2017/06/19 by M. Morillo, Morillo, M., J. M. Casado +1
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Applied mathematics #Attractor #Bifurcation #Context (archaeology) #Differential equation #Dynamical systems theory #FOS: Physical sciences #Fokker–Planck equation #Langevin dynamics #Langevin equation #Mathematical analysis #Mathematics #Maxima #Maxima and minima #Nonlinear system #Physics #Probability density function #Quantum mechanics #Statistical Mechanics (cond-mat.stat-mech) #Statistical physics #Statistics #Stochastic processes and statistical mechanics #cond-mat.stat-mech #stochastic dynamics and bifurcation
paper · pdf · doi:10.48550/arxiv.1706.05866
published in arXiv (Cornell University) (Cornell University)
arxiv created 2017/06/19 · openalex publication_date 2017/06/19 · arxiv updated 2017/06/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The usual Langevin approach to describe systems driven by noise fails to describe the long time behavior of systems with multiple attractors. The solution of the associated linear Fokker-Planck equation is always unique, even though it might show one or more maxima. In this context, it is customary to call transitions to changes in the shape of the equilibrium distribution function and relate the maxima to the attractors. Some years ago, a theory was developed for a system with interacting elements or subunits that, starting from the Langevin description of all the variables, leads to bifurcating \textquotedblleft one-particle\textquotedblright distribution functions when the number of elements tends to infinity. In this paper, a mean-field hypothesis has been used to deal with systems with a finite number of elements. We carry out numerical simulations yielding bifurcation solutions for the probability density of a collective variable. We also compare the results of the mean-field hypothesis with those obtained with the Langevin approach for finite systems.