2008/04/21 by Veit Schwämmle, Fernando Nobre, Fernando D. Nobre +1
Chemistry · Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Chemistry #Diffusion equation #Exponential function #Gaussian #Hamiltonian (control theory) #Kurtosis #Mathematical analysis #Mathematical physics #Mathematics #Partial differential equation #Physics #Porosity #Porous medium #Quantum mechanics #Relaxation (psychology) #Spectroscopy and Quantum Chemical Studies #Statistical Mechanics and Entropy #Statistical physics #Statistics #cond-mat.stat-mech
paper · pdf · doi:10.1140/epjb/e2008-00451-y
20 pages, 6 figures
arxiv created 2008/04/21 · openalex publication_date 2008/12/01 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
The stability of q-Gaussian distributions as particular solutions of the linear diffusion equation and its generalized nonlinear form, \pderivP(x,t)t = D \pderiv2 [P(x,t)]2-qx2, the porous-medium equation, is investigated through both numerical and analytical approaches. It is shown that an initial q-Gaussian, characterized by an index qi, approaches the final, asymptotic solution, characterized by an index q, in such a way that the relaxation rule for the kurtosis evolves in time according to a q-exponential, with a relaxation index q\rm rel ≡ q\rm rel(q). In some cases, particularly when one attempts to transform an infinite-variance distribution (qi ≥ 5/3) into a finite-variance one (q<5/3), the relaxation towards the asymptotic solution may occur very slowly in time. This fact might shed some light on the slow relaxation, for some long-range-interacting many-body Hamiltonian systems, from long-standing quasi-stationary states to the ultimate thermal equilibrium state.