Growth of loops Coding of loops LCS Conclusions References Braids of entangled particle trajectories Jean-Luc Thiffeault 1 Michael Allshouse 2 1 Department of Mathematics University of Wisconsin – Madison 2 Department of Mechanical Engineering MIT Workshop on Braids and Their Applications Centro di Ricerca Matematica Ennio De Giorgi, Pisa, Italy 22 June 2011 1 / 29
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Growth of loops Coding of loops LCS Conclusions References
Braids of entangled particle trajectories
Jean-Luc Thiffeault1 Michael Allshouse2
1Department of MathematicsUniversity of Wisconsin – Madison
2Department of Mechanical EngineeringMIT
Workshop on Braids and Their ApplicationsCentro di Ricerca Matematica Ennio De Giorgi, Pisa, Italy
Growth of loops Coding of loops LCS Conclusions References
Growth of curves on a diskOn a disk with 3 punctures (rods), we can also look at the growthof curves:
�1
�1�2-1
�1�2-1�1
�2-1�1�2
-1�1
initial
We use the braid generator notation: σi means the clockwiseinterchange of the ith and (i + 1)th rod. (Inverses arecounterclockwise.)
The motion above is denoted σ1σ−12 .
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Growth of loops Coding of loops LCS Conclusions References
Growth of curves on a disk (2)
The rate of growth h = log λ is called the topological entropy.
But how do we find the rate of growth of curves for motions onthe disk?
For 3 punctures it’s easy: the entropy for σ1σ−12 is h = logϕ2,
where ϕ is the Golden Ratio!
For more punctures, use Moussafir iterative technique (2006).
[Thiffeault, Phys. Rev. Lett. (2005); Chaos (2010); Gouillart et al., Phys.
Rev. E (2006) ‘ghost rods’]
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Growth of loops Coding of loops LCS Conclusions References
Iterating a loop
It is well-known that the entropy can be obtained by applying themotion of the punctures to a closed curve (loop) repeatedly, andmeasuring the growth of the length of the loop (Bowen, 1978).
The problem is twofold:
1. Need to keep track of the loop, since its length is growingexponentially;
2. Need a simple way of transforming the loop according to themotion of the punctures.
However, simple closed curves are easy objects to manipulate in2D. Since they cannot self-intersect, we can describe themtopologically with very few numbers.
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Growth of loops Coding of loops LCS Conclusions References
Solution to problem 1: Loop coordinates
What saves us is that a closed loop can be uniquely reconstructedfrom the number of intersections with a set of curves. For instance,the Dynnikov coordinates involve intersections with vertical lines:
2
30 0
14 4 4
2 2
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Growth of loops Coding of loops LCS Conclusions References
Crossing numbers
Label the crossing numbers:
1 2 i-1 i i+1 nn-1
�1
�2
�1
�2i-5
�2i-4
�2i-3
�2i-2
�2i-1
�2i
�2n-5
�2n-4
�i-2 �i-1 �i �n-1... ...
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Growth of loops Coding of loops LCS Conclusions References
Dynnikov coordinates
Now take the difference of crossing numbers:
ai = 12 (µ2i − µ2i−1) ,
bi = 12 (νi − νi+1)
for i = 1, . . . , n − 2.
The vector of length (2n − 4),
u = (a1, . . . , an−2, b1, . . . , bn−2)
is called the Dynnikov coordinates of a loop.
There is a one-to-one correspondence between closed loops andthese coordinates: you can’t do it with fewer than 2n− 4 numbers.
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Growth of loops Coding of loops LCS Conclusions References
Intersection number
A useful formula gives the minimum intersection number with the‘horizontal axis’:
L(u) = |a1|+ |an−2|+n−3∑i=1
|ai+1 − ai |+n−1∑i=0
|bi | ,
For example, the loop on theleft has L = 12.
The crossing number growsproportionally to the thelength.
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Growth of loops Coding of loops LCS Conclusions References
Solution to problem 2: Action on coordinates
Moving the punctures according to a braid generator changes somecrossing numbers:
�1-1
There is an explicit formula for the change in the coordinates!
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Growth of loops Coding of loops LCS Conclusions References
Action on loop coordinates
The update rules for σi acting on a loop with coordinates (a,b)can be written
a′i−1 = ai−1 − b+i−1 −
(b+
i + ci−1
)+,
b′i−1 = bi + c−i−1 ,
a′i = ai − b−i −(b−i−1 − ci−1
)−,
b′i = bi−1 − c−i−1 ,
wheref + := max(f , 0), f − := min(f , 0).
ci−1 := ai−1 − ai − b+i + b−i−1 .
