Mesoscopic theory for fluctuating active nematics

The term active nematics designates systems in which apolar elongated particles spend energy to move randomly along their axis and interact by inelastic collisions in the presence of noise. Starting from a simple Vicsek-style model for active nematics, we derive a mesoscopic theory, complete with effective multiplicative noise terms, using a combination of kinetic theory and It\^o calculus approaches. The stochastic partial differential equations thus obtained are shown to recover the key terms argued in EPL \textbf{62} (2003) 196 to be at the origin of anomalous number fluctuations and long-range correlations. Their deterministic part is studied analytically, and is shown to give rise to the long-wavelength instability at onset of nematic order (see arXiv:1011.5408). The corresponding nonlinear density-segregated band solution is given in a closed form.


Introduction
The study of collective properties of systems of interacting active particles [1,2,3] is currently attracting a great deal of interest. In active matter, particles extract energy from their surrounding and dissipate