The purpose of this work is to study the Vicsek model for self-driven articles, in particular its time-continuous version. We will study its mean-field limit, as well as the large-scale behaviour of the model. In particular, we find that the equilibrium distributions change according to whether the density is above or below a given threshold. Below this value, the only equilibrium distribution is isotropic in velocity direction and is stable; moreover, the convergence to this equilibrium is exponentially fast. When the density is above the threshold, we have a second class of anisotropic equilibria, formed by Von-Mises-Fischer distributions of arbitrary orientation. In this case, the isotropic equilibria become unstable and there is exponentially fast convergence to the anisotropic ones.
Collective motion of living organisms: the Vicsek model
Zass, Alexander
2017/2018
Abstract
The purpose of this work is to study the Vicsek model for self-driven articles, in particular its time-continuous version. We will study its mean-field limit, as well as the large-scale behaviour of the model. In particular, we find that the equilibrium distributions change according to whether the density is above or below a given threshold. Below this value, the only equilibrium distribution is isotropic in velocity direction and is stable; moreover, the convergence to this equilibrium is exponentially fast. When the density is above the threshold, we have a second class of anisotropic equilibria, formed by Von-Mises-Fischer distributions of arbitrary orientation. In this case, the isotropic equilibria become unstable and there is exponentially fast convergence to the anisotropic ones.File | Dimensione | Formato | |
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https://hdl.handle.net/20.500.12608/27267