Microscopic field theory for structure formation in systems of self-propelled particles with generic torques

Active matter encompasses a large number of systems that constantly convert internal energy into autonomous motion. They, thus, evolve far from equilibrium, such that the understanding of their collective states resulting from the combination of different interactions between their constituents, in the presence of dissipation and fluctuations violating the standard rules of equilibrium dynamics, still raises interesting challenges. Lifting constraints imposed by equilibrium, such as the fluctuation–dissipation theorem, active matter can reach novel spatiotemporal structures absent in passive matter.^1,2 Living organisms constitute obvious examples of active matter, although artificially self-propelled objects, such as granular^3–6 or colloidal^7–11 particles, have become popular in soft matter physics labs over the last decade.

In order to somehow grasp the physics of collections of active particles, quite some efforts have been devoted to the study of simplified models.^12,13 In this context, much progress has been achieved in the fundamental understanding of two widespread collective phenomena in active particle systems: on one hand, the emergence of collective motion, or flocking, and on the other hand, particle clustering in the absence of attractive interactions.

The Vicsek model was the first attempt to describe the former phenomenon, i.e., the emergence of collective motion, as a symmetry breaking phase transition, due to the competition between noise and local alignment of the particles’ (birds) self-propulsion direction.^14–16 Its continuum counterpart, the Toner–Tu theory, provides a hydrodynamic description incorporating the same main fundamental ingredients as the agent-based model.^17,18 Such a continuum approach allows for an understanding of the generic mechanisms controlling the large-scale properties of systems of aligning self-propelled particles, hoping them to be, to some extent at least, universal. Since then, further efforts have been put into deriving continuum equations in a consistent way, by explicitly coarse-graining the stochastic dynamics of self-propelled particles.^19–22 One of the advantages of constructing a field theory starting from the microscopic dynamics is that it allows for a direct comparison between the continuum and particle-based results, something which is missing in the original Toner–Tu approach, based on symmetry and conservation laws assumptions. Besides the Vicsek model (and its variants), which prescribes a velocity-aligning torque between otherwise non-interacting agents, flocking behavior can also arise from excluded-volume interactions between elongated self-propelled particles^23–27 or from other more complex mechanisms involving the coupling with the environment, or the specificities of the particles’ self-propulsion mechanism.^3,28–31

The other salient phenomenon generically encountered in active systems is the spontaneous aggregation in the absence of attractive interactions.^8,9,32 At the level of simple models, this phenomenon is thought to arise from the mere competition between persistent motion along a given direction and excluded-volume interactions. For persistent enough particles and at high enough densities, one can eventually observe the system phase separates into a macroscopic dense cluster surrounded by a dilute gas-like phase, a phenomenon known as motility-induced phase separation (MIPS).^33,34 MIPS has been reported in numerical studies of minimal models, like Active Brownian Particles (ABP), consisting of persistent Brownian spherical particles, interacting with, typically, an isotropic short-range repulsive potential.^34–40 Here as well, there have been several attempts to construct a continuum field theory to describe such non-equilibrium phase transition, exploiting symmetry and conservation laws, or trying to make a smooth connection with the dynamics of the microscopic models^41–49—in all cases under strong assumptions hard to put into test.

Besides the achiral active particles considered by the Vicsek and the ABP models, self-propelled chiral particles, whose propulsion direction turns at a given rate, are also common to encounter at different scales, constituting yet another class of active particle systems. Examples include micro-organisms showing autonomous rotation, as is the case of E. Coli near a wall^50–52 and sperm cells,^53,54 L-shaped Janus colloids,⁵⁵ or chiral grains,⁶ among others. The study of such circle swimmers from minimal models, usually thought of as extensions of the Vicsek and ABP models including an intrinsic frequency, is attracting increasing attention over the past few years,^56–64 laying the groundwork for an analysis of the interplay between chirality and aligning interactions or excluded-volume effects, respectively. Systems of chiral active particles might feature both macrophase separation and microphase separation, depending on the rate at which their heading direction turns.^57,60,63

