The Interaction of Collinear Gaps of Arbitrary Charge in a Two Dimensional Dimer System

The correlation of gaps in dimer systems was introduced by Fisher and Stephenson (Phys Rev 132:1411–1431, 1963), who looked at the interaction of two monomers generated by the rigid exclusion of dimers on the closely packed square lattice. In previous work we considered the analogous problem on the hexagonal lattice, and we extended the set-up to include the correlation of any finite number of monomer clusters. For fairly general classes of monomer clusters we proved that the asymptotics of their correlation is given, for large separations between the clusters, by a multiplicative version of Coulomb’s law for 2D electrostatics. However, our previous results required that the monomer clusters consist (with possibly one exception) of an even number of monomers. In this paper we determine the asymptotics of general defect clusters along a lattice diagonal in the square lattice (involving an arbitrary, even or odd number of monomers), and find that it is given by the same Coulomb law. We also obtain a conceptual interpretation for the multiplicative constant as the product of the correlations of the individual clusters.     

The Semi-Infinite q-Boson System with Boundary Interaction

Upon introducing a one-parameter quadratic deformation of the q-boson algebra and a diagonal perturbation at the end point, we arrive at a semi-infinite q-boson system with a two-parameter boundary interaction. The eigenfunctions are shown to be given by Macdonald’s hyperoctahedral Hall–Littlewood functions of type BC. It follows that the n-particle spectrum is bounded and absolutely continuous and that the corresponding scattering matrix factorizes as a product of two-particle bulk and one-particle boundary scattering matrices.     

A New Boundary Problem for the Two Dimensional Navier-Stokes System

We formulate a new boundary value problem for the 2D Navier-Stokes system on the unit square. Under some suitable assumptions on the initial velocity, we obtain quantitative decay estimates of the Fourier modes of both the vorticity and the velocity. It is found that in one direction the Fourier modes decay exponentially and along the other direction their decay is only power like.     

A Free Boundary Problem with Non Local Interaction

We prove local existence for classical solutions of a free boundary problem which arises in one of the biological selection models proposed by Brunet and Derrida, (Phys. Rev. E 56, 2597D2604, 1997) and Durrett and Remenik, (Ann. Probab. 39, 2043–2078, 2011). The problem we consider describes the limit evolution of branching brownian particles on the line with death of the leftmost particle at each creation time as studied in De Masi et al. (2017). We use extensively results in Cannon (1984) and Fasano (2008).     

二维荷电粒子维格纳晶格:边界效应

采用一种非线性的优化方法,研究了处于硬壁限制势下二维带电多粒子系统的基态,分析不同形状边界对系统基态构型的影响.由于圆形边界对称性高,基态结构和抛物限制势下情况相似.在正方形边界下,当系统粒子数N<66时,荷电粒子形成方形晶格;当N≥66时,由于边界影响被削弱,内层粒子形成六角维格纳晶格.进一步分析了椭圆和矩形边界对维格纳晶格的影响.     

Free Molecular Flow with Boundary Effect

We study the dissipative effect of the boundary condition in the kinetic theory. We focus our study on the simplest situation of the free molecular flow with diffuse reflection boundary condition and constant boundary temperature, T _*. The geometry is also chosen to be the simplest ones, a bounded symmetric domain in ${\mathbb{R}^d}$ : an interval for d = 1, a disk for d = 2, and a ball for d = 3. It is shown that the solution converges to the global Maxwellian with the given boundary temperature T _*. We obtain the optimal convergence rates of (t + 1)^?d . The stochastic formulation of Shih-Hsien Yu is refined and generalized for our analysis.     

The Two Dimensional Kondo Model with Rashba Spin-Orbit Coupling

We investigate the effect that Rashba spin-orbit coupling has on the low energy behaviour of a two dimensional magnetic impurity system. It is shown that the Kondo effect, the screening of the magnetic impurity at temperatures T<T _K , is robust against such spin-orbit coupling, despite the fact that the spin of the conduction electrons is no longer a conserved quantity. A proposal is made for how the spin-orbit coupling may change the value of the Kondo temperature T _K in such systems and the prospects of measuring this change are discussed. We conclude that many of the assumptions made in our analysis invalidate our results as applied to recent experiments in semi-conductor quantum dots but may apply to measurements made with magnetic atoms placed on metallic surfaces.     

A Universal Construction of Calogero-Moser System with Two Types Spins Interaction

We present universal construction for the Calogero-Moser system with two types spins interaction of trigonometric potential based on the root system of semi-simple Lie algebra. In this formalism, we successfully build up the correct Lax pair as well as the R-matrix for this generalized Calogero-Moser models. Moreover using the property of root system, we make a concise explanation that in the quantized model, the R-matrix takes the same form as the classical one, which is the main new result of this paper.     

Discrete Charges on a Two Dimensional Conductor

We investigate the electrostatic equilibria of N discrete charges of size 1/N on a two dimensional conductor (domain). We study the distribution of the charges on symmetric domains including the ellipse, the hypotrochoid and various regular polygons, with an emphasis on understanding the distributions of the charges, as the shape of the underlying conductor becomes singular. We find that there are two regimes of behavior, a symmetric regime for smooth conductors, and a symmetry broken regime for “singular” domains. For smooth conductors, the locations of the charges can be determined, to within $O\left( {\sqrt {\log {N \mathord{\left/ {\vphantom {N {N^2 }}} \right. \kern-0em} {N^2 }}} } \right)$ by an integral equation due to Pommerenke [ Math. Ann., 179: 212–218, (1969)]. We present a derivation of a related (but different) integral equation, which has the same solutions. We also solve the equation to obtain (asymptotic) solutions which show universal behavior in the distribution of the charges in conductors with somewhat smooth cusps. Conductors with sharp cusps and singularities show qualitatively different behavior, where the symmetry of the problem is broken, and the distribution of the discrete charges does not respect the symmetry of the underlying domain. We investigate the symmetry breaking both theoretically, and numerically, and find good agreement between our theory and the numerics. We also find that the universality in the distribution of the charges near the cusps persists in the symmetry broken regime, although this distribution is very different from the one given by the integral equation.     

