The aim of this paper is to discuss the mathematical modeling of Brownian active particle systems, a recently popular paradigmatic system for self-propelled particles. We present four microscopic models with different types of repulsive interactions between particles and their associated macroscopic models, which are formally obtained using different coarse-graining methods. The macroscopic limits are integro-differential equations for the density in phase space (positions and orientations) of the particles and may include nonlinearities in both the diffusive and advective components. In contrast to passive particles, systems of active particles can undergo phase separation without any attractive interactions, a mechanism known as motility-induced phase separation (MIPS). We explore the onset of such a transition for each model in the parameter space of occupied volume fraction and Péclet number via a linear stability analysis and numerical simulations at both the microscopic and macroscopic levels. We establish that one of the models, namely, the mean-field model which assumes long-range repulsive interactions, cannot explain the emergence of MIPS. In contrast, MIPS is observed for the remaining three models that assume short-range interactions that localize the interaction terms in space.


  1. self-propelled particles
  2. phase separation
  3. excluded-volume interactions
  4. stability analysis

MSC codes

  1. 35Q84
  2. 35R09
  3. 35B35
  4. 60J70
  5. 35C20

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Supplementary Material

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Index of Supplementary Materials

Title of paper: Phase Separation in Systems of Interacting Active Brownian Particles

Authors: M. Bruna, M. Burger, A. Esposito and S. M. Schulz

File: supplement.pdf

Type: PDF

Contents: Derivation and linear stability analyis of the crowded Goldstein-Taylor model. Contains figures showing phase separation patters from the PDE models and the stochastic microscopic model associated to Model 4 (lattice-based model).

File: ABM_animations1.zip

Type: Compressed Animation Files

Contents: Animation videos showing associations to plots shown in the Figures.

File: ABM_animations2.zip

Type: Compressed Animation Files

Contents: Animation videos showing associations to plots shown in the Figures.

File: PDE_animations.zip

Type: Compressed Animation Files

Contents: Animation videos showing associations to plots shown in the Figures.


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Information & Authors


Published In

cover image SIAM Journal on Applied Mathematics
SIAM Journal on Applied Mathematics
Pages: 1635 - 1660
ISSN (online): 1095-712X


Submitted: 18 October 2021
Accepted: 24 May 2022
Published online: 30 August 2022


  1. self-propelled particles
  2. phase separation
  3. excluded-volume interactions
  4. stability analysis

MSC codes

  1. 35Q84
  2. 35R09
  3. 35B35
  4. 60J70
  5. 35C20



Funding Information

Alexander von Humboldt-Stiftung https://doi.org/10.13039/100005156
Deutsche Forschungsgemeinschaft https://doi.org/10.13039/501100001659 : CRC TR 154
H2020 European Research Council https://doi.org/10.13039/100010663 : 883363
Royal Society https://doi.org/10.13039/501100000288 : URF/R1/180040, RGF/EA/181043

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