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Notion of Random Domino Automaton revisited To cite this article: Arpan Bagchi 2018 J. Phys.: Conf. Ser. 965 012007
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ISQS25 IOP Conf. Series: Journal of Physics: Conf. Series 965 (2018) 1234567890 ‘’“”012007
IOP Publishing doi:10.1088/1742-6596/965/1/012007
Notion of Random Domino Automaton revisited Arpan Bagchi Institute of Geophysics, Polish Academy of Sciences, ul. Ks. Janusza 64, 01-452 Warszawa, Poland E-mail:
[email protected] Abstract. Inspired by a need of effective simulations of a system of Random Domino Automaton type defined for Bethe lattice, new variables are introduced. Main results obtained for 1-dimensional system — including a set of equations describing stationary state and relation to Motzkin numbers — are investigated in this new notion.
1. Introduction Random Domino Automaton (RDA) was introduced as a straightforward totally discrete stochastic dynamical framework serving as a toy model of earthquakes [9, 10]. It may be considered as an extension of 1 dimensional Drossel and Schwabl forest fire model [14, 13, 17]. However, contrary to the Drossel-Schwabl model where parameters are constants, RDA allows some parameters to be functions of cluster’s size (forests’ size) and a good mathematical structure of the equations of the model is preserved. As a consequence, RDA generates bigger variety of distributions depending on respective parameters — form exponential distributions to inversepower ones [10] and also quasi-periodic for a finite version [5]. Moreover, it is remarkable that the inverse problem of finding parameters (that define the dynamics of the system) from the given distribution can be solved [10, 4]. That last property is useful for explaining statistical properties of occurrence of earthquakes [6]. The extension also leads to a significant link to the Motzkin number recurrence [3, 7]. The published results are related to 1-dimensional RDA. Construction of RDA type systems on more complex geometries leads to more complicated equations, mainly due to loss of one-toone correspondence between a size of a cluster and a size of its boundary. For 1-dimensional systems there are always just two end cells of a cluster, and the cluster may be enlarged only through them. For 2-dimensional systems the perimeter depends on a shape of a cluster. Thus, in order to extend the geometry of RDA preserving its good mathematical structure, one may choose Bethe lattice — an infinite loop free graph with fixed coordination number K, introduced by Hans Bethe in 1935 [12, 19, 15, 11]. In Bethe lattice there is a fixed relation between a cluster size and the size of its perimeter. This property allows to derive a set of equations using meanfield approximation in a stationary state of RDA type system on Bethe lattice [2]. Bethe lattice is infinite and homogenous everywhere, so any of its point can be regarded as an origin. However, when considering a finite part of a Bethe Lattice — for example in simulations — one may notice that majority of cells within that part are placed close to its boundary — see FIG. 1, where the boundary is created by fixing a certain radius. It follows, one can find many
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ISQS25 IOP Conf. Series: Journal of Physics: Conf. Series 965 (2018) 1234567890 ‘’“”012007
IOP Publishing doi:10.1088/1742-6596/965/1/012007
clusters which cross the boundary and extend outside. When an avalanche happen, i.e. a cluster is removed from the system (or equivalently all cells of a clusters become empty) one can not estimate how many cells are actually responsible for the avalanche by observing that finite part only. The definition of RDA (and other similar systems) on Bethe lattice in natural way leads to the difficulty described above. To avoid this difficulty, a new variable Si (for i = 1, 2, . . .) is introduced as follows [8] ∞ X Si = nj , j=i
where nj is the number of clusters of size j (i.e. with exactly j occupied cells) and (Si ) is the number of clusters of size ≥ i. In the FIG. 1 it is visible, that the finite part contains only 10 cells in which 8 are occupied and 2 are empty. There are five clusters and they are extended through the boundary. So it is not possible to find out information about the distribution of clusters (ni ), because there is no cluster which lay just inside the finite part. By the definition of new variables one can say all 5 clusters contribute to S1 and at least one cluster contribute to S4 . In the present paper, this new variable is investigated for 1-dimensional RDA. For example, FIG. 2 presents a finite section of line where two clusters cross left and right boundaries of fixed finite part of the line. Thus it is impossible to know which ni they contribute to, but it is clear that they both contribute to S2 and at least one to S3 . Main results obtained for 1-dimensional RDA are rewritten in the new notion. Study of the RDA on Bethe lattice — using both ”old” and ”new” notions — is a matter of a forthcoming paper.
