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Phase transitions of McKean-Vlasov SDEs in Multi-well Landscapes

Alexander Alecio

Abstract

Phase transitions and critical behaviour of a class of MV-SDEs, whose concomitant non-local Fokker-Planck equation includes the Granular Media equation with quadratic interaction potential as a special case, is studied. By careful analysis of an implicit auxiliary integral equation, it is shown for a wide class of potentials that below a certain `critical threshold' there are exactly as many stationary measures as extrema of the potential, while above another the stationary measure is unique, and consequently phase transition(s) between. For symmetric bistable potentials, these critical thresholds are proven to be equal and a strictly increasing function of the aggregation parameter. Additionally, a simple condition is provided for symmetric multi-well potentials with an arbitrary number of extrema to demonstrate analogous behaviour. This answers, with considerably more generality, a conjecture of Tugaut [Stochastics, 86:2, 257-284]. To the best of our knowledge many of these results are novel. Others simplify the proofs of known results whilst greatly increasing their applicability.

Phase transitions of McKean-Vlasov SDEs in Multi-well Landscapes

Abstract

Phase transitions and critical behaviour of a class of MV-SDEs, whose concomitant non-local Fokker-Planck equation includes the Granular Media equation with quadratic interaction potential as a special case, is studied. By careful analysis of an implicit auxiliary integral equation, it is shown for a wide class of potentials that below a certain `critical threshold' there are exactly as many stationary measures as extrema of the potential, while above another the stationary measure is unique, and consequently phase transition(s) between. For symmetric bistable potentials, these critical thresholds are proven to be equal and a strictly increasing function of the aggregation parameter. Additionally, a simple condition is provided for symmetric multi-well potentials with an arbitrary number of extrema to demonstrate analogous behaviour. This answers, with considerably more generality, a conjecture of Tugaut [Stochastics, 86:2, 257-284]. To the best of our knowledge many of these results are novel. Others simplify the proofs of known results whilst greatly increasing their applicability.
Paper Structure (9 sections, 17 theorems, 61 equations)

This paper contains 9 sections, 17 theorems, 61 equations.

Table of Contents

  1. Set-up
  2. Outline and relation to other works
  3. Generic Multi-well Potentials
  4. Definitions and Assumptions
  5. Symmetric Effective Potentials
  6. Bistable Potential
  7. Multi-Well Potential
  8. Definitions and Assumptions
  9. Conclusion

Key Result

Proposition 3

The self-consistency equation is equivalent to to the first moment evolution equation

Theorems & Definitions (48)

  • Definition 1
  • Definition 2
  • Proposition 3
  • proof
  • Definition 1.1
  • Proposition 1.2: Convergence of $F_\sigma(m)$ (\ref{['sfcn']})
  • proof
  • Corollary 1.3
  • proof
  • Remark 1.4
  • ...and 38 more