcanonical measure actually establishes the thermodynamic limit for the entropy density defined with the proper microcanonical measure. At this stage a key formula is still derived from the regularized measures. We also show that with only minor changes in the proof the regularization of the microcanonical measure is actually not needed at all.

math-ph↗
arXiv2008

Statistical Equilibrium Dynamics

The mean-field thermodynamic limit is studied for a class of isolated Newtonian N-body systems whose Hamiltonian admits several invariants of motion. It is shown that the macrostates of individual members of a statistical equilibrium ensemble are not necessarily themselves in a state of global thermal equilibrium in the strict sense. Yet they are always locally in thermodynamic equilibrium, and always global maximizers of the pertinent maximum entropy principle.

math-ph↗
arXiv2007

Quantum Abraham models with de Broglie-Bohm laws of quantum motion

We discuss a class of quantum Abraham models in which the N-particle spinor wave function of N electrons solves a Pauli respectively Schroedinger equation, featuring regularized classical electromagnetic potentials which solve the semi-relativistic Maxwell-Lorentz equations for regularized point charges, which move according to some de Broglie-Bohm law of quantum motion. Thus there is a feedback loop from the actual particle motions to the wave function. The electrons have a bare charge and positive bare mass different from their empirical charge and mass due to renormalization by the self-fields. In the classical limit the various models reduce to the Hamilton-Jacobi version of corresponding Abraham models of classical electron theory.

physics.atom-ph↗
arXiv2006

Nonperturbative calculation of Born-Infeld effects on the Schroedinger spectrum of the hydrogen atom

We present the first nonperturbative numerical calculations of the nonrelativistic hydrogen spectrum as predicted by first-quantized electrodynamics with nonlinear Maxwell-Born-Infeld field equations. We also show rigorous upper and lower bounds on the ground state. When judged against empirical data our results significantly restrict the range of viable values of the new electromagnetic constant which is introduced by the Born-Infeld theory. We assess Born's own proposal for the value of his constant.

math-ph↗
arXiv2005

The Vlasov limit and its fluctuations for a system of particles which interact by means of a wave field

In two recent publications [Commun. PDE, vol.22, p.307--335 (1997), Commun. Math. Phys., vol.203, p.1--19 (1999)], A. Komech, M. Kunze and H. Spohn studied the joint dynamics of a classical point particle and a wave type generalization of the Newtonian gravity potential, coupled in a regularized way. In the present paper the many-body dynamics of this model is studied. The Vlasov continuum limit is obtained in form equivalent to a weak law of large numbers. We also establish a central limit theorem for the fluctuations around this limit.

math-ph↗
arXiv2004

On the master equation approach to kinetic theory: linear and nonlinear Fokker--Planck equations

We discuss the relationship between kinetic equations of the Fokker-Planck type (two linear and one non-linear) and the Kolmogorov (a.k.a. master) equations of certain N-body diffusion processes, in the context of Kac's "propagation of chaos" limit. The linear Fokker-Planck equations are well-known, but here they are derived as a limit N->infty of a simple linear diffusion equation on (3N-C)-dimensional N-velocity spheres of radius sqrt(N) (with C=1 or 4 depending on whether the system conserves energy only or energy and momentum). In this case, a spectral gap separating the zero eigenvalue from the positive spectrum of the Laplacian remains as N->infty,so that the exponential approach to equilibrium of the master evolution is passed on to the limiting Fokker-Planck evolution in R^3. The non-linear Fokker-Planck equation is known as Landau's equation in the plasma physics literature. Its N-particle master equation, originally introduced (in the 1950s) by Balescu and Prigogine (BP), is studied here on the (3N-4)-dimensional N-velocity sphere. It is shown that the BP master equation represents a superposition of diffusion processes on certain two-dimensional sub-manifolds of R^{3N} determined by the conservation laws for two-particle collisions. The initial value problem for the BP master equation is proved to be well-posed and its solutions are shown to decay exponentially fast to equilibrium. However, the first non-zero eigenvalue of the BP operator is shown to vanish in the limit N->infty. This indicates that the exponentially fast approach to equilibrium may not be passed from the finite-N master equation on to Landau's nonlinear kinetic equation.

math-ph↗
arXiv2004

Electromagnetic field theory without divergence problems: 1. The Born Legacy

A fully consistent classical relativistic electrodynamics with spinless point charges is constructed. The classical evolution of the electromagnetic fields is governed by the nonlinear Maxwell--Born--Infeld field equations, the classical evolution of the point charges by a many-body Hamilton--Jacobi law of motion. The Pauli principle for bosons can be incorporated in the classical Hamilton--Jacobi formalism. The Cauchy problem is explained and illustrated with examples. The question of charge-free field solitons is addressed also and it is shown that if they exist, their peak field strengths must be enormous. The value The value of Born's constant is shown to be a subtle open issue.

math-ph↗