International Forum for Green Asia 2020

Collective motion in Ising-type Vicsek model
Kazuya Ishibashi and Hidetsugu Sakaguchi
Department of Applied Science for Electronics and Materials,
Interdisciplinary Graduate School of Engineering Science, Kyushu University
Kasuga-koen, Kasuga, Fukuoka 816-8580, Japan

Abstract
This study shows collective phenomena of self-propelled particles using two-dimensional
Ising-type Vicsek model, that we propose, by direct numerical simulation. Multiple particles
can occupy one lattice point; self-propelled particles move one of eight directions in this model;
particles change directions stochastically. They form a solitary wave at a high probability foraligning. Solitary waves suddenly change directions like a flip motion.
1. Introduction
Collectively moving animals and insects
are fascinating. A simple model was proposedby Vicsek’s group to study collective motionof self-propelled particles.[1] Agent-basedmodels have been proposed and numericallystudied to investigate dynamical behaviors ofa large population of self-propelled
particles.[2] Experimental study is also
conducted, which is analyzed with Vicseklike model.[3]In Vicsek model, motion directions ofparticles align with neighborhood at highparticle density. However, particles move
randomly at low particle density. Then,directions of motion do not align for strong
noise, which is for alignment direction.
Therefore, order–disorder transition appears
in this model.
Chaté’s group found another state using
Vicsek-like model with vector noise insteadof scalar noise.[4] The state takes a band
structure of dense particles on disorder state,
which is like a solitary wave. We have shownthat the solitary wave state appears innonlinear Kramers equation; it is for theprobability distribution of the position andvelocity of self-propelled particles.[5]Furthermore, we have proposed a simple
model of one-dimensional solitary wave
state; the linear instability of the uniform
disordered state leads to the formation of asolitary wave state.[6] Kuwayama’s group has
experimentally found biological soliton.[7]Experimental studies are expected to clarify
the formation mechanism of a solitary wave.
Solon and Tailleur have proposed theactive Ising model, in which self-propelled
particles undergo a diffusion motion biased in
the right or left direction and the updating is
randomly performed like the Monte-Carlosimulation in the Ising model. Reversal
phenomenon appear in the one-dimensionalmodel.[8] The studies of the model are
conducted,[9, 10] but the mechanism is unclear.
We have proposed Ising-type Vicsek
model, which shows flip motions in one- andtwo-dimensional
system.[11]Twodimensional Ising-type Vicsek model isconstrained by moving rule for simplicity,
because particles move in only right or left
directions. We show numerical result withgeneralized model and discuss themechanism.
2. Ising-type Vicsek model
Ising model is the model describing
phase transition of ferromagnet. One of plusspin and minus-spin can occupy on a lattice
point in the model. However, Ising-type
Vicsek model takes multiple states, that is
multiple particles can occupy one lattice
point. The total number of particles isexpressed as N. The lattice size is Lx and Ly inthe x and y directions. Periodic boundary
conditions are assumed. The number ofparticles at the x-y coordinate (i, j) isexpressed as ni,j. The motion direction of a1International Forum for Green Asia 2020

R1 (a)
2π0.5Rθm01 (b)

0.5
θ=0
θ=π/4

θ=π/2
θ=3π/4θ=πθ=5π/4πθ=3π/2θ=7π/40100 1104 2104 3104 4104 5104 6104 7104 8104 9104 11050t

0100 1104 2104 3104 4104 5104 6104 7104 8104 9104 1105tFigure 1. Time evolutions of alignment
directions ratio of self-propelled particles at (a)g = 1 and (b) g = 2 for N = 100000, Lx × Ly =200 × 200. Colors show alignment directions.
All particles move in the same direction at R =1. The lower limit of R is 1/8, because themotion direction of the particle takes one of theeight directions.

Figure 2. Time evolution of the mean motion
direction of self-propelled particles at g = 2, N
= 100000, Lx × Ly = 200 × 200. The change of3( from 7π/4 to 0 means the direction changesby π/4.

particle is expressed as !! , where k isparticle number. The particle number withmoving direction ! at (i, j) is expressed as#",$,% and the particle number with moving
direction ! at the neighboring sites aroundthe (i, j) site is expressed as $",$,% , that is,neighboring particles with high probability p,
however, the particle can change the direction
with probability (1 − %) . The parallel
updating is performed in this model, that is
all particle move at the same time. Our model
is similar to the Vicsek model in thatcooperative interaction and random forces
are incorporated. The main difference is thatthe majority decision rule is adopted in ourmodel.
Figure 1 shows that the moving particles
form a cluster and the moving directionchanges occasionally. The change of movingdirection occurs suddenly. On the other hand,a steadily moving solitary-wave state isobserved but the sequential flipping ofmoving direction is not reported in theoriginal Vicsek model and the twodimensional active Ising model. The
alignment directions rate increases withincreasing probability parameter g. The
reversal time intervals increase withdecreasing probability parameter g. The
alignment directions ratio R is!!,#,$ = #!%&,#%&,$ + #!%&,#,$ + #!%&,#'&,$ +#!,#%&,$ + #!,#,$ + #!,#'&,$ +#!'&,#%&,$ + #!'&,#,$ + #!'&,#'&,$ . (1)The direction $&'( taking the largest$",$,% is calculated at each site. A particle
located at the (i, j) site moves at the next stepto the direction !&'( with probability % =e) ⁄(e) + e*) ). Moreover, the particle canmove to the direction !&'( ± ./4 withprobability (1 − %)/3 , to the direction!&'( ± ./2 with probability (1 − %)/6 .
For example, when !&'( is 0, A particle
located at the (i, j) site moves to x-axis
forward (! = 0) with probability % ; theparticle can move to top-right direction (! =./4) or bottom-right direction (! = 7.⁄4)
with probability (1 − %)/3 , to y-axis
forward direction (! = .⁄2) or y-axis
downward direction (! = 3.⁄2) withprobability (1 − %)/6. That, is each particle
moves toward the majority direction of

