Journal of Statistical Physics

, Volume 159, Issue 4, pp 972–986 | Cite as

Existence of Ground States of Nonlocal-Interaction Energies

Article

Abstract

We investigate which nonlocal-interaction energies have a ground state (global minimizer). We consider this question over the space of probability measures and establish a sharp condition for the existence of ground states. We show that this condition is closely related to the notion of stability (i.e. \(H\)-stability) of pairwise interaction potentials. Our approach uses the direct method of the calculus of variations.

Keywords

Ground states Global minimizers H-stability Pair potentials Self-assembly Aggregation 

Mathematics Subject Classification

49J45 82B21 82B05 35R09 45K05 

Notes

Acknowledgments

The authors are thankful to Rustum Choksi for initiating this collaboration and to José Antonio Carrillo for a valuable discussion. The authors would like to thank the Center for Nonlinear Analysis of the Carnegie Mellon University for its support, and hospitality during IT’s visit. RS was supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the Carnegie Mellon Portugal Program under Grant SFRH/BD/33778/2009. DS is grateful to NSF (Grant DMS-1211760) and FCT (Grant UTA CMU/MAT/0007/2009). IT was also partially supported by the Applied Mathematics Laboratory of the Centre de Recherches Mathématiques. The research was also supported by NSF PIRE Grant OISE-0967140.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Robert Simione
    • 1
  • Dejan Slepčev
    • 1
  • Ihsan Topaloglu
    • 2
  1. 1.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA
  2. 2.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

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