Proceedings of International Forum for Green Asia 2019
5th International Exchange and Innovation Conference on Engineering & Sciences (IEICES 2019)

GA02 / P14
Directional reversal of solitary wave in Ising type Vicsek model
Kazuya Ishibashi*, Hidetsugu Sakaguchi
Department of Applied Science for Electronics and Materials, Interdisciplinary Graduate School of Engineering
Sciences, Kyushu University, 6-1 Kasuga-koen, Kasuga, Fukuoka 816-8580, Japan
E-mail: k.ishibashi.567@s.kyushu-u.ac.jp
Abstract: We study collective phenomena of self-propelled particles using one-dimensional Ising type Vicsek model bydirect numerical simulation. The velocity takes 1 or -1 and multiple particles can occupy one lattice point. We finddirection of propagating solitary wave consisted of them reverses stochastically. The peak of the solitary wave is highest
immediately after reversal; The wave becomes smaller and wider with time. The reversal phenomenon occurs by colliding
broad wave and adverse wave of very small amplitude.
Keywords: Self-propelled particle; Collective motion; Vicsek model; Ising model; Solitary wave.
1. INTRODUCTION
The collective motions of self-propelled particles such as
school of fish and flocks of birds are found in nature.
They have been intensively studied since Vicsek et al.
proposed simple model [1]. Various agent-based models
similar to the Vicsek model have been proposed andnumerically studied to investigate the dynamical
behaviors of a large population of self-propelled particles[2]. Collective motion of biological systems was foundexperimentally and has been studied [3, 4]. In theseexperiments, large structures were formed out of small
traveling objects.
In the Vicsek model (similarly VM hereafter), selfpropelled particles with a constant absolute velocity
interact with other particles. The particles can go in any
direction in plane. At next time step, the particle goes tothe average direction of motion of other particles within
a distance from the particle. In addition, a noise is added
for the average direction. In Vicsek-type models, order–disorder transition is seen. When the noise effect isdominant, the particles move randomly, that is thedisorder state. When the interaction for alignment
dominates, the particles move almost in the samedirection, that is the order state. In addition, it was foundpropagating localized regions of high density with aconstant velocity similarly to solitary waves [5]. Thisphenomenon appears near transition line. The spatial
distribution of the solitary wave state in two-dimensional
system is nonuniform for one axis, but uniform for theother. Here, disorder state is appeared as the background.
(a) wave state appeared in one-dimensional
The solitary
(b)
(a)

3

N-2 N-1

N1

<v>

2

2. ISING TYPE VICSEK MODEL
Ising model is famous as a model of phase transition offerromagnet. The state in Ising model have either upward
or downward spin on lattice. The spin on the latticeinteracts with its nearest neighbors and changes the spin
direction at next time step. Here, we propose Ising type
Vicsek model to study behavior of collective motion ofself-propelled particles. Multiple particles which aremoving right or left can stay on a lattice point in thismodel as shown in Fig. 1(a). The velocity of particle is 1
or -1 on one-dimensional lattice and the particle interacts
with other particles on the lattice point and itsneighboring points. At next time step, each particle
travels in the direction determined by majority rule ofparticles on the lattice points with high probability p.
Then, p is probability providing p = 𝑒 $ /(𝑒 $ + 𝑒 ($ ) ,where g is parameter. At g = 0, particles move randomly
without aligning. For the majority rule of the direction, aparameter Q is used.
𝑄+ =𝑃+ − 𝑀+ +1/2(𝑃+(1 − 𝑀+(1 )+1/2(𝑃+21 − 𝑀+21 ),(1)where Pi and Mi are number of particles moving to rightt1system [6]. The solitary wave state appears away fromtransition line. This phenomenon was arisen owing to aninstability from spatial uniform state in the nonlinear
Kramers eqation. In addition, A reversal of localized
cluster of self-propelled particles appears in active Isingmodel [7].