The “Two and One–Half Dimensional” Relativistic Vlasov Maxwell System

The motion of a collisionless plasma is modeled by solutions to the Vlasov–Maxwell system. The Cauchy problem for the relativistic Vlasov–Maxwell system is studied in the case when the phase space distribution function f = f(t,x,v) depends on the time t, and . Global existence of classical solutions is obtained for smooth data of unrestricted size. A sufficient condition for global smooth solvability is known from [12]: smooth solutions can break down only if particles of the plasma approach the speed of light. An a priori bound is obtained on the velocity support of the distribution function, from which the result follows. Received: 18 March 1996/Accepted: 29 July 1996     

The Spectral Gap of the 2-D Stochastic Ising Model with Mixed Boundary Conditions

We establish upper bounds for the spectral gap of the stochastic Ising model at low temperatures in an l×l box with boundary conditions which are not purely plus or minus; specifically, we assume the magnitude of the sum of the boundary spins over each interval of length l in the boundary is bounded by l, where <1. We show that for any such boundary condition, when the temperature is sufficiently low (depending on ), the spectral gap decreases exponentially in l.     

Sedimentation of a Low-Concentration Suspension of Stokes Particles in a Stirred Layer with a Movable Free Boundary

Technical Physics - The problem of sedimentation of polydisperse low-concentration Stokes particles in conditions of stirring of the carrier medium in a plane layer with a moving free surface is...     

Boundary of Chaos for the Gap Map

By means of kneading theory the gap map is studied based on the admissibility conditions.A criterion for the existence of chaos is derived. The structure of the skeleton is analyzed by using two transformations for symbols. The boundary of chaos for the gap map is described.     

Interaction of Solute Impurity with Grain Boundary: The Impurity Drag Effect

An equation of grain boundary motion in a binary polycrystal is derived. The derivation is based on minimization of free energy of the total systems. The equation takes into account an impurity segregation at the grain boundary, grain boundary curvature and energy.As an example, we apply this equation to the analysis of the impurity drag effect problem. It is shown, that the sign of the impurity effect on grain boundary velocity (delay or acceleration) does not depend on kinetic coefficients. The sign of the effect is determined by a thermodynamic function which combines the grain boundary segregation coefficient, the derivative of grain boundary energy with respect to absorbed impurity concentration, and the derivative of bulk free energy with respect to bulk impurity concentration.     

Metastability in the Two-Dimensional Ising Model with Free Boundary Conditions

We investigate metastability in the two dimensional Ising model in a square with free boundary conditions at low temperatures. Starting with all spins down in a small positive magnetic field, we show that the exit from this metastable phase occurs via the nucleation of a critical droplet in one of the four corners of the system. We compute the lifetime of the metastable phase analytically in the limit T 0, h 0 and via Monte Carlo simulations at fixed values of T and h and find good agreement. This system models the effects of boundary domains in magnetic storage systems exiting from a metastable phase when a small external field is applied.     

Thermal Dissipation in Two Dimensional Relativistic Fermi Gases with a Relaxation Time Model

Journal of Statistical Physics - The thermal transport properties of a two dimensional Fermi gas are explored, for the full range of temperatures and densities. The heat flux is established by...     

Well-Posedness for the Linearized Motion of a Compressible Liquid with Free Surface Boundary

 We study the motion of a compressible perfect liquid body in vacuum. This can be through of as a model for the motion of the ocean or a star. The free surface moves with the velocity of the liquid and the pressure vanishes on the free surface. This leads to a free boundary problem for Euler's equations, where the regularity of the boundary enters to highest order. We prove linearized stability in Sobolev space assuming a ``physical condition', related to the fact that the pressure of a fluid has to be positive. Received: 23 September 2002 / Accepted: 2 December 2002 Published online: 14 April 2003 RID="⋆" ID="⋆" The author was supported in part by the National Science Foundation. Communicated by P. Constantin     

Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary

We study the motion of a compressible perfect liquid body in vacuum. This can be thought of as a model for the motion of the ocean or a star. The free surface moves with the velocity of the liquid and the pressure vanishes on the free surface. This leads to a free boundary problem for Euler's equations, where the regularity of the boundary enters to highest order. We prove local existence in Sobolev spaces assuming a ``physical condition', related to the fact that the pressure of a fluid has to be positive. The author was supported in part by the National Science Foundation.     

The Anisotropic Free Energy of the Solid-Liquid Phase Boundary in Al

We have performed large-scale molecular dynamics simulations of solid-liquid interfaces in aluminum, in order to calculate their interfacial stiffness and free energy. The anisotropy of these properties, though small, is key for determining the steady-state solutions for solidification in three dimensions. We find that the interfacial free energy, (), can be well represented by the form () = ₀(1 + cos 4, where measures the angle relative to the [100] interface, rotating about a [001] direction. We find values of ₀ = 120 ± 6 mJ/m² and = 1.3 ± 0.4%, in agreement with previous estimates and current experimental results.     

Nonlinear Instability in Two Dimensional Ideal Fluids: The Case of a Dominant Eigenvalue

It is proved that any steady 2 dimensional ideal fluid flow is nonlinearly unstable with respect to L ² growth in the velocity, provided there exists an eigenvalue for the linearised Euler equation with Re>. Here is the maximal Lyapunov exponent of the steady flow.