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Concurrent Coupling of Atomistic and Continuum Models

Mei Xu

in Multiscale Methods: Bridging the Scales in Science and Engineering

This chapter presents a review of concurrent multiscale methods for coupling continuum, molecular, and quantum mechanics with a particular emphasis on the Bridging Domain Method. An expanded taxonomy ... More

This chapter presents a review of concurrent multiscale methods for coupling continuum, molecular, and quantum mechanics with a particular emphasis on the Bridging Domain Method. An expanded taxonomy of multiscale methods is presented which highlights the role and characteristics of concurrent multiscale methods. Several popular 0 K methods are summarized: master-slave coupling, ONIOM, the Bridging Domain Method, the Bridging Scale Method and the Quasicontinuum method. The topic of ghost forces in the Bridging Domain Method is studied and the stability properties of Lagrange multiplier coupling methods are reviewed. Several methods for coupling molecular dynamics with continua are described: master-slave and handshake methods, the Bridging Domain Method, and the Bridging Scale Method. The conservation properties of the Bridging Domain Method are demonstrated. Numerical examples are presented which focus on the reduction of spurious phonon reflections from the continuum/molecular interfaces in the Bridging Domain Method. Examples of the simulation of cracks and defects in graphene using a modified ONIOM method to couple continuum, molecular, and quantum subdomain are also given.Less

Introduction

Neil Tennant

in Changes of Mind: An Essay on Rational Belief Revision

This chapter distinguishes logic as a theory of belief‐statics from our sought account of belief dynamics. The various kinds of belief change are classified. These are: surrendering, adopting or ... More

This chapter distinguishes logic as a theory of belief‐statics from our sought account of belief dynamics. The various kinds of belief change are classified. These are: surrendering, adopting or switching individual beliefs; and thereby contracting, expanding or revising one’s system of beliefs. Our account of the epistemic norms involved is agent-centric. The idealized figure of the logical paragon (as opposed to the completely fictional figure of the ‘logical saint’) is introduced as the guiding model of a rational agent who is thoroughly competent in matters of belief change. The chapter discusses what a theory of belief change needs to characterize or make feasible. Two key constraints are formulated: both minimal mutilation and minimal bloating of systems of belief undergoing contractions and revisions needs to be explicated (and ensured). The explicit goal is to provide a computationally implementable account of belief change. The chapter foreshadows welcome results to be proved about the computational complexity of the contraction problem. It stresses that our account of belief dynamics will be able to cope with differences among different schools, or ‘‐isms’, in epistemology, regarding permissible global patterns of support or justification among beliefs. The chapter includes an important discussion of methodology, invoking the contrast between propositional and first-order logic as a case study, in order to highlight the virtues of simplicity in formal modeling. This chapter promises to be an account of belief change under judiciously chosen simplifying assumptions that nevertheless allow a rich structure to come into focus, and challenging problems to emerge.Less

A Simple Equilibrium in Performance

Geoffrey Brennan and Philip Pettit

in The Economy of Esteem: An Essay on Civil and Political Society

How A scores in the esteem stakes depends on the assessment of A’s performance relative to standards. The higher the standards the greater the incentive to perform well, other things equal. But ... More

How A scores in the esteem stakes depends on the assessment of A’s performance relative to standards. The higher the standards the greater the incentive to perform well, other things equal. But standards themselves reflect prevailing practice. So there is a feedback loop between performance and standards in any estimable domain. The aim of this chapter is to explore this feedback loop by constructing a simple equilibrium in the performance domain. Even if esteem is not ‘positional’ in the sense that it is awarded on the basis of one’s ranking, esteem exhibits zero-sum properties in the simplest cases. This fact raises questions about the normative properties of esteem-seeking behaviour.Less

The Uses of Place

Martin Meisel

in How Plays Work: Reading and Performance

Duration and location are features of performance; time and place are intrinsic to the play. Of the latter two, place would seem the simpler matter, as less essentially ‘dramatic’. The dramatic uses ... More

Duration and location are features of performance; time and place are intrinsic to the play. Of the latter two, place would seem the simpler matter, as less essentially ‘dramatic’. The dramatic uses of time are manifest: in the familiar notion of suspense, for example, where we have to wait and see; in the melodramatic (now cinematic) race against the clock; in the one-thing-leading-to-another succession of events, culminating in revelation or denouement. In contrast, the work of place in the modern theatre tends to be equated with ‘setting’, which suggests torpidity and fixity. This chapter discusses the following topics: statics, dynamics, and place-time.Less

