Journal of Statistical Physics

, Volume 138, Issue 1–3, pp 351–380 | Cite as

The McKean–Vlasov Equation in Finite Volume

Open Access
Article

Abstract

We study the McKean–Vlasov equation on the finite tori of length scale L in d-dimensions. We derive the necessary and sufficient conditions for the existence of a phase transition, which are based on the criteria first uncovered in Gates and Penrose (Commun. Math. Phys. 17:194–209, 1970) and Kirkwood and Monroe (J. Chem. Phys. 9:514–526, 1941). Therein and in subsequent works, one finds indications pointing to critical transitions at a particular model dependent value, θ of the interaction parameter. We show that the uniform density (which may be interpreted as the liquid phase) is dynamically stable for θ<θ and prove, abstractly, that a critical transition must occur at θ=θ. However for this system we show that under generic conditions—L large, d≥2 and isotropic interactions—the phase transition is in fact discontinuous and occurs at some \(\theta_{\text{T}}<\theta^{\sharp }\) . Finally, for H-stable, bounded interactions with discontinuous transitions we show that, with suitable scaling, the \(\theta_{\text{T}}(L)\) tend to a definitive non-trivial limit as L→∞.

Keywords

Phase transitions Mean-field approximation Kirkwood–Monroe equation H-stability 

References

  1. 1.
    Bertozzi, A.L., Carrillo, J.A., Laurent, T.: Blowup in multidimensional aggregation equations with mildly singular interaction kernels. Nonlinearity 22, 683–710 (2009) MATHCrossRefMathSciNetADSGoogle Scholar
  2. 2.
    Bertozzi, A.L., Laurent, T.: The behavior of solutions of multidimensional aggregation equations with mildly singular interaction kernels. Chin. Ann. Math. (2010, to appear) Google Scholar
  3. 3.
    Bolley, F., Guillin, A., Malrieu, F.: Trend to equilibrium and particle approximation for a weakly self-consistent Vlasov–Fokker–Planck equation. Preprint (2009). arXiv:0906.1417
  4. 4.
    Buttà, P., Lebowitz, J.L.: Local mean field models of uniform to nonuniform density fluid-crystal transitions. J. Phys. Chem. B 109, 6849–6854 (2005) CrossRefGoogle Scholar
  5. 5.
    Buttà, P., Lebowitz, J.L.: Hydrodynamic limit of Brownian particles interacting with short and long range forces. J. Stat. Phys. 94(3/4), 653–694 (1999) MATHCrossRefGoogle Scholar
  6. 6.
    Carlen, E.A., Carvalho, M.C., Esposito, R., Lebowitz, J.L., Marra, R.: Free energy minimizers for a two-species model with segregation and liquid-vapor transition. Nonlinearity 16, 1075–1105 (2003) MATHCrossRefMathSciNetADSGoogle Scholar
  7. 7.
    Carrillo, J.A., D’Orsogna, M.R., Panferov, V.: Double milling in self-propelled swarms from kinetic theory. Kinet. Relat. Models 2(2), 363–378 (2009) CrossRefMathSciNetGoogle Scholar
  8. 8.
    Chuang, Y.-L., D’Orsogna, M.R., Marthaler, D., Bertozzi, A.L., Chayes, L.: State transitions and the continuum limit for a 2d interacting, self-propelled particle system. Physica D 232, 33–47 (2007) MATHCrossRefMathSciNetADSGoogle Scholar
  9. 9.
    Constantin, P.: The Onsager equation for corpora. J. Comput. Theor. Nanosci. (2010, to appear). arXiv:0803.4326
  10. 10.
    D’Orsogna, M.R., Chuang, Y.-L., Bertozzi, A.L., Chayes, L.S.: Self-propelled particles with soft-core interactions: patterns, stability, and collapse. Phys. Rev. Lett. 96, 104302 (2006) CrossRefADSGoogle Scholar
  11. 11.
    Esposito, R., Guo, Y., Marra, R.: Phase transition in a Vlasov–Boltzmann binary mixture (2009). arXiv:0904.0791v1 [math-ph]
  12. 12.
    Gates, D.J.: Rigorous results in the mean-field theory of freezing. Ann. Phys. 71, 395–420 (1972) CrossRefADSGoogle Scholar
  13. 13.
