A Solvable Many-Body Problem in the Plane

Abstract

A solvable many-body problem in the plane is exhibited. It is characterized by rotation-invariant Newtonian (“acceleration equal force”) equations of motion, featuring one-body (“external”) and pair (“interparticle”) forces. The former depend quadratically on the velocity, and nonlinearly on the coordinate, of the moving particle. The latter depend linearly on the coordinate of the moving particle, and linearly respectively nonlinearly on the velocity respectively the coordinate of the other particle. The model contains 2n ² arbitrary coupling constants, n being the number of particles. The behaviour of the solutions is outlined; special cases in which the motion is confined (multiply periodic), or even completely periodic, are identified.     

Solutions of the Einstein-Maxwell equations with many black holes

In Newtonian gravitational theory a system of point charged particles can be arranged in static equilibrium under their mutual gravitational and electrostatic forces provided that for each particle the charge,e, is related to the mass,m, bye=G ^1/2 m. Corresponding static solutions of the coupled source free Einstein-Maxwell equations have been given by Majumdar and Papapetrou. We show that these solutions can be analytically extended and interpreted as a system of charged black holes in equilibrium under their gravitational and electrical forces.We also analyse some of stationary solutions of the Einstein-Maxwell equations discovered by Israel and Wilson. If space is asymptotically Euclidean we find that all of these solutions have naked singularities.Alfred P. Sloan Research Fellow, supported in part by the National Science Foundation.     

Relativistic equations with some point potentials

Exact solutions have been obtained for relativistic one-dimensional integral equations that describe the scattering of two particles with potentials of the “delta function n-th derivative” type for n = 1, 2, 3. Based on the solutions, the transmission and reflection coefficients have been found and some their properties have been investigated.     

Influence of interatomic repulsion on the structure of liquids at melting

Radial distribution functions and liquid structure factors for fluids of particles interacting through 1/rⁿ repulsive potentials are computed ‘exactly’ and systematically compared at crystallization for several values of n ranging between the two extremes of hard spheres (n = ∞) and Coulomb forces (n = 1). Striking similarities are pointed out in the large r behaviour (beyond the first peak) of the various radial distribution functions, and in the small k behaviour (including the first peak) of the various structure factors. Some information about the steepness of the repulsive potential in liquid rare gases and in liquid alkalis is gained from a comparison of the damping of successive peaks of the structure factors with experimental data.     

Algebraic and Geometric Properties of Matrix Solutions of Nonlinear Wave Equations

We improve the construction of exact matrix solutions for nonlinear wave equations by using unitary anti-Hermitian and anticommuting matrices. We prove the theorem that constructs the matrix functions u _n satisfying the nonlinear wave equation for a set of special potentials. In this case, the graph of complex solution u ₁ has a soliton-like form with a finite number of coils. Exponential representation of matrix solutions u _n is associated with continuous rotations that can be used for describing intrinsic rotations and state changes of elementary particles. We also prove the theorem on the decomposition of continuous rotation (described by solution u ₂) onto three simultaneous rotations about coordinate vectors. Each of the three constructed matrix solutions u ₃ is also decomposed into the triplet of elementary matrix solutions.     

The runaway effect in a Lorentz gas

The Lorentz gas of charged particles in a constant and uniform electric field is studied. The gas flows through the medium of immobile, randomly distributed scatterers. Particles with velocity v suffer collisions with frequency proportional to ¦v¦ ⁿ . Forn < 0 runaway of the gas is forced by the field: the mean velocity of the flow increases without bounds. By a simple physical argument an integral relation is established between the probability of collisionless motion and the velocity distribution. It is then shown that whenn < –1 a fraction of particles moves as if the scattering centers were absent. The detailed discussion of this uncollided runaway is presented. Some qualitative features of the velocity distribution are illustrated on rigorous solutions in one dimension.     

