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

Abstract

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::

• phase transitions
• McKean–Vlasov diffusions
• invariant probabilities
• interacting particles systems
• free-energy

2000 Ams Classification number::

• primary 60H10
• secondary 60G10
• 60J60
• 65C05
• 82C22

Acknowledgements

Most of the ideas of this work have been found while I was at the Institut Élie Cartan in Nancy. And so I wanted to mention that I would not have been able to write it without the hospitality I received from the beginning especially from Samuel Herrmann. I also would like to thank Florent Malrieu for one of the remarks he made to me on Wednesday 23 June 2010 and which provided me the ideas of the convexity method. I also thank Jesper for the english revision. Vrlo velika hvala Marini za sve (Many thanks also to Marina for everything). Finalement, un très grand merci à Manue et à Sandra pour tout (Finally, a very big thank you to Manue and to Sandra for everything). This work was supported by the DFG-funded CRC 701, Spectral Structures and Topological Methods in Mathematics at the University of Bielefeld.