Michael K.-H. Kiessling

Rotational Symmetry of Solutions of Some Nonlinear Problems in Statistical Mechanics and in Geometry

The method of moving planes is used to establish a weak set of conditions under which the nonlinear equation −Δu(x)=V(|x|)eu(x),x∈ℝ2 admits only rotationally symmetric solutions. Additional structural invariance properties of the equation then yield another set of conditions which are not originally covered by the moving plane technique. For instance, nonmonotonicV can be accommodated. Results for −Δu(y)=V(y)eu(y)−c, withy∈S2, are obtained as well. A nontrivial example of broken symmetry is also constructed. These equations arise in the context of extremization problems, but no extremization arguments are employed. This is of some interest in cases where the extremizing problem is neither manifestly convex nor monotone under symmetric decreasing rearrangements. The results answer in part some conjectures raised in the literature. Applications to logarithmically interacting particle systems and geometry are emphasized.

On the equilibrium statistical mechanics of isothermal classical self-gravitating matter

The canonical ensemble is investigated for classical self-gravitating matter in a finite containerΛ[d]⊂ℝd,d=3 and 2. Starting with modified gravitational interactions (smoothed-out singularity), it is proven by explicit construction that, in thew*-topology, the canonical equilibrium measure converges to a superposition of Dirac measures when the limit of exact Newtonian gravitational interactions between classical point particles is taken. The consequences of this result for more realistic classical systems are evaluated, and the existence of a gravitational phase transition is proven. The results are discussed with view toward applications in astrophysics and space science. Some attention is paid also to the problem of founding thermodynamics by means of statistical mechanics.

A NOTE ON THE EIGENVALUE DENSITY OF RANDOM MATRICES

Abstract:The distribution of eigenvalues of N×N random matrices in the limit N→∞ is the solution to a variational principle that determines the ground state energy of a confined fluid of classical unit charges. This fact is a consequence of a more general theorem, proven here, in the statistical mechanics of unstable interactions. Our result establishes the eigenvalue density of some ensembles of random matrices which were not covered by previous theorems.

SELF-SIMILAR GRAVITATIONAL COLLAPSE IN STELLAR DYNAMICS

Self-similar gravitational collapse of a globular cluster is studied using the nonlinear kinetic standard model of stellar dynamics consisting of the Fokker-Planck-Vlasov transport equation coupled self-consistently to Poisson’s equation for the Newtonian gravitational potential. It is shown rigorously that any locally integrable self-similar solution to these equations must approach a mass density profile , , in the final 2a r(r, t) / r a p 3 stage of the collapse. The discrepancy between the exact value and previous results in the range a p 32 ! obtained from the orbit-averaged approximation to the kinetic model raises some questions about the a ! 2.5 validity of this popular approximation. Subject headings: celestial mechanics, stellar dynamics — galaxies: star clusters — globular clusters: general

Phase transitions in gravitating systems and the formation of condensed objects

Abstract The thermal equilibrium of a self-gravitating classical fluid with local equation of state corresponding to a system of hard spheres is studied numerically. When the volume effectively occupied by the particles is much smaller than the accessible volume, a phase transition occurs at which the system can shuttle between a quasi-uniform state and one in which a highly condensed nucleus is immersed in a dilute atmosphere. Under isothermal contact conditions, the two states have different energies; under isoenergetic conditions, they have different temperatures. The isothermal transition bridges a region of negative specific heat in the family of isoenergetic systems. The phase transitions mark nonlinear fluid stability thresholds. These can differ by orders of magnitude from the traditional linear ones, i.e. the gravothermal catastrophe and Jeans instability, which only mark the stability limits of thermally metastable regions. It is discussed how phase transitions may give the proper onset criteria for the formation of condensed objects from the size of planetoids up to stars.

Electromagnetic Field Theory Without Divergence Problems 1. The Born Legacy

Borns quest for the elusive divergence problem-free quantum theory of electromagnetism led to the important discovery of the nonlinear Maxwell–Born–Infeld equations for the classical electromagnetic fields, the sources of which are classical point charges in motion. The law of motion for these point charges has however been missing, because the Lorentz self-force in the relativistic Newtonian (formal) law of motion is ill-defined in magnitude and direction. In the present paper it is shown that a relativistic Hamilton–Jacobi type law of point charge motion can be consistently coupled with the nonlinear Maxwell–Born–Infeld field equations to obtain a well-defined relativistic classical electrodynamics with point charges. Curiously, while the point charges are spinless, the Pauli principle for bosons can be incorporated. Borns reasoning for calculating the value of his aether constant is re-assessed and found to be inconclusive.

