Cosmological simulations of structure formation and the Vlasov equation

In cosmology numerical simulations of structure formation are now of central importance, as they are the sole instrument for providing detailed predictions of current cosmological models for a whole class of important constraining observations. These simulations are essentially molecular dynamics simulations of N (≫1, now up to of order several billion) particles interacting through their self-gravity. While their aim is to produce the Vlasov limit, which describes the underlying (“cold dark matter”) models, the degree to which they actually do produce this limit is currently understood, at best, only very qualitatively, and there is an acknowledged need for “a theory of discreteness errors”. In this talk I will describe, for non-cosmologists, both the simulations and the underlying theoretical models, and will then focus on the issue of discreteness, describing some recent progress in addressing this question quantitatively.     

Detecting irregular orbits in gravitational N-body simulations

We present a qualitative diagnostic based on the continuous wavelet transform, for the detection of irregular behaviour in time series of particle simulations. We apply the method to three qualitatively different gravitational 3-body encounters. The intrinsic irregular behaviour of these encounters is well reproduced by the presented method, and we show that the method accurately identifies the irregular regime in these encounters. We also provide an instantaneous quantification for the degree of irregularity in these simulations. Furthermore we demonstrate how the method can be used to analyse larger systems by applying it to simulations with 100-particles. It turns out that the number of stars on irregular orbits is systematically larger for clusters in which all stars have the same mass compared to a multi-mass system. The proposed method provides a quick and sufficiently accurate diagnostic for identifying stars on irregular orbits in large scale N-body simulations.     

The essential spectrum of N-body systems with asymptotically homogeneous order-zero interactions

We study the essential spectrum of N-body Hamiltonians with potentials defined by functions that have radial limits at infinity. The results extend the HVZ theorem which describes the essential spectrum of usual N-body Hamiltonians. The proof is based on a careful study of algebras generated by potentials and their cross-products. We also describe the topology on the spectrum of these algebras, thus extending to our setting a result of A. Mageira. Our techniques apply to more general classes of potentials associated with translation invariant algebras of bounded uniformly continuous functions on a finite-dimensional vector space X.     

On the spectral analysis of many-body systems

We describe the essential spectrum and prove the Mourre estimate for quantum particle systems interacting through k-body forces and creation-annihilation processes which do not preserve the number of particles. For this we compute the “Hamiltonian algebra” of the system, i.e. the C^∗-algebra C generated by the Hamiltonians we want to study, and show that, as in the N-body case, it is graded by a semilattice. Hilbert C^∗-modules graded by semilattices are involved in the construction of C. For example, if we start with an N-body system whose Hamiltonian algebra is CN and then we add field type couplings between subsystems, then the many-body Hamiltonian algebra C is the imprimitivity algebra of a graded Hilbert CN-module.     

Second-order corrections to mean field evolution of weakly interacting Bosons. II

We study the evolution of an N-body weakly interacting system of Bosons. Our work forms an extension of our previous paper Grillakis, Machedon, and Margetis (2010) [13], in which we derived a second-order correction to a mean-field evolution law for coherent states in the presence of small interaction potential. Here, we remove the assumption of smallness of the interaction potential and prove global existence of solutions to the equation for the second-order correction. This implies an improved Fock-space estimate for our approximation of the N-body state.     

Do-it-yourself computational astronomy

We overview our GRAPE (GRAvity PipE) and GRAPE-DR project to develop dedicated computers for astrophysical N-body simulations. The basic idea of GRAPE is to attach a custom-build computer dedicated to the calculation of gravitational interaction between particles to a general-purpose programmable computer. By this hybrid architecture, we can achieve both a wide range of applications and very high peak performance. GRAPE-6, completed in 2002, achieved the peak speed of 64 Tflops. The next machine, GRAPE-DR, will have the peak speed of 2 Pflops and will be completed in 2008. We discuss the physics of stellar systems, evolution of general-purpose high-performance computers, our GRAPE and GRAPE-DR projects and issues of numerical algorithms.     

