Kinetic equation for Smoluchowski's diffusion

Not scheduled
15m
poster

Speaker

Ryszard Wojnar (Instytut Podstawowych Problemów Techniki Polskiej Akademii Nauk)

Description

Smoluchowski's diffusion, [1], it is a diffusion in an external field is studied, and a general kinetic equation valid within the framework of classical statistical mechanics is found. $ $ We apply Stecki's projection operator method, [2], to obtain a kinetic equation for an auxiliary distribution function f(k,v_1,t). This function yields the intermediate scattering function I_s(k,t), which is a Fourier transform of van Hove's function G_s(r,t). The function G_s(r,t) gives the probability of finding in a equilibrium system a given (marked) particle at (r,t), if it was known to be at the position r=0 at the time t=0. A kinetic equation of the convolution type for the time evolution of the distribution function f(k,v_1,t) is given. $ $ Diffusion problems studied in the time scale comparable with time of particles collision lead to kinetic equations which for step-wise potentials are functional equations in the velocity space. $ $ We start, as usually in statistical physics from Liouville's theorem, which is a key theorem in classical statistical and Hamiltonian mechanics. The gas is composed of N particles. We pay a particular attention to Lorentz’ gas with N-1 immovable particles-scatterers, between which a light particle number 1 is moving, and to Brownian diffusion (one heavy–marked particle among N-1 light particles). The second case is a natural generalization of Smoluchowski’s equation. $ $ [1] Subrahmanyan Chandrasekhar, Mark Kac, Roman Smoluchowski, Marian Smoluchowski - his life and scientific work, ed. by R. S. Ingarden, Seria: Polish Men of Science, PWN - Polish Scientific Publishers, Warszawa 1999. [2] Jan Stecki, On the kinetic equation nonlocal in time for the generalized self-diffusion process, Journal of Computational Physics 7 (3) 547–553 (1971).

Primary author

Ryszard Wojnar (Instytut Podstawowych Problemów Techniki Polskiej Akademii Nauk)

Presentation Materials

There are no materials yet.
Your browser is out of date!

Update your browser to view this website correctly. Update my browser now

×