Segregation in the Asymmetric Hubbard Model

We study the “asymmetric” Hubbard model, where hoppings of electrons depend on their spin. For strong interactions and sufficiently asymmetric hoppings, it is proved that the ground state displays phase separation away from half-filling. This extends a recent result obtained with Freericks and Lieb for the Falicov–Kimball model. It is based on estimates for the sum of lowest eigenvalues of the discrete Laplacian in arbitrary domains.     

Minority Population in the One-Dimensional Schelling Model of Segregation

Schelling models of segregation attempt to explain how a population of agents or particles of two types may organise itself into large homogeneous clusters. They can be seen as variants of the Ising model. While such models have been extensively studied, unperturbed (or noiseless) versions have largely resisted rigorous analysis, with most results in the literature pertaining models in which noise is introduced, so as to make them amenable to standard techniques from statistical mechanics or stochastic evolutionary game theory. We rigorously analyse the one-dimensional version of the model in which one of the two types is in the minority, and establish various forms of threshold behaviour. Our results are in sharp contrast with the case when the distribution of the two types is uniform (i.e. each agent has equal chance of being of each type in the initial configuration), which was studied in Brandt et al. (in: STOC ’12: proceedings of the 44th symposium on theory of computing, pp. 789–804, 2012) and Barmpalias et al. (in: 55th Annual IEEE symposium on foundations of computer science, Oct 18–21, Philadelphia, FOCS’14, 2014).

     

Segregation in arch formation

We report new segregation phenomena in the clogging arches formed during the discharge of granular piles. Results from molecular dynamics simulations show segregation effects with respect to both size and density ratios used in piles built with bidisperse mixtures of grains. The clogging arch is preferentially constituted of large grains when size bidisperse piles were discharged, whereas for density bidisperse mixtures there is a predominance of light grains in the arch for large orifice widths but, for small widths, an inversion in the preference is observed, with a slightly higher incidence of heavy grains forming the arches. We present arguments based on the reverse buoyancy effect and the statistics collected for the avalanche size distributions to explain how these effects can be understood as a crossover between two different segregation mechanisms acting independently at small and large orifice width limits.     

Segregation in reaction-diffusion systems

Segregation of reactants in reaction-diffusion systems is a spatial structure that can be formed either as a result of a dynamical process or as an initially prepared system. In this paper we review our recent results on both such systems. First we study the dynamic segregation at a single trap, in particular in the presence of fields and disorder. Then we study properties of the dynamic reaction front produced due to initial segregation of the reactants in the A + B→C system. Both systems are shown to exhibit anomalous kinetic properties.     

Segregation in thin films

The longitudinal and transverse components of the image force acting on a moving charge near a metal surface are calculated. In particular, it is found that the transverse force is finite at the surface in contrast to the classical image force. The transverse force leads to deviations in trajectory in grazing incidence reflection inelastic electron scattering experiments, which are estimated. A consequence is the existence of a limiting maximum probability for excitation of surface plasmons.     

Segregation studies in Pd-Ag

Auger studies of the surface composition of a Pd/Ag alloy ($\text{60}\text{40 wt\%)}$ display reversible, temperature dependent enhancement of silver in the absence of sulfur. Co-segregation of sulfur effects a strong depletion of silver in the detected volume. No evidence could be found for the surface formation of a bulk palladium sulfide. In the absence of compound formation, the silver depletion can be qualitatively understood in terms of a recent ternary solution model which incorporates solute—solute interactions.     

Self-organized Segregation on the Grid

We consider an agent-based model with exponentially distributed waiting times in which two types of agents interact locally over a graph, and based on this interaction and on the value of a common intolerance threshold \(\tau \), decide whether to change their types. This is equivalent to a zero-temperature ising model with Glauber dynamics, an asynchronous cellular automaton with extended Moore neighborhoods, or a Schelling model of self-organized segregation in an open system, and has applications in the analysis of social and biological networks, and spin glasses systems. Some rigorous results were recently obtained in the theoretical computer science literature, and this work provides several extensions. We enlarge the intolerance interval leading to the expected formation of large segregated regions of agents of a single type from the known size \(\epsilon >0\) to size \(\approx 0.134\). Namely, we show that for \(0.433< \tau < 1/2\) (and by symmetry \(1/2<\tau <0.567\)), the expected size of the largest segregated region containing an arbitrary agent is exponential in the size of the neighborhood. We further extend the interval leading to expected large segregated regions to size \(\approx 0.312\) considering “almost segregated” regions, namely regions where the ratio of the number of agents of one type and the number of agents of the other type vanishes quickly as the size of the neighborhood grows. In this case, we show that for \(0.344 < \tau \le 0.433\) (and by symmetry for \(0.567 \le \tau <0.656\)) the expected size of the largest almost segregated region containing an arbitrary agent is exponential in the size of the neighborhood. This behavior is reminiscent of supercritical percolation, where small clusters of empty sites can be observed within any sufficiently large region of the occupied percolation cluster. The exponential bounds that we provide also imply that complete segregation, where agents of a single type cover the whole grid, does not occur with high probability for \(p=1/2\) and the range of intolerance considered.     

