Properties of the diffusion equation
Éric Akkermans, Gilles Montambaux · Cambridge University Press eBooks · 2007
Introduction In Chapter 4, we established general expressions for the probability of quantum diffusion P d ( r , r′ , ω) and for the structure factor Γ ω ( r , r ). With the aid of these quantities, which are solutions of the integral equations (4.24) and (4.25), we can describe all the physical phenomena studied in this book. It is therefore useful to present these solutions for commonly encountered geometries. Moreover, we have shown that for an infinite medium, in the regime of slow variations, P d and Γ ω are solutions of a diffusion equation. In this chapter, we shall study the solutions of this equation for certain geometries. The validity of the diffusion equation will be discussed for the cases of infinite and semi-infinite media in Appendices A5.1 and A5.3. We shall take particular interest in the Laplace transform P γ ( r , r′ ) of the probability P ( r , r′ , t ). This measures the sum of the contributions to the probability from multiple scattering trajectories between r and r′ for times less than 1/γ. From this quantity we shall define a characteristic time, the recurrence time , which describes the total time spent around an arbitrary point in the medium. It depends on the space dimensionality and on the geometry of the system. For a finite system, typically a cube of side L , there is a natural characteristic time scale τ D defined by L 2 = D τ D . It separates the short time regime, where the role of boundary conditions may be neglected, from the long time regime where these conditions become essential. The inverse of this time defines the so-called Thouless frequency 1/τ D or the Thouless energy E c = ħ/τ D .