This is called a piecewise-linear action.Easy to code up (see for example Thiffeault (2010)).
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Growth of loops Coding of loops LCS Conclusions References
Growth of L
For a specific rod motion, say as given by the braidσ−1
3 σ−12 σ−1
3 σ2σ1, we can easily see the exponential growth of Land thus measure the entropy:
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Growth of loops Coding of loops LCS Conclusions References
Growth of L (2)
0 10 20 30 40 50
0.76
0.78
0.8
0.82
m
(log
L)/
m
m is the number of times the braid acted on the initial loop.
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Growth of loops Coding of loops LCS Conclusions References
Oceanic float trajectories
72 oW
54 oW 36
oW 18
oW
0o
42 oN
48 oN
54 oN
60 oN
66 oN
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Growth of loops Coding of loops LCS Conclusions References
Oceanic floats: Data analysis
What can we measure?
• Single-particle dispersion (not a good use of all data)
• Correlation functions (what do they mean?)
• Lyapunov exponents (some luck needed!)
Another possibility:
Compute the σi for the float trajectories (convert to a sequence ofsymbols), then look at how loops grow. Obtain a topologicalentropy for the motion (similar to Lyapunov exponent).
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Growth of loops Coding of loops LCS Conclusions References
Oceanic floats: Entropy10 floats from Davis’ Labrador sea data:
0 100 200 30010
0
101
102
t (days)
L(u
) entropy = 0.0171
crossings = 126
Floats have an entanglement time of about 50 days — timescalefor horizontal stirring.
Source: WOCE subsurface float data assembly center (2004)18 / 29
Growth of loops Coding of loops LCS Conclusions References
Lagrangian Coherent Structures
• There is a lot more information inthe braid than just entropy;
• For instance: imagine there is anisolated region in the flow thatdoes not interact with the rest,bounded by Lagrangian coherentstructures (LCS);
• Identify LCS and invariant regionsfrom particle trajectory data bysearching for curves that growslowly or not at all.
• For now: regions are not ‘leaky.’
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Growth of loops Coding of loops LCS Conclusions References
Sample system: Modified Duffing oscillator
−2 −1 0 1 2 3−4
−3
−2
−1
0
1
2
3
x
y
1
2
3
4
5
6
7
x = y + α cosωt,
y = x(1− x2) + γ cosωt − δy ,
+ rotation to further hide two regions. α = .1, γ = .14, δ = .08, ω = 1.
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Growth of loops Coding of loops LCS Conclusions References
Growth of a vast number of loops
0 50 100 150 200 25010
0
105
1010
1015
time
inte
rsec
tion
s
0 50 100 150 200 2500
20
40
60
80
100
time
inte
rsec
tion
sLeft: semilog plot; Right: linear plot of slow-growing loops.
Clearly two types of loops!
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Growth of loops Coding of loops LCS Conclusions References
What do the slowest-growing loops look like?
(a)
(b)
(c)
1 2 3 4 5 6 7
−2 −1 0 1 2 3−4
−3
−2
−1
0
1
2
3
x
y
1
2
3
4
5
6
7
[(c) appears because the coordinates also encode ‘multiloops.’]
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Growth of loops Coding of loops LCS Conclusions References
Computational complexity
Here’s the bad news:
• There are an infinite number of loops to consider.
• But we don’t really expect hyper-convoluted initial loops (nordo we care so much about those).
• Even if we limit ourselves to loops with Dynnikov coordinatesbetween −1 and 1, this is still 32n−4 loops.
• This is too many. . . can only treat about 10–11 trajectoriesusing this direct method.
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Growth of loops Coding of loops LCS Conclusions References
An improved method: Pair-loops
The biggest problem is that we only look at whether a loop growsor not. But there is a lot more information to be found in how aloop entangles the punctures as it evolves.
(1,2)
(a)
(1,3)
(4,2) (4,5)
{1, 2, 3, 4, 5}
(b)
{1, 3}
{2, 4, 5} {2, 4, 5}
Consider loops that enclose two punctures at once. More involvedanalysis, but scales much better with n.
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Growth of loops Coding of loops LCS Conclusions References