# One-dimensional classical diffusion in a random force field with weakly concentrated absorbers

### Christophe Texier 1, Christian Hagendorf 2

#### Europhysics Letters (EPL) 86 (2009) 37011

A one-dimensional model of classical diffusion in a random force field with a weak concentration $\rho$ of absorbers is studied. The force field is taken as a Gaussian white noise with $\mean{\phi(x)}=0$ and $\mean{\phi(x)\phi(x')}=g \delta(x-x')$. Our analysis relies on the relation between the Fokker-Planck operator and a quantum Hamiltonian in which absorption leads to breaking of supersymmetry. Using a Lifshits argument, it is shown that the average return probability is a power law $\smean{P(x,t|x,0)}\sim{}t^{-\sqrt{2\rho/g}}$ (to be compared with the usual Lifshits exponential decay $\exp{-(\rho^2t)^{1/3}}$ in the absence of the random force field). The localisation properties of the underlying quantum Hamiltonian are discussed as well.

• 1. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
CNRS : UMR8626 – Université Paris XI - Paris Sud
• 2. Laboratoire de Physique Théorique de l'ENS (LPTENS),
CNRS : UMR8549 – Université Paris VI - Pierre et Marie Curie – Ecole Normale Supérieure de Paris - ENS Paris