MATH.EN.JEANS
Avalanches simulator
I.S.I.S.S. ”MARCO CASAGRANDE”
Pieve di Soligo, Italy
Cluj-Napoca, 07/04/17
I.S.I.S.S. MARCO CASAGRANDE
1
The problem statement
• Let’s consider a n x n grid with a whole number of snow flakes
• If in a moment a cell contains at least 4 flakes, it is unstable
• It avalanches giving all its flakes to the 4 neighbouring cells
• If the avalanching cell is on the border, the flakes that would fall outside the grid are
considered lost
Cluj-Napoca, 07/04/17
I.S.I.S.S. MARCO CASAGRANDE
2
Our hypothesis
• The matrix is initialized putting all flakes in the central cell
• The matrix is large enough to avoid border effects
5
4,5
4
3,5
3
2,5
2
1,5
1
0,5
0
Cluj-Napoca, 07/04/17
I.S.I.S.S. MARCO CASAGRANDE
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Simulation tools
• Spreadsheets with Visual Basic’s macro
• C++ algorithm
• Java algorithm
Cluj-Napoca, 07/04/17
I.S.I.S.S. MARCO CASAGRANDE
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Analyzed variables
N vs r
• N - the number of snow flakes inserted in the central cell
• r - the radius of the matrix, defined as the maximum
number of occupied cells next to the central one
Cluj-Napoca, 07/04/17
I.S.I.S.S. MARCO CASAGRANDE
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N vs r
r<5
N = 8 ∙ r2 - 12 ∙ r + 8
This parabola’s
coefficients are
perfectly integer
numbers.
Cluj-Napoca, 07/04/17
I.S.I.S.S. MARCO CASAGRANDE
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N vs r
r≥5
N = 8 ∙ r2 - 12 ∙ r + 8
As the radius
increasis, the next
points don’t fit
anymore on the
parabola.
Cluj-Napoca, 07/04/17
I.S.I.S.S. MARCO CASAGRANDE
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r vs N
r=-4∙
10-6
∙
N2
+0,0143 ∙ N + 2,371
r = 0,5417 ∙ N0,451
We tried to reverse
the relation between
N and r
The fit wasn’t
acceptable yet.
The better to
approximate the points
we used a power law
function
Cluj-Napoca, 07/04/17
I.S.I.S.S. MARCO CASAGRANDE
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Power law
𝑟 =𝑡∙𝑁
𝑠
This functions provides the following
advantages:
 Best fit
 Low sensibility of coefficients t
and s on N
 Almost monotonous variations of
t and s vs N
Cluj-Napoca, 07/04/17
I.S.I.S.S. MARCO CASAGRANDE
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s asymptotic behaviour
In order to give an extimate of s expected value we considered the
inverse relation of the parabola
−𝑏 + 𝑏 2 − 4𝑎(𝑐 − 𝑁)
𝑟(𝑁) =
2𝑎
𝑁 𝑟 = 𝑎 ∙ 𝑟2 + 𝑏 ∙ 𝑟 + 𝑐
For 𝑁 → ∞, 𝑟 𝑁 ~
Cluj-Napoca, 07/04/17
𝑁
𝑎
=𝑡
∙ 𝑁𝑠
I.S.I.S.S. MARCO CASAGRANDE
1
𝑠→
2
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s vs t
This is a linear relation
𝑠 =𝑚∙𝑡+𝑞
So we can express r with a
single parameter
𝑠−𝑞
𝑟=
∙ 𝑁𝑠
𝑚
With the best fit for this
coefficient given by:
t=
Cluj-Napoca, 07/04/17
I.S.I.S.S. MARCO CASAGRANDE
𝑠−𝑞
≈ 0,44
𝑚
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Geometric interpretation
This figure represents the flakes’
stability distrubution
Different shades of grey are used to
represent the occupation values of
cells in the final state with the
darkest corresponding to 3 flakes
and white for 0.
The radius of the figure is equal to
the matrix’s radius.
Cluj-Napoca, 07/04/17
I.S.I.S.S. MARCO CASAGRANDE
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Cylinder Volume
The initial volume of the central cell is
equal to the number of snow flakes in
it. Furthermore we supposed that the
matrix has no limits, so this value
must be the same in the final state.
𝑉=𝑁
The volume of the cylinder is equal to
V= 𝜋 ∙ 𝑟 2 ∙ ℎ
So
𝑁≤
3𝜋𝑟 2
⇒𝑟≥
Cluj-Napoca, 07/04/17
1
3𝜋
𝑁 = 0,3 𝑁
I.S.I.S.S. MARCO CASAGRANDE
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t asymptotic behaviour
Numerical simulation outcomes:
• the number of cells with h = 3 is significantly
larger than the ones with h = 0
• for large values of N
1
ℎ𝑎𝑣𝑔 → 2
𝑟(𝑁)~
∙ 𝑁 ~ 0,4 ∙ 𝑁
2𝜋
We can’t provide a rigorous proof of the result…
nevertheless we’re confident of it!
Cluj-Napoca, 07/04/17
I.S.I.S.S. MARCO CASAGRANDE
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Conclusions
 We studied the problem putting all the flakes in the
central cell of an open matrix
 We simulated the avalanche using different softwares
 We showed that for 𝑟 < 5 𝑜𝑟 𝑁 < 144, N(r) can be fitted
with a parabola
 We showed that for 𝑁 → ∞, 𝑟(N) can be fitted with a
power law, 𝑟 𝑁 = 𝑡 ∙ 𝑁 𝑠
 We provided an estimate of the asymptotic values of s
1
and t parameters: 𝑠 → , 𝑡 → 0,4
2
Cluj-Napoca, 07/04/17
I.S.I.S.S. MARCO CASAGRANDE
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THANK YOU
■ TEAM:
– Barisan Paolo
– Breda Leonardo
– Carollo Marco
– De Biasi Anna
– Grotto Giusi
– Metaliu Klara
– Micheletto Marco
– Piccin Erica
Cluj-Napoca, 07/04/17
■
–
–
–
–
PROFESSORS:
Breda Fabio
Cardano Francesco Maria
Meneghello Alberto
Zampieri Francesco
I.S.I.S.S. MARCO CASAGRANDE
■ MATH.EN.JEANS:
– Zanardo Alberto
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