Stefania Patrizi and Enrico Valdinoci
Relaxation times for atom dislocations in crystals
(742K, pdf)

ABSTRACT.  We study the relaxation times 
for a parabolic differential equation 
whose solution represents the atom dislocation in a crystal. 
The equation that we consider comprises the classical 
Peierls-Nabarro model as a particular case, 
and it allows also long range interactions. 
It is known that 
the dislocation function of such a model 
has the tendency to concentrate at single points, 
which evolve in time according to 
the external stress and a singular, long range potential. 
Depending on the orientation of the dislocation function 
at these points, the potential may be either attractive or 
repulsive, hence collisions may occur in the latter case 
and, at the collision time, the dislocation function does not 
disappear. 
The goal of this paper is to provide accurate 
estimates on the relaxation times of the system after collision. 
More precisely, 
we take into account the case of two and three colliding points, 
and we show that, after a small transition time subsequent to the 
collision, the dislocation function relaxes exponentially fast 
to a steady state. 
We stress that the exponential decay is somehow exceptional 
in nonlocal problems (for instance, the spatial decay in this case 
is polynomial). The exponential time decay is due to the coupling 
(in a suitable space/time scale) between the evolution term and 
the potential induced by the periodicity of the crystal.