Motility-induced phase separation of active particles in the presence of velocity alignment
Self-propelled particle (SPP) systems are intrinsically out of equilibrium systems, where each individual particle converts energy into work to move in a dissipative medium. When interacting through a velocity alignment mechanism, and with the medium acting as a momentum sink, even momentum is not conserved. In this scenario, a mapping into an equilibrium system seems unlikely. Here, we show that an entropy functional can be derived for SPPs with velocity alignment and density-dependent speed, at least in the (orientationally) disordered phase. This non-trivial result has important physical consequences. The study of the entropy functional reveals that the system can undergo phase separation before the orientational-order phase transition known to occur in SPP systems with velocity alignment. Moreover, we indicate that the spinodal line is a function of the alignment sensitivity and show that density fluctuations as well as the critical spatial diffusion, that leads to phase separation, dramatically increase as the orientational-order transition is approached.
Examples of biological interacting self-propelled particle (SPP) systems include animal groups cavagna10 (); bhattacharya10 (), insect swarms buhl06 (); romanczuk09 (), bacteria zhang10 (); peruani2012 (); starruss2012 (), and, at the microcellular scale, microtubules driven by molecular motors schaller10 (). Even though most examples of SPPs come from biology, there exist non-living SPP systems. There are several ways of fabricating artificial SPPs. The self-propulsion of such particles typically requires an asymmetry in the particle: two distinct friction coefficients kudrolli2008 (); deseigne2010 (); weber2013 (), light absorption coefficients jiang2010 (); golestanian2012 (); theurkauff2012 (); palacci2013 (), or catalytic properties paxton2004 (); mano2005 (); rucker2007 (); howse2007 (); golestanian2007 () depending on whether energy injection is done through vibration, light emission, or chemical reaction, respectively. Interestingly, this asymmetry does not need to be an intrinsic particle property. Self-propelled Quincke rollers bartolo2013 () as well as actively moving drops thutupalli2011 () are remarkable examples where the asymmetry results from a spontaneous symmetry breaking that sets the particle to move in a given direction.
At a theoretical level, we have learned in the recent years that the large-scale properties of SPP systems depend on few microscopic details. The symmetry associated to the self-propulsion mechanism of the particles, which can be either polar vicsek1995 (); gregoire2004 (); peruani2008 () or apolar chate2006 (), as well as the symmetry of the particle-particle interactions, that often occur via a velocity alignment mechanism, which can be either ferromagnetic vicsek1995 (); gregoire2004 () or nematic chate2006 (); peruani2008 (); peruani2006 (), play a fundamental role in the resulting self-organized patterns. Equally important is the dimension of the space where particles move, whether this space is continuous (off-lattice) vicsek1995 (); gregoire2004 (); chate2006 (); peruani2008 () or discrete (on lattice) bussemaker1997 (); csahok1995 (); loan1999 (); raymond2006 (), and whether particles move on a homogeneous or heterogeneous medium chepizkho2013a (); chepizkho2013b (); reichhardt2014 (); quint2014 ().
Another aspect of key importance, and central to the present study, is whether there exists a coupling between the speed and the density of the SP particles. Notice that the importance lies on the existence of such coupling and not on the mere fact that the speed may fluctuate. In the context of SPPs with a velocity alignment, it has been shown first with a lattice model peruani2011 () and later on with an off-lattice model tailleur2012 () that such a coupling induces spontaneous phase separation and a zoology of complex patterns. The most evident physical mechanism that can introduce a coupling between speed and density is simple volume exclusion as showed with simple lattice models by adding exclusion process rules in the absence of particle-particle alignment in thompson2011 () and with alignment in peruani2011 (), and with an off-lattice model of self-propelled disks interacting by a soft-core repulsion fily2012 (). The observed non-equilibrium phase separation can be traced back to the non-equilibrium Motility Induced Phase Separation (MIPS) introduced in the context of interacting run-and-tumble particles by J. Tailleur and M.E. Cates in tailleur2008 (); tailleur2013 (). In absence of an alignment mechanism, MIPS is a generic feature of active particles interacting by volume exclusion as shown in simulations with self-propelled disks fily2012 (); redner2013 (); fily2014 () and spheres valeriani2013 (); wysocki2013 (), and argued theoretically in thompson2011 (); cates2013 (); speck2013 (). One exciting aspect of the MIPS, as first pointed out in thompson2011 (); tailleur2008 (); tailleur2013 (), is the remarkable similarity with equilibrium phase-separation, which allows the mapping between these non-equilibrium active systems with the analogous equilibrium systems.
