Asymptotics of thermal dispersion in periodic media

Using a central limit theorem we will describe a wide class of thermal dispersive phenomena occuring in macrohomogeneous systems. More precisely, we will focus on the diffusions $\bm{X}_t$ generated by an operator $L$ having periodic coefficients. The central limit theorem asserts that $\lambda^{-\frac{1}{2}}(\bm{X}_{\lambda t}-\lambda U_0\ol{\bm{b}}t)$ $t\geq 0$, converges in distribution to Brownian motion as $\lambda\to\infty$. Here $\ol{\bm{b}}$ is the mean of $\bm{b}(\bm{x})$. In the present contribution the functional dependence of the dispersion matrix $\ol{\bms{D}}$ of this limiting Brownian motion on the velocity parameter $U_0$ and the period $a$ is analysed. We will give precise analytical conditions imposed on the geometry of functions $b_i$ which determine the asymptotic behavior of elements $\ol{D}_{ij}$ as functions of $U_0$. Specific examples are given to illustrate computation of the macroscale coefficients as functions of the comparable microscale data.