The interplay between both excluded-volume and aligning interparticle interactions, or torques, has been addressed in a series of works,^{11,25,27,65–75} focused on activity-induced aggregation. Following the approach first introduced for pure ABP systems in Ref. 45, a hydrodynamic description has been derived for specific types of polar and nematic (Vicsek-like) alignment rules,⁷⁴ for chiral ABP⁶³ and for dipolar ABP.⁷⁶ However, a general theoretical framework encompassing both chiral and achiral self-propelled particles interacting via generic central forces and aligning torques is still missing. Besides its formal interest, establishing a theory incorporating torques would allow us to address several questions on general grounds. For instance, why, despite their fundamental difference, both circle swimmers and spinning particles (with no self-propulsion) suppress phase separation for large enough rotational frequencies. On one hand, it is now known that intrinsic torques generically interrupt MIPS^63,72,75 in systems of chiral self-propelled particles, giving rise to clusters of self-limited sizes, both in systems with and without alignment interactions.^57,58,63 On the other hand, ferromagnetic colloidal particles, spinning at a given frequency imposed by an external magnetic field, have a tendency to condense as a result of mutual attractive interactions, but this phase separation is arrested at large enough spinning frequencies, giving rise, again, to finite-sized clusters.⁷⁷ To what extent a similar mechanism might explain finite-size clustering in both set-ups remains to be an interesting open problem.

Here, we establish a general framework that allows us to systematically derive continuum hydrodynamic equations describing systems of self-propelled particles subjected to generic torques acting on the self-propulsion direction, under the assumption of isotropy and homogeneity (we will study later on the linear stability of such a homogeneous state). These torques can be intrinsic to the particle or resulting from interparticle interactions. Interestingly, we can define a set of parameters that capture, at the microscopic level of pairwise interactions, the effect of both the activity and the torques considered. This paves the way for a mean-field analysis of the destabilization of the homogeneous phase, which might lead to different scenarios depending on the origin and type of torques considered. Within this framework, one can show that the activity triggers a spinodal-like long-wavelength instability associated with MIPS (the location of which is affected by the self-propulsion mechanism and the different interactions), while self-rotations trigger a short-wavelength instability, introducing a characteristic length ℓ. This characteristic length appears to scale as the inverse of the turning rate $ℓ∼ω0−1$⁠, as found in a simple model of chiral active particles with polar alignment⁵⁷ and in suspensions of spinning hematite colloids,⁷⁷ suggesting a common mechanism underlying these two a priori unrelated physical systems.

The paper is organized as follows: In Sec. II, starting from the microscopic dynamics, we derive the (mean-field) continuum hydrodynamic equations that govern the evolution of the density and the polarization fields. We then go on to analyze the linear stability of the homogeneous and isotropic phases in Sec. III. We formally show that adding both an intrinsic frequency of rotation and interparticle torques to the dynamical equations qualitatively changes the phase behavior predicted by the linear stability analysis. We discuss the different cases in Secs. III B 1 and III B 2.

The function ψ_N gives the probability to find N particles of the system at N given positions in a 2D space, r_i(t) = (x_i, y_i), and with N given orientations φ_i. The system is composed of particles that self-propel at constant speed, v₀, along e_i = (cos φ_i, sin φ_i), and rotate at an intrinsic frequency, ω₀. They are also subjected to thermal and rotational noise, characterized by the diffusion constants D₀ and D_r, respectively. From now on, we set D₀ = 1 without loss of generality.

For the sake of generality, we do not specify the functional form of $U$⁠. Consequently, we are deriving a framework to describe systems of self-propelled particles whose interactions depend on the center-to-center distance between pairs and on their inner orientation.

Forces come from the spatial dependency of the pair potential. Typically, one considers excluded-volume interactions, which set the particles’ characteristic finite size. Conversely, torques result from the orientation dependency in Eq. (2) and, thus, act on the direction of self-propulsion of particles. Depending on the type of aligning potential considered, torques can lead to different scenarios. Some particular cases have been already studied in the literature. Vicsek-like aligning rules, which are decoupled from the excluded-volume forces, are considered in Ref. 74. On the contrary, dipolar interactions between permanent point dipoles, which couple spatial and angular degrees of freedom, are analyzed in Ref. 76.

