L-2

Stochastic Interfaces and growth processes

The physical properties of many materials are controlled by the interfaces embedded in it. This is the case of the dislocations in a crystal, the domain walls in a ferromagnet or the vortices in a supercoductors. In the next lecture we will discuss how impurities affect the behviour of these interfaces. Today we focus on thermal fluctuations and introduce two important equations for the interface dynamics: the Edward Wilkinson euqation and the Kardar Parisi Zhang equation.

An interface at Equilibrium: the Edward Wilkinson equation

Consider domain wall $h(r,t)$ fluctuating at equilibrium at the temparature $T$. Here $t$ the time, $r$ defines the d-dimensional coordinate of the interface and $h$ is scalar height field. The domain wall separating two phases in a film has $d=1,r\in {\cal {R}}$, in a solid instead $d=2,r\in {\cal {{R}^{2}}}$. Two assumptions are done:

• Overhangs, pinch-off are neglected, so that $h(r,t)$ is a scalar univalued function.
• The dynamics is overdamped, so that we can neglect the inertial term.

The Langevin equation of motion is