Collective Dynamics of Active Particles

MATH 858F, Fall 2024

Course Information

 

Course description

Ants form colonies, birds flock, mobile networks synchronize, and a consensus may emerge from the interaction of diverse human opinions. These are simple examples of collective dynamics in which small scale interactions lead to emergence of larger-scale patterns.
We will survey recent mathematical developments in collective dynamics. Three levels of descriptions are considered: agent-based dynamics on graphs, Vlasov-type kinetic description, and large-crowd hydrodynamics. The dynamics is governed by different protocols of pairwise interactions. Different models for such protocols, based on attraction alignment and repulsion, go back to the influential works of Reynolds, Krause, Vicsek and Cucker & Smale.
A main question of interest is how different protocols affect the large-time, large-crowd dynamics, and the emergence of large-scale patterns.





Syllabus

1     Agent-based models: slides [ 10 examples ] lecture notes [ Part I ]

1.1  First-order models for aggregation • Consensus . . . . . . . . . . . . . . . . . . . .
       Bounded confidence model • Bearing only model . . . . . . . . . . . . . . . . . . .
       Clustering • Consensus-based mean-shift . . . . . . . . . . . . . . . . . . . . . . . .
       Transformers and the geometry of emerging clusters . . . . . . . . . . . . . . . .
1.2  Second-order models for alignment • Flocking • Swarming . . . . . . . . . . . .
1.2.1 Cucker-Smale model • Vicsek model . . . . . . . . . . . . . . . . . . . . . . . . . . .
       Extensions: Motch-Tadmor model • Tendency • p-alignment • PTWA model
1.2.2 Communication kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
       Metric vs. topological • Short- vs. long-range tails • Regular vs. singular head
1.3  3Zone model: Attraction • Repulsion • Alignment . . . . . . . . . . . . . . . . . . .
       Anticipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4  Extensions • External forcing • Rayleigh-type friction • Time delay . . . . . . .
1.5  Synchronization • Kuramoto and related models . . . . . . . . . . . . . . . . . . . .

2    Large-time emerging behavior lecture notes [ Part II ]

2.1  First-order models • Consensus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2  Second-order models • Flocking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
      ℓ^∞-diameter -- coefficient of ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . .
      ℓ²-diameter -- graph connectivity • spectral gap . . . . . . . . . . . . . . . . . . . .
      Short-range kernels: propagation of connectivity . . . . . . . . . . . . . . . . . . .
      Homophilious vs. heterophilious dynamics. . . . . . . . . . . . . . . . . . . . . . . .

Assignment #1 [ pdf file ]

3    Large-crowd behavior -- hydrodynamic description

3.1  Mean-field limit -- First-order aggregation models . . . . . . . . . . . . . . . . . . .
3.2  Mean-field limit -- second-order models • . . . . . . . . . . . . . . . . . .

Related material:
Kennard (1938) Derivation of Liouville eq
P. E. Jabin [A review on mean field limit]
V. Nguyen and R. Shvydkoy [propagation of chaos for heavy-tailed Cucker-Smale]

4    Large-crowd behavior -- hydrodynamic description

4.1 The closure of entropic pressure
4.2 Regularity and critical thresholds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Related material:
Critical thersholds: Bounded kernels [1D] [2D] [1D Singular kernel]

4.3  Large-time emerging behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
       Spectral gap • Energy Flctuations • Entropy . . . . . . . . . . . . . . . . . . . . . .
4.4  Multi-species • Anticipation • External forcing . . . . . . . . . . . . . . . . . . . . .

4     Multi-scale descriptions

4.1  Multi-Flocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2  Multi-species. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Assignment #2 [ pdf file ]


References:

• N. Bellomo, J. Carrillo, P. Degond & E. Tadmor (eds), ``Active Particles'', Vol 1 (2017), Vol 2 (2019)
  Vol 3 (2022), Vol 4 (2024), Birkhäuser.
• I. Couzin & N. Franks, Self-organized lane formation and optimized traffic flow in army ants,
  Proc. R. Soc. Lond. B, 270 (2003) 139-146.
• F. Cucker & S. Smale, On the math. of emergence, Japan. J. Math. 2 (2007) 197-227.
• F. Cucker & S. Smale, Emergent behavior in flocks. IEEE Trans. Automat. Control 52 (2007).
• S.-Y. Ha & E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking,
  Kinetic and Related Models 1(3) (2008) 415-435.
• R. Hegselmann & U. Krause, Opinion dynamics and bounded confidence: models, analysis simulation,
  J. Artificial Soc. and Social Simul. 5(3) (2002).
• S. Motsch & E. Tadmor, A new model for self-organized dynamics and its flocking behavior,
  JSP 144(5) (2011) 923-947.
• C. Reynolds, Flocks, herds and schools: A distributed behavioral model,
  ACM SIGGRAPH 21 (1987) 25-34.
• R. Shu & E. Tadmor, Flocking hydrodynamics with external potentials, ARMA 238 (2020) 347-381.
• R. Shu & E. Tadmor, Anticipation breeds alignment, ARMA 240 (2021) 203-241.
• R. Shvydkoy, Dynamics and Analysis of Alignment Models of Collective Behavior, Springer, 2021
  (lecture notes).
• R. Shvydkoy & E. Tadmor, Topologically-based fractional diffusion and emergent dynamics with
  short-range interactions, SIMA 52(6) (2020) 5792-5839.
• E. Tadmor, Swarming: hydrodynamic alignment with pressure, Bulletin AMS 60(3) (2023) 285-325.
• T. Vicsek, Czirók, E. Ben-Jacob, I. Cohen & O. Schochet, Novel type of phase transition in a system of
  self-driven particles, PRL 75 (1995) 1226-1229.
• T. Vicsek & A. Zefeiris, Collective motion, Physics Reprints, 517 (2012) 71-140.


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Eitan Tadmor