To proceed, we introduce a change in variables. As we only focus on small deviations from homogeneity, one defines the vector distance r₁₂ = r₂ − r₁ = r₁₂(cos ω, sin ω) and the orientations φ₁ and φ₂. Thus, in the lab frame of reference, the set of independent variables is (r₁₂, ω, φ₁, φ₂), as depicted in Fig. 1. We note that the integrals in Eqs. (4) and (5) are over r₂ and φ₂, while r₁ and φ₁ are kept fixed. This allows us to express orientations as a function of φ₁, and, therefore, we define φ₁₂ = φ₂ − φ₁. Employing a body-fixed frame, one can express the directions along the plane relative to e₁. We thus introduce θ₁ = φ₁ − ω and θ₂ = φ₂ − ω. However, θ₂ can be expressed as a function of θ₁ and φ₁₂, θ₂ = φ₁₂ − θ₁. We can, therefore, use r₁₂, θ₁, and φ₁₂ as our set of independent variables, without loss of generality.

In the remainder of the paper, we shall drop the subscripts for clarity. We will now group the two-body terms in the force and torque’s expression in single scalar coefficients.

This factor κ is linked to the spatial and orientational correlations encoded in $Gr,θ,φ,t$⁠. If one considers the passive-particle limit, i.e., v₀ = 0, then, in a homogeneous suspension of isotropic disks, it is equally probable to find a particle at any distance from the tagged particle, r, in any in-plane direction, θ, and with any relative orientation φ. Therefore, the correlation function fulfils the head–tail symmetry (θ → θ + π) and the symmetry against exchange of particles’ position (θ → −θ), as well as the parallel–antiparallel symmetry (φ → φ + π) and the symmetry against exchange of orientations (φ → −φ). Introducing activity breaks the symmetry θ → θ + π. This implies that it is more likely to find another particle in front of the tagged particle than behind of it, a signature of the self-trapping mechanism that leads to MIPS.

Keeping these symmetries in mind, it is straightforward to argue the cases in which κ has a non-zero value. The alignment mechanisms usually studied in the field enter in the interaction potential with an even dependency in φ (e.g., Vicsek-like alignment interactions). Alternatively, one may think of more complex interactions also leading to effective alignment, like dipole–dipole interactions, which also involve an even dependency in θ (i.e., dipolar interactions involve a dependency on both relative orientations and relative positions in space). All in all, this results in $∂φur,φ$ having an odd dependency in φ and, depending on the specific interaction considered, also in θ. Thus, the product of $∂φur,φ$ times a correlation function $Gr,θ,φ,t$⁠, which fulfils the symmetries against exchange of particles’ position and exchange of orientations (i.e., it is even in φ and θ), results in κ = 0 upon integration, Eq. (10). This scenario does not change in the presence of activity. We thus state that κ remains identically zero as long as the symmetry φ → −φ and/or θ → −θ are not broken.

The first term on the right-hand side (RHS) of Eq. (11) is the component of the force acting along the direction of self-propulsion. We can interpret this component as the one quantifying the imbalance between the self-propulsion of the tagged particle and its arrest induced by collisions with neighboring particles.

In a system of passive colloids, ζ = 0, which can be simply understood applying the same symmetry arguments given earlier and, thus, it is independent of the interparticle potential. This further means that, in the present construction, ζ is irrelevant for a standard spinodal decomposition in an equilibrium system of attractive particles at low enough temperatures. Our approach is particularly tailored for activity-induced aggregation, exploiting the preferred direction of motion e to decompose the effective force, Eq. (7). In order to account for equilibrium phase separation, one could invoke a mean-field approximation and split the two-body distribution function as a product of one-body ones. This will lead to an effective diffusivity at the level of the one-body Smoluchowski equation (see below) that will change sign when the homogeneous state becomes unstable, signaling a spinodal long-wavelength instability.

