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Equality of the Jellium and Uniform Electron Gas next-order asymptotic terms for Coulomb and Riesz potentials

[Moved Online] Hot Topics: Optimal transport and applications to machine learning and statistics May 04, 2020 - May 08, 2020

May 06, 2020 (11:00 AM PDT - 12:00 PM PDT)
Speaker(s): Codina Cotar (University College)
Location: MSRI: Online/Virtual
Tags/Keywords
  • Coulomb and Riesz gases

  • N-marginals optimal transport with Coulomb and Riesz costs

  • next order term

  • Hohenberg-Kohn functional

  • optimal Lieb-Oxford bound

  • Jellium

  • Uniform Electron Gas

  • equality of next-order constants

  • Fefferman-Gregg decomposition

  • Abrikosov conjecture

  • screening

  • multi-scale decomposition

  • Density Functional Theory (DFT)

Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC
Video

Equality Of The Jellium And Uniform Electron Gas Next-Order Asymptotic Terms For Coulomb And Riesz Potentials

Abstract

We consider two sharp next-order asymptotics problems, namely the asymptotics for the minimum energy for optimal point configurations and the asymptotics for the many-marginals Optimal Transport, in both cases with Riesz costs with inverse power-law long range interactions. The first problem describes the ground state of a Coulomb or Riesz gas, while the second appears as a semiclassical limit of the Density Functional Theory energy modelling a quantum version of the same system. Recently the second-order term in these expansions was precisely described, and corresponds respectively to a Jellium and to a Uniform Electron Gas model. The present work shows that for inverse-power-law interactions with power s∈[d−2,d) in d dimensions, the two problems have the same minimum value asymptotically. For the Coulomb case in d=3, our result verifies the physicists' long-standing conjecture regarding the equality of the second-order terms for these two problems. Furthermore, our work implies that, whereas minimum values are equal, the minimizers may be different. Moreover, provided that the crystallization hypothesis in d=3 holds, which is analogous to Abrikosov's conjecture in d=2, then our result verifies the physicists' conjectured ≈1.4442 lower bound on the famous Lieb-Oxford constant. Our work also rigorously confirms some of the predictions formulated by physicists, regarding the optimal value of the Uniform Electron Gas second-order asymptotic term. We also show that on the whole range s∈(0,d), the Uniform Electron Gas second-order constant is continuous in s.

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Equality Of The Jellium And Uniform Electron Gas Next-Order Asymptotic Terms For Coulomb And Riesz Potentials

H.264 Video 928_28395_8323_Equality_of_the_Jellium_and_Uniform_Electron_Gas_Next-Order_Asymptotic_Terms_for_Coulomb_and_Riesz_Potentials.mp4
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