Effective diffusion in the Focker-Planck equation
type: Item
instance of: scholarly article
title: Effective diffusion in the Focker-Planck equation
zbMATH Open document ID: 0687.47039
DOI: 10.1007/BF01157929
published in: Mathematical Notes
publication date: 1989-01-01T00:00:00Z
review text: The paper deals with the Focker-Planck diffusion equation: \[ (\partial /\partial t){\mathcal P}=A^*{\mathcal P},\quad A^*=\mu \Delta_ p+\nabla_ q\cdot H_ p+\alpha \nabla_ pH_ p-\nabla_ p\cdot H_ q, \] where \(\mu\) is the molecular diffusion, \(\alpha\) the friction, \(H=p^ 2/2m+U(q)\) hamiltonian, V(q) a periodic potential. An investigation of general properties of spectra of such operator is made as well as full homogenization theory. The problem of the behavior of effective diffusion \(\sigma\) (\(\mu)\) when \(\mu\) \(\to 0\), \(n=1\) is also under consideration. With the means of WKB theory it is found the leading term of the exponentially small \(\sigma\) (\(\mu)\) and made a comparison with the same object in Einstein-Smolukhovskij approximation. Under the same assumptions it is found an asymptotic of spectra of the Focker-Planck operator on \(T^ n\times {\mathbb{R}}^ n\) when \(\mu\) \(\to 0\).
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 35J99
Mathematics Subject Classification ID: 35K99
zbMATH DE Number: 4126135
zbMATH Keywords: Focker-Planck diffusion equation
zbMATH Keywords: molecular diffusion
zbMATH Keywords: homogenization
zbMATH Keywords: effective diffusion
zbMATH Keywords: WKB theory
zbMATH Keywords: Einstein-Smolukhovskij approximation
zbMATH Keywords: asymptotic of spectra of the Focker-Planck operator
author: Serguei M. Kozlov
MaRDI profile type: Publication
cites work: Q4329425
cites work: The transfer of radiation in stellar atmospheres
cites work: Boundary layers and homogenization of transport processes
cites work: Q3746966
cites work: Ohrnstein—uhlenbeck process in a random potential
cites work: Q3721536
cites work: Geometric aspects of averaging
cites work: ON THE ANALYTICITY OF SOLUTIONS OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS
cites work: Brownian motion in a field of force and the diffusion model of chemical reactions
cites work: On The Asymptotic Equivalence of the Fokker-Planck and Diffusion Equations
cites work: Q3818026
cites work: Q3896271
full work available at URL: https://doi.org/10.1007/bf01157929
OpenAlex ID: W1981596834
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description: scientific article; zbMATH DE number 4126135