As soon as the activity enters the systems, the θ → θ + π symmetry is broken and, thus, ζ ≠ 0, due to the cos θ term stemming from the projection of the force (see the Gram–Schmidt orthonormalization in  Appendix A). We note that in the active case, ζ will also have contributions from the aligning potential, evidencing that in the model we have derived, alignment interactions modify the force imbalance arising from the collision persistence and captured by ζ.

The second term on the RHS of Eq. (11) can be interpreted as an effective diffusion acting along the gradient of the one-body probability distribution.

The first two terms on the RHS of Eq. (14) correspond to the advection and diffusion, respectively, of the spatial degrees of freedom. Here, the translational advection term sets an effective self-propulsion speed, $vρ̄$⁠, Eq. (15), which decays with the mean density, $ρ̄$⁠, at a rate given by ζ and which can thus be interpreted as a translational friction coefficient, accounting for the arrest of particles in crowded environments. $D$ is an effective many-body diffusivity. The third and fourth terms on the RHS of Eq. (14) correspond to the advection and diffusion of the orientations. In the advective term, $ερ̄$ is the effective frequency of rotation, where ω₀ is the intrinsic frequency of rotation, as stated earlier, and κ can be interpreted, by analogy with ζ, as a rotational friction coefficient, stemming from interparticle aligning torques. Note the equivalent role played by (⁠$vρ̄$⁠, v₀, ζ) and (⁠$ερ̄$⁠, ω₀, κ), Eq. (15). Finally, the rotational diffusion coefficient is D_r.

The microscopic information of the one-body equation just derived is captured by ζ, κ, and $D$⁠, which link the one-body distribution to higher order ones. To proceed, we make the assumption that ζ, κ, and $D$ are independent of the tagged particle’s position, which is valid as long as the system is in (close to) an homogeneous state. We, therefore, close the hierarchy of coupled equations by considering these parameters as constants. This is a central approximation of our approach, which allows us to derive effective hydrodynamic equations.

We emphasize that, opposed to top-down approaches that base the derivation of the effective hydrodynamic equations on symmetry arguments and conservation laws,⁴² our approach directly coarse-grains the microscopic dynamics. Thus, the coefficients we define are not phenomenological but stem from interparticle interactions. They, indeed, take specific numerical values in particle-based models, whose calculation allows for a direct quantitative comparison between the microscopic model and the coarse-grained theory. Another relevant feature of this approach^45,74 is the effective speed $vρ̄$ decaying at increasing density (a signature of MIPS), which here is an outcome of the derivation and not introduced as an hypothesis.

Here, ⊥ indicates a rotation corresponding to $p⊥=Rp$ and $∇⊥=R∇$ with $R=0−110$⁠. The time evolution equation of each moment is linearly coupled to the next order moment. Therefore, the time evolution of the polarization is coupled to the nematic tensor, Q. To close the set of hydrodynamic equations, we drop the dependency of Eq. (19) on Q. As we show in  Appendix C, after dropping the dependency in Q and performing an adiabatic approximation to the hydrodynamic equations (i.e., ∂_tp = 0), we still capture the relevant information on the linearly unstable modes at any wave vector. This proves that ρ(r, t) is the slowest moment of the probability distribution and the higher order moments, i.e., p and Q, are enslaved to it. This, in turn, justifies cutting the hierarchy of hydrodynamic equations to Q.

The hydrodynamic equations above have an isotropic homogeneous steady-solution $(ρ(r,t)=ρ̄,p(r,t)=0)$ but do not admit a polar steady-solution, as continuum theories of flocking. The present theory does not provide a symmetry breaking term à la Landau, as in the Toner–Tu theory, and, therefore, it is limited to the description of non-polar states.

Our goal is to study how torques affect the structure formation in systems of self-propelled particles, which result from the competition between self-propulsion and interparticle collisions. In the mean-field model we have derived, the effect of torques (intrinsic or due to interparticle alignment) is captured in $ερ̄$⁠, while ζ quantifies the collision persistence. Thus, to explore the phenomenology of our model, we need to scan a set of three parameters: v₀, $ερ̄$⁠, and ζ. From now on, we note $ερ̄=ε$⁠.

