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Mean Field Limits for Interacting Diffusions in a Two-Scale Potential.

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  • 1Department of Mathematics, Imperial College London, London, SW7 2AZ UK.

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Summary
This summary is machine-generated.

The study reveals that mean field and homogenization limits for interacting diffusions do not commute in the long term, impacting stationary state analysis. Bifurcation diagrams differ based on the order of these limits, affecting stability and solution counts.

Keywords:
Bifurcation diagramCurie–Weiss modelDesai–Zwanzig modelInteracting particlesMcKean–Vlasov equationMultiscale diffusionsPhase transitions

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Area of Science:

  • Mathematical Physics
  • Statistical Mechanics
  • Stochastic Processes

Background:

  • Investigating systems of weakly interacting diffusions in complex potentials.
  • Understanding the interplay of mean field and homogenization limits is crucial for analyzing long-term behavior.

Purpose of the Study:

  • To analyze the combined mean field and homogenization limits for diffusions in a two-scale periodic potential.
  • To determine if these limits commute and how their order affects stationary states.

Main Methods:

  • Analysis of weakly interacting diffusions.
  • Mathematical treatment of mean field and homogenization limits.
  • Construction and analysis of bifurcation diagrams for stationary states.

Main Results:

  • The mean field and homogenization limits commute for finite times but not in the long time limit.
  • The order of taking these limits influences the bifurcation diagrams of stationary states.
  • The study characterizes the impact of multiple local minima on solution stability.

Conclusions:

  • The non-commutativity of limits in the long time regime has significant implications for modeling complex systems.
  • The potential landscape's local minima critically affect the number and stability of equilibrium solutions.