Kinetic theory for systems of self-propelled particles with metric-free interactions

Chou, Yen-Liang; Wolfe, Rylan; Ihle, Thomas

doi:10.1103/physreve.86.021120

Cited by 53 publications

(98 citation statements)

References 41 publications

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“…The connection between the Vicsek model and the Toner-Tu continuum equations was missing until the equations were derived from a microscopic model of particles with binary collisions using a Boltzmann approach [45][46][47][48]. The method was generalized later by considering multiple collisions using an Enskog-type kinetic theory [49][50][51][52]. Similarly, instead of formulating the Boltzmann equation for the probability distribution density equations, one can derive continuum equations using the Fokker-Planck equation [34,53,54]; however, it does not drastically change * ejtehadi@sharif.edu the result [55].…”

Section: Introductionmentioning

confidence: 99%

Gaussian theory for spatially distributed self-propelled particles

2016

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Obtaining a reduced description with particle and momentum flux densities outgoing from the microscopic equations of motion of the particles requires approximations. The usual method, we refer to as truncation method, is to zero Fourier modes of the orientation distribution starting from a given number. Here we propose another method to derive continuum equations for interacting selfpropelled particles. The derivation is based on a Gaussian approximation (GA) of the distribution of the direction of particles. First, by means of simulation of the microscopic model we justify that the distribution of individual directions fits well to a wrapped Gaussian distribution. Second, we numerically integrate the continuum equations derived in the GA in order to compare with results of simulations. We obtain that the global polarization in the GA exhibits a hysteresis in dependence on the noise intensity. It shows qualitatively the same behavior as we find in particles simulations. Moreover, both global polarizations agree perfectly for low noise intensities. The spatio-temporal structures of the GA are also in agreement with simulations. We conclude that the GA shows qualitative agreement for a wide range of noise intensities. In particular, for low noise intensities the agreement with simulations is better as other approximations, making the GA to an acceptable candidates of describing spatially distributed self-propelled particles.

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Section: Introductionmentioning

confidence: 99%

Gaussian theory for spatially distributed self-propelled particles

2016

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“…VM-like models without a coupling between density and order, such as the Vicsek-model with topological interactions, Refs. [54,59,60], do not show this instability. In this Appendix, the mathematical details of the stability analysis for the standard VM will be given.…”

Section: Appendix a Linear Stability Analysismentioning

confidence: 99%

“…This is typically not the case in the VM, unless directly at the order-disorder transition point. The condition λ = 1 is identical to the condition for the mean-field bifurcation of a homogeneous disordered solution into an ordered solution [13,15,54]. For a given fixed noise η, one can find a critical mean particle number M R,crit by the condition λ(η, M R,crit ) = 1.…”

Section: Reevaluation and Closure Of The Moment Equationsmentioning

confidence: 99%

Chapman–Enskog expansion for the Vicsek model of self-propelled particles

Ihle¹

2016

J. Stat. Mech.

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Abstract. Using the standard Vicsek model, I show how the macroscopic transport equations can be systematically derived from microscopic collision rules. The approach starts with the exact evolution equation for the N-particle probability distribution, and after making the mean-field assumption of Molecular Chaos leads to a multi-particle Enskog-type equation. This equation is treated by a non-standard Chapman-Enskog expansion to extract the macroscopic behavior. The expansion includes terms up to third order in a formal expansion parameter ǫ, and involves a fast time scale. A selfconsistent closure of the moment equations is presented that leads to a continuity equation for the particle density and a Navier-Stokes-like equation for the momentum density. Expressions for all transport coefficients in these macroscopic equations are given explicitly in terms of microscopic parameters of the model. The transport coefficients depend on specific angular integrals which are evaluated asymptotically in the limit of infinitely many collision partners, using an analogy to a random walk. The consistency of the Chapman-Enskog approach is checked by an independent calculation of the shear viscosity using a Green-Kubo relation.

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“…Of course, to fully explore the competition of polar, nematic and other apolarly ordered states, a comprehensive stability analysis similar to Refs. [32,46], and more simulations are needed. This is beyond the scope of this paper.…”

Section: Nematic Solutions and Tricritical Pointsmentioning

confidence: 99%

Tricritical points in a Vicsek model of self-propelled particles with bounded confidence

2014

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We study the orientational ordering in systems of self-propelled particles with selective interactions. To introduce the selectivity we augment the standard Vicsek model with a bounded-confidence collision rule: a given particle only aligns to neighbors who have directions quite similar to its own. Neighbors whose directions deviate more than a fixed restriction angle α are ignored. The collective dynamics of this systems is studied by agent-based simulations and kinetic mean field theory. We demonstrate that the reduction of the restriction angle leads to a critical noise amplitude decreasing monotonically with that angle, turning into a power law with exponent 3/2 for small angles. Moreover, for small system sizes we show that upon decreasing the restriction angle, the kind of the transition to polar collective motion changes from continuous to discontinuous. Thus, an apparent tricritical point with different scaling laws is identified and calculated analytically. We investigate the shifting and vanishing of this point due to the formation of density bands as the system size is increased. Agent-based simulations in small systems with large particle velocities show excellent agreement with the kinetic theory predictions. We also find that at very small interaction angles the polar ordered phase becomes unstable with respect to the apolar phase. We derive analytical expressions for the dependence of the threshold noise on the restriction angle. We show that the mean-field kinetic theory also permits stationary nematic states below a restriction angle of 0.681 π. We calculate the critical noise, at which the disordered state bifurcates to a nematic state, and find that it is always smaller than the threshold noise for the transition from disorder to polar order. The disordered-nematic transition features two tricritical points: At low and high restriction angle the transition is discontinuous but continuous at intermediate α. We generalize our results to systems that show fragmentation into more than two groups and obtain scaling laws for the transition lines and the corresponding tricritical points. A novel numerical method to evaluate the nonlinear Fredholm integral equation for the stationary distribution function is also presented. This method is shown to give excellent agreement with agent-based simulations, even in strongly ordered systems at noise values close to zero.

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Kinetic theory for systems of self-propelled particles with metric-free interactions

Cited by 53 publications

References 41 publications

Gaussian theory for spatially distributed self-propelled particles

Gaussian theory for spatially distributed self-propelled particles

Chapman–Enskog expansion for the Vicsek model of self-propelled particles

Tricritical points in a Vicsek model of self-propelled particles with bounded confidence

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