The eigenvalues correspond to the dispersion relations quantifying the growth of a perturbation with dimensionless wave vector q and allow us to explore the onset of linear instabilities. As mentioned before, the parameter space of our model is conformed by $v0v*$⁠, ζ, and ɛ. In Sec. III A, we briefly discuss the torque-free case ɛ = 0, which has been extensively studied in Refs. 45 and 78 and which we add for completeness. Here, the two relevant parameters controlling the system’s destabilization are $v0v*$ and ζ. We then move on to introduce ɛ and show that the predictions of the linear stability analysis qualitatively change in the presence of effective torques.

To obtain ɛ ≠ 0, one can think of a functional form of the alignment potential involving odd dependencies in the angular variables, which would automatically lead to κ ≠ 0 upon integration. It is worth mentioning, though, that alignment interactions leading to either parallel or antiparallel alignment involve even dependencies in the angular variables. Alternatively, one can consider a nonreciprocal pairwise alignment interaction that breaks the symmetry under exchange of orientations φ → −φ and/or under exchange of particles’ position θ → −θ and that, thus, results in κ ≠ 0. Besides, chiral active particles self-rotate at an intrinsic frequency ω₀, adding a constant (non-zero) contribution to ɛ.

For systems with no effective torques, ɛ = 0. In this case, originally considered in Ref. 45, the phase behavior is exclusively controlled by the competition between activity and interparticle collisions. Alternatively, effective torques can also lead to ɛ = 0 in some particular microscopic achiral models, such as systems with Vicsek-like alignment⁷⁴ or dipole–dipole interactions,⁷⁶ where the symmetry of the alignment interaction together with the symmetry of the angular correlation leads to a rotational friction coefficient that is identically zero upon integration (see the discussion on angular symmetries in Sec. II).

Such LW instability, coming from an increase in the effective friction ζ along the direction of self-propulsion as the self-propulsion speed increases, is associated with MIPS.⁴⁵ 

We now study the impact of a non-vanishing ɛ in the stability of the homogeneous phase. We thus explore the parameter space $(v0v*,ζ,ε)$ and show that taking ɛ into account qualitatively changes the linear stability of the homogeneous isotropic state. We recall that one way of realizing this is by considering self-turning chiral particles.

Out of the three eigenvalues, only λ₁ gives rise to a growing instability, just as in the ɛ = 0 case, as shown in Sec. III A. We also note that λ₁ does not have a complex part. On the contrary, λ₂ and λ₃ are complex conjugated numbers. We, moreover, point out that their functional form does not reduce to that in Eq. (26) when ɛ = 0. This is due to the fact that introducing an effective frequency of rotation ɛ changes the nature of the solutions of the characteristic polynomial, Eq. (B1). For an arbitrary value of ɛ, we show in  Appendix B that, near q = 0, the solutions of the characteristic polynomial lead to one real and two complex conjugated eigenvalues, resulting in Eq. (28) after Taylor expanding Eq. (B7) close to q = 0. Nevertheless, for small values of ɛ below the threshold value $εt=+972−8318$ (computed in  Appendix B), the eigenvalues correspond to three different real numbers. In the particular case ɛ = 0, their expression is given in Eq. (25), which leads, in turn, to the Taylor expansion in Eq. (26).

Increasing ɛ leads to a shift in the LW instability region to higher values of the self-propulsion speed, as shown in Fig. 3. This result from the linear stability analysis is consistent with the phase behavior of chiral active particles reported in previous studies, showing that active rotation generically hinders motility-induced phase separation.^60,62,63 As put forward in Ref. 60, this can be understood as a result of the faster reorientation of chiral active particles, which opposes the self-trapping mechanism responsible for cluster formation. In our formalism, the hindrance of MIPS is evidenced by the shift of the instability region to higher values of $v0v*$⁠, at increasing ɛ. According to our theory, one needs larger self-propulsion speed to eventually destabilize the homogeneous state and reach a condensate of chiral active particles.