Next time step

123400500

(c)

0-1 0
0×10

5×104
xt
Fig. 1. (a) Ising type Vicsek model. (b) Reversal of solitary wave. (c) Average velocity of all particles.
N-2 N-1

N-3-1×105

(a)(b)

3×104

8×103

53.6exp(2g)

T1000

1×104

00×100

5×104

1×1050×100(c)

6×103

2×104
TPmax / Qmax

2000

4×103

15.9exp(2g)2×10301(4b)
Figure 2(b) shows average reversal time T in this model at
N = 5000, L = 500. The cross points are numerical results
and solid line is the fitting curve, which is 𝑇(𝑔) =53.6exp(2𝑔). This numerical result agrees well with the-4-30×1000
123ggtFig. 2. (a) Maximum number of particles moving to right or left. (b) Average reversal time in this model. (c) Decay timein Eq. (2).

and left of ith element. All particles N obey above the ruleon the one-dimensional lattice L applied periodic
boundary condition. The Monte Carlo simulation isperformed from disorder state as initial condition.
With time, if probability is high, groups of particles
propagating to the same direction are formed out of thedisorder state where particles are randomly moving. The
groups repeat reversal and form a solitary wave of which
peak is highest immediately after reversal. Figure 1(b)shows the time evolution of the maximum number ofparticles moving to right or left at N = 5000, L = 500, g =2.5. The wave (dashed line) which is smaller amplitude
and wider with time reverses by colliding with adverse
wave (solid line) which is very small amplitude. The same
parameter values are used for Fig. 1(b), (c), and Fig. 2(a).
Figure 1(c) shows time series of the average velocity ofall particles 〈𝑣+ 〉 = (1/𝑁) ∑ 𝑣+ . At < v > = 0, particles
move randomly. At < v > = 1, all particles move to right.
The reversal of solitary wave occurs rapidly and theinterval between two reversals is not constant. Figure 2(a)shows time series of maximum number of particles
moving to right or left, which are denoted by solid anddashed line, respectively. At first, the peaks are attenuatedquickly, and the decrements are gradually slow for latertime. The peaks correspond to reversals of the averagevelocity between -1 and 1 in Fig. 1(c).
Here, we think the movements of particles at next step inthis model. The number of particles moving to right 𝑃+921
and left 𝑀+921 are expressed as99𝑃+921 = 𝑝𝑃+(1+ (1 − 𝑝)𝑀+(1,(2a)921
99𝑀+ = (1 − 𝑝)𝑃+21 + 𝑝𝑀+21,(2b)where p is constant despite of the majority rule. If theprobability p is high and the solitary wave state for rightis strong, by using the condition: 𝑀+9 ~0 , Eq. (2a) issatisfied
9𝑃+921 = 𝑝𝑃+(1.
(3)
The peak is decreased like 𝑝9 with time n. If the reversal
occurs at α < 1, the following relation is given as
𝑝9 = 𝛼.
(4a)
?9 @
?9 @
?9 @
n = ?9 A = ?9{C D/(C D2C ED)} = − ?9(12C EGD) ~(− ln 𝛼)𝑒 I$ .

2

approximation in Fig. (4b). For g > 2.6, there is deviation
for the average owing to short of number of reversals.
Figure 2(c) shows a decay time T to reach 1/10Pmax in Eq.
(2) at N = 2000, L = 500, 𝑃S/I = 𝑃S⁄I21 = 1000. The fitting
curve is 𝑇(𝑔) = 15.9exp(2g). This result is close to theresult in Fig. 2(b) in Ising type Vicsek model.
3. SUMMARY
We have studied the Ising type Vicsek model for onedimensional self-propelled particles. The direction of
propagating solitary wave reverses stochastically. The
peak and the reversal time are not constant. The reversal
time is approximated to 𝑒 I$ . We would like to generalize
this model, taking effect of broader interaction or in twodimensional system for further understanding thephenomenon.
4. 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, Nature 483, 448(2012).
[4] H. Kuwayama, S. Ishida, Sci Rep. 3 2272 (2013).
[5] H. Chaté, F. Ginelli, G. Grégoire, F. Peruani, and F.
Raynaud, Eur. Phys. J. B 64, 451 (2008).
[6] H. Sakaguchi and K. Ishibashi, J. Phys. Soc. Jpn. 86,114003 (2017).
[7] A. P. Solon and J. Tailleur, Phys. Rev. Lett. 111,078101 (2013).