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

Ihle, Thomas

doi:10.1088/1742-5468/2016/08/083205

Cited by 27 publications

(53 citation statements)

References 51 publications

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“…Using the same procedure as explained in Ref. [24] but keeping more terms in the expansions, the critical condition Q 1 = 1 for the noise becomes Fig. 2(a).…”

Section: Shift In the Critical Noisementioning

confidence: 99%

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Crossover between parabolic and hyperbolic scaling, oscillatory modes, and resonances near flocking

Bonilla

Trenado

2018

Phys. Rev. E

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A stability and bifurcation analysis of a kinetic equation indicates that the flocking bifurcation of the two-dimensional Vicsek model exhibits an interplay between parabolic and hyperbolic behavior. For box sizes smaller than a certain large value, flocking appears continuously from a uniform disordered state at a critical value of the noise. Because of mass conservation, the amplitude equations describing the flocking state consist of a scalar equation for the density disturbance from the homogeneous particle density (particle number divided by box area) and a vector equation for a current density. These two equations contain two time scales. At the shorter scale, they are a hyperbolic system in which time and space scale in the same way. At the longer, diffusive, time scale, the equations are parabolic. The bifurcating solution depends on the angle and is uniform in space as in the normal form of the usual pitchfork bifurcation. We show that linearization about the latter solution is described by a Klein-Gordon equation in the hyperbolic time scale. Then there are persistent oscillations with many incommensurate frequencies about the bifurcating solution, they produce a shift in the critical noise and resonate with a periodic forcing of the alignment rule. These predictions are confirmed by direct numerical simulations of the Vicsek model. arXiv:1811.06455v2 [cond-mat.soft]

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“…Using the same procedure as explained in Ref. [24] but keeping more terms in the expansions, the critical condition Q 1 = 1 for the noise becomes Fig. 2(a).…”

Section: Shift In the Critical Noisementioning

confidence: 99%

“…In the limit as n → ∞, the central limit theorem implies that we can replace n − 1 integrals (with n − 1 ∼ n) in J (n, j) by [24] J (n, j) = 1 2π…”

Section: Final Remarksmentioning

confidence: 99%

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Crossover between parabolic and hyperbolic scaling, oscillatory modes, and resonances near flocking

Bonilla

Trenado

2018

Phys. Rev. E

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“…This was verified independently by using the mean-field prediction for this coefficient from Refs. [29,60], evaluating the cubic term by hand and observing that it is negligible compared to the linear terms.…”

Section: Hydrodynamic Theory For Vicsek-like Modelsmentioning

confidence: 99%

Transport coefficients of self-propelled particles: Reverse perturbations and transverse current correlations

Nikoubashman

Ihle

2019

Phys. Rev. E

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The reverse perturbation method [Phys. Rev. E 59, 4894 (1999)] for shearing simple liquids and measuring their viscosity is extended to systems of self-propelled particles with time-discrete stochastic dynamics. For verification, this method is first applied to Multi-Particle Collision Dynamics (MPCD) [J. Chem. Phys. 110, 8605 (1999)], a momentum-conserving solvent. An extension to the Vicsek-model (VM) of self-propelled particles [Phys. Rev. Lett. 75, 1226(1995 shows a phenomenon that is similar to the skin effect of an alternating electric current: momentum that is fed into the boundaries of a layer decays mostly exponentionally towards the center of the layer. It is shown how two transport coefficients, i.e. the shear viscosity ν and the momentum amplification coefficient λ, can be obtained by fitting this decay with an analytical solution of the hydrodynamic equations for the VM. As for the MPCD case, the viscosity of the VM consists of two parts, the kinetic and the collisional viscosity. An analytical expression for the collisional part is derived by an Enskog-like kinetic theory. In the following paper (Part II), reasonable quantitative agreement between agent-based simulations and predictions by kinetic theory is observed. In Part II, transverse current correlations and a Green-Kubo relation are used to obtain λ and ν, in excellent agreement with the reverse perturbation results. PACS numbers:87.10.-e, 05.20.Dd, 64.60.Cn, 02.70.Ns I. INTRODUCTION During the past two decades, there has been a large interest in active matter systems, such as bird flocks [1], swarming bacteria [2, 3], active colloids [4, 5], microtubule mixtures [6] and actin networks [7] driven by molecular motors. These systems display interesting behaviors such as pattern formation, collective motion and non-equilibrium phase transitions [8,9]. Some of these features already occur in one of the simplest models for active matter, the Vicsek-model (VM) of self-propelled particles [10-12] and its variants [13][14][15][16][17][18][19][20]. Because of the simplicity of its interaction rules and the existence of a non-standard transition to a collective state of polar order, the VM became an archetype of active matter.Due to the many degrees of freedom, theoretical studies of active matter systems are often based on coarse-grained macroscopic transport equations for the slow variables such as density or momentum. Originally, the general forms of these equations were postulated by symmetry and renormalization group arguments, such as in the seminal Toner-Tu theory [21][22][23] for polar active matter. However, this approach leaves the coefficients of the terms in the transport equation largely undetermined. Furthermore, memory and other nonlocal terms are usually not considered, although for particular models there is evidence on their relevance [24]. These shortcomings motived many researchers to derive macroscopic transport equations directly from the microscopic interactions and to obtain explicit expressions for the occurring coefficients [2...

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“…A kinetic theory taking into account multi-particle interactions, as present in the Vicsek model [110], has also been considered [114,115]. However, this approach still neglects correlations between particles prior to collisions, assuming a factorized form for the pre-collisional N-body distribution.…”

Section: Self-propelled Particles With Aligning Binary Collisionsmentioning

confidence: 99%

Theoretical approaches to the steady-state statistical physics of interacting dissipative units

Bertin¹

2017

J. Phys. A: Math. Theor.

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Abstract. The aim of this review is to provide a concise overview of some of the generic approaches that have been developed to deal with the statistical description of large systems of interacting dissipative 'units'. The latter notion includes, e.g., inelastic grains, active or self-propelled particles, bubbles in a foam, low-dimensional dynamical systems like driven oscillators, or even spatially extended modes like Fourier modes of the velocity field in a fluid for instance. We first review methods based on the statistical properties of a single unit, starting with elementary mean-field approximations, either static or dynamic, that describe a unit embedded in a 'selfconsistent' environment. We then discuss how this basic mean-field approach can be extended to account for spatial dependences, in the form of space-dependent mean-field Fokker-Planck equations for example. We also briefly review the use of kinetic theory in the framework of the Boltzmann equation, which is an appropriate description for dilute systems. We then turn to descriptions in terms of the full N -body distribution, starting from exact solutions of one-dimensional models, using a Matrix Product Ansatz method when correlations are present. Since exactly solvable models are scarce, we also present some approximation methods that can be used to determine the N -body distribution in a large system of dissipative units. These methods include the Edwards approach for dense granular matter and the approximate treatment of multiparticle Langevin equations with coloured noise, which models systems of self-propelled particles. Throughout this review, emphasis is put on methodological aspects of the statistical modeling and on formal similarities between different physical problems, rather than on the specific behaviour of a given system.

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

Cited by 27 publications

References 51 publications

Crossover between parabolic and hyperbolic scaling, oscillatory modes, and resonances near flocking

Crossover between parabolic and hyperbolic scaling, oscillatory modes, and resonances near flocking

Transport coefficients of self-propelled particles: Reverse perturbations and transverse current correlations

Theoretical approaches to the steady-state statistical physics of interacting dissipative units

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