So far, we have focused on LW instabilities that signal the onset of a phase separation. However, the formalism we have derived allows us to study instabilities happening at any wave vector q, meaning λ₁(q) > 0. In fact, our analysis predicts a short-wavelength (SW) instability for ɛ ≠ 0 over a broad range of parameter values. A finite q* > 0 indicates the growth of some structure, or pattern, with a characteristic length scale ℓ ∼ 1/q*. Therefore, in the SW instability region of the parameter space, a phase separation (i.e., MIPS) is not expected, but rather the formation of smaller finite-sized clusters, which, according to the prediction to linear order, will not coarsen to form a macroscopic structure.

We numerically compute the eigenvalues from the expressions in Eq. (B7) and plot λ₁(q) for two representative cases in Fig. 4(a). In one case, λ₁(q) is always positive irrespective of q (red curve), while in the other case, it only becomes positive above a certain threshold q* (blue curve). The dependency of q* on $v0v*$ at fixed ɛ and ζ is shown in Fig. 4(b).

We observe that as soon as ɛ ≠ 0, a SW instability appears, as depicted in Fig. 4(a). For ɛ = 4, we plot in Fig. 5(a) the LW instability region in dark blue together with the SW instability region in light blue. We also represent in Fig. 5(b) the SW and LW unstable regions for ɛ = 8 with a color map showing the value of q*. As evidenced by the comparison between Figs. 5(a) and 5(b), the extent of the SW instability region grows by increasing ɛ. Note that the closer a $(v0v*,ζ)$-point is to the long-wavelength (dark blue) instability region, the smaller the value of q* is until it becomes identically zero inside of it. In other words, the characteristic length of the SW instabilities continuously grows when approaching the LW instability region, until it becomes infinite (spanning all the system’s size) inside it.

As we have shown in colored dashed curves in Fig. 5(b), Eq. (30) successfully predicts both the LW and SW limits of stability. Thus, the dashed curves can be interpreted as “iso-q lines” along which the instability will have the same characteristic length scale.

Our analysis also allows for a quantification of the finite wavelength instability q* as a function of the self-propulsion speed $v0v*$⁠. We show such dependency, $q*(v0v*)$⁠, in Fig. 4(b) at fixed ζ = 4.6, for which the system never enters the LW instability region [see also the dashed–dotted line in Fig. 5(b)]. We report a decrease in q* as the system penetrates into the instability region, until it reaches a minimum and then monotonically grows before exiting towards the stable region.

In Fig. 6(a), we look further at the dependency of ℓ = 1/q* with ɛ. We now fix ζ = 45 and $v0v*=60$⁠, allowing us to explore a broad range of ɛ values. We recall that the bigger the ɛ is, the larger the area of SW instability in the $(v0v*,ζ)$ plane. We find that ℓ decays as 1/ɛ over a broad range of parameter values. Consistently, in the limit of ɛ → ∞, the unstable eigenvalue λ₁ ∼ɛ [see Eq. (B7)]. Therefore, the limit of stability at finite wave vector, $λ1(v0v*,ζ,ε,q*)=0$⁠, leads to q* ∼ ɛ in the limit of large ɛ. These results predict that the rotational frequency of chiral active particles controls the selection of a characteristic length scale, which decreases with increasing ɛ. Furthermore, in Fig. 6(b), we plot ℓ as a function of the inverse rotational frequency for different fixed values of $v0v*$ and ζ, which all lay on the critical-point line defined in  Appendix D, Eq. (D1), and represented by a dotted line in Fig. 5(b). We observe a linear dependency $ℓ∝v0v*ε$⁠, indicating that the selected length scale is proportional to the radius of the individual circular trajectory of a chiral active particle, or circle swimmer. Interestingly, a similar scaling has been found in suspensions of spinning magnetic rotors⁷⁷ and model systems of polar chiral active particles.⁵⁷ However, there are important differences between these systems and the present model, which are worthy to be mentioned. First, magnetic colloids self-spin without net self-propulsion, while in our framework, no instability can take place in the limit v₀ → 0. Second, in Ref. 57, rotations trigger a SW instability of the homogeneous polar (or flocking) state. A symmetry breaking has to occur in this case to give rise to microflocks of typical size ℓ ∝ v₀/ω₀, while the continuum theory derived here does not admit solutions with global polar order. One has, thus, to be cautious when making connections between these different systems, although the fact that, in all cases, one finds a typical length scale $ℓ∼ω0−1$ certainly deserves to be highlighted, as it suggests that a common general pattern formation mechanism might be at play in systems of chiral particles. Moreover, previous numerical simulations of chiral active particles⁶⁰ have shown that finite-sized clusters collectively rotate in the opposite direction of free chiral active particles.