Mean field dynamics of fermions and the time-dependent Hartree–Fock equation

The time-dependent Hartree–Fock equations are derived from the N-body linear Schrödinger equation with the mean-field scaling in the limit N→+∞ and for initial data that are close to Slater determinants. Only the case of bounded, symmetric binary interaction potentials is treated in this work. We prove that, as N→+∞, the first partial trace of the N-body density operator approaches the solution of the time-dependent Hartree–Fock equations (in operator form) in the sense of the trace norm.     

Periodic solutions of the N-body problem

This paper proves the existence of six new classes of periodic solutions to the N-body problem by small parameter methods. Three different methods of introducing a small parameter are considered and an appropriate method of scaling the Hamiltonian is given for each method. The small parameter is either one of the masses, the distance between a pair of particles or the reciprocal of the distances between one particle and the center of mass of the remaining particles. For each case symmetric and non-symmetric periodic solutions are established. For every relative equilibrium solution of the (N ? 1)-body problem each of the six results gives periodic solutions of the N-body problem. Under additional mild non-resonance conditions the results are roughly as follows. Any non-degenerate periodic solutions of the restricted N-body problem can be continued into the full N-body problem. There exist periodic solutions of the N-body problem, where N ? 2 particles and the center of mass of the remaining pair move approximately on a solution of relative equilibrium and the pair move approximately on a small circular orbit of the two-body problems around their center of mass. There exist periodic solutions of the N-body problem, where one small particle and the center of mass of the remaining N ? 1 particles move approximately on a large circular orbit of the two body problems and the remaining N ? 1 bodies move approximately on a solution of relative equilibrium about their center of mass. There are three similar results on the existence of symmetric periodic solutions.     

Rate of Convergence Towards Semi-Relativistic Hartree Dynamics

We consider the semi-relativistic system of N gravitating Bosons with gravitation constant G. The time evolution of the system is described by the relativistic dispersion law, and we assume the mean-field scaling of the interaction where N → ∞ and G → 0 while GN = λ fixed. In the super-critical regime of large λ, we introduce the regularized interaction where the cutoff vanishes as N → ∞. We show that the difference between the many-body semi-relativistic Schrödinger dynamics and the corresponding semi-relativistic Hartree dynamics is at most of order N ^?1 for all λ, i.e., the result covers the sub-critical regime and the super-critical regime. The N dependence of the bound is optimal.     

Requirements of a Package for N-Body Simulations of the Solar System

N-body simulations of the Solar System form a challenging set of initial value problems for numerical integrators. The challenge comes from the variety of problems and their size – one recent simulation had 300,015 second order equations and required 9×10¹⁰ integration steps. A number of packages for specific types of simulations are available. I discuss what is required of a package intended to efficiently perform a wide range of N-body simulations.     

Analytic continuation of the resolvent of the Laplacian on symmetric spaces of noncompact type

Let (M,g) be a globally symmetric space of noncompact type, of arbitrary rank, and Δ its Laplacian. We introduce a new method to analyze Δ and the resolvent (Δ-σ)^-1; this has origins in quantum N-body scattering, but is independent of the ‘classical’ theory of spherical functions, and is analytically much more robust. We expect that, suitably modified, it will generalize to locally symmetric spaces of arbitrary rank. As an illustration of this method, we prove the existence of a meromorphic continuation of the resolvent across the continuous spectrum to a Riemann surface multiply covering the plane. We also show how this continuation may be deduced using the theory of spherical functions. In summary, this paper establishes a long-suspected connection between the analysis on symmetric spaces and N-body scattering.     

Spectral properties of Schrödinger operators on radial N-dimensional infinite trees

We study the discreteness of the spectrum of Schrödinger operators which are defined on a class of radial N-dimensional rooted trees of a finite or infinite volume, and are subject to a certain mixed boundary condition. We present a method to estimate their eigenvalues using operators on a one-dimensional tree. These operators are called width-weighted operators, since their coefficients depend on the section width or area of the N-dimensional tree. We show that the spectrum of the width-weighted operator tends to the spectrum of a one-dimensional limit operator as the sections width tends to zero. Moreover, the projections to the one-dimensional tree of eigenfunctions of the N-dimensional Laplace operator converge to the corresponding eigenfunctions of the one-dimensional limit operator.     