Segregation surface in binary solid solutions

Analytic equations describing the concentration profile, surface energy, and adsorption for a binary solid solution with the emergence of one of the components to the surface (surface segregation) were obtained using a unique approach, a modified van der Waals approximation. The calculation results were compared with the experimental secondary ion mass spectrometry data on Cu-Ge and Cu-Al metallic binary solutions; agreement was satisfactory.     

Segregation in fluidized versus tapped packs

We compare the predictions of two different statistical mechanics approaches, corresponding to different physical measurements, proposed to describe binary granular mixtures subjected to some external driving (continuous shaking or tap dynamics). In particular we analytically solve at a mean field level the partition function of a simple hard sphere lattice model under gravity and focus on the phenomenon of size segregation. We find that the two approaches lead to similar results and seem to coincide in the limit of very low shaking amplitude. However, they give different predictions of the crossovers from Brazil nut effect to reverse Brazil nut effect with respect to the shaking amplitude, which could be detected experimentally.     

Segregation transitions in wet granular matter

We report the effect of interstitial fluid on the extent of segregation by imaging the pile that results after bidisperse color-coded particles are poured into a silo. Segregation is sharply reduced and preferential clumping of small particles is observed when a small volume fraction of fluid V(f) is added. We find that viscous forces in addition to capillary forces have an important effect on the extent of segregation s and the angle of repose straight theta. We show that the sharp initial change and the subsequent saturation in s and straight theta occurs over similar V(f). We also find that a transition back to segregation can occur when the particles are completely immersed in a fluid at low viscosities.     

Segregation of Solutes in Two-Phase Mixtures

Fractions of indium solutes in each phase of a mixture of two binary phases were measured using perturbed angular correlation of gamma rays. Measurements of phase fractions were made on Pd₃Ga₇–PdGa, PdGa–Pd₅Ga₃, and FeAl₂–FeAl mixtures as a function of composition. The phase fractions were analyzed using a thermodynamic model that takes into account differences between energies of solute atoms in the two phases. From the model, segregation coefficients were obtained for the systems studied. Also, earlier measurements on Ni₂Al₃–NiAl were reanalyzed. Large differences are found among the segregation coefficients. This revised version was published online in September 2006 with corrections to the Cover Date.     

Segregation in binary mixtures under gravity

We employ kinetic theory for a binary mixture to study segregation by size and/or mass in a gravitational field. Simple segregation criteria are obtained for spheres and disks that are supported by numerical simulations.     

Segregation at stepped surfaces

A theoretical model of segregation at a (120) stepped surface with ($\text{1}$10) monoatomic steps and terraces on the (110) plane of a body centered cubic alloy is presented. The model is based on pairwise interactions between nearest neighbours only and includes long and short range order effects. Results are presented for the concentration at different sites on the stepped surface as well as for the short range order parameters as functions of temperature.     

Segregation at alloy surfaces

We present a theory for segregation at alloy surfaces, where both the bond breaking and strain theories have been combined into one. The formulation can be applied to alloys of any concentration. In addition to it we present some ideas which should be considered while dealing with polycrystalline samples. Good agreement is obtained with Auger electron spectroscopy data on a Ni-Au system.     

Segregation of iron inCuAuFe alloys

TheCuAuFe system has been studied by room temperature Mössbauer spectroscopy in order to examine the influence of varying both iron (Cu_94?yAu₆Fe_y; y=0.2–1.0 at%) and gold (Cu_99?xAu_xFe₁; x=6–50.7 at%) concentrations on the clustering of iron atoms. Samples were examined in different metallurgical states, as rolled, fast quenched and melt spun with similar degrees of clustering being observed. The isomer shift is found to vary inversely with atomic volume for the Cu_99?xAu_xFe₁ alloys.     

Segregation of copper in the ITER cable and ramp rate limitation

The quench of a superconducting cable containing a defect and subjected to a current increase is studied. The cable is represented by a model which consists of a strand with a local decrease in the critical current I_c and the rest of the cable lumped together. It is assumed that the quench takes place when the electric field in the defect reaches the instability level. An important feature of the process is the noticeable portion of current expelled from the defect to the rest of the cable prior to the quench. It is shown that given the strong inequality in the strand currents during the ramp, the influence of the defect on the quench current is proportional to I_c⁴. This criterion is applied to the problem of copper separation in the triplets of the ITER cables.     

Segregation instabilities of moving interfaces

The segregation of solute particles on a moving interface leads to the appearance of two types of instabilities near competing velocity thresholds. This behavior is shown to occur in a variety of exactly solvable models where the interface motion is coupled to a diffusion process of the solute particles. These models directly apply to the propagation of internal domain walls, but can also be generalized to surfaces of growing crystals in the kinetics-limited regime.     

Segregation of evaporated gallium in laser doped silicon

Silicon wafers of n-type covered with a thin evaporated layer of pure gallium have been exposed to Q-switched Nd : YAG laser pulses (32 ns; 280, 410, $\text{560}\text{MW}\text{cm}{}^2$). Rutherford backscattering measurements with a 3 MeV He⁺ beam show that a significant fraction of the gallium is incorporated in the silicon substrate. A drastic segregation accumulates the Ga atoms near the surface.