Stationary Modes and Localized Metastable States in a Triangular Lattice of Active Particles

Received 06 April 2018

2018, Vol. 14, no. 2, pp. 195-207

Author(s): Sergeev K. S., Dmitriev S. V., Korznikova E. A., Chetverikov A. P.

The dynamics of a triangular lattice consisting of active particles is studied. Particles with nonlinear friction interact via nonlinear forces of Morse potential. Nonlinear friction slows down fast particles and accelerates slow ones. Each particle interacts mainly with the nearest neighbors due to the choice of the cut-off radius.
Stationary modes (attractors) and metastable states of the lattice are studied by methods of numerical simulation.
It is shown that the main attractor of the system under consideration is the so-called translational mode — the state with equal and unidirectional velocities of all particles. For some parameter values translational modes with defects in the form of vacancies and interstitial particles are possible.
Metastable localized states are presented by the plane soliton-like waves (M-solitons) with inherent velocity and density maxima. The lifetime of such states depends on the lattice parameters and the wavefront width. All metastable states transform into the translational mode after a transient process.

Keywords: lattices, active particles, solitons, Morse potential

Citation: Sergeev K. S., Dmitriev S. V., Korznikova E. A., Chetverikov A. P., Stationary Modes and Localized Metastable States in a Triangular Lattice of Active Particles, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 2, pp. 195-207

