Journal of Statistical Physics

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

The McKean–Vlasov Equation in Finite Volume

Open Access


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→∞.


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


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© The Author(s) 2010

Authors and Affiliations

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

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