Schell Model Research Articles

Grating interferometer for light-efficient spatial coherence measurement of arbitrary sources.

We present a theoretical analysis and experimental verification of a z-scanning double-grating interferometer for spatial coherence measurements in space-frequency and space-time domains. This interferometer permits the measurement of spatial coherence between an arbitrary pair of points along a one-dimensional line, and in favorable conditions, it has a high light efficiency compared to the classical Young's two-pinhole experiment. The scheme is applicable to both quasi-monochromatic and broadband sources that need not obey the Schell model. We first provide experimental results with several narrowband primary and secondary sources, and then apply the technique to broadband sources with discrete and continuous spectra. In the latter case, the complex degree of (time-domain) spatial coherence is retrieved from spectrally resolved measurements using the Friberg-Wolf theorem [Opt. Lett.20, 623 (1995)OPLEDP0146-959210.1364/OL.20.000623]. We compare all results to those obtained with Young's interferometer realized using a digital micromirror device.

Twisted partially coherent bosonic matter wave and its orbital angular momentum

Orbital angular momentum (OAM) is a general characteristic of coherent bosonic matter waves. Partially coherent bosonic matter waves (PCMWs) can also have OAM according to a vortex or twisted structure of the correlation phase of the first-order correlation function. We theoretically prove that a twist phase exists under partial coherence, and this phase contributes to the OAM flux of the PCMW field. We also present an example which illustrates that a twist phase can be produced by illuminating a Gaussian Schell model cold atomic cloud with a sequence of cylindrical lens pulses.

The coherent radiation fraction of low-emittance synchrotrons.

In this work the coherence properties of the synchrotron radiation beam from an X-ray undulator in a fourth-generation storage ring are analyzed. A slightly focused X-ray beam is simulated using a wavefront propagation through a non-redundant array of slits and the mutual coherence function is directly obtained and compared with the Gaussian-Schell approximation. The numerical wave propagation and the approximate analytical approaches are shown to agree qualitatively, and it is also shown that, when the coherent fraction is selected by a finite aperture before the focusing element, even achromatic focusing systems like total reflection mirrors become slightly chromatic. This effect is also well accounted for in the Gaussian-Schell model. The wavefront propagation simulation through the non-redundant array was repeated with an imperfect mirror demonstrating that, although the wavefront is distorted, its coherent length is practically unchanged.

School segregation and the identification of tipping behavior

We develop a method to empirically implement the Schelling model of segregation, and use it to study racial segregation in Los Angeles schools from 1995 to 2012. Our two-step method combines the estimation of parents' preferences for their children's peers with a counterfactual simulation analysis. We find substantial heterogeneity in the existence and locations of tipping points and stable equilibria. Schools are observed on equilibrium trajectories, but many remained out of equilibrium by 2012. We also introduce novel instrumental variables to identify preferences for endogenous peer groups that require no additional data and can be used in other educational settings.

Modeling random screens for predefined electromagnetic Gaussian-Schell model sources.

In a previous paper [Opt. Express22, 31691 (2014)] two different wave optics methodologies (phase screen and complex screen) were introduced to generate electromagnetic Gaussian Schell-model sources. A numerical optimization approach based on theoretical realizability conditions was used to determine the screen parameters. In this work we describe a practical modeling approach for the two methodologies that employs a common numerical recipe for generating correlated Gaussian random sequences and establish exact relationships between the screen simulation parameters and the source parameters. Both methodologies are demonstrated in a wave-optics simulation framework for an example source. The two methodologies are found to have some differing features, for example, the phase screen method is more flexible than the complex screen in terms of the range of combinations of beam parameter values that can be modeled. This work supports numerical wave optics simulations or laboratory experiments involving electromagnetic Gaussian Schell-model sources.

EMERGENT DENSE SUBURBS IN A SCHELLING METAPOPULATION MODEL: A SIMULATION APPROACH

The Schelling model describes the formation of spatially segregated clusters starting from individual preferences based on tolerance. To adapt this framework to an urban scenario, characterized by several individuals sharing very close physical spaces, we propose a metapopulation version of the Schelling model defined on the top of a regular lattice whose cells can be interpreted as a bunch of buildings or a district containing several agents. We assume the model to contain two kinds of agents relocating themselves if their individual utility is smaller than a tolerance threshold. While the results for large values of the tolerances respect the common sense, namely coexistence is the rule, for small values of the latter we obtain two non-trivial results: first we observe complete segregation inside the cells, second the population redistributes highly heterogeneously among the available places, despite the initial uniform distribution. The system thus converges toward a complex heterogeneous configuration after a long quasi-stationary transient period, during which the population remains in a well mixed phase. We identify three possible global spatial regimes according to the tolerance value: microscopic clusters with local coexistence of both kinds of agents, macroscopic clusters with local coexistence (hereafter called soft segregation) and macroscopic clusters with local segregation but homogeneous densities (hereafter called hard segregation).

