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Phase transitions of McKean–Vlasov processes in double-wells landscape

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The aim of this work is to establish the results for a particular class of inhomogeneous processes, the McKean–Vlasov diffusions. Such diffusions correspond to the hydrodynamical limit of an interacting particle system. In convex landscapes, existence and uniqueness of the invariant probability is a well-known result. However, previous results state the nonuniqueness of the invariant probabilities under nonconvexity assumptions. Here, we prove that there exists a phase transition. Below a critical value, there are exactly three invariant probabilities and above another critical value, there is exactly one. Under simple assumptions, these critical values coincide and it is characterized by a simple implicit equation. We also investigate other cases in which phase transitions occur. Finally, we provide numerical estimations of the critical values.
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Keywords: 60J60; 65C05; 82C22; McKean–Vlasov diffusions; free-energy; interacting particles systems; invariant probabilities; phase transitions; primary 60H10; secondary 60G10

Document Type: Research Article

Affiliations: Fakultät für Mathematik, Universität Bielefeld, D-33615,Bielefeld, Germany

Publication date: March 4, 2014

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