The goal of the present study is to look at MIPS in the context of SPPs with a velocity alignment mechanism. Specifically, we want to understand the role played by the alignment mechanism in the phase separation process. Let us recall that SPP systems with a velocity alignment mechanism exhibit a phase transition from a disordered to an ordered phase. In the disordered phase, the large-scale behavior of the particles is diffusive as occurs for SPPs without a velocity alignment. Thus, we may hope that a mapping to an equilibrium scenario, as the one performed in tailleur2008 (); tailleur2013 (), remains possible. We push for such an analogy as far as possible. To be exact, to the onset of the ordered phase.
Before starting, let us review briefly some of the most relevant theoretical results for (dried) SPP systems with velocity alignment and in the absence of density-dependent speed. The first hydrodynamical equations were derived based on symmetry arguments and contained all allowed terms by symmetry toner1995 (); toner1998 (). These initial studies provided a theoretical basis to understand the emergence of long-range order (LRO) in two dimensional systems with continuum symmetry as well as the presence of giant number fluctuations in the ordered phase. The drawback of these initial approaches is the impossibility of connecting the parameters of the hydrodynamic equations with those of the microscopic models. In bertin2009 (), the macroscopic equations were derived from given microscopic equations. Such an approach revealed that the “parameters” of the hydrodynamic theory are in fact non-linear functions of the density. This has allowed to understand the emergence of macroscopic structures, such as bands, in this type of SPP systems mishra2010 (); caussin2014 (). For a detailed review, we refer the reader to marchetti2013 (). Here, we just mention that macroscopic equations have been derived for ferromagnetic toner1995 (); toner1998 (); bertin2009 (); marchetti2013 () and nematic velocity alignment peshkov2012 () in the dilute approximation and close to the order-disorder transition, with the exception of ihle2011 () and degond2008 (). Even though we have now a fairly good qualitative understanding of the hydrodynamics of SPP systems (in homogeneous media), many open questions and fundamental problems remain unsolved.
The paper is organized as follows: we start by introducing the microscopic model we are interested in. The following sections are devoted to the rather long computation which starts from the microscopic model and ends with the entropy functional describing the spatial density in the system. The main steps of the computation are outlined at the beginning of the third section. Finally, we draw some physical conclusions from the derived coarse-grained equations.
We consider a system of active particles, , moving in a two-dimensional space. The position of the i-th particle is given by and what we refer to as its active velocity (AV) by , where defines the direction of the AV and its norm, with a constant and a function that depends on . The term refers to the local density around the i-th particle. More specifically, where the function , with defines the interaction range. Finally, we consider an over-damped dynamics for the evolution of the and such that the equations of motion of the i-th particle take the form:
and for are white, gaussian and uncorrelated noises with unit covariance. These noises represents a “bath” with very short time correlations or memory. The alignment sensitivity is not directly related to the fluctuation amplitude through an Einstein relation. A suitable option for the function is to take for and otherwise, definition by which defines the interaction range. Notice that the model definition, in particular , implies a mesoscale description; i.e., we assume that there is a microscopic physical mechanism that leads to . Furthermore, we require to be a differentiable function, which is not necessary applicable to lattice models with strict exclusion rules soto2014 ().
It is convenient to write the equations of motion in adimensional form. Calling the box size, we write , . We also rescale the time , and adopt as the velocity scale. By using and , we arrive to:
where we have introduced the dimensionless parameters , , , and , and so . We notice that is the ratio between the passive () and active (]) diffusion coefficient, which we consider to be of order . From now on, we work with the adimensional equations and drop the to simplify the notation. In the following, we will assume .
3 Main computation
As is clear from the scaling introduced through the parameter , we are interested in the situation where the dynamics over the angles is fast with respect to the spatial dynamics. Furthermore, we want to study the density of particles and their local mean orientation over large length scales. Our ultimate goal is to obtain a static large deviation principle yielding an entropy functional, which describes the fluctuations of the spatial empirical density, along the lines of tailleur2008 (); tailleur2013 (), but accounting for the existence of a velocity alignment mechanism. To make the forthcoming computations easier to follow, we outline here the general scheme:
Write an effective equation for the phase space empirical density (that is in the variables and ), keeping the finite fluctuations. This leads to a stochastic PDE with a noise term of order .
Take advantage of the time-scale separation to obtain a closed effective dynamics for the empirical density of the slow spatial variables .
Write a functional Fokker-Planck equation for the macroscopic density field .
Look for the stationary probability density of the macroscopic field , under the asymptotic form and solve for at leading order in .
3.1 Dynamical equation for the phase space empirical density
We assume that is large, but finite, and denote by the empirical density of particles in the 3D space given but . The temporal evolution of is expressed in terms of the following stochastic partial differential equation:
where and are gaussian noises, delta-correlated in time and space, and is the empirical spatial density defined by:
This can be shown at a formal level by following Dean96 ().