We have presented a general continuum description of self-propelled particles subjected to generic torques, derived by explicitly coarse-graining the microscopic dynamics. As a consequence, the parameters of the hydrodynamic model are linked to the microscopic interactions as well as to the inter-particle spatial and angular correlations. Thus, we can interpret them based on the specific interactions that might come into play. This particular feature does not constraint the field equations but, on the contrary, it allows us to describe a wide variety of particle-based models, where the force needs only be central, and torques can both be intrinsic to the particle (chiral) and derive from an alignment interaction of a general functional form.

At the mean-field level, the linear stability analysis of the field equations unveils different instabilities of the homogeneous and isotropic state. We observe that torques tend to oppose a long-wavelength instability, which we interpret as motility-induced phase separation. This is in agreement with previous numerical studies⁶⁰ as well as analytical descriptions of chiral active particles.^62,63 Moreover, effective torques lead to a finite wavelength instability, which suggests the formation of finite-sized structures. Our analysis predicts a linear dependency between the characteristic cluster size and the average radius of the trajectory of a single chiral active particle, ℓ ∝ v₀/ɛ. This result echoes the $ℓ∼ω0−1$ behavior found in systems of self-spinning colloids and polar chiral active particles.^57,77

Our analytical approach constitutes a powerful tool to investigate the effect that different interactions have on the destabilization of MIPS. Besides, it is not constraint to a particular model but can be systematically applied to a number of dry active particle models by just fine-tuning their mutual interactions. Since the field theory is derived from the microscopic dynamics, it also allows for a direct quantitative comparison with particle-based simulations.

An interesting continuation to this work would be to envision a particle-based model that breaks the angular symmetries φ → −φ and θ → −θ yielding κ ≠ 0. This would allow us to test the predictions given by the coarse-grained mean-field model in a system with effective aligning torques beyond chiral particles with a self-torque.

E.S.-S. and I.P. acknowledge the Swiss National Science Foundation Project No. 200021-175719. D.L. acknowledges MCIU/AEI/FEDER for financial support under Grant Agreement No. RTI2018-099032-J-I00. I.P. acknowledges support from Ministerio de Ciencia, Innovación y Universidades MCIU/AEI/FEDER for financial support under Grant Agreement No. PGC2018-098373-B-100 AEI/FEDER-EU and from Generalitat de Catalunya under Project No. 2017SGR-884.

The authors have no conflicts to disclose.

Elena Sesé-Sansa: Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal). Demian Levis: Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal). Ignacio Pagonabarraga: Investigation (equal); Writing – original draft (supporting); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

We would like to decompose the force in the vector basis {e, ∇ψ₁}. In order to ensure that this basis is orthonormal, we perform a Gram–Schmidt orthonormalization, as in Refs. 45, 74, and 78. A detailed description of the steps to follow, first introduced in Ref. 74, is subsequently given.

The parameters A, B, and C_i, which in turn group combinations of parameters a, b, c, and d, have been introduced, again, for ease of notation.

We have thus identified the threshold value ɛ₁ ≡ ɛ_t above which λ₂ and λ₃ are complex conjugated numbers, while for 0 ≤ ɛ < ɛ_t, the three eigenvalues in Eq. (B7) are different real numbers. These findings are in accordance with the torque-free case, ɛ = 0, discussed in Sec. III A. Here, one can directly see from Eq. (B9) that D < 0 at q = 0 and, thus, the eigenvalues in this case are three different real numbers, see Eq. (25).