Active Matter and the Vicsek Model of Flocking


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A short introduction to active matter and the Vicsek model of flocking.

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Active Matter and the Vicsek Model of Flocking

  1. 1. What is Active Matter?• As opposed to passive matter, it is any system consisting of active units with the capability of taking in, employing and dissipating energy.• These often lead to large scale organization.• They can be considered as a state of matter and some condensed matter theory and statistics can be applied to them.• Examples: cytoskeleton with molecular motors, schools of fish, flocks of birds.
  2. 2. Criteria for Active Matter1. The energy input takes place directly at the scale of each active particle, and is thus homogeneously distributed through the bulk of the system, unlike bulk fluids or matter where energy is applied at the boundaries.2. Self-propelled motion is force-free: the forces that particle and fluid exert on each other cancel.3. The direction of self-propelled motion is set by the orientation of the particle itself, not fixed by an external field.
  3. 3. Flocking: The Vicsek Model Vicsek et al, PRL 1995.• Flock of moving particles. Continuous in space, discrete in time.• Every particle has the same fixed speed but different directions of motion.• At each time step a particle assumes the average direction of motion of the particles in its neighborhood of radius r.• Some random perturbation added to this alignment.• Does this satisfy all criteria of active matter?
  4. 4. Vicsek Model: some more specifics• Starts with random density and directions and periodic boundary conditions.• At every time step, the positions are updated:• And the directions of motion:• Statistically, the flow is continuous in space.• Parameters: η, v (0.003 – 0.3), ρ = N/L2.
  5. 5. Simulation (N=300, v=0.03)L=7, η=2.0, t=0 L=25, η=0.1after some time L=5, η=0.1
  6. 6. Von-Mises Distribution of Velocity• It is the distribution of a drift-diffusion system on a circle with a harmonic potential.• Parameters: mean velocity direction and density.
  7. 7. Phase Transition• Net momentum is not conserved for the flow.• Order parameter:• Rises from 0 to 1 as we go from perfectly random to perfectly coherent flow.
  8. 8. Change ofOrder Parameter• Fixed density: reaches 1 with lowering η (chessboard problem).• Fixed noise: does not reach 1.
  9. 9. Analogy with a Ferromagnetism model• Similar tendency to locally align spins in the same direction.• Random part of alignment can be connected to thermal noise.• The difference is the motion in the case of the flock: equilibrium in ferromagnetism is a static uniform alignment, for the flock it is a fixed and uniform direction of flow for all particles.
  10. 10. Advantages of the Vicsek Model• One of the simplest models of a self-driven system showing cooperative behaviour.• Although self-driven systems are unusual in Physics, they are common in living systems.• Transitions have been observed in traffic models (cars are self-driven units)• Behaviour of the order parameter suggests that theoretical methods for equilibrium critical phenomena can be applied to self-propelled far- from-equilibrium systems
  11. 11. Limitations of the Vicsek Model Analysis• It is too minimalistic to model complex living systems and requires more control terms.• Does not explain how living units effect the averaging that is key to the working of the model.• The averaging neighbourhood radius R has not been tuned (traffic model). This might affect the range of observed cooperative flow and thus the order parameter.
  12. 12. References• The Mechanics and Statistics of Active Matter Sriram Ramaswamy, arXiv 1004.1993v1• Novel Type of Phase Transition in a System of Self-Driven Particles Vicsek et al, Phys. Rev. L. 1995 v 75 no 6• Webpage of Pierre Degond: Self-Organized Hydrodynamics