Application of the trial equation method for solving some nonlinear evolution equations arising in mathematical physics

In this paper some exact solutions including soliton solutions for the KdV equation with dual power law nonlinearity and the K(m, n) equation with generalized evolution are obtained using the trial equation method. Also a more general trial equation method is proposed.     

Self-Similar Asymptotics for the Boltzmann Equation with Inelastic and Elastic Interactions

We consider some questions related to the self-similar asymptotics in the kinetic theory of both elastic and inelastic particles. In the second case we have in mind granular materials, when the model of hard spheres with inelastic collisions is replaced by a Maxwell model, characterized by a collision frequency independent of the relative speed of the colliding particles. We first discuss how to define the n-dimensional (n = 1,2,...) inelastic Maxwell model and its connection with the more basic Boltzmann equation for inelastic hard spheres. Then we consider both elastic and inelastic Maxwell models from a unified viewpoint. We prove the existence of (positive in the inelastic case) self-similar solutions with finite energy and investigate their role in large time asymptotics. It is proved that a recent conjecture by Ernst and Brito devoted to high energy tails for inelastic Maxwell particles is true for a certain class of initial data which includes Maxwellians. We also prove that the self-similar asymptotics for high energies is typical for some classes of solutions of the classical (elastic) Boltzmann equation for Maxwell molecules. New classes of (not necessarily positive) finite-energy eternal solutions of this equation are also studied.     

N = 4 mechanics,WDVV equations and polytopes

N = 4 superconformal n-particle quantum mechanics on the real line is governed by two prepotentials, U and F, which obey a system of partial nonlinear differential equations generalizing the Witten—Dijkgraaf—Verlinde—Verlinde (WDVV) equation for F. The solutions are encoded by the finite Coxeter systems and certain deformations thereof, which can be encoded by particular polytopes. We provide A _n and B ₃ examples in some detail. Turning on the prepotential U in a given F background is very constrained for more than three particles and nonzero central charge. The standard ansatz for U is shown to fail for all finite Coxeter systems. Three-particle models are more flexible and based on the dihedral root systems.     

A Solvable N-body Problem of Goldfish Type Featuring N2 Arbitrary Coupling Constants

A N-body problem “of goldfish type” is introduced, the Newtonian (“acceleration equal force”) equations of motion of which describe the motion of N pointlike unit-mass particles moving in the complex z-plane. The model—for arbitrary N—is solvable, namely its configuration (positions and velocities of the N “particles”) at any later time t can be obtained from its configuration at the initial time by algebraic operations. It features specific nonlinear velocity-dependent many-body forces depending on N² arbitrary (complex) coupling constants. Sufficient conditions on these constants are identified which cause the model to be isochronous—so that all its solutions are then periodic with a fixed period independent of the initial data. A variant with twice as many arbitrary coupling constants, or even more, is also identified.     

Combustion of alane and aluminum with water for hydrogen and thermal energy generation

The combustion of alane and aluminum with water in its frozen state has been studied experimentally and theoretically. Both nano and micron-sized particles are considered over a broad range of pressure. The linear burning rate and chemical efficiency are obtained using a constant-pressure strand burner and constant-volume cell, respectively. The effect of replacing nano-Al particles by micron-sized Al and alane particles are examined systematically with the additive mass fraction up to 25%. The equivalence ratio is fixed at 0.943. The pressure dependence of the burning rate follows the power law, r_b = aPⁿ, with n ranging from 0.41 to 0.51 for all the materials considered. The burning rate decreases with increasing alane concentration, whereas it remains approximately constant with cases containing only Al particles. The chemical efficiency ranged from 32% to 83%, depending on the mixture composition and pressure. Thermo-chemical analyses are conducted to provide insight into underlying causes of the decreased burning rate of the alanized compositions. A theoretical model is also developed to explore the detailed flame structure and burning properties. Reasonably good agreement is achieved with experimental observations.     