    Gates, D.J., Penrose, O.: The van der Waals limit for classical systems III. Deviation from the van der Waals–Maxwell theory. Commun. Math. Phys. 17, 194–209 (1970) CrossRefMathSciNetADSGoogle Scholar
  14. 14.
    Grewe, N., Klein, W.: The Kirkwood–Salsburg equations for a bounded stable kac potential II. Instability and phase transitions. J. Math. Phys. 18(9), 1735–1740 (1977) CrossRefADSGoogle Scholar
  15. 15.
    Haskovec, J., Schmeiser, C.: Stochastic particle approximation for measure valued solutions of the 2D Keller–Segel system. J. Stat. Phys. 135(1), 133–151 (2009) MATHCrossRefMathSciNetADSGoogle Scholar
  16. 16.
    Kac, M.: On the partition function of a one-dimensional gas. Phys. Fluids 87, 8–12 (1959) CrossRefADSGoogle Scholar
  17. 17.
    Kac, M., Uhlenbeck, G.E., Hemmer, P.C.: On the van der Waals theory of the vapor-liquid equilibrium. I. Discussion of a one-dimensional model. J. Math. Phys. 2(1), 216–228 (1963) CrossRefMathSciNetADSGoogle Scholar
  18. 18.
    van Kampen, N.G.: Condensation of a classical gas with long-range attraction. Phys. Rev. 135, A362–A369 (1964) CrossRefGoogle Scholar
  19. 19.
    Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970) CrossRefGoogle Scholar
  20. 20.
    Kirkwood, J.G., Monroe, E.J.: Statistical mechanics of fusion. J. Chem. Phys. 9, 514–526 (1941) CrossRefADSGoogle Scholar
  21. 21.
    Lebowitz, J., Penrose, O.: Rigorous treatment of the van der Waals–Maxwell theory of the liquid-vapour transition. J. Math. Phys. 7, 98–113 (1966) CrossRefMathSciNetADSGoogle Scholar
  22. 22.
    Levine, H., Rappel, W.J., Cohen, I.: Self-organization in systems of self-propelled particles. Phys. Rev. E 63, 017101 (2000) CrossRefADSGoogle Scholar
  23. 23.
    Malrieu, F.: Logarithmic Sobolev inequalities for some nonlinear PDE’s. Stoch. Process. Appl. 95(1), 109–132 (2001) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Manzi, G., Marra, R.: Phase segregation and interface dynamics in kinetic systems. Nonlinearity 19, 115–147 (2006) MATHCrossRefMathSciNetADSGoogle Scholar
  25. 25.
    Martzel, N., Aslangul, C.: Mean-field treatment of the many-body Fokker–Planck equation J. Phys. A Math. Gen. 34, 11225–11240 (2001) MATHCrossRefMathSciNetADSGoogle Scholar
  26. 26.
    McKean, H.P. Jr.: Propagation of chaos for a class of non-linear parabolic equations. In: Stochastic Differential Equations. Lecture Series in Differential Equations, vol. 7, pp. 41–57 (1967) Google Scholar
  27. 27.
    Ruelle, D.: Statistical Mechanics, Rigorous Results. Benjamin, New York (1969) MATHGoogle Scholar
  28. 28.
    Shigeo, S., Tamura, Y.: Gibbs measures for mean field potentials. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31, 223–245 (1984) MATHMathSciNetGoogle Scholar
  29. 29.
    Stevens, A.: The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems. SIAM J. Appl. Math. 61(2), 183–212 (2000) MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Sznitman, A.S.: Topics in propagation of chaos. In: École d’Été de Probabilités de Saint-Flour. Lecture Notes in Math., vol. 1464, pp. 165–251. Springer, Berlin (1991) Google Scholar
  31. 31.
    Tamura, Y.: On asymptotic behaviors of the solution of a nonlinear diffusion equation. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31, 195–221 (1984) MATHMathSciNetGoogle Scholar
  32. 32.
    Thompson, C.J.: Validity of mean-field theories in critical phenomena. Prog. Theor. Phys. 87, 535–559 (1992) CrossRefADSGoogle Scholar
  33. 33.
    Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics, vol. 58. AMS, Providence (2003) MATHGoogle Scholar

Copyright information

© The Author(s) 2010

Authors and Affiliations

  1. 1.Department of MathematicsUCLALos AngelesUSA
  2. 2.Department of MathematicsCSUNNorthridgeUSA

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