Beyond Gross-Pitaevskii Mean Field Theory

A large number of effects related to the phenomenon of Bose-Einstein Condensation (BEC) can be understood in terms of lowest order mean field theory, whereby the entire system is assumed to be condensed, with thermal and quantum fluctuations completely ignored. Such a treatment leads to the Gross-Pitaevskii Equation (GPE) used extensively throughout this book. Although this theory works remarkably well for a broad range of experimental parameters, a more complete treatment is required for understanding various experiments, including experiments with solitons and vortices. Such treatments should include the dynamical coupling of the condensate to the thermal cloud, the effect of dimensionality, the role of quantum fluctuations, and should also describe the critical regime, including the process of condensate formation. The aim of this Chapter is to give a brief but insightful overview of various recent theories, which extend beyond the GPE. To keep the discussion brief, only the main notions and conclusions will be presented. This Chapter generalizes the presentation of Chapter 1, by explicitly maintaining fluctuations around the condensate order parameter. While the theoretical arguments outlined here are generic, the emphasis is on approaches suitable for describing single weakly-interacting atomic Bose gases in harmonic traps. Interesting effects arising when condensates are trapped in double-well potentials and optical lattices, as well as the cases of spinor condensates, and atomic-molecular coupling, along with the modified or alternative theories needed to describe them, will not be covered here.Comment: Review Article (19 Pages) - To appear in 'Emergent Nonlinear Phenomena in Bose-Einstein Condensates: Theory and Experiment', Edited by P.G. Kevrekidis, D.J. Frantzeskakis and R. Carretero-Gonzalez (Springer Verlag

A theory of antitrust limits

Antitrust doctrine is under heavy fire in the academic literature. Modern criticism of antitrust doctrine attacks three ‘limits’ that would excessively constrain enforcement of the law: (i) the consumer welfare standard, (ii) the rule of reason, and (iii) a self-imposed neglect of labor markets in the exercise of prosecutorial discretion. These limits would be particularly prevalent in monopolization law and, to a lesser extent, in merger law. In calling to relax the limits that constrain antitrust power, modern critics give considerate attention to their foundations. These voices point to a simple reason: ideology. Antitrust limits arguably result from a body of doctrine shaped since the 1970s by opinions favorable to laissez-faire, neo-liberal economics, and the Chicago School of antitrust. This Article proposes that (1) ideology is a lazy answer; (2) the present limits in US and EU antitrust law rest mostly on practical considerations; and (3) the question of setting better limits is empirical and context-dependent. The point is that besides ideology, practicality matters. Unfortunately, the issue of the foundations of antitrust limits has not been systematically treated. This Article addresses this problem by supplying a theory of the limits of antitrust law. Part I shows that the limits of antitrust are not ideological. Instead, the process of setting limits to antitrust law is an inherent development in any system of enforcement and judicial application. To establish this, a careful review of the case-law of the Supreme Court in pre-Chicagoan times is undertaken. Part II then turns to the next, harder question: how are the inherent limits of antitrust law set? The thesis advanced is that the foundations of antitrust limits are diverse, discrediting again the role of ideology but also casting doubt on other mainstream explanations. The study of US and EU antitrust law suggests that multiple institutional, formal, and developmental reasons combined to produce the limits of antitrust. Last, Part III leaves the descriptive world to address a normative question: how to deal with the problem of the obsolescence of the limits of antitrust in contexts of socio-economic change? This Article argues that reforming the limits of antitrust law requires an empirical assessment of the costs of monopoly compared to the costs of judicial errors that goes beyond simplistic presuppositions about the efficiency of market processes and inefficiencies of the legal system. A realistic appraisal of legal institutions, technology, and markets should guide the assessment