DOI:10.20537/nd180204

Download File
PDF, 13.59 Mb


References

[1]	Romanczuk, P., Bar, M., Ebeling, W., Lindner, B., and Schimansky-Geier, L., “Active Brownian Particles: From Individual to Collective Stochastic Dynamics”, Eur. Phys. J. Special Topics, 202:1 (2012), 1–162
[2]	Saintillan, D. and Shelley, M. J., “Instabilities and Pattern Formation in Active Particle Suspensions: Kinetic Theory and Continuum Simulations”, Phys. Rev. Lett., 100:17 (2008), 178103, 4 pp.
[3]	Farrell, F. D. C., Marchetti, M. C., Marenduzzo, D., and Tailleur, J., “Pattern Formation in Self-Propelled Particles with Density-Dependent Motility”, Phys. Rev. Lett., 108:24 (2012), 248101, 5 pp.
[4]	Schweitzer, F., Brownian Agents and Active Particles: Collective Dynamics in the Natural and Social Sciences, Springer, Berlin, 2003, XVI, 421 pp.
[5]	Bechinger, C., Di Leonardo, R., Löwen, H., Reichhardt, Ch., Volpe, Giorgio, and Volpe, Giovanni, “Active Particles in Complex and Crowded Environments”, Rev. Modern Phys., 88:4 (2016), 045006, 50 pp.
[6]	Marchetti, M., Fily, Ya., Henkes, S., Patch, A., and Yllanes, D., “Minimal Model of Active Colloids Highlights the Role of Mechanical Interactions in Controlling the Emergent Behavior of Active Matter”, Curr. Opin. Colloid Interface Sci., 21 (2016), 34–43
[7]	Balboa Usabiaga, F., Kallemov, B., Delmotte, B., Bhalla, A. P. S., Griffith, B. E., and Donev, A., “Hydrodynamics of Suspensions of Passive and Active Rigid Particles: A Rigid Multiblob Approach”, Commun. Appl. Math. Comput. Sci., 11:2 (2016), 217–296
[8]	Alarcón, F. and Pagonabarraga, I., “Spontaneous Aggregation and Global Polar Ordering in Squirmer Suspensions”, J. Mol. Liq., 185 (2013), 56–61
[9]	Romensky, M., Scholz, D., and Lobaskin, V., “Hysteretic Dynamics of Active Particles in a Periodic Orienting Field”, J. R. Soc. Interface, 12:108 (2015), 20150015
[10]	Wensink, H. H., Kantsler, V., Goldstein, R. E., and Dunkel, J., “Controlling Active Self-Assembly through Broken Particle-Shape Symmetry”, Phys. Rev. E, 89:1 (2014), 010302(R), 5 pp.
[11]	Sergeev, K. S., Vadivasova, T. E., and Chetverikov, A. P., “Noise-Induced Transition in a Small Ensemble of Active Brownian Particles”, Tech. Phys. Lett., 40:11 (2014), 976–979 ; Pis'ma Zh. Tekh. Fiz., 40:21 (2014), 88–96
[12]	Suchkov, S. V., Sukhorukov, A. A., Huang, J., Dmitriev, S. V., Lee, C., and Kivshar, Yu. S., “Nonlinear Switching and Solitons in PT-Symmetric Photonic Systems”, Laser Photonics Rev., 10:2 (2016), 177–213
[13]	Barashenkov, I. V., Suchkov, S. V., Sukhorukov, A. A., Dmitriev, S. V., and Kivshar, Yu. S., “Breathers in PT-Symmetric Optical Couplers”, Phys. Rev. A, 86:5 (2012), 053809, 12 pp.
[14]	Saadatmand, D., Borisov, D. I., Kevrekidis, P. G., Zhou, K., and Dmitriev, S. V., “Resonant Interaction of $ϕ^{4}$ $\phi^4$ Kink with PT-Symmetric Perturbation with Spatially Periodic Gain/Loss Coefficient”, Commun. Nonlinear Sci. Numer. Simul., 56 (2018), 62–76
[15]	Borisov, D. I. and Dmitriev, S. V., “On the Spectral Stability of Kinks in 2D Klein – Gordon Model with Parity-Time-Symmetric Perturbation”, Stud. Appl. Math., 138:3 (2017), 317–342
[16]	Saadatmand, D., Dmitriev, S. V., Borisov, D. I., Kevrekidis, P. G., Fatykhov, M. A., and Javidan K., “Effect of the $ϕ^{4}$ $\phi^4$ Kink's Internal Mode at Scattering on a PT-Symmetric Defect”, JETP Lett., 101:7 (2015), 497–502 ; Pis'ma v Zh. Èksper. Teoret. Fiz., 101:7 (2015), 550–555
[17]	Saadatmand, D., Dmitriev, S. V., Borisov, D. I., Kevrekidis, P. G., Fatykhov, M. A., and Javidan K., “Kink Scattering from a Parity-Time-Symmetric Defect in the $ϕ^{4}$ $\phi^4$ Model”, Commun. Nonlinear Sci. Numer. Simul., 29:1–3 (2015), 267–282
[18]	Saadatmand, D., Dmitriev, S. V., Borisov, D. I., and Kevrekidis, P. G., “Interaction of sine-Gordon Kinks and Breathers with a Parity-Time-Symmetric Defect”, Phys. Rev. E, 90:5 (2014), 052902, 10 pp.
[19]	Takatori, S. C. and Brady, J. F., “A Theory for the Phase Behavior of Mixtures of Active Particles”, Soft Matter, 11:40 (2015), 7920–7931
[20]	Mallory, S. A., Valeriani, C., and Cacciuto, A., “Anomalous Dynamics of an Elastic Membrane in an Active Fluid”, Phys. Rev. E, 92:1 (2015), 012314, 6 pp.
