Journal of Statistical Physics

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

Existence of Ground States of Nonlocal-Interaction Energies



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.


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

Mathematics Subject Classification

49J45 82B21 82B05 35R09 45K05 



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.


  1. 1.
    Au Yeung, Y., Friesecke, G., Schmidt, B.: Minimizing atomic configurations of short range pair potentials in two dimensions: crystallization in the Wulff shape. Calc. Var. Partial Differ. Equ. 44(1–2), 81–100 (2012)Google Scholar
  2. 2.
    Balagué, D., Carrillo, J.A., Laurent, T., Raoul, G.: Dimensionality of local minimizers of the interaction energy. Arch. Ration. Mech. Anal. 209, 1055–1088 (2013)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Balagué, D., Carrillo, J.A., Laurent, T., Raoul, G.: Nonlocal interactions by repulsive–attractive potentials: radial ins/stability. Phys. D 260, 5–25 (2013)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Balagué, D., Carrillo, J.A., Yao, Y.: Confinement for attractive–repulsive kernels. Discret. Contin. Dyn. S. (to appear)Google Scholar
  5. 5.
    Benedetto, D., Caglioti, E., Pulvirenti, M.: A kinetic equation for granular media. RAIRO Modél. Math. Anal. Numér. 31(5), 615–641 (1997)MATHMathSciNetGoogle Scholar
  6. 6.
    Bernoff, A.J., Topaz, C.M.: A primer of swarm equilibria. SIAM J. Appl. Dyn. Syst. 10(1), 212–250 (2011)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Bertozzi, A.L., Carrillo, J.A., Laurent, T.: Blow-up in multidimensional aggregation equations with mildly singular interaction kernels. Nonlinearity 22(3), 683–710 (2009)CrossRefADSMATHMathSciNetGoogle Scholar
  8. 8.
    Bertozzi, A.L., Laurent, T.: Finite-time blow-up of solutions of an aggregation equation in \(\mathbf{R}^n\). Commun. Math. Phys. 274(3), 717–735 (2007)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Bertozzi, A.L., Laurent, T., Flavien, L.: Aggregation and spreading via the Newtonian potential: the dynamics of patch solutions. Math. Model. Methods Appl. Sci. 22(Supp. 1), 1140005 (2012)CrossRefGoogle Scholar
  10. 10.
    Bertozzi, A.L., Laurent, T., Rosado, J.: \({L}^p\) theory for the multidimensional aggregation equation. Commun. Pur. Appl. Math. 64(1), 45–83 (2011)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Billingsley, P.: Weak Convergence of Measures: Applications in Probability. Society for Industrial and Applied Mathematics, Philadelphia (1971)CrossRefMATHGoogle Scholar
  12. 12.
    J.A. Cañizo, Carrillo, J.A. and Patacchini F.S.: Existence of compactly supported global minimizers for the interaction energy. preprint, 2014Google Scholar
  13. 13.
    Calogero, F., Simonov, YuA: Simple upper bound to the ground-state energy of a many-body system and condition on the two-body potential necessary for its stability. Phys. Rev. 183, 869–872 (1969)CrossRefADSGoogle Scholar
  14. 14.
    Carrillo, J.A., Chipot, M., and Huang Y.: On global minimizers of repulsive-attractive power-law interaction energies. preprint, 2014Google Scholar
  15. 15.
    Carrillo, J.A., Delgadino, M.G., and Mellet, A.: Regularity of local minimizers of the interaction energy via obstacle problems. preprint, 2014Google Scholar
  16. 16.
    Carrillo, J.A., DiFrancesco, M., Figalli, A., Laurent, T., Slepčev, D.: Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations. Duke Math. J. 156(2), 229–271 (2011)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Carrillo, J.A., DiFrancesco, M., Figalli, A., Laurent, T., Slepčev, D.: Confinement in nonlocal interaction equations. Nonlinear Anal. 75(2), 550–558 (2012)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Carrillo, J.A., McCann, R.J., Villani, C.: Contractions in the 2-Wasserstein length space and thermalization of granular media. Arch. Ration. Mech. Anal. 179(2), 217–263 (2006)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Choksi, R., Fetecau, R.C., and Topaloglu, I.: On minimizers of interaction functionals with competing attractive and repulsive potentials. Ann. Inst. H. Poincaré Anal. Non Linéaire.( to appear)Google Scholar