Thomas C. Schelling and James M. Sakoda: The Intellectual, Technical, and Social History of a Model

The Journal of Mathematical Sociology (JMS) started in 1971. The second issue contained its most cited article: Thomas C. Schelling, “Dynamic Models of Segregation†. In that article, Schelling presented a family of models, one of which became a canonical model. To date it is called the Schelling model—an eponym that affixes the inventor’s name to the invention, one of the highest forms of scientific recognition. In the very first issue of JMS, James Minoru Sakoda published an article entitled “The Checkerboard Model of Social Interaction†. Sakoda’s article more or less went unrecognized. Yet, a careful comparison demonstrates that in a certain sense the Schelling model is just an instance of Sakoda’s model. A precursor of that model was already part of Sakoda’s 1949 dissertation submitted to the University of California at Berkeley. A substantial amount of evidence indicates that in the 1970s Sakoda was well known and recognized as a computational social scientist, whereas Schelling was an unknown in the field. A generation later, the pattern of recognition almost completely reversed: Sakoda had become the unknown, while Schelling was the well-known inventor of the pioneering Schelling model. This article explains this puzzling pattern of recognition. Technical and social factors play a decisive role. Some contrafactual historical reflection suggests that the final result was not inevitable.

Measurement of the spatial coherence of hard synchrotron radiation using a pencil beam

We present a simple method to measure the spatial coherence of hard x-ray beams. Based on the convolution of Gaussian functions, we analyze the diffraction patterns of a grating irradiated by partially coherent hard x rays with a constrained beam diameter. The spatial coherence properties of an x-ray beam are obtained from the width of the diffraction peaks with high accuracy. The results of experiments conducted by combining a pinhole with a grating show a good agreement with our calculation using the Gaussian–Schell model.

Dynamics of Transformation from Segregation to Mixed Wealth Cities.

We model the dynamics of a variation of the Schelling model for agents described simply by a continuously distributed variable—wealth. Agent movement is not dictated by agent choice as in the classic Schelling model, but by their wealth status. Agents move to neighborhoods where their wealth is not lesser than that of some proportion of their neighbors, the threshold level. As in the case of the classic Schelling model, we find here that wealth-based segregation occurs and persists. However, introducing uncertainty into the decision to move—that is, with some probability, if agents are allowed to move even though the threshold condition is contravened—we find that even for small proportions of such disallowed moves, the dynamics no longer yield segregation but instead sharply transition into a persistent mixed wealth distribution, consistent with empirical findings of Benenson, Hatna, and Or. We investigate the nature of this sharp transformation, and find that it is because of a non-linear relationship between allowed moves (moves where threshold condition is satisfied) and disallowed moves (moves where it is not). For small increases in disallowed moves, there is a rapid corresponding increase in allowed moves (before the rate of increase tapers off and tends to zero), and it is the effect of this non-linearity on the dynamics of the system that causes the rapid transition from a segregated to a mixed wealth state. The contravention of the tolerance condition, sanctioning disallowed moves, could be interpreted as public policy interventions to drive de-segregation. Our finding therefore suggests that it might require limited, but continually implemented, public intervention—just sufficient to enable a small, persistently sustained fraction of disallowed moves so as to trigger the dynamics that drive the transformation from a segregated to mixed equilibrium.

Evolution properties of multi-Gaussian Schell model beams propagating in uniaxial crystal orthogonal to the optical axis

Evolution properties of multi-Gaussian Schell model beams propagating in uniaxial crystal orthogonal to the optical axis

Unperturbed Schelling Segregation in Two or Three Dimensions

Schelling's model of segregation, first described in 1969, has become one of the best known models of self-organising behaviour. While Schelling's explicit concern was to understand the mechanisms underlying racial segregation in large cities from a game theoretic perspective, the model should be seen as one of a family, arising in fields as diverse as statistical mechanics, neural networks and the social sciences, and which are concerned with interacting populations situated on network structures. Despite extensive study, however, the (unperturbed) Schelling model has largely resisted rigorous analysis, prior results in the literature generally pertaining to variants of the model in which noise is introduced into the dynamics of the system, the resulting model then being amenable to standard techniques from statistical mechanics or stochastic evolutionary game theory. A series of recent papers (one by Brandt, Immorlica, Kamath, and Kleinberg, and two by the authors), has seen the first rigorous analysis of the one dimensional version of the unperturbed model. Here we provide the first rigorous analysis of the two and three dimensional unperturbed models, establishing most of the phase diagram, and answering a challenge from a recent paper by Brandt, Immorlica, Kamath, and Kleinberg.

