The Brownian mean field model

Abstract

Abstract. We discuss the dynamics and thermodynamics of the Brownian Mean Field (BMF) model which is a system of N Brownian particles moving on a circle and interacting via a cosine potential. It can be viewed as the canonical version of the Hamiltonian Mean Field (HMF) model. The BMF model displays a second order phase transition between a homogeneous phase and an inhomogeneous phase below a critical temperature Tc = 1/2. We first complete the description of this model in the mean field approximation valid for N → +∞. In the strong friction limit, the evolution of the density towards the mean field Boltzmann distribution is governed by the mean field Smoluchowski equation. For T < Tc, this equation describes a process of self-organization from a non-magnetized (homogeneous) phase to a magnetized (inhomogeneous) phase. We obtain an analytical expression for the temporal evolution of the magnetization close to Tc. Then, we take fluctuations (finite N effects) into account. The evolution of the density is governed by the stochastic Smoluchowski equation. From this equation, we derive a stochastic equation for the magnetization and study its properties both in the homogenous and inhomogeneous phases. We show that the fluctuations diverge close to the critical point so that the mean field approximation ceases to be valid. Actually, the limits N → +∞ and T → Tc do not commute. The validity of the mean field approximation requires N (T − Tc) → +∞ so that N must be larger and larger as T approaches Tc. We show that the direction of the magnetization changes rapidly close to Tc but its amplitude takes a long time to relax. We also indicate that, for systems with long-range interactions, the lifetime of metastable states scales as e N except close to a critical point. The BMF model shares many analogies with other systems of Brownian particles with long range interactions such as self-gravitating Brownian particles, the Keller-Segel model describing the chemotaxis of bacterial populations, and the Kuramoto model describing the collective synchronization of coupled oscillators.

Kuramoto model of synchronization: equilibrium and nonequilibrium aspects

“…The resulting Brownian mean-field (BMF) model has thus a canonical ensemble dynamics given by [24,25].…”

“…that corresponds to canonical equilibrium, with r st determined self-consistently [25], see equation (50). For σ = 0, the incoherent stationary state is [20] …”

“…With the noise, but without the external torques, the resulting model is the so-called Brownian mean-field (BMF) model [24,25], introduced as a generalization of the celebrated Hamiltonian meanfield (HMF) model that serves as a prototype to study statics and dynamics of long-range interacting systems [26,27]. In recent years, there has been a surge in interest in studies of systems with long-range interactions.…”

Generalized Stochastic Fokker-Planck Equations

“…When finite N effects are taken into account, the deterministic mean field FP equations are replaced by stochastic FP equations that take fluctuations into account [15]. Examples of Brownian systems with long-range interactions include self-gravitating Brownian particles [16], the Brownian mean field (BMF) model [17], and bacterial populations undergoing chemotaxis [18].…”

Origin and growth of the disorder within an ordered state of the spatially extended chemical reaction model

“…В работе [Chavanis, 2014] рассмотрены динамика и термодинамика модели броуновского среднего поля, в [Cēbers, 2014] получено решение кинетических уравнений в MFT для связан-ного через поток поступательного и вращательного движения частиц, в [Kudrnovsky et al, 2014] проведена оценка температуры Кюри в сплавах Fe3Al, Fe3Si, в [Bricmont, Bosch, 2015] показано существование фазовых переходов для вероятностных клеточных автоматов. Кроме того, MFT нашло широкое применение в исследованиях квантовых явлений [Ayik, 2008;Horvath et al, 2008;Graefe et al, 2008;Akerlund et al, 2013;Sowinski, Chhajlany, 2014;Bighin, Salasnich, 2014;Hayami, Motome, 2015;Hinschberger et al, 2015;Leeuw et al, 2015;Ishihara, Nasu, 2015;Rosati et al, 2014;Serreau, Volpe, 2014;Vermersch, Garreau, 2015;Yilmaz et al, 2014;Yilmaz et al, 2015], в частности для описания динамики ядер [Ayik, 2008], получения динамики среднего поля на сфере Блоха для эрмитова и неэрмитова случаев [Graefe et al, 2008], изучения свойств обобщенной модели Бозе-Хаббарда, включающей трехмерные взаи-модействия при нулевой температуре [Sowinski, Chhajlany, 2014], получения наиболее обоб-щенных уравнений эволюции, описывающих распространение в среде (анти)нейтрино [Serreau, Volpe, 2014], исследования эволюции системы взаимодействия ультрахолодных бо-зонов, которые демонстрируют нелинейное хаотическое поведение в пределе большого числа частиц [Vermersch, Garreau, 2015].…”