A Journey Around the Different Scales Involved in the Description of Matter and Complex Systems by Francisco Chinesta & Emmanuelle Abisset-Chavanne

Author:Francisco Chinesta & Emmanuelle Abisset-Chavanne
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(3.41)

the particle balance, also known as the diffusion equation, is given by

(3.42)

The integration of this equation assuming that all the particles are localized at the origin at the initial time, , leads to

(3.43)

whose second order moment (variance) scales with the time

(3.44)

that is, the mean squared displacement scales with the elapsed time t, and the diffusion coefficient D.

By applying the Fourier transform with respect to the space coordinate to the diffusion equation (3.42), and denoting by the Fourier transform of the particle density , i.e., with k as the so-called wave number, it results that

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