Figure 1. New variable in Bethe lattice with coordination number 3
Figure 2. New variable in line
2
ISQS25 IOP Conf. Series: Journal of Physics: Conf. Series 965 (2018) 1234567890 ‘’“”012007
IOP Publishing doi:10.1088/1742-6596/965/1/012007
2. Evolution rules of extended RDA We shortly recall evolution rules of RDA. Assume that any cell may be in one of two states: empty or occupied by a ball (portion energy). The state of automaton in the discrete time t is characterized by instantaneous density ρt equal to number of occupied cells in a given time t divided by the size of the system N. Rules of evolution include parameters c and µ and are as follows. In each time step t one cell is chosen (we assume that each has the same probability). The probability that the chosen cell is empty is equal to (1−ρt ). Then the empty cell becomes occupied with the probability c, or it remains empty (the ball is rebounded) with probability (1 − c) and the state of automaton is unchanged. If the chosen cell is already occupied by energy (the respective probability is ρt ), then either the energy will be rebounded with the probability (1 − µ) or it triggers an avalanche with the probability µ. The avalanche size is equal to the number of cells changing their state. Then the update procedure repeats in next time step. ↓
time = t
···
•
•
•
• •
time = t + 1
···
•
↓ •
↓ •
↓ •
c
c
c
c
c
c
c •
•
•
···
↓ •
•
•
···
3. Equations for RDA 3.1. Equations for RDA with the variable ni The stationary state of the system may be described by the distribution of clusters ni . The number of all clusters n and and the density ρ are n=
X
ni ,
ρ=
i≥1
1 X ini . N
(1)
i≥1
The set of equations for the distribution of clusters ni and the distribution of empty clusters n0i for i = 1, 2, . . . in stationary state is [10] n1 =
µ1 c
1 (1 − ρ)N − 2n + n01 , +2
(2)
2n − 2n01 , P 4 + c02n i≥1 µi ini " # i−2 1 n01 X n01 ni = µ i 2ni−1 1 − + 2 nk ni−1−k , for i ≥ 3 n n c i+2
n2 =
k=1
3
(3)
(4)
ISQS25 IOP Conf. Series: Journal of Physics: Conf. Series 965 (2018) 1234567890 ‘’“”012007
IOP Publishing doi:10.1088/1742-6596/965/1/012007
and n01 = n02 =
3+
2 cn
2n P
i≥1 µi ini
,
(5)
2n − 2n01 , P 4 + c02n i≥1 µi ini
2(n − n0k =
k−1 X i=1
(6)
k−2 k−1−j 0 X X µj n0l nk−j−l + jnj c n n j=1 l=1 , P 2 2 + k + cn i≥1 µi ini
n0i )
for k ≥ 3.
(7)
The balance equation for the total number of clusters n and for the density ρ reads [10] (1 − ρ)N − 2n =
X µi c
i≥1
(1 − ρ)N =
X µi i≥1
c
ni i,
ni i2 .
(8)
(9)
3.2. Equations for RDA with the variable Si Some properties of Si : 1. By the basic definition of Si , S1 can be written as the total number of clusters. X S1 = n = ni .
(10)
i≥1
2. Si is decreasing function i.e. S1 ≥ S2 ≥ S3 ≥ ......
(11)
3. ni can be witten as the difference between Si and Si+1 ni = Si − Si+1 .
