2009/10/24 by L. Chayes, Lincoln Chayes, V. Panferov +1
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Bounded function #Critical exponent #Geometry #Isotropy #Limit (mathematics) #Markov Chains and Monte Carlo Methods #Mathematical Biology Tumor Growth #Mathematical analysis #Mathematical physics #Mathematics #Phase transition #Physics #Quantum mechanics #Scaling #Scaling limit #Statistical physics #Torus #math-ph #math.MP #msc:35Q82 #msc:82C26
paper · pdf · doi:10.1007/s10955-009-9913-z
arxiv created 2009/10/24 · openalex publication_date 2010/01/07 · arxiv updated 2015/05/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We study the McKean–Vlasov equation on the finite tori of length scale L in d-dimensions. We derive the necessary and sufficient conditions for the existence of a phase transition, which are based on the criteria first uncovered in Gates and Penrose (Commun. Math. Phys. 17:194–209, 1970) and Kirkwood and Monroe (J. Chem. Phys. 9:514–526, 1941). Therein and in subsequent works, one finds indications pointing to critical transitions at a particular model dependent value, θ ♯ of the interaction parameter. We show that the uniform density (which may be interpreted as the liquid phase) is dynamically stable for θ<θ ♯ and prove, abstractly, that a critical transition must occur at θ=θ ♯ . However for this system we show that under generic conditions—L large, d≥2 and isotropic interactions—the phase transition is in fact discontinuous and occurs at some θT<θ\sharp . Finally, for H-stable, bounded interactions with discontinuous transitions we show that, with suitable scaling, the θT(L) tend to a definitive non-trivial limit as L→∞.