1% = max$ ) , #!,#,$ - ,+(2)!,#where max$ ( ) is the normalized sum of#!,#,$ in !&'( . Figure 2 shows the mean2

International Forum for Green Asia 2020R

1 (a)
2×π

0.5Rθm01 (b)

1×π

0.5
θ=0
θ=π/4

θ=π/2
θ=3π/4θ=πθ=5π/40×πθ=3π/2θ=7π/40100 1104 2104 3104 4104 5104 6104 7104 8104 9104 1105

0010011042104

3104

4104

5104

6104

7104

8104

9104

t

1105

t

Figure 3. Time evolutions of alignment
directions ratio of self-propelled particles with
the next-nearest-neighbor couplings at (a) g =1 and (b) g = 2 for N = 100000, Lx × Ly = 200 ×200.

Figure 4. Time evolution of the mean motion
direction of self-propelled particles with thenext-nearest-neighbor couplings at g = 2, N =100000, Lx × Ly = 200 × 200.

motion direction of all particle. The directionchanges correspond with the changes in Fig.
1. The time evolution in the mean direction!& seems to be discontinuous, but it iscontinuous.
We have shown the case of the nearestneighbor interaction so far; however, similar
flip motions of solitary waves are observed
even in a model with both the nearestneighborandnext-nearest-neighborcouplings. Hereafter, We show results of themodel with the next-nearest-neighborcouplings. The interaction range is expanded.
The particle number at the next-nearestneighboring sites !′!,#,$ is

3. Summary
We have proposed Ising-type Vicsek
model for two-dimensional self-propelled
particles. We have found the sudden directionchange of self-propelled particles in twodimensional system. An inversely-moving
solitary wave appears from the edge of themain solitary wave because of the collision
between inverse moving wave with small
amplitude and the edge of solitary wave. We
have generalized Ising-type Vicsek model toreproduce the sudden cooperative change ofmoving direction.
References[1] T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen,
and O. Shochet, Phys. Rev. Lett. 75, 1226
(1995).
[2] T. Vicsek and A. Zafeiris, Phys. Rep. 517, 71(2012).
[3] Y. Sumino, K. H. Nagai, Y. Shitaka, D. Tanaka,
K. Yoshikawa, H. Chaté, and K. Oiwa, Nature483, 448 (2012).
[4] H. Chaté, F. Ginelli, G. Grégoire, F. Peruani,and F. Raynaud, Eur. Phys. J. B 64, 451 (2008).
[5] H. Sakaguchi and K. Ishibashi, J. Phys. Soc. Jpn.
86, 114003 (2017).
[6] H. Sakaguchi and K. Ishibashi, J. Phys. Soc. Jpn.
87, 064001 (2018).
[7] H. Kuwayama and S. Ishida, Sci Rep. 3 2272
(2013).
[8] A. P. Solon and J. Tailleur, Phys. Rev. Lett. 111,078101 (2013).
[9] A. P. Solon and J. Tailleur, Phys. Rev. E 92,!′!,#,$ = !!,#,$ + #!%),#%),$ + #!%),#%&,$ +#!%),#,$ + #!%),#'&,$ + #!%),#'),$ +#!%&,#%),$ + #!%&,#'),$ + #!,#%),$ +#!,#'),$ + #!'&,#%),$ + #!'&,#'),$ +#!'),#%),$ + #!'),#%&,$ + #!'),#,$ +#!'),#'&,$ + #!'),#'),$ .
(3)

Figure 3 shows that the moving particles
form a cluster more decidedly compared toFig. 1. Figure 4 shows the frequency of flip
motion is smaller compared to Fig. 2. The
direction of motion of particles is easier toalign with the next-nearest-neighborcouplings.

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International Forum for Green Asia 2020042119 (2015).
[10] A. P. Solon, H. Chaté, and J. Tailleur, Phys. Rev.
Lett. 114, 068101 (2015).
[11] H. Sakaguchi and K. Ishibashi, Phys. Rev. E

100, 052113 (2019).

Email: k.ishibashi.567@s.kyushu-u.ac.jp

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