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Jamming and tiling in aggregation of rectangles
Eli Ben-NaimLos Alamos National Laboratory
Talk, publications available from: http://cnls.lanl.gov/~ebn
APS March Meeting, Boston MA, March 5, 2019
with: Daniel Ben-Naim (UC Santa Barbara) & Paul Krapivsky (Boston Univ.)
arXiv://1808.03714 J. Phys. A: Mathematical & Theoretical 51, 455002 (2018)
Jamming and tiling in two-dimensionsPick two neighboring rectangles at random
Merge them if they are compatible
System reaches a jammed stateNo two neighboring rectangles are compatible
The jammed stateno two neighbors share a common side
Kastelyan 61Fisher 61Lieb 67Baxter 68Kenyon 01
Features of the jammed state
•Local alignment•Finite rectangle density
•Finite tile density
•Finite stick density
•Finite square density
•Area distribution of rectangles with width w
⇢ = 0.1803
T = 0.009949
S = 0.1322
H = 0.02306
0 20 40 60 80 100 120 140
ω2
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
mω
m! ⇠ exp
��const.⇥ !2
�
No theoretical framework!
Jamming: mean-field version
• Start with N 1x1 tiles (elementary building blocks)
• Pick two rectangles at random
• Pick an orientation at random (vertical or horizontal)
• Merge rectangles if they are perfectly compatible
• System is jammed when f rectangles have: f distinct horizontal sizes and f distinct vertical sizes
+
(i1, j) + (i2, j) ! (i1 + i2, j)
(i, j1) + (i, j2) ! (i, j1 + j2)
System reaches a jammed state
An example of a jammed state
• Characterize rectangle by horizontal and vertical size
• Characterize rectangle by maximal and minimal size
• Width = minimal size, Length = maximal length
• Ordered widths of f=13 rectangles for N=10,000
Width sequence has gaps!
(i, j)
! = min(i, j) ` = max(i, j)
(!, `)
{1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 7, 9}
Number of jammed rectangles•Average Number of rectangles grows algebraically with N
•Nontrivial exponent
•Typical width of rectangles grows algebraically with N
•Area density of rectangles of width w decays as a power law
A single exponent characterizes the jammed state
F ⇠ N↵
↵ = 0.229± 0.002
! ⇠ N↵
m! ⇠ !�� with � = ↵�1 � 2
Numerical simulations
100
101
102
103
104
105
106
107
108
109
N
100
101
102
F
Eq. (1)
0.00 0.02 0.04 0.06 0.08 0.10
1/F
0.22
0.23
0.24
0.25
α
F ⇠ N↵
100
101
102
ω
10-6
10-5
10-4
10-3
10-2
10-1
100
mω
Eq. (7)
N=108
N=109
m! ⇠ !��
! 1 2 3 4 5 6m! 0.622 0.182 0.0694 0.0365 0.0214 0.0139M! 0.622 0.804 0.873 0.910 0.931 0.945
Rectangles with finite width are macroscopic!Rectangles with width 1,2,3,4,5 contain 95% of area
Still, the area distribution has a broad power-law tail!
Kinetic theory•Straightforward generalization of ordinary aggregation
•Allows calculation of the density of sticks
•Simple decay for the stick density and jamming time
• Jammed state properties give density decay and width growth
Jamming exponent characterizes the kinetics, too
dRi,j
dt=
X
i1+i2=i
Ri1,jRi2,j � 2Ri,j
X
k�1
Rk,j +X
j1+j2=j
Ri,j1Ri,j2 � 2Ri,j
X
k�1
Ri,k
dS
dt= �S2 � 2
X
i,j
R1,jRi,j
S ' t�1 =) ⌧ ⇠ N
⇢ ⇠ t↵�1 and w ⇠ t↵
Smoluchowski 1917
Numerical validation
Numerics validate approximationSuggest two aggregation modes: elongating and widening
100
101
102
103
104
105
106
107
108
t
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
ρ
Eq. (13)SEq. (11)
⇢ ⇠ t↵�1
S ⇠ t�1
Primary aggregation: elongation•Aggregation between two rectangles of same width
•Ordinary aggregation equation (example: sticks)
•Length distribution as in d=1, length grows linearly l~t
•Behavior extends to all rectangles with finite width
+
dR1,`
dt=X
i+j=`
R1,iR1,j�2SR1,`�2
X
i
Ri,`
!R1,`
R!,`(t) ' t
�2�!
�` t
�1�
with �!(x) = (2!/m!) exp(�2!x/m!)
R1,` ' (2/m1t2) exp(�2`/m1 t)
Finite width: problem reduces to one-dimensional aggregationHowever, total mass for each width is not known
Numerical validation
Exponential scaling functiontotal mass set by the jammed state
0 1 2 3 4 5x
10-6
10-5
10-4
10-3
10-2
10-1
100
101
Φ1
Eq. (17)
t=103
t=104
t=105
0 0.2 0.4 0.6 0.8 1x
10-6
10-4
10-2
100
102
Φ2
eq.(23)
t=103
t=104
t=105
Secondary aggregation: widening•Aggregation between two rectangles of same length
•The area fraction is coupled to the size distribution
•Insights about relaxation toward jammed state
+
dm!
dt=
1
2
X
i+j=!
X
`
!`Ri,`Rj,`�X
j
X
`
!`Rj,`R!,`
m!(t)�m!(1) ' C! t�1 with C! = �2!X
i+j=!
µiµj
(µi + µj)2+ 4!
X
j
µ!µj
(µ! + µj)2
µ! =2!
m!
Closure & theoretical determination of remains elusive↵
Conclusions• Random aggregation of compatible rectangles
• Process reaches a jammed state where all rectangles are incompatible
• Number of jammed rectangle grows as power-law
• Area distribution decays as a power law
• A single, nontrivial, exponent characterize both the jammed state and the time-dependent behavior
• Primary aggregation: rectangles of same width
• Secondary aggregation: rectangles of same length
• Slow transfer of “mass” from thin to wide rectangles
• Kinetic theory successfully describes primary aggregation process only
