Effective diffusion in the Focker-Planck equation
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 \(\mu\) \(\to 0\).
asymptotic of spectra of the Focker-Planck operator
scientific article; zbMATH DE number 4126135
scientific article