[21]	Huepe, C., Ferrante, E., Wenseleers, T., and Turgut, A. E., “Scale-Free Correlations in Flocking Systems with Position-Based Interactions”, J. Stat. Phys., 158:3 (2015), 549–562
[22]	Ferrante, E., Turgut, A. E., Dorigo, M., and Huepe, C., “Collective Motion Dynamics of Active Solids and Active Crystals”, New J. Phys., 15 (2013), 095011, 20 pp.
[23]	Jones, J. E., “On the Determination of Molecular Fields: 2. From the Equation of State of a Gas”, Proc. Roy. Soc. London Ser. A, 106:738 (1924), 463–477
[24]	Morse, Ph. M., “Diatomic Molecules According to the Wave Mechanics: 2. Vibrational Levels”, Phys. Rev., 34:1 (1929), 57–64
[25]	Chetverikov, A. P., Ebeling, W., and Velarde, M. G., “Localized Nonlinear, Soliton-Like Waves in Two-Dimensional Anharmonic Lattices”, Wave Motion, 48:8 (2011), 753–760
[26]	Ikeda, K., Doi, Y., Feng, B.-F., and Kawahara, T., “Chaotic Breathers of Two Types in a Two-Dimensional Morse Lattice with an On-Site Harmonic Potential”, Phys. D, 225:2 (2007), 184–196
[27]	Chetverikov, A. P., Ebeling, W., and Velarde, M. G., “Soliton-Like Excitations and Solectrons in Two-Dimensional Nonlinear Lattices”, Eur. Phys. J. B, 80:2 (2011), 137–145
[28]	Chetverikov, A. P., Ebeling, W., and Velarde, M. G., “Properties of Nano-Scale Soliton-Like Excitations in Two-Dimensional Lattice Layers”, Phys. D, 240:24 (2011), 1954–1959
[29]	Korznikova, E. A., Fomin, S. Yu., Soboleva, E. G., and Dmitriev, S. V., “Highly Symmetric Discrete Breather in a Two-Dimensional Morse Crystal”, JETP Lett., 103:4 (2016), 277–281 ; Pis'ma v Zh. Èksper. Teoret. Fiz., 103:4 (2016), 303–308
[30]	Chetverikov, A. P., Shepelev, I. A., Korznikova, E. A., Kistanov, A. A., Dmitriev, S. V., and Velarde, M. G., “Breathing Subsonic Crowdion in Morse Lattices”, Comput. Condens. Matter, 13 (2017), 59–64
[31]	Dmitriev, S. V., Medvedev, N. N., Chetverikov, A. P., Zhou, K., and Velarde, M. G., “Highly Enhanced Transport by Supersonic $N$ $N$ -Crowdions”, Phys. Status Solidi RRL, 11:12 (2017), 1700298, 5 pp.
[32]	Dmitriev, S. V., Korznikova, E. A., and Chetverikov, A. P., “Supersonic $N$ $N$ -Crowdions in a Two-Dimensional Morse Crystal”, JETP, 126:3 (2018), 347–352 ; Zh. Èksper. Teoret. Fiz., 153:3 (2018), 417–423 (Russian)
[33]	Korznikova, E. A., Bachurin, D. V., Fomin, S. Yu., Chetverikov, A. P., and Dmitriev, S. V., “Instability of Vibrational Modes in Hexagonal Lattice”, Eur. Phys. J. B, 90:2 (2017), 23, 8 pp.
[34]	Korznikova, E. A., Kistanov, A. A., Sergeev, K. S., Shepelev, D. A., Davletshin, A. R., Bokii, D. I., and Dmitriev, S. V., “The Reason for Existence of Discrete Breathers in 2D and 3D Morse Crystals”, Letters on Materials, 6:3 (2016), 221–226 (Russian)
[35]	Dmitriev, S. V., Korznikova, E. A., Baimova, Yu. A., and Velarde, M. G., “Discrete Breathers in Crystals”, Physics-Uspekhi, 59:5 (2016), 446–461 ; Uspekhi Fiz. Nauk, 186:5 (2016), 471–488 (Russian)
[36]	Makarov, V. A., del Rio, E., Ebeling, W., Velarde, M. G., “Dissipative Toda – Rayleigh Lattice and Its Oscillatory Modes”, Phys. Rev. E, 64:3 (2001), 036601, 14 pp.
[37]	Ebeling, W., Landa, P. S., and Ushakov, V. G., “Self-Oscillations in Ring Toda Chains with Negative Friction”, Phys. Rev. E, 63:4 (2001), 046601, 8 pp.
[38]	del Rio, E., Makarov, V. A., Velarde, M. G., and Ebeling, W., “Mode Transitions and Wave Propagation in a Driven-Dissipative Toda – Rayleigh Ring”, Phys. Rev. E, 67:5 (2003), 056208, 9 pp.
[39]	Sergeev, K. S. and Chetverikov, A. P., “Metastable States in the Morse – Rayleigh Chain”, Nelin. Dinam., 12:3 (2016), 341–353 (Russian)
[40]	Velarde, M. G., “Solitons as Dissipative Structures”, Int. J. Quant. Chem., 98:2 (2004), 272–280
[41]	Chetverikov, A. P., Ebeling, W., Velarde, M. G., “Solitons and Clusters in One-Dimensional Ensembles of Interacting Brownian Particles”, Izv. SGU. Novaya Seriya. Fizika, 6:1–2 (2006), 28–41 (Russian)

This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License

Stationary Modes and Localized Metastable States in a Triangular Lattice of Active Particles

References