Dissipative stationary plasmas: Kinetic modeling, Bennett’s pinch and generalizations

The structure of the self‐consistent electromagnetic fields, E, B, and one‐particle distribution functions, fs, of a stationary dissipative unbounded S‐species plasma, which satisfy a system of Maxwell–Vlasov–Fokker–Planck–Boltzmann equations with velocity‐independent (effective) dissipation coefficients and applied constant electric field E, is studied. It is proven that when the plasma is invariant along the direction of E, then (a) the fs are given by S drifting Maxwell–Boltzmann distributions, with densities satisfying self‐consistent Poisson–Boltzmann equations of the type first considered by Bennett for S=2; (b) all smooth radial current profiles with normalized particle densities satisfy a generalized Bennett relation; (c) Bennett’s current profile is the only fully conformally invariant one; (d) there exist other radial solutions with profiles different from Bennett’s.

Statistical mechanics approach to some problems in conformal geometry

A weak law of large numbers is established for a sequence of systems of N classical point particles with logarithmic pair potential in Rn, or Sn,n∈N, which are distributed according to the configurational microcanonical measure δ(E−H), or rather some regularization thereof, where H is the configurational Hamiltonian and E the configurational energy. When N→∞ with non-extensive energy scaling E=N2e, the particle positions become i.i.d. according to a self-consistent Boltzmann distribution, respectively a superposition of such distributions. The self-consistency condition in n dimensions is some nonlinear elliptic PDE of order n (pseudo-PDE if n is odd) with an exponential nonlinearity. When n=2, this PDE is known in statistical mechanics as Poisson–Boltzmann equation, with applications to point vortices, 2D Coulomb and magnetized plasmas and gravitational systems. It is then also known in conformal differential geometry, where it is the central equation in Nirenbergs problem of prescribed Gaussian curvature. For constant Gauss curvature it becomes Liouvilles equation, which also appears in two-dimensional so-called quantum Liouville gravity. The PDE for n=4 is Paneitz’ equation, and while it is not known in statistical mechanics, it originated from a study of the conformal invariance of Maxwells electromagnetism and has made its appearance in some recent model of four-dimensional quantum gravity. In differential geometry, the Paneitz equation and its higher order n generalizations have applications in the conformal geometry of n-manifolds, but no physical applications yet for general n. Interestingly, though, all the Paneitz equations have an interpretation in terms of statistical mechanics.

CLASSICAL ELECTRON THEORY AND CONSERVATION LAWS

Abstract It is shown that the traditional conservation laws for total charge, energy, linear and angular momentum, hold jointly in classical electron theory if and only if classical electron spin is included as dynamical degree of freedom.

Surfaces with prescribed Gauss curvature

where dx denotes Lebesgue measure on R2, is called the integral curvature of the surface (sometimes called total curvature). We say that Sg is a classical surface over R2 if u ∈ C2(R2). Clearly, K ∈ C0(R2) in that case. The inverse problem, namely, to prescribe K and to find a surface Sg pointwise conformal to R2 for which K is the Gauss curvature, renders (1.2) a semilinear elliptic partial differential equation (PDE) for the unknown function u. The problem of prescribing Gaussian curvature thus amounts to studying the existence, uniqueness or multiplicity, and classification of solutions u of (1.2) for the given K . A particularly interesting aspect of the classification problem is the question under which conditions radial symmetry of the prescribed Gauss curvature function K implies radial symmetry of the classical surface Sg = (R2,g) and under which conditions radial symmetry is broken. Notice that the inverse problem may not have a solution. In particular, when considered on S2 instead of R2, there are so many obstructions to finding a solution u to (the analog of) (1.2) for the prescribed K that Nirenberg was prompted many years ago to raise the question: Which real-valued functions K are Gauss curvatures of some surface