- 15-29 Stefania Patrizi and Enrico Valdinoci
- Relaxation times for atom dislocations in crystals
Mar 31, 15
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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
The goal of this paper is to provide accurate
estimates on the relaxation times of the system after collision.
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.