3.2 Averaging step - Time scale separation
We now take Fourier components of of increasing order, stopping the expansion as soon as possible. This is a standard strategy, see e.g. bertin2009 (). Notice that finite fluctuations are taken into account. We use the following notations: , with
Integrating Eq. (5) respectively over and , we obtain:
where the noises are defined by
By construction, the noises are gaussian, delta-correlated in time and space. Furthermore
Consistently with our approximation, we have dropped in the noise correlation all Fourier coefficients beyond the first. Notice that in Eq. (9) we have neglected higher order Fourier coefficients since we are interested in characterizing the system dynamics in the disordered phase – i.e. when collective motion is not observed. This implies that our approximation is only valid below the onset of collective motion. Moreover, without higher Fourier components, Eq. (9) predicts that grows unboundedly for . To obtain a system of equations that is physically well-behaved in the ordered phase, we have to go at least one component further in Fourier. Such extra Fourier component is connected to the nematic order, while to polar order. To understand the disordered phase, which is our objective here, we insist that it is enough to develop up to polar order. Furthermore, we stress that Eq. (9) is consistent with a small expansion.
If smaller than 2 and not too close to 2, very quickly reaches its stationary value and remains small: particle motion is locally disordered, since the interaction promoting alignment is not strong enough to create a local orientational order. In this regime, we can take the l.h.s. of Eq. (9) to be in order to determine the stationary value of . Neglecting terms of order and , we obtain:
These computations are formal, and could in general lead to incorrect results: one should in particular be cautious about the meaning of the noise term, which is multiplicative. However, since we will eventually take a small noise limit (large ), this formal approach will correctly yield the leading order in . We insert Eq.(17) into (8) and look for the long-time behavior of by introducing a new time-scale :
where again we have dropped the and replace by . Notice that due to the involved change of time-scale, all ’s have disappeared of the final equation, and both noise terms give a contribution. The expansion in powers of is formally consistent, in the sense that further Fourier components would contribute terms which are formally of higher order. This means that one can hope that Eq. (18) is in some sense exact in the limit .
Eq. (18) can be expressed in a more compact notation in the following way:
We have combined here the two independent gaussian noise terms into a single one.
3.3 Functional Fokker-Planck equation
Note that we have here assumed an interpretation of the noise corresponding to Ito’s convention. This has no consequence at leading order in . We look for a stationary solution taking the asymptotic form
and compute at leading order in . The drift and the noise correlation depend on . However, we see that the relevant terms at leading order in are obtained when the functional derivatives with respect to act on rather than on or . This leads to the following equation for :
By comparison and using the expression (20) for , one sees that a sufficient condition to find is to solve the equation
Formally, it can be shown that this expression represents the equilibrium condition for , that is the condition for a zero-flux solution of the functional Fokker-Planck equation, Eq. (22). Equivalently, this corresponds to the reversibility of the dynamics given by Eq. (19) with respect to the density . We look for a solution , which is a local functional of , and of the form:
where the entropy density is a real function to be determined. One finds
When the integration of Eq. (27) is possible, the function can be explicitly retrieved. This is the main result of this article. In the absence of alignment, i.e. , Eq. (27) leads to the same results derived in tailleur2013 (). For instance, it is straightforward to see that if in addition we make , the spinodal line is given by the condition as explained in tailleur2008 (); tailleur2013 ().
Notice that the procedure followed to arrive to the free energy, consisted in deriving a stochastic equation for the empirical density, and using Ito’s calculus to obtain an expression for the density field , in turn connected to the free energy . We stress that the described procedure is fundamentally different from recent approaches speck2013 () used to describe phase separation in non-aligning active particles, where a free energy is obtained by deriving first a non-fluctuating BBGKY hierarchy of equations, performing a perturbation expansion, and making a direct analogy between the derived equation for the density field at second leading order and the Cahn-Hilliard equation, whose free energy is well known. Our derivation, on the contrary, contains finite fluctuations, going beyond mean-field, and allows us to obtain directly a free-energy-like functional form, which is not necessary a Cahn-Hilliard free energy with a local cubic term as in speck2013 (). In the following section, we discuss the physical meaning of equations here derived.
4 Physical discussion and final remarks
Let us review the results we obtained. The equations of motion (1) and (2) were our starting point. We required to be a differentiable function, which implies that our derivation is, in principle, not adequate to describe sharp interfaces as the ones observed in lattice models with strict exclusion rules soto2014 (). Under these assumptions, we derived an equation for the empirical density, Eq. (5), following Dean96 (). This equation becomes an exact description only in the limit of infinite and infinite densities. Thus, for finite but large , Eq. (5) should provide a good description for the particle density of a system whose microscopic dynamics is given by Eqs. (1) and (2). Our goal has been to derive an entropy-like functional for the particle density. In order to do that, we made an expansion in Fourier of Eq. (5), given by Eqs. (8) and (9), up to polar order , and made use of the fast relaxation of with respect of the temporal evolution of , to express as function of and its gradients. Such expansion up to polar order, as well as the separation of time-scales between the temporal evolution of and , are exclusively valid in the disordered phase. From Eq. (9), we can easily see that the onset of local (orientational) order occurs for , that is, when the angular diffusion is such that . This means that the entropy functional, given by Eqs. (26) and (27), is valid for , see solid vertical line in Fig. 1.
