2009/07/11 by Marcelo R. Ubriaco · 1 citation
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Anomalous diffusion #Differential equation #Differential operator #Entropy (arrow of time) #Fisher information #Fractional Differential Equations Solutions #Function (biology) #Mathematical analysis #Mathematical physics #Mathematics #Mean squared displacement #Measure (data warehouse) #Physics #Quantum mechanics #Statistical Mechanics and Entropy #Statistical physics #Statistics #cond-mat.stat-mech
paper · pdf · doi:10.1016/j.physleta.2009.08.064
13 pages,3 figures
arxiv created 2009/07/11 · openalex publication_date 2009/09/04 · arxiv updated 2015/05/13 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Starting with the relative entropy based on a previously proposed entropy function Sq[p]=∫ dx p(x)(-ln p(x))q, we find the corresponding Fisher's information measure. After function redefinition we then maximize the Fisher information measure with respect to the new function and obtain a differential operator that reduces to a space coordinate second derivative in the q→ 1 limit. We then propose a simple differential equation for anomalous diffusion and show that its solutions are a generalization of the functions in the Barenblatt-Pattle solution. We find that the mean squared displacement, up to a q-dependent constant, has a time dependence according to <x2>∼ K1/qt1/q, where the parameter q takes values q=(2n-1)/(2n+1) (superdiffusion) and q=(2n+1)/(2n-1) (subdiffusion), ∀ n≥ 1.