A comparative study between two models of active cluster crystals

We study a system of active particles with soft repulsive interactions that lead to an active cluster-crystal phase in two dimensions. We use two different modelizations of the active force - Active Brownian particles (ABP) and Ornstein-Uhlenbeck particles (AOUP) - and focus on analogies and differences between them. We study the different phases appearing in the system, in particular, the formation of ordered patterns drifting in space without being altered. We develop an effective description which captures some properties of the stable clusters for both ABP and AOUP. As an additional point, we confine such a system in a large channel, in order to study the interplay between the cluster crystal phase and the well-known accumulation near the walls, a phenomenology typical of active particles. For small activities, we find clusters attached to the walls and deformed, while for large values of the active force they collapse in stripes parallel to the walls.


Description of Movie 1
The Movie is realized with N = 2 × 10 3 particles interacting with a GEM-3 potential, given by Eq.(1) of the main text with α = 3. The Movie is composed of four panels each realized with an independent simulation with a different set of parameters. Top panels are obtained with the AOUP model, where the self-propulsion evolves with Eq.(4) of the main text, while bottom panels are realized with the ABP active force dynamics, given by Eq.(3) of the main text. The left column and right one are obtained by fixing D r = 1 (or τ = 1) and D r = 0.1 (or τ = 10), respectively. In each panel, we draw the average active force of each cluster with a green vector in the middle of each cluster. Instead, the black arrow in the middle of the box is the average active force of the whole system. The other parameters involved in the simulations are: U 0 = 2, R = 10 −1 , L = 1, γ = 1, = 1, T = 10 −4 U 2 0 /γD r . The Movie shows that both ABP and AOUP active forces give rise to a stable drifting pattern, whose typical rate of change in drift direction increases with τ (or equivalently with 1/D r ).

Derivation of Eqs.(12) and (13) of the main text
To derive Eqs. (12) and (13) of the main text, it is convenient to switch from the differential stochastic equation (2) of the main text in the presence of the ABP active force, to the associated Fokker Planck equation for the probability distribution, P(x, θ), which reads: being the external potential U a harmonic trap of the form: U (x, y) = k 2 (x 2 + y 2 ) .
Because of the radial symmetry of U we change from Cartesian to Polar coordinates (x, y, θ) → (r, φ, θ), being r = x 2 + y 2 and φ = arctan (y/x), in such a way that the Fokker-Planck equation for the probability distribution functionP(r, φ, θ) becomes: Finding an exact solution of such a partial differential equation is not so easy. Assuming that θ ∼ φ, we can approximate sin (φ − θ) ≈ φ−θ and cos (φ − θ) ≈ 1, neglecting terms of order (φ−θ) 2 and higher. This approximation can hold only in the limit D r → 0. Assuming that the radial flux (i.e. the first square brackets in Eq. (2)) is zero, we can find an approximate stationary solution for the radial component ofP, under the assumption thatP is factorized as the product of a radial component, ρ(r), and an angular component, f (θ, φ). The vanishing of the first square bracket in Eq.
(2) leads toP where f (θ, φ) is to be determined. The radial part of this expression is proportional to ρ(r), and thus p(x) (switching to Cartesian components) is a Gaussian with variance T /k centered around a ring of radius r * = γU 0 /k.
Since circular symmetry of the clusters implies translational invariance of the angular components, it is straighforward to derive that f (θ, φ) = g(φ − θ). Approximating r ≈ r * in the remaining terms of Eq. (2) and using that sin (φ − θ) ≈ φ − θ, we find with N a normalization factor. Eq.(4) shows that φ is distributed as a Gaussian centered at θ, whose variance is Var(φ − θ) = T + γD r (r * ) 2 γU 0 r * , and demonstrates Eq.(13) of the main text. Combining both results we get an approximate solution of the Fokker-Planck equation, whose validity is restricted to the limit D r → 0.