Let T_L = Z/L Z be the one-dimensional torus with L points. For α>0, let g: N→ R_+ be given by g(0)=0, g(1)=1
Nearby in the stack
,
g(k)=[k/(k−1)]α
,
k≥2
. Consider the totally asymmetric zero range process on
T_L
in which a particle jumps from a site
x
, occupied by
k
particles, to the site
x+1
at rate
g(k)
. Let
N
stand for the total number of particles. In the stationary state, if
α>1
, as
N↑∞
, all particles but a finite number accumulate on one single site. We show in this article that in the time scale
N1+α
the site which concentrates almost all particles evolves as a random walk on
T_L
whose transition rates are proportional to the capacities of the underlying random walk, extending to the asymmetric case the results obtained in bl3 for reversible zero-range processes on finite sets.