Phase transitions of McKean–Vlasov processes in double-wells landscape
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.
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: 04 March 2014
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