Theorem on the Distribution of Short-Time Particle Displacements with Physical Applications

The distribution of the initial short-time displacements of particles is considered for a class of classical systems under rather general conditions on the dynamics and with Gaussian initial velocity distributions, while the positions could have an arbitrary distribution. This class of systems contains canonical equilibrium of a Hamiltonian system as a special case. We prove that for this class of systems the nth order cumulants of the initial short-time displacements behave as the 2n-th power of time for all n > 2, rather than exhibiting an nth power scaling. This has direct applications to the initial short-time behavior of the Van Hove self-correlation function, to its non-equilibrium generalizations the Green's functions for mass transport, and to the non-Gaussian parameters used in supercooled liquids and glasses. PACS Number: 05.20.-y, 02.30.Mv, 66.10.-x, 78.70.Nx, 05.60.Cd     

Non-Abelian Vortices on Compact Riemann Surfaces

We consider the vortex equations for a U(n) gauge field A coupled to a Higgs field f{\phi} with values on the n × n matrices. It is known that when these equations are defined on a compact Riemann surface Σ, their moduli space of solutions is closely related to a moduli space of τ-stable holomorphic n-pairs on that surface. Using this fact and a local factorization result for the matrix f{\phi} , we show that the vortex solutions are entirely characterized by the location in Σ of the zeros of det f{\phi} and by the choice of a vortex internal structure at each of these zeros. We describe explicitly the vortex internal spaces and show that they are compact and connected spaces.     

Interlaced Particle Systems and Tilings of the Aztec Diamond

Motivated by the problem of domino tilings of the Aztec diamond, a weighted particle system is defined on N lines, with line j containing j particles. The particles are restricted to lattice points from 0 to N, and particles on successive lines are subject to an interlacing constraint. It is shown that this particle system is exactly solvable, to the extent that not only can the partition function be computed exactly, but so too can the marginal distributions. These results in turn are used to give new derivations within the particle picture of a number of known fundamental properties of the tiling problem, for example that the number of distinct configurations is 2 ^N(N+1)/2, and that there is a limit to the GUE minor process, which we show at the level of the joint PDFs. It is shown too that the study of tilings of the half Aztec diamond—not known from earlier literature—also leads to an interlaced particle system, now with successive lines 2n−1 and 2n (n=1,…,N/2−1) having n particles. Its exact solution allows for an analysis of the half Aztec diamond tilings analogous to that given for the Aztec diamond tilings.     

Grain growth controlled by triple junctions: Effect of number of topological elements

In addition to driving forces due to curvature of grain boundaries there are driving forces acting on triple junctions which also contribute to grain growth. Equations are derived for the rate of change, due to the triple junction forces, of the average area or average volume of 2D and 3D grains, respectively, with a fixed number of topological elements (edges in 2D and faces in 3D). The equations derived are compared with the von Neumann-Mullins equation for 2D curvature driven grain growth and to the extension of that equation to 3D grain growth. In triple junction controlled grain growth, the effect ofn orF is qualitatively the same as in curvature driven growth, with a threshold atn or –F between shrinkage and growth. However, the rates are in general not linear onn orF, and there is a size effect which has a repercursion on the overall growth kinetics.     

Stochastic dynamics of two-dimensional infinite-particle systems

The time evolution of an open system of infinitely many two-dimensional classical particles is investigated. Particles are interacting by a singular pair potentialU, and each particle is connected to a heat bath of temperatureT. The heat baths are represented by independent white noise forces and Langevin damping terms. Existence of strong solutions to the corresponding infinite system of stochastic differential equations is proved for initial configurations with a logarithmic order of energy fluctuations. Gibbs states forU at temperatureT are invariant under time evolution.     