On the final evolution of the n-body problem

The Newtonian n-body problem is studied. The main result gives the asymptotic properties of the distances between particles as time (t) approaches infinity. It is shown how particles can separate into subsystems between which the distances are asymptotic to constant multiples of time, and within which the mutual distances are at most of the order $\text{t}{}^{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}$. Each subsystem behaves asymptotically like a pure n-body problem in the sense that an energy and an angular momentum relationship are asymptotically satisfied. These techniques allow some three-body results found by Birkhoff and by Sundman to be extended to the n-body problem. Finally, some n-body results are derived for a motion which expands faster than time.     

On the existence of the N-body Efimov effect

In this paper, we prove the existence of the Efimov effect for N-body quantum systems with N?4. Under the conditions that the bottom of the essential spectrum, E₀, of the N-body operator is attained by the spectra of a unique three-cluster Subhamiltonian and its three associated two-cluster Subhamiltonians, and that at least two of these two-cluster Subhamiltonians have a resonance at the threshold E₀, we give a lower bound of the form C₀|log(E₀−λ)| for the number of eigenvalues on the left of , where C₀ is a positive constant depending only on the reduced masses in the three-cluster decomposition. We also obtain a lower bound on the number of discrete eigenvalues in coupling constant perturbation.     

Fast directional algorithms for the Helmholtz kernel

This paper presents a new directional multilevel algorithm for solving N-body or N-point problems with highly oscillatory kernels. We address the problem by first proving that the interaction between a ball of radius r and a well-separated region has an approximate low rank representation, as long as the well-separated region belongs to a cone with a spanning angle of O(1/r) and is at a distance which is at least O(r²) away from the ball. Based on this representation, our algorithm organizes the high frequency computation using a multidirectional and multiscale strategy. Our algorithm is proved to have an optimal O(NlogN) computational complexity for any given accuracy when the points are sampled from a two-dimensional surface.     

The pyramidal configurations of N+1 bodies cannot rotate

For the (N+1)-body problem, we assume that N bodies are at the vertices of a unit regular polygon and the (N+1)st body is along the vertical line normal to the plane formed by the former N bodies. If N bodies rotate at the unit circle and the (N+1)st body oscillates along the vertical line of the plane formed by the former N bodies and passing through the geometrical center, then we prove that the (N+1)st body must locate at the geometrical center of unit regular polygon.     

Parker-Sochacki method for solving systems of ordinary differential equations using graphics processors

The Parker-Sochacki method, which is used for solving systems of ordinary differential equations, and implementation of this method on graphics processors are described. The solution to the classical N-body problem is considered as a test. The algorithm makes it possible to effectively use massive parallel graphics processors and provides acceptable accuracy with multiple time reduction, as compared to processors of a conventional architecture.     

Morse index properties of colliding solutions to the N-body problem

We study a singular Hamiltonian system with an α-homogeneous potential that contains, as a particular case, the classical N-body problem. We introduce a variational Morse-like index for a class of collision solutions and, using the asymptotic estimates near collisions, we prove the non-minimality of some special classes of colliding trajectories under suitable spectral conditions provided α is sufficiently away from zero. We then prove some minimality results for small values of the parameter α.     

Derivation of the Schrödinger–Poisson equation from the quantum N-body problemJustification de l'équation de Schrödinger–Poisson à partir du problème quantique à N corps

We derive the time-dependent Schrödinger–Poisson equation as the weak coupling limit of the N-body linear Schrödinger equation with Coulomb potential. To cite this article: C. Bardos et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 515–520.     

Absence of Positive Eigenvalues for Hard-Core N-Body Systems

We show absence of positive eigenvalues for generalized N-body hard-core Schrödinger operators under the condition of bounded obstacles with connected exterior. A particular example is atoms and molecules with the assumption of infinite mass and finite extent nuclei.