2009
DOI: 10.3934/krm.2009.2.363
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Abstract: We present a kinetic theory for swarming systems of interacting, self-propelled discrete particles. Starting from the Liouville equation for the many-body problem we derive a kinetic equation for the single particle probability distribution function and the related macroscopic hydrodynamic equations. General solutions include flocks of constant density and fixed velocity and other non-trivial morphologies such as compactly supported rotating mills. The kinetic theory approach leads us to the identification of … Show more

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Cited by 227 publications
(296 citation statements)
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References 34 publications
(74 reference statements)
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“…Some continuum models in the literature [41,42,7] include attraction-repulsion mechanisms and spatial diffusion to deal with random effects. Other continuum models are based on hydrodynamic descriptions [13,10] derived from mean-field particle limits. In fact, as usually done in statistical physics, there is a middle ground in modeling between particle and hydrodynamic descriptions given by the mesoscopic kinetic equations describing the probability of finding particles in phase space.…”
Section: Introductionmentioning
confidence: 99%
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“…Some continuum models in the literature [41,42,7] include attraction-repulsion mechanisms and spatial diffusion to deal with random effects. Other continuum models are based on hydrodynamic descriptions [13,10] derived from mean-field particle limits. In fact, as usually done in statistical physics, there is a middle ground in modeling between particle and hydrodynamic descriptions given by the mesoscopic kinetic equations describing the probability of finding particles in phase space.…”
Section: Introductionmentioning
confidence: 99%
“…In fact, as usually done in statistical physics, there is a middle ground in modeling between particle and hydrodynamic descriptions given by the mesoscopic kinetic equations describing the probability of finding particles in phase space. Kinetic models of swarming has recently been proposed [22,10,11] and the connection between the above IBMs and the continuum models via kinetic theory has been tackled very recently in [21,10,9]. The interest of the kinetic theory models is to give a rigorous tool to connect IBMs and hydrodynamic descriptions as well as to interpret certain patterns as solutions of a given model, as in the case of the double mills [10].…”
Section: Introductionmentioning
confidence: 99%
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“…Working with reduced units and using the standard parameter values [19] for a passive DPD fluid, we set k B T = 1.0, r c = 1.0, A = 25.0, γ = 4.5, and ρ 2D = N L 2 = 2.5 (two-dimensional (2D) analog of ρ 3D = 4.0), where L is the dimensionless edge length of the square computational domain. Apart from standard DPD forces, we incorporate selfpropulsion through a flocking termwith α ≥ 0 the constant self-propulsion force parameter and β ≥ 0 the constant Rayleigh friction parameter [20][21][22][23][24][25]. To model the features of a mixture, we apply f i F only on the fraction φ of active agents and the random contribution of the DPD interactions f ij R only between pairs of the fraction 1 − φ of passive agents.…”
mentioning
confidence: 99%
“…with α ≥ 0 the constant self-propulsion force parameter and β ≥ 0 the constant Rayleigh friction parameter [20][21][22][23][24][25]. To model the features of a mixture, we apply f i F only on the fraction φ of active agents and the random contribution of the DPD interactions f ij R only between pairs of the fraction 1 − φ of passive agents.…”
mentioning
confidence: 99%