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boundary is made of. In those circles the boundary
condition is time–dependent on a periodic schedule,
more precisely the positive time axis is subdivided
in disjoint intervals (periodic cycles) of equal length
and any of such intervals is subdived into two disjoint
parts. The boundary condition on the circles is then
assumed to be homogeneous Dirichlet into the first
part of each of these time intervals and homogeneous
Neumann in the second part. (More general shapes
than circles are actually considered in (Andreucci and
Bellaveglia, 2012).)
If the field is interpreted as the density of indi-
viduals in the playgroud, the boundary condition in
(Andreucci and Bellaveglia, 2012) can be described
as follows: the boundary is always reflecting except
for the small circles which are reflecting only in the
second part of each of the time intervals considered
above, while the individuals are allowed to exit the
playground through these circles in the first part of
each of these intervals. In other words the small cir-
cles are doors of the playground and those doors are
open only in the first part of each of the time intervals.
The time periodic micro–structured boundary
conditions suggest to approach the problem from
the homogenization theory point of view (Bensous-
san et al., 1978). With this approach in (Andreucci
and Bellaveglia, 2012) it is proven that, provided the
length of the open time is suitably small with respect
to the length of the cycle, the ratio between the out-
going flux and the field on the small circles (the door)
is not trivial, in the sense that it tends to a real num-
ber when the length of the periodic cycles tends to
zero. This constant ratio is explicitely computed in
(Andreucci and Bellaveglia, 2012) and is proven to
depend on the way in which each time interval is sub-
divided into two parts, that is to say on the length of
the open door and on that of the closed door time sub–
intervals. This result is, in this context, an answer
to the question that opened the paper, namely, to the
question about the relation between the outgoing flux
and the local density of individuals close to the exit.
The present paper has a two–fold aim. In the one–
dimensional case we setup a Monte Carlo simulation
aiming to (i) test numerically the homogenizationlim-
iting result (in the spirit for example of (Haynes et al.,
2010)), (ii) compute the ratio between the outgoing
flux and the local density close to the exit when the
length of the periodic cycles is finite.
This project is realized by introducing a one–
dimensional discrete space model on which indepen-
dent particles perform symmetric random walks. The
space is a finite interval on Z with a boundary point
which is reflecting, whereas the other periodically
changes its status from absorbing to reflecting and
viceversa. We tune the parameters so that the dis-
crete and the continuum space models have equiv-
alent behaviors. Moreover, in the thermodynamics
limit, namely, when the number of site of the discrete
space model tends to infinity, the homogeneization re-
sult proven in the framework of the continuum space
model is recovered. This is not proven rigorously, but
it is demonstrated via heuristc arguments and Monte
Carlo simulations.
The paper is organized as follows. In Sec-
tion 2 we summarize the homogenization results
found in (Andreucci and Bellaveglia, 2012) in the
one–dimensional case. In Section 3 the discrete space
model is introduced and its behavior is discussed on
heuristic grounds. This model is studied via Monte
Carlo simulations in Section 4, where all the numeri-
cal results are discussed. Section 5 is finally devoted
to some brief conclusions.
2 A CONTINUUM SPACE MODEL
In this section we approach the problem via a con-
tinuum space model. We summarize, in the one–
dimensional case, the results found in (Andreucci and
Bellaveglia, 2012). We first introduce the mathemati-
cal model and then discuss its physical interpretation.
Pick the two reals τ ≥ σ ≥ 0, the integer m, and
the function u
0
∈ L
2
([0,L]). Set T = (m + 1)τ and
consider the boundary value problem consisting in the
diffusion equation
u
t
−Du
xx
= 0 on (0,L) ×(0,T) (1)
with D > 0 the diffusion coefficient, the initial condi-
tion
u(x,0) = u
0
(x) ∀x ∈(0,L) (2)
and the boundary conditions
u
x
(0,t) = 0 ∀t ∈[0,T) (3)
and
u(L,t) = 0 ∀t ∈ A and u
x
(L,t) = 0 ∀t ∈C (4)
where
A =
m
τ
[
k=0
[kτ,kτ + σ) and C =
m
τ
[
k=0
[kτ+ σ,kτ+ τ).
According to the discussion in Section 1, the
model above can be interpreted as follows: the field
u is the density of individuals in the playground, m
is the number of the door opening/closing cycles, τ is
the length of each cycle, σ is the length of the time
interval in each cycle during which the door is open,
and, finally, A and C are, respectively, the parts of the
FluxthroughaTime-periodicGate-MonteCarloTestofaHomogenizationResult
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