(12)
4. Density can be written interms of Si as follows. ∞ ∞ 1 X 1 X ini = Si . ρ= N N
(13)
∞ ∞ ∞ X 1 X 1 X 2 i ni = 2 iSi − Si . N N
(14)
i=1
i=1
5. 2nd order moment of ni is
i=1
i=1
4
i=1
ISQS25 IOP Conf. Series: Journal of Physics: Conf. Series 965 (2018) 1234567890 ‘’“”012007
IOP Publishing doi:10.1088/1742-6596/965/1/012007
3.3. Equations for extended RDA with the variable Si The system of the equations ((2)-(4)) can be reconstructed in terms of Si as follows. X µi ini + 2S1 = (1 − ρ)N, c i≥1 X µi ini + 2S2 = 2S1 − n01 , c i≥2 ! X µi 2S2 − 1 n01 , ini + 2S3 = 2S2 − c S1 i≥3 " # j−2 X µi n01 X n01 ini + 2Sj = (2 − )Sj−1 + 2 Sk − Sk+1 Sj−1−k for i ≥ 4. c S1 S1 i≥j
(15) (16)
(17)
(18)
k=1
In general it is very difficult to solve the equation (18) because for the 1st term. Here we will investigate the solution for special choice of rebound parameter.
4. Special case: µi = δi , where θ = constant 4.1. Solution for ni with special setting of rebound parameter Choosing this setting of rebound parameters, equations (2)-(4) gives n1 = n2 = ni =
1 (1 − ρ)N − 2n + n01 , θ+2 2 n01 1− n1 , θ+2 n # " i−2 1 n01 X n01 nk ni−1−k + 2 2ni−1 1 − θ+2 n n
(19) (20) (21)
k=1
for i ≥ 3, where θ = δc , θ ∈ [0, ∞). We define new variables Mi for i = 0, 1, . . . , by Mi =
β αi+1
ni+1 ,
(22)
where n0
α =
2(1 − n1 ) , θ+2 n01 n
β =
2(1 −
n01 n )n
(23)
.
(24)
Then, the equation (21) can be rewritten in the form Mm+2 = Mm+1 +
m X k=0
5
Mk Mm−k .
(25)
ISQS25 IOP Conf. Series: Journal of Physics: Conf. Series 965 (2018) 1234567890 ‘’“”012007
IOP Publishing doi:10.1088/1742-6596/965/1/012007
which is valid for m ≥ 0 (m = i − 2). Initial data M0 and M1 are easily obtained from equations (19)-(20), when it is transformed according the rule of equation (22), namely M0 = M1 =
n01 [(1 n2
− ρ)N − 2n + n01 ] 4(1 −
n01 2 n )
=
n01 n (θ
+
4(1 −
n01 n )
n01 2 n )
.
(Using(8))
(26)
Notice, M0 and M1 are equal. The above equation (25) has the form of Motzkin numbers recurrence [18, 1, 16, 3]. From the Motzkin numbers: 1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, . . ., etc. we can find the condition M0 = M1 = 1 which gives, n01 2 n01 0 [(1 − ρ)N − 2n + n ] = 4(1 − ) = 1. 1 n2 n
(27)
The solution of the equation (25) is
Mm
b m+2 2 c (2M0 − 12 )j 2(m − j) + 1 m − j + 2 1 X . = 2 (m − j + 2)2m−j m−j+1 j
(28)
j=0
Thus formula (22) gives the explicit solution of the equation (19)-(21) for the distribution of ni for any values of θ. By the above choice of µi (8) and (9) give the values of ρ, n and n01 [10] as follows. 1 , θ+1 Nθ , n= (θ + 1)(θ + 2) 2n n01 = . 2θ + 3 ρ=
(29) (30) (31)
4.2. Solution for Si with special setting of rebound parameter By this chosen setting of rebound parameter, the set of equations (15)-(16) transform into the following set of equations. (θ + 2)S1 = (1 − ρ)N, (θ + 2)S2 = 2S1 −
(32)
n01 ,
(33) !