Effective hydrogenation and surface damage induced by MW-ECR plasma of fine-grained polycrystalline silicon

This work reports the investigations on the effects of the hydrogenation process of thin film polycrystalline n⁺pp⁺ mesa silicon cells using MW-ECR plasma in a conventional PECVD system. Different operating parameters such as MW-ECR power, annealing temperature and the doping level of the emitter region were varied. The n⁺-type emitter regions were obtained by phosphorus diffusion in a conventional furnace using an oxide doping source containing phosphorus (P507 or P509 solutions, from Filmtronics Inc.). The MW hydrogenation was carried out at a sample temperature of 400°C for 60 min. Both types of emitters formed from P507 and P509 showed V _oc of 155 mV and 206 mV, which increased linearly to 305 mV and 331 mV, respectively, after hydrogenation when the MW power varied from 200 to 650 W. However, the sheet resistances of the n⁺ emitter region showed a slight increase depending upon hydrogenation power because of its etching. In a further study, hydrogenated samples were annealed in neutral or forming gas (FG) and we observed interesting results on V _oc in the presence of FG. The FG annealing temperature study revealed a strong dependence of V _oc on MW power, which affected the etching level of emitter and emitter dopant concentration, which controls the diffusion of hydrogen ions during post-hydrogenation step. The results were explained in detail by combining the effects of MW power and dopant level of the emitter.     

On the configuration of classical Yang-Mills fields with topological charge <Emphasis Type="Italic">k</Emphasis> = 4

The solutions of the nonlinear matrix equation in the Atiyah-Hitchin-Drifeld-Manin (AHDM) construction that determine the Yang-Mills self-dual fields with topological charge k = 4 for symplectic gauge groups are discussed. In the case of Sp(n), n > 2, it is possible to use a procedure that was proposed earlier for generating solutions with k = 3. It is shown that for SU(2) = Sp(1) the AHDM matrix can be generated by using cubic equation solutions with coefficients that depend on 8k — 3 parameters.     

Gauge theories of weak and strong gravity

A review of some recent papers on gauge theories of weak and strong gravity is presented. For weak gravity, SL(2, C) gauge theory along with tetrad formulation is described which yields massless spin-2 gauge fields (quanta gravitons). Next a unified SL(2n,C) model is discussed along with Higgs fields. Its internal symmetry is SU(n). The free field solutions after symmetry breaking yield massless spin-1 (photons) and spin-2 (gravitons) gauge fields and also massive spin-1 and spin-2 bosons. The massive spin-2 gauge fields are responsible for short range superstrong gravity. Higgs-fermion interaction can lead to baryon and lepton number non-conservation. The relationship of strong gravity with other forces is also briefly considered.     

Precipitation kinetics of W2B5 in (Ti0.4W0.5Cr0.1)B2 solid solutions

The isothermal precipitation kinetics of W₂B₅ secondary phase from supersaturated polycrystalline (Ti_0.4W_0.5Cr_0.1)B₂ solid solutions were investigated with X-ray diffractometry and scanning electron microscopy in the temperature range between 1500 and 1700°C. The precipitate formation is described by a modified Johnson–Mehl–Avrami–Kolmogorov (JMAK) model, where W₂B₅ particles nucleate preferentially at grain boundaries and subsequently grow into the volume by a two-dimensional process controlled by volume diffusion of the transition metals. Numerical calculations are used to describe quantitatively the time dependence of the precipitated fraction and to determine a differential JMAK exponent n _diff which gives information on the nucleation and growth modes. n _diff decreases during the precipitation process from 2 to about 0.8 for all temperatures investigated. The first limit corresponds to the classical JMAK model (two-dimensional diffusional growth and constant nucleation rate) and the decrease in n _diff is the consequence of an impingement of the nucleating and growing particles in the late stages of the process. Nucleation and growth rates are determined as functions of reciprocal temperature, where the first quantity shows a non-monotonic behaviour with a maximum at about 1650°C and the second quantity exhibits an Arrhenius behaviour with an activation enthalpy of 3.6?eV. From this it can be concluded that the overall precipitate formation is dominated by the kinetics of atomic motion at low temperatures and by the thermodynamics of nucleation at high temperatures.