Statistical physics · Active matter · Collective motion

Self-Propelled Particles and Collective Behavior

Self-propelled particle models make it possible to investigate how simple local interactions can generate collective motion, organized patterns, and transitions between ordered and disordered states.

This research line uses computational simulations of self-propelled particle systems to study phenomena inspired by biological groups such as fish schools, bird flocks, swarms, and other aggregates. The studies examine the effects of noise, speed, density, interaction range, and different alignment rules on collective organization.

What are self-propelled particles?

A self-propelled particle is an agent capable of moving by consuming energy and changing its direction through interactions with other agents. In the Vicsek model, each particle moves at approximately constant speed and updates its orientation from the average direction of neighboring particles, with an added noise term.

\[\mathbf{r}_i(t+\tau)=\mathbf{r}_i(t)+\mathbf{v}_i(t)\tau\]

\[\theta_i(t+\tau)=\operatorname{arg}\!\left[\sum_{j\in \mathcal{N}_i}e^{i\theta_j(t)}\right]+\eta\,\varepsilon_i(t)\]

Despite the simplicity of the microscopic rules, the system can display complex macroscopic behavior, including collectively ordered states and phase transitions.

Important parameters

Noise \(\eta\) represents orientational perturbations. Speed, density, and interaction range modify the frequency and strength of interactions among individuals.

Central idea: relatively small changes in these parameters can significantly alter the collective state of the system.

Order, disorder, and phase transitions

The global degree of alignment can be measured by the order parameter \(\psi\), defined from the normalized mean velocity of the particle ensemble.

\[\psi=\frac{1}{Nv}\left|\sum_{i=1}^N\mathbf{v}_i\right|\]

When \(\psi\) is close to zero, directions are predominantly disordered. When \(\psi\) approaches one, the particles exhibit strong collective alignment. Increasing noise can drive the system from an ordered state to a disordered state.

Order parameter as a function of noise for Metropolis and Vicsek rules — Order parameter as a function of noise intensity for different system sizes, comparing the Metropolis and Vicsek rules. Figure based on *Collective dynamics in a Vicsek model with Metropolis-based interactions*, Physica A 678 (2025) 130958.

Collective behavior patterns

Depending on the interaction rules and model parameters, simulations can generate different collective states. These include disordered motion, coherent collective motion, collective rotation, and formation flight.

Four collective behavior states in self-propelled particle groups — Four collective states observed in simulations: disordered motion, coherent collective motion, collective rotation, and formation flight. Figure from *Collective behavior states in animal groups*, Modern Physics Letters B 31 (2017) 1750054.

Collective rotation and the milling pattern

In the collective rotation state, individuals move coherently around a central region, forming a ring-like structure. This behavior is commonly referred to as milling.

A measure used to characterize this state is the angular-momentum parameter \(M\), which quantifies the degree of rotation of the group around its center of mass.

\[M=\frac{\left|\sum_{i=1}^N \mathbf{r}_i\times\mathbf{v}_i\right|}{\sum_{i=1}^N |\mathbf{r}_i|\,|\mathbf{v}_i|}\]

Values close to one indicate strong collective rotation. The studies show that milling formation can depend on parameters such as speed, density, and particle number.

Collective milling pattern — Milling pattern obtained in simulation. Figure from *The milling pattern in animal groups and its dependence on the density and on the number of particles*, Physica A 507 (2018) 289–293.

Noise and different interaction rules

Another part of this research line investigates how the way noise is introduced changes collective dynamics. Angular noise, vectorial noise, and hybrid combinations have been studied. In more recent work, the local alignment rule of the Vicsek model was also compared with an interaction mechanism inspired by the Metropolis criterion.

These extensions make it possible to compare different microscopic mechanisms and examine how they affect spatial correlations, the order parameter, susceptibility, and the Binder cumulant.

Related publications

Effects of Two Distinct Types of Noise on the Directional Alignment of a Self-Propelled Particles System

Chinese Journal of Physics 53 (2015). DOI: 10.6122/CJP.20150616

Critical exponents of a self-propelled particles system

Physica A 444 (2016) 582–588. DOI: 10.1016/j.physa.2015.10.075

Collective behavior states in animal groups

Modern Physics Letters B 31 (2017) 1750054. DOI: 10.1142/S0217984917500543

Behavioral transitions induced by speed and noise in animal aggregates

International Journal of Modern Physics B 31 (2017) 1750073. DOI: 10.1142/S0217979217500734

The milling pattern in animal groups and its dependence on the density and on the number of particles

Physica A 507 (2018) 289–293. DOI: 10.1016/j.physa.2018.05.111

Vicsek model of self-propelled particles with hybrid noise

Modern Physics Letters B 34 (2020) 2050144. DOI: 10.1142/S0217984920501444

Collective dynamics in a Vicsek model with Metropolis-based interactions

Physica A 678 (2025) 130958. DOI: 10.1016/j.physa.2025.130958