(θ + 2)S3 = 2S2 −
2S2 − 1 n01 , S1
(34)
" # j−2 n01 n01 X (θ + 2)Sj = (2 − )Sj−1 + 2 Sk − Sk+1 Sj−1−k S1 S1
for i ≥ 4
(35)
k=1
To solve (35) we define new variables Mi∗ for i = 0, 1, . . . , by Mi∗ =
B Si+1 , Ai+1 6
(36)
ISQS25 IOP Conf. Series: Journal of Physics: Conf. Series 965 (2018) 1234567890 ‘’“”012007
IOP Publishing doi:10.1088/1742-6596/965/1/012007
where A =
B =
n01 S1
2−
θ+2
,
(37)
n01 S1
S1 (θ + 2)
.
(38)
Then, the equation (35) can be rewritten in the form m
∗ ∗ Mm+2 = Mm+1 +
m
X 1 X ∗ ∗ ∗ ∗ Mk Mm−k − Mk+1 Mm−k . A k=0
(39)
k=0
This recurrence is equivalent to Motzkin number recurrence [3]. Initial data M0∗ and M1∗ are easily obtained from equations (32)-(33). M0∗ = M1∗ =
n01 S1
2−
n01 S1
.
(40)
Notice, M0∗ and M1∗ are equal. We will solve (39) by using generating function [3]. Let us consider the generating function X ∗ m D(z) = Mm z . (41) m≥0
By this generating function the recurrence (39) gives ! ! z 2 ∗ − 1 D (z) + z + zM0 − 1 D(z) + z A
! (M1∗
−
M0∗ )z
+
M0∗
= 0.
Since M0∗ = M1∗ it gives, ! z z − 1 D2 (z) + A
! z+
zM0∗
− 1 D(z) + M0∗ = 0.
(42)
Let D(z) be the solution of (42). Then, D(z) =
−(−1 + z + zM0∗ ) − 2zM0∗
z A
p 1 − 2z(p + qz) ! ,
(43)
−1
where, p = (1 − M0∗ ), q=
1 2
(44) !
4M0∗ − (1 + M0∗ )2 . A
We choose negative sign in order D(z) to be a power series of z. 7
(45)
ISQS25 IOP Conf. Series: Journal of Physics: Conf. Series 965 (2018) 1234567890 ‘’“”012007
IOP Publishing doi:10.1088/1742-6596/965/1/012007
We are interested to find out the coefficient of z m of D(z). "
q 1 X (−1) 2n − 3 n D(z) = 1 − z− z (p + qz)n 2M0∗ 2M0∗ n2n−2 n − 1
#
n≥2
n 2n − 3 n−1 X n q q 1 X pn z = 1− z− ∗ ∗ n−2 2M0 2M0 n2 n−1 j p "
!j # zj ×
j=0
n≥2
z 1− A " X n≥0
!−1
z A
!n # .
(46)
So, coefficient of z m is ∗ Mm
=
m+1 X
+
x=b
m+3 2
c
" b m+1 c x 2 X X
!y px 2x − 3 x q 1 x2x−2 x − 1 y p Am+1−x−y x=2 y=0 !y # m+1−x X px 2x − 3 x q 1 , x2x−2 x − 1 y p Am+1−x−y
M0∗ q 1 − − Am 2Am−1 2
(47)
y=0
where
p = (1 − M0∗ ), ! 4M0∗ − (1 + M0∗ )2 , A
1 q= 2 A=
2−
n01 S1
θ+2
.
By the above formula of ρ, n and n01 ((29)-(31)) we can calculate the values of Si , for i = 1, 2, . . . as follows.
