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Fig. 3.
Phase diagram for the damage found numerically by considering the evolution
starting from uncorrelated configurations with initial density equal to 0.5.
domain is stable regardless of the initial density. On the line
p
2
= 0 the critical
point of the density and that of the damage spreading coincide.
3.2 Reconstruction of the Potential
An important point in the study of systems exhibiting absorbing states is the
formulation of a coarse-grained description using a
Langevin equation
. It is gen-
erally accepted that the directed percolation universal behavior is represented
by
∂c
(
x,t
)
∂t
=
ac
(
x,t
)
− bc
2
(
x,t
)+
∇
2
c
(
x,t
)+
c
(
x,t
)
α
(
x,t
)
,
where
c
(
x,t
) is the density field,
a
and
b
are control parameters and
α
is a Gaus-
sian noise with correlations
α
(
x,t
)
α
(
x
,t
)
=
δ
x,x
δ
t,t
. The diffusion coe
9
cient
has been absorbed into the parameters
a
and
b
and the time scale.
It is possible to introduce a zero-dimensional approximation to the model by
averaging over the time and the space, assuming that the system has entered a
metastable state. In this approximation, the size of the original systems enters
through the renormalized coe
9
cients
a
,
b
,
∂c
(
x,t
)
∂t
=
ac
(
x,t
)
− bc
2
(
x,t
)+
c
(
x,t
)
α
(
x,t
)
,
where also the time scale has been renormalized.
The associated Fokker-Planck equation is
∂
2
∂c
2
cP
(
c,t
)
,
∂P
(
c,t
)
∂t
=
−
∂
∂c
(
ac − bc
2
)
P
(
c,t
)+
1
2
