Analysis in Kantrovich geometric space for quasi-stable patterns in

2D-OV model

Ryosuke Ishiwata† and Yuki Sugiyama

Department of Complex Systems Science Nagoya University

TGF2015, on Oct. 29

ishiwata@phys.cs.is.nagoya-u.ac.jp

Equation of motion

1-1. Two-dimensional Optimal Velocity Model An organism moves to maintain an optimal velocity which depends on distances to others. Basic Idea

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Attractive

i j

Repulsive i

k

r j i : x j -x i a : sensitivity c : control parameter of the threshold of attractive and repulsive interactions

i

j

Attractive region

Repulsive region

θji xi

k

1-2. Pattern formations of 2D-OV model

(a) a=5.0, c=1.0, N=50

Attractive interaction

(a) a=5.0, c=0.0, N=30

Attractive + Repulsive mixed Interaction

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String-like flow Cluster flow

c=1.0 c=0.0

2-1. Optimal patterns of Collective bio-motion(Amoeboid) and Flow of self-driven particles(2D-OV) - e.g. Maze solving -

Nakagaki, T., Yamada, H., & Tóth, A. (2000). Maze-solving by an amoeboid organism. Nature, 407(6803), 470.

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Amoeboid 2D-OV

• Amoeboid organism organize the optimal path between the two locations of foods in the maze.

• 2D-OV particles organize optimal flow pattern between two gates connected with the periodic boundary condition in the maze.

Is there any quantity to estimate the stability of the optimal flow pattern?

Inital state (randam motions)

Intermediate state (creating strings)

i) Soluition of periodic boundary ii) Solution of elastic boundary

（a=20.0,N=128）

Solutions of optimal path

Formation of solutions for a simple maze

Periodic boundary

Elastic boundary

2-2. Search for the quantity of stability of a pattern - e.g. Simple maze -

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Trajectory for the changing of flow patterns in Kantorovich metric space

Motion of 2D-OV particles in real space

Kantorovich metric space

One pattern of particles in real space

One point in the Kantorovich metric space

Measurement of the similarity between patterns in real space

Ψ2

Ψ1

Kantorovich Metric (measurement of affinity, similarity, resemblance)

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Distance of kr (Pi, Pj)

Pattern Pi Pattern Pj

・ ・ ・ ・ ・

・

x1 x2 x3 xm

・ ・ ・ ・ ・

・

x’1 x’2 x’3 x’m

Transportation of points from Pi to Pj : several plans

x・ : point

kr(Pi, Pj) := min { Cost of transportation (Pi, Pj) } ⇔ “Optimal Transportation problem”

Time sequence of patterns : P(t1) → P(t2) → P(t3) → → P(tn) “Affinity matrix” B := Kr( P(t1), P(t1)) Kr( P(t1), P(t2)) ・・・・ Kr( P(t1), P(tn))

Kr( P(t2), P(t1)) Kr( P(t2), P(t2)) ・・・・ Kr( P(t2), P(tn)) ・ Kr( P(tn), P(t1)) Kr( P(tn), P(t2)) ・・・・ Kr( P(tn), P(tn))

“Kantorovich space” : eigen vector : { Ψ1, Ψ2, ・・・, Ψr } r = rank B eigen value : λ1 ＞ λ2＞ ・・・＞ λr

a pattern P(ti) ⇒ c1Ψ1 + c2Ψ2 + ・・・ : a vector in Kantorovich space ↓ pi ∈ vector in Eucledean space R^r ~ R^2 dimensional reduction

Transition of patterns : P(t1) → P(t2) → P(t3) → → P(tn) ⇒ Motion of a point in Kantorovich space: p1 → p2 → p3 → → pn

2-3. Time evolution of the forming optimal flow pattern in the Kantorovich metric space

Optimal flow 1 Optimal flow 2

Kantorovich metric space

Localized region in the Kantorovich metric space

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initial pattern

Optimal flow

Conclusions and Discussions

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• 2D-OV particles create several flow patterns for a given control parameter.

• Optimal flow of 2D-OV particles form an optimal pattern in a maze. Continuous deformation of pattern

• Optimal flows in the maze can be represented as each localized region in a low dimensional Kantorovich metric space.

● The high degeneracy of ground state of “thermo dynamical potential” of a few macroscopic variables Ψ1, Ψ2 ? in Kantorovich metric space.

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Ψ１

Ψ２

(Non-equlibutium steady state) Jam flow

Homogenous flow

Initial condition (random)

Change parameter a=1.1 →1.5

Change parameter a=1.5 →1.8

Ψ1 axis is important !

Flow in Kantorovich space

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Changing of the coefficient of Ψ1

Changing of the variance of headway σ

What variable is corresponding to Ψ1 axis ?

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● A trial to more difficult maze -1 (a=20, c=1）

Solution

# of particles 330 • Soluiton is formed as a queasi stable state. • Optimal # of particles is needed.

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● Another trial to more difficult maze -2

360=N stable

unstable

Life time and frequency of appearance of patterns

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http://www.es.hokudai.ac.jp/labo/cell/ • Y.Sugiyama, A.Nakayama and E Yamada: ”Phase diagram of group

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