Si =
Nθ , θ+1 θ+2 4Nθ , 2 θ+2
i=2
2θ+3
(i−1) 2(2i−1) N θ θ+1 ∗ i−1 Mi−1 i θ+2
i=1
(48) i≥3
2θ+3
Now we will find out the distribution of ni by (22) and (36) and finally we will show that both distributions are same. From the equation (22) we get, ni =
αi Mi−1 β n0 1 n θ+ n , θ+2 i+1 " i+2 = n0 bX # 1) 2 c 1 j 2(1− (2M − ) 2(i − j) − 1 i − j + 1 n 0 2 i . n01 (i − j + 1)2i−j−1 i−j j 2
n2
θ+2
j=0
8
i=1 (49) i≥2
ISQS25 IOP Conf. Series: Journal of Physics: Conf. Series 965 (2018) 1234567890 ‘’“”012007
IOP Publishing doi:10.1088/1742-6596/965/1/012007
Again from the equation (12) and (36) we get
ni = Si − Si−1 " # Ai ∗ ∗ = Mi−1 − Mi B A 1 − A M0∗ , i=1 B " !i+1−k # i+1 X = Ai+1 2k − 3 k q pk . i≥2 2B k−2 k−1 i+1−k p k2 i+2 k=b 2 c 0 n1 n θ+ S 1 , i=1 θ+2 = n0 i+1 " !i+1−k # i+1 k 2− S1 X p 2k − 3 k q 1 i . i≥2 n0 p i+1−k k2k−2 k − 1 1 2 2 θ+2 i+2 k=b 2 c S1
(50)
We need to show that (49) and (50) are equivalent. If we substitute k = i + 1 − j and S1 = n in the equation (50) then we get !j # n0 i+1 " b i+1 c 2 2 − n1 X q pi+1 2(i − j) − 1 i − j + 1 ni = 0 . i (i − j + 1)2i−j−1 i−j j p2 n j=0 2 n12 θ + 2
for i ≥ 2
(51) Now if we proof the following hypothesis then we can tell that (49) and (50) are equivalent. Hypothesis: n0 i+1 q j n0 i+1 1 2i+1 1 − 1 (2M0 − )j = pi+1 2 − 1 . ∀ j n 2 n p2 !i+1 n0 j 2(1 − n1 ) 1 j i+1 q (2C − ⇒ ) = p . ∀ j 0 n0 2 p2 2 − n1
(52)
We need to prove p=
2(1 − 2−
and
n01 n
θ+
1 q (2M0 − ) = 2 ⇒ 2 p 2 1−
n01 n
n01 n ) . n01 n
−
n01 n )
1 1 = 2 2
(53)
! 4M0∗ A1 − 1 2 − 1 . 1 − M0∗
(54)
By the values of p,M0∗ and A form the equation (44), (37) and (40) we can easily prove (53) and (54).
9
ISQS25 IOP Conf. Series: Journal of Physics: Conf. Series 965 (2018) 1234567890 ‘’“”012007
IOP Publishing doi:10.1088/1742-6596/965/1/012007
5. Conclusions We pointed out a difficulty in analysis of finite part of the Random Domino Automaton related to clusters crossing boundary of the system. The problem is present for any geometry, but essential for Bethe lattice — a natural candidate for extension RDA P beyond dimension 1. Following the remark [8] we introduce a new variable Si = j≥i nj and repeated main results obtained for 1-dimensional Random Domino Automaton using the new notion. Obtained formulas are in the form convenient to comparisons with numerical simulations. Moreover, we solve the system by choosing special rebound parameters by generating function method which produce a counterpart of Motzkin number recurrence. Obtained results will be useful for extension and investigation of Random Domino Automaton type system defined on Bethe lattice.
Acknowledgments This work was partially supported by Institute of Geophysics, Polish Academy of Science , project 500-10-28E and partially supported within statutory activities No. 3841/E-41/S/2017 of the Ministry of Science and Higher Education of Poland. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
[10]
[11] [12] [13] [14] [15] [16] [17] [18] [19]
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