A Very Social Darwinist: Herbert Spencer’s Lamarckian Radicalism

Piers J. Hale

in Political Descent: Malthus, Mutualism, and the Politics of Evolution in Victorian England

Following recent scholarship it is clear that Herbert Spencer was no social Darwinist, rather his Lamarckian politics reflected an appreciation of the radicalism of William Godwin and Erasmus Darwin. ... More

Following recent scholarship it is clear that Herbert Spencer was no social Darwinist, rather his Lamarckian politics reflected an appreciation of the radicalism of William Godwin and Erasmus Darwin. Unlike many Whigs Spencer was ambivalent about Malthus's claim that population would outstrip resources. In his “Theory of Population” Spencer explained that the struggle for existence would drive the development of human intelligence as people exercised and thereby increased their mental capacities. In doing so they would deplete the neurine in their brains. Citing respected contemporary physiologists Spencer believed neurine was the same chemical substance that fuelled sexual desire, thus solving the Malthusian dilemma. In Social Statics and other early works Spencer articulated limited government as the basis of a progressive evolution towards a utopian socialist future, it was only later, when contemporary socialists embraced statist solutions to social problems, that he drew back from these conclusions.Less

Matching by Categories

Pierre-André Chiappori

in Matching with Transfers: The Economics of Love and Marriage

This chapter considers matching by categories, beginning with a discussion of a specific but empirically very relevant family of models in order to provide a richer representation of heterogeneity ... More

This chapter considers matching by categories, beginning with a discussion of a specific but empirically very relevant family of models in order to provide a richer representation of heterogeneity between individuals that would account not only for economic aspects such as income or education, as well as more subjective (and less easily observable) ones, such as idiosyncratic preferences for marriage in general and for specific types of spouses in particular. The chapter explores a simple model that describes matching on income or education, the separability assumption, how separability can be justified, and the dual structure under separability. It also provides an overview of the Choo-Siow model, focusing on its basic structure, the matching function, heteroskedasticity, comparative statics, testability and identifiability, and the Galichon and Salanié's cupid framework as an extension of the model.Less

A Text Rhetorically Designed to Unite Competing Fields

in Shaping Science with Rhetoric: The Cases of Dobzhansky, Schrodinger, and Wilson

This chapter discusses some of the rhetorical maneuvers that can be discovered through a study of Dobzhansky's text in its social and intellectual context. The tools that Dobzhansky used to amplify ... More

This chapter discusses some of the rhetorical maneuvers that can be discovered through a study of Dobzhansky's text in its social and intellectual context. The tools that Dobzhansky used to amplify his explanations were thought experiments and the description of experimental or field reports that corroborated the results of the mathematical theories. The rhetorical strategy that Dobzhansky used to introduce these data is called prolepsis, or “anticipatory refutation,” in which an opposition argument is anticipated and directly countered by the rhetoric. Dobzhansky used prolepsis to bring out into the open, and then refute with data from experiments, the arguments against a new evolutionary synthesis that would be offered by skeptical naturalists and geneticists. The semantic distinctions between statics and dynamics show that the dimensions must be combined to create a complete understanding of biology. The former concerns the forces that bring about change in the genetic composition of a population, the latter the interaction of forces in the formation and disintegration of races and species.Less

Squiggly Economics

Robert G. Chambers

in Competitive Agents in Certain and Uncertain Markets

Three generic economic optimization problems (expenditure (cost) minimization, revenue maximization, and profit maximization) are studied using the mathematical tools developed in Chapters 2 and 3. ... More

Three generic economic optimization problems (expenditure (cost) minimization, revenue maximization, and profit maximization) are studied using the mathematical tools developed in Chapters 2 and 3. Conjugate duality results are developed for each. The resulting dual representations (E(q;y), R(p,x), and π‎(p,q)) are shown to characterize all of the economically relevant information in, respectively, V(y), Y(x), and Gr(≽(y)). The implications of different restrictions on ≽(y) for the dual representations are examined.Less

The Principle of Virtual Work

Jennifer Coopersmith

in The Lazy Universe: An Introduction to the Principle of Least Action

The meaning behind the mysterious Principle of Virtual Work is explained. Some worked examples in statics (equilibrium) are given, and the method of Virtual Work is compared and contrasted with the ... More