  20. 20.
    Dobrushin, R.: Investigation of conditions for the asymptotic existence of the configuration integral of Gibbs’ distribution. Theory Probab. Appl. 9(4), 566–581 (1964)CrossRefGoogle Scholar
  21. 21.
    Fellner, K., Raoul, G.: Stable stationary states of non-local interaction equations. Math. Model. Methods Appl. Sci. 20(12), 2267–2291 (2010)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Fetecau, R.C., Huang, Y.: Equilibria of biological aggregations with nonlocal repulsive–attractive interactions. Phys. D 260, 49–64 (2013)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Fetecau, R.C., Huang, Y., Kolokolnikov, T.: Swarm dynamics and equilibria for a nonlocal aggregation model. Nonlinearity 24(10), 2681–2716 (2011)CrossRefADSMATHMathSciNetGoogle Scholar
  24. 24.
    Fisher, Michael E., Ruelle, David: The stability of many-particle systems. J. Math. Phys. 7, 260–270 (1966)CrossRefADSMathSciNetGoogle Scholar
  25. 25.
    Haile, J.M.: Molecular Dynamics Simulation: Elementary Methods. Wiley, New York (1992)Google Scholar
  26. 26.
    Holm, D.D., Putkaradze, V.: Aggregation of finite-size particles with variable mobility. Phys. Rev. Lett. 95, 226106 (2005)CrossRefADSGoogle Scholar
  27. 27.
    Holm, D.D., Putkaradze, V.: Formation of clumps and patches in selfaggregation of finite-size particles. Phys. D. 220(2), 183–196 (2006)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Kolokolnikov, T., Huang, Y., Pavlovski, M.: Singular patterns for an aggregation model with a confining potential. Phys. D 260, 65–76 (2013)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Kolokolnikov, T., Sun, H., Uminsky, D., Bertozzi, A.L.: A theory of complex patterns arising from 2D particle interactions. Phys. Rev. E, Rapid. Phys. Rev. E Rapid Commun. 84, 015203(R) (2011)CrossRefADSGoogle Scholar
  30. 30.
    Laurent, T.: Local and global existence for an aggregation equation. Commun. Partial Differ. Equ. 32(10–12), 1941–1964 (2007)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Leverentz, A.J., Topaz, C.M., Bernoff, A.J.: Asymptotic dynamics of attractive–repulsive swarms. SIAM J. Appl. Dyn. Syst. 8(3), 880–908 (2009)CrossRefADSMATHMathSciNetGoogle Scholar
  32. 32.
    Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The locally compact case. I. Ann. Inst. H. Poincaré Anal. Non Linéaire. 1(2):109–145 (1984)Google Scholar
  33. 33.
    Mogilner, A., Edelstein-Keshet, L.: A non-local model for a swarm. J. Math. Biol. 38, 534–570 (1999)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Radin, C.: The ground state for soft disks. J. Stat. Phys. 26(2), 365–373 (1981)CrossRefADSMathSciNetGoogle Scholar
  35. 35.
    Ruelle, D.: Statistical Mechanics: Rigorous Results. W. A. Benjamin Inc., New York-Amsterdam (1969)MATHGoogle Scholar
  36. 36.
    Struwe, M.: Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 3rd edn. Springer-Verlag, Berlin (2000)CrossRefGoogle Scholar
  37. 37.
    Sütő, A.: Ground state at high density. Commun. Math. Phys. 305(3), 657–710 (2011)CrossRefADSGoogle Scholar
  38. 38.
    Theil, F.: A proof of crystallization in two dimensions. Commun. Math. Phys. 262(1), 209–236 (2006)CrossRefADSMATHMathSciNetGoogle Scholar
  39. 39.
    Toscani, G.: One-dimensional kinetic models of granular flows. M2AN Math. Model. Numer. Anal. 34(6), 1277–1291 (2000)CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    van der Vaart, A.W., Wellner, J.: Weak Convergence and Empirical Processes: With Applications to Statistics. Springer Series in Statistics. Springer, New York (1996)CrossRefGoogle Scholar
  41. 41.
    von Brecht, J.H., Uminsky, D., Kolokolnikov, T., Bertozzi, A.L.: Predicting pattern formation in particle interactions. Math. Model. Methods Appl. Sci. 22(Suppl. 1), 1140002 (2012)CrossRefMathSciNetGoogle Scholar

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

Personalised recommendations