To understand the physical meaning of the derived equations, let us adopt a concrete functional form for , e.g. . Notice that the qualitative features discussed below do not depend on the precise functional form for , provided it is decreasing. Phase separation occurs below the binodal line defined by the double tangent construction on the curve. Easier to detect and compute is the spinodal line, below which an homogeneous phase cannot be stable. The spinodal can be found solving , using expression Eq. (27). One finds
For such that , Eq. (28) is represented by the red dashed line in Fig. 1. Above this line, the homogeneous solution is stable; below it, spinodal decomposition (Motility-Induced Phase Separation) occurs. Notice that this line, which corresponds to a dynamical instability, can also be obtained by linearizing Eq. (19), without noise, around a spatially homogeneous solution. Eq. (27) also gives access to the metastable regions around the spinodal line. Figure 1 shows that the critical spatial diffusion below which the homogeneous solution is unstable strongly depends on : the divergence of in our approximation is connected to the term in in Eq. (20). This high sensitivity of the spinodal line to the alignment strength, i.e. , is consistent with results obtained in simulations peruani2011 (). As expected, the entropy is also affected by as shown in Fig. 2(a).
Thus, our results indicate that MIPS as described in tailleur2008 (); tailleur2013 () also occurs in the presence of alignment interactions in the disordered phase, with affecting the spinodal (as well as binodal) line as shown in Fig. 2(b). Notice that Eq. (28) allows us to draw the phase diagram , which is the counterpart of the classical gas-liquid phase diagram . From this expression we can obtain the so-called critical point (, ), with and for the chosen functional form of . For , Eq. (28) has no solution, which implies that in this case, the homogeneous phase is stable for any density and value of . In short, MIPS has disappeared above . Hence, the critical value of the spatial diffusion needed to destroy the MIPS strongly increases when the alignment interactions increase. Moreover, in the current approximation, it diverges as order-disorder transition is approached, which suggests that there is no critical point in the ordered phase.
Finally, we stress that at this level of approximation, (i.e., the original, dimensional, spatial diffusion constant) does not affect the orientational order transition point (more precisely the instability of the homogeneous disordered phase), while and play a role on both, the disordered and ordered phase. Now, let us turn to the analysis of density fluctuations related to the noise term present in Eq. (19). From Eq. (27), we learn that in the homogeneous phase, an alignment interaction makes smaller in absolute value, and thus it increases the density fluctuations. Notice that density fluctuations diverge for . This occurs on the spinodal line, as well as in the MIPS phase, as we approach the instability of the (homogeneous) disordered phase, i.e. . It is worth noticing that for , has no influence on the phase diagram shown in Fig. 1, while it still has on the density fluctuations. A note of caution is in order here. According to the proposed approach, , , as well as density fluctuations diverge as the disorder-order transition is approached. As explained above, our approximation is not valid in the limit of . While we can be sure that , and density fluctuation increase as we approach the onset of collective motion, we cannot ensure that the system behavior at the disorder-order transition or in its vicinity is as predicted by the present approach.
In summary, our calculations indicate that phase separation can occur in the disorder phase with the alignment strength – more specifically with the “distance” to the instability of the (homogeneous) disordered phase – controlling the position of the spinodal line involved in the MIPS as well as the size of density fluctuations. In short, we have generalized the approach of tailleur2008 (); tailleur2013 () in order to account for the presence of a velocity alignment mechanism. While the nature of the described phase separation remains a MIPS as observed in non-aligning systems fily2012 (); redner2013 (); fily2014 (); valeriani2013 (); wysocki2013 (); cates2013 (); speck2013 (), this does not exclude that in the orientationally ordered phase, phase separation can be of a different nature as suggested in peruani2013 ().
acknowledgements We acknowledge enlightening discussions with O. Dauchot, P. Degond, and J. Tailleur and financial support from the PEPS-PTI “Anomalous fluctuations in the collective motion of self-propelled particles”. The suggestion by J. Tailleur of adding a white noise to Eq.(1) has proved to be very fruitful for the present study. FP thanks the Kavli Institute for Theoretical Physics (University of California, Santa Barbara) and the organizers of the bioacter14 program for hospitality and financial support.
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