Spatial attraction in migrants' settlement patterns in the city of Catania

(ProQuest: ... denotes formulae omitted.)1. IntroductionResidential location influences individuals' proximity to important resources (such as schools, hospitals, child care facilities, labor markets, and employment opportunities) and to potential risks, including environmental threats and social hazards (such as exposure to crime and violence) (Reardon 2006). Furthermore, it impacts access to social networks and other forms of social capital; overall, it shapes human interaction and the demographic processes that originate from it, such as mortality, fertility and mobility (Almquist and Butts 2012).A minority ethnic group is spatially clustered when the spatial arrangement of minority households departs from expectations based upon a random spatial allocation (Freeman, Pilger, and Alexander 1971).In broad terms, and apart from ethnic discriminatory rules enforced by law or traditions in some places and at some times, we may distinguish between two sources of spatial clustering. One source is spatial inhomogeneity or apparent contagion. Typically, the different parts of a city exhibit large variations in the price of residential property, in the accessibility of low cost public infrastructures, and in the availability of certain types of jobs; these inhomogeneities may lead to a mostly economically induced segregation. As Schelling (1971) observes, ethnicity is often correlated with income, and income with residence; so even if residential choices were unconstrained by ethnic discrimination, the different ethnic groups would not be randomly distributed among residences.The second source is spatial attraction or true contagion. Survey data on the ideal neighborhood composition for different ethnic groups in the USA, reported in Clark and Fossett (2008), show that all groups prefer living in areas where their group is a majority or near-majority. These preferences have complex origins and may reflect attachment to group identity and culture (e.g., language, religion, customs, etc.). Newly arrived minority migrants may benefit from positive spillovers in settling close to their compatriots, in terms of reciprocal acceptance, common language, and support. Transnational social networks play an important role in channeling arriving migrants into specific neighborhoods and also into particular occupations (Gelderblom and Adams 2006).However, regardless of what the basis of the individual preferences for coethnic contact is, they produce identical patterns of residential segregation (Clark and Fossett 2008). The Schelling (1971) model provides an analysis of the implications of individual preferences and shows that when a household enters a neighborhood, that neighborhood becomes more attractive to members of the household's own group and less attractive to members of other groups. In other words, the presence of a household in a given area increases the probability of others of the same group to locating nearby.It is relevant in social research to be able to distinguish between these two sources of clustering. Whereas economic induced segregation might explain some initial degree of segregation and raises questions of social equity, the Schelling model highlights the importance of individually motivated segregation and posits that even mild preferences for living with similar neighbors carry the potential of being strong determinants for residential segregation (Clark and Fossett 2008). The spatial distribution of households may be represented by a point pattern, i.e.,a set of points in a map. Ripley's K-function (Ripley 1981) is widely used to detect clustering in point processes. The inhomogeneous K-function is a version of Ripley's K-function conceived for assessing the effects of spatial attraction (or inhibition), while adjusting for the effects of spatial inhomogeneity. In other words, this approach allows us to distinguish between the two sources of clustering, by assessing clustering above and beyond that due to apparent contagion. …

Average intensity and directionality of partially coherent model beams propagating in turbulent ocean.

We studied Gaussian beams with three different partially coherent models, including the Gaussian-Schell model (GSM), Laguerre-Gaussian Schell model (LGSM), and Bessel-Gaussian Schell model (BGSM), propagating through oceanic turbulence. The expressions of average intensity, beam spreading, and beam wander for GSM, LGSM, and BGSM beams in the paraxial channel are derived. We make a contrast for the three models in numerical simulations and find that the GSM beam has smaller spreading than the others, and the LGSM beam needs longer propagation distance to transform into a well-like profile of average intensity than the BGSM beam in the same conditions. The salinity fluctuation has a greater contribution to the wander of LGSM and BGSM beams than that of the temperature fluctuation. Our results can be helpful in the design of an optical wireless communication link operating in oceanic environment.

Spreading and wandering of Gaussian–Schell model laser beams in an anisotropic turbulent ocean

The effect of anisotropic turbulence on the spreading and wandering of Gaussian–Schell model (GSM) laser beams propagating in an ocean is studied. The long-term spreading of a GSM beam propagating through the paraxial channel of a turbulent ocean is also developed. Expressions of random wander for such laser beams are derived in an anisotropic turbulent ocean based on the extended Huygens–Fresnel principle. We investigate the influence of parameters in a turbulent ocean on the beam wander and spreading. Our results indicate that beam spreading and random beam wandering are smaller without considering the anisotropy of turbulence in the oceanic channel. Salinity fluctuation has a greater contribution to both the beam spreading and beam wander than that of temperature fluctuations in a turbulent ocean. Our results could be helpful for designing a free-space optical wireless communication system in an oceanic environment.

