Renewal and memory properties in the random growth of surfaces Page: 2
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variable y and in Section IV we study the memory properties of the variable ((t). In Section V we shall point out the
result of this paper.
II. BALLISTIC MODEL
We plan to shed light into the issues mentioned in Section I with the model of Ballistic Deposition (BD), which,
is, in fact, a paradigmatic model for complexity. We shall limit our discussion to the (1+1)-dimension case, and as
a consequence, Fig.1 is fully adequate to illustrate how the model works. At any time step, n 1, 2,... - we select
randomly one of the L columns, and we drop a particle on it. The particles fall down in sequence till to settle either
at the bottom of the column or at the top of an earlier selected particle that by chance fell down in the same column.
However, if the height of one of the two nearest neighbor columns is higher than the selected column, the particle sticks
to the side of the highest particle of this neighbor column. There is also periodic boundary conditions to decrease the
effect of limited size of surface.
Actually, this side sticking action corresponds to a transverse transport of information, through which the column
under study is informed about the height of the surrounding columns. Thus, examining the time evolution of a single
column is equivalent to studying the behavior of a single individual and to assessing to what an extent it reflects the
properties of the whole society. We plan to prove that the cooperation among the different columns generates memory
and this memory generates renewal effects.
It is well known  that the time distance between the arrival of one particle in this column and the next has the
Poisson time distribution
(T) = Aexp(-AT),
Of course, when one particle arrives, the height of the column increase by a quantity that can be also much greater
than 1, thank to the side-sticking effect. We denote by ((t) the height increase of the column, and this quantity is 0
when no particle arrives, and a number equal to 1 or larger when a particle arrives.
X = incoming particles
X' = deposited particles
i=li=2 . . . . . . . . . i=L
FIG. 1: Model of ballistic deposition. The particle B settles at the top of an earlier particle of the ame coloum, given the fact
that there are no particles at a higher level in the two nearest neighbor columns. The particle A sticks to the right side of the
left nearest neighbor colomn rather than at the top of a particle in the same colomn at a lower level.
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Cakir, Rasit; Grigolini, Paolo & Ignaccolo, Massimiliano. Renewal and memory properties in the random growth of surfaces, article, February 4, 2008; (digital.library.unt.edu/ark:/67531/metadc132977/m1/2/: accessed October 17, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.