 99171 Patrick Grosfils,Jean Pierre Boon,E.G.D. Cohen,L.A. Bunimovich
 Propagation and organization in lattice random media
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May 14, 99

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Abstract. We show that a signal can propagate in a particular direction through
a model random medium regardless of the precise state of the medium.
As a prototype, we consider a point particle moving on a onedimensional
lattice whose sites are occupied by scatterers with the following
properties:
(i) the state of each site is defined by its {\em spin} (up or down);
(ii) the particle arriving at a site is scattered forward (backward)
if the spin is up (down);
(iii) the state of the site is modified by the passage of the particle,
i.e. the spin of the site where a scattering has taken place, flips
($\uparrow \Leftrightarrow \downarrow $).
We consider one dimensional and triangular lattices, for which we
give a microscopic description of the dynamics, prove the propagation
of a particle through the scatterers, and compute analytically its
statistical properties. In particular we prove that, in one dimension,
the average propagation velocity is $\langle c(q) \rangle = 1/(32q)$,
with $q$ the probability that a site has a spin $\uparrow$, and, in
the triangular lattice, the average propagation velocity is independent
of the scatterers distribution: $\langle c \rangle = 1/8$. In both
cases, the origin of the propagation is a blocking mechanism,
restricting the motion of the particle in the direction opposite to
the ultimate propagation direction, and there is a specific
reorganization of the spins after the passage of the particle.
A detailed mathematical analysis of this phenomenon is, to the
best of our knowledge, presented here for the first time.
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