On spectral and temporal coherence of x-ray free-electron laser beams.

A model for the coherence properties of free-electron lasers (FELs) in time and frequency domains is introduced within the framework of classical second-order coherence theory of nonstationary light. An iterative phase-retrieval algorithm is applied to construct an ensemble of field realizations in both domains, based on single-pulse spectra measured at the Linac Coherent Light Source (LCLS) in self-amplified spontaneous emission mode. Such an ensemble describes the specific FEL pulse train in a statistically averaged sense. Two-time and two-frequency correlation functions are constructed, demonstrating that the hard X-ray free-electron laser at LCLS in this case behaves as a quasistationary source with low spectral and temporal coherence. We also show that the Gaussian Schell model provides a good description of this FEL.

Can topology reshape segregation patterns?

We consider a metapopulation version of the Schelling model of segregation over several complex networks and lattices. We show that the segregation process is topology independent and hence it is intrinsic to the individual tolerance. The role of the topology is to fix the places where the segregation patterns emerge. In addition we address the question of the time evolution of the segregation clusters, resulting from different dynamical regimes of a coarsening process, as a function of the tolerance parameter. We show that the underlying topology may alter the early stage of the coarsening process, once large values of the tolerance are used, while for lower ones a different mechanism is at work and it results to be topology independent.

Diversity and Community Can Coexist.

We examine the (in)compatibility of diversity and sense of community by means of agent-based models based on the well-known Schelling model of residential segregation and Axelrod model of cultural dissemination. We find that diversity and highly clustered social networks, on the assumptions of social tie formation based on spatial proximity and homophily, are incompatible when agent features are immutable, and this holds even for multiple independent features. We include both mutable and immutable features into a model that integrates Schelling and Axelrod models, and we find that even for multiple independent features, diversity and highly clustered social networks can be incompatible on the assumptions of social tie formation based on spatial proximity and homophily. However, this incompatibility breaks down when cultural diversity can be sufficiently large, at which point diversity and clustering need not be negatively correlated. This implies that segregation based on immutable characteristics such as race can possibly be overcome by sufficient similarity on mutable characteristics based on culture, which are subject to a process of social influence, provided a sufficiently large "scope of cultural possibilities" exists.

A state equation for the Schelling’s segregation model

An aspatial version for the famous Schelling’s segregation model has recently been proposed, which is still in an agent-based format like the original Schelling model. In the present study, we propose a new, state equation version of the Schelling model. The new equation is based on the aspatial version and is derived in terms of a set of discrete maps. Fixed point solutions for the new equation are found analytically and confirmed numerically. Especially, we show that the extremely simple state equations can reasonably reveal the essence of the Schelling dynamics: integration, segregation and tipping. In addition to the fixed point solutions, periodic solutions were also identified and conditions of the limit cycles were derived analytically.

Continuous utility factor in segregation models.

We consider the constrained Schelling model of social segregation in which the utility factor of agents strictly increases and nonlocal jumps of the agents are allowed. In the present study, the utility factor u is defined in a way such that it can take continuous values and depends on the tolerance threshold as well as the fraction of unlike neighbors. Two models are proposed: in model A the jump probability is determined by the sign of u only, which makes it equivalent to the discrete model. In model B the actual values of u are considered. Model A and model B are shown to differ drastically as far as segregation behavior and phase transitions are concerned. In model A, although segregation can be achieved, the cluster sizes are rather small. Also, a frozen state is obtained in which steady states comprise many unsatisfied agents. In model B, segregated states with much larger cluster sizes are obtained. The correlation function is calculated to show quantitatively that larger clusters occur in model B. Moreover for model B, no frozen states exist even for very low dilution and small tolerance parameter. This is in contrast to the unconstrained discrete model considered earlier where agents can move even when utility remains the same. In addition, we also consider a few other dynamical aspects which have not been studied in segregation models earlier.

House exchange and residential segregation in networks

This paper considers a Schelling model in an arbitrary fixed network where there are no vacant houses. Agents have preferences either for segregation or for mixed neighborhoods. Utility is non-transferable. Two agents exchange houses when the trade is mutually beneficial. We find that an allocation is stable when for two agents of opposite-color each black (white) agent has a higher proportion of neighbors who are black (white). This result holds irrespective of agents’ preferences. When all members of both groups prefer mixed neighborhoods, an allocation is also stable provided that if an agent belongs to the minority (majority), then any neighbor of opposite-color is in a smaller minority (larger majority).