The meaning behind the mysterious Principle of Virtual Work is explained. Some worked examples in statics (equilibrium) are given, and the method of Virtual Work is compared and contrasted with the method of Newtonian Mechanics. The meaning of virtual displacements is explained very carefully. They must be ‘small’, happen simultaneously, and do not cause a force, result froma force, or take any time to occur. Counter to intuition, not all the actual displacements can be allowed as virtual displacements. Some examples worked through are: Feynman’s pivoting (cantilever) bar, a “black box,” a weighted spring, a ladder, a capacitor, a soap bubble, and Atwood’s machine. The links between mechanics and geometry are demonstrated, and it is shown how the reaction or constraint forces are always perpendicular to the virtual displacements. Lanczos’s Postulate A and its astounding universality are explained.Less

Rigid Beams and the Foundations of Physics

Jeffrey K. McDonough

in A Miracle Creed: The Principle of Optimality in Leibniz's Physics and Philosophy

This chapter explores Leibniz’s ingenious treatment of the breaking strength of rigid bodies. In 1684, in a seminal paper published in the Acta Eruditorum, Leibniz offered a model of the behavior of ... More

This chapter explores Leibniz’s ingenious treatment of the breaking strength of rigid bodies. In 1684, in a seminal paper published in the Acta Eruditorum, Leibniz offered a model of the behavior of rigid bodies that crucially takes it for granted that rigid beams must satisfy his Principle of Optimality. In doing so, he successfully arrived at an improved formula for calculating the strength of a cubic beam. Beyond its scientific results, however, Leibniz’s approach to the problem of determining the breaking strength of rigid beams had profound implications for the structure and nature of matter itself. This chapter draws out the implications of Leibniz’s account for his views on the nature of forces and material bodies. It concludes by considering the implications of Leibniz’s thinking about forces and bodies for his understanding of the relationship between monads and space.Less

Hanging Chains and Monadic Agency

Jeffrey K. McDonough

in A Miracle Creed: The Principle of Optimality in Leibniz's Physics and Philosophy

This chapter looks to Leibniz’s work on catenaries—freely hanging cords suspended at both ends—for insight into his understanding of monadic agency. After discussing Leibniz’s understanding of ... More

This chapter looks to Leibniz’s work on catenaries—freely hanging cords suspended at both ends—for insight into his understanding of monadic agency. After discussing Leibniz’s understanding of catenaries as instances of optimal form, the chapter argues that catenaries provide an especially elegant model for Leibniz’s understanding of monadic agency, a model that helps to resolve otherwise outstanding problems in Leibniz’s thinking about how immaterial mind-like substances may unfold in a goal-directed manner. As a whole, the chapter aims to show how specific features of optimal form may be invoked to provide Leibniz’s views on monadic agency a coherence and elegance that they would otherwise lack.Less

Falling Bodies and the Rise of Variational Mechanics

Jeffrey K. McDonough

in A Miracle Creed: The Principle of Optimality in Leibniz's Physics and Philosophy

This chapter discusses Leibniz’s efforts with respect to the famous problem of the brachistochrone—that is, the problem of determining the shape of the path of quickest descent between two points in ... More

This chapter discusses Leibniz’s efforts with respect to the famous problem of the brachistochrone—that is, the problem of determining the shape of the path of quickest descent between two points in a vertical plane. After setting the historical stage, the chapter argues that the rational mechanics of the eighteenth century and beyond—developed by the likes of Leonhard Euler, Joseph-Louis Lagrange, and William Rowan Hamilton—served to vindicate Leibniz’s optimality approach to problems in optics, mechanics, and statics. The chapter thus aims to show that Leibniz’s Principle of Optimality has proven itself to be a miracle creed not only in virtue of its odd epistemological standing—in its being fully grounded neither in reason nor experience—but also in having “been borne out to an amazing extent by the development of science.”Less

The rise of mv²

Robert T. Hanlon

in Block by Block: The Historical and Theoretical Foundations of Thermodynamics

Galileo connected the distance of a body’s free fall with the increase in the square of the body’s speed. This experimental finding was later used by Huygens and then Leibniz to establish mv2 as an ... More

Galileo connected the distance of a body’s free fall with the increase in the square of the body’s speed. This experimental finding was later used by Huygens and then Leibniz to establish mv2 as an important quantity in physics.Less

Equilibrium models and explanation

Lawrence A. Boland

in Equilibrium Models in Economics: Purposes and Critical Limitations

This chapter examines the explanatory purpose of building equilibrium models and the need to consider dynamics and disequilibria. It examines Marshall’s two ‘Principles’ of explanation, the ... More

This chapter examines the explanatory purpose of building equilibrium models and the need to consider dynamics and disequilibria. It examines Marshall’s two ‘Principles’ of explanation, the ‘Principle of Substitution’ (essentially the usual neoclassical premise that every decision maker is a maximizer) and the ‘Principle of Continuity’ (that using the assumption of maximization as a basis for explanation is not possible without a continuous range of options to choose among). Marshall’s main mode of explanation using these Principles is his comparative statics analysis. His version of comparative statics introduces a role for time by distinguishing his long-run from short-run equilibria. With this in mind, the chapter goes further to explain why an equilibrium state implies recognition of disequilibrium dynamics and why equilibrium models must recognize the knowledge necessary for the dynamics of equilibrium attainment.Less

The Parallelogram of Forces and the Autonomy of Statics

Marc Lange

in Because Without Cause: Non-Causal Explanations in Science and Mathematics

This chapter concludes the account of “explanation by constraint,” an important variety of non-causal explanation. It shows how that account applies to the longstanding controversy regarding the ... More

This chapter concludes the account of “explanation by constraint,” an important variety of non-causal explanation. It shows how that account applies to the longstanding controversy regarding the explanation of the law of nature that governs the composition of forces (the “parallelogram of forces”). Various rival proposed explanations of this law are presented. One rival is Isaac Newton’s and William Thomson’s dynamical explanation. Another rival is a non-causal “explanation by constraint” from statics that appeals to symmetry principles and that was defended by Simeon Denis Poisson, Willliam Whewell, and James Clerk Maxwell. The chief questions addressed are what makes these proposals incompatible rivals (as physicists take them to be) and what would make one of these rivals rather than another correct. The account given in previous chapters of counterfactuals reflecting stronger and weaker grades of necessity underlying explanations by constraint nicely applies to this case. This account of explanations by constraint is compared to accounts of “structural explanations.”Less

The Rise of the Human Sciences

Christopher J. Berry

in Scottish Philosophy in the Eighteenth Century, Volume I: Morals, Politics, Art, Religion

This chapter examines a key focal characteristic of the Scottish Enlightenment, namely, its delineation of how a ‘science of man’ can inform and structure an account of ‘society’. The key ... More

This chapter examines a key focal characteristic of the Scottish Enlightenment, namely, its delineation of how a ‘science of man’ can inform and structure an account of ‘society’. The key contribution of the Scots to the rise of the human sciences lies in a conception of society as a set of interlocked institutions and behaviours. The Scots provided an analysis of both social statics and social dynamics, which shifted the focus away from the individualism that characterized early modern jurisprudence. Humans as social beings are best understood in society and not as monadic individuals. The Scottish analysis also sidelined the centrality traditionally allotted to the political. Humans are social as well as political animals. Political institutions are simply one kind of institution among several, to be given no greater priority than the rest.Less

Fluid Statics and Buoyancy

David Jon Furbish

in Fluid Physics in Geology: An Introduction to Fluid Motions on Earth’s Surface and Within its Crust

Fluid statics concerns the behavior of fluids that possess no linear acceleration within a global (Earth) coordinate system. This includes fluids at rest as well as fluids possessing steady motion ... More

Fluid statics concerns the behavior of fluids that possess no linear acceleration within a global (Earth) coordinate system. This includes fluids at rest as well as fluids possessing steady motion such that no net forces exist. Such motions may include steady linear motion within the global coordinate system as well as rotation with constant angular velocity about a fixed vertical axis. In this latter case, centrifugal forces must be balanced by centripetal forces (which arise, for example, from a pressure gradient acting toward the axis of rotation). Moreover, we assert that no relative motion between adjacent fluid elements exists. Fluid motion, if present, is therefore like that of a rigid body. In addition, we neglect molecular motions that lead to mass transport by diffusion. Thus, the idea of a static fluid is a macroscopic one. The developments in this chapter clarify how pressure varies with coordinate position in a static fluid. Both compressible and incompressible fluids are treated. In the simplest case in which the density of a fluid is constant, we will see that pressure varies linearly with vertical position in the fluid according to the hydrostatic equation. In addition, we will consider the possibility that fluid density is not constant. Then, variations in density must be taken into account when computing the pressure at a given position in a fluid column; the pressure arising from the weight of the overlying fluid no longer varies linearly with depth. In the case of an isothermal fluid, whose temperature is constant throughout, any variation in density must arise purely from the compressible behavior of the fluid in response to variations in pressure. In the case where temperature varies with position, fluid density may vary with both pressure and temperature. We will in this regard consider the case of a thermally stratified fluid whose temperature varies only with the vertical coordinate direction. Because fluid statics requires treating how fluid temperature, pressure, and density are related, the developments below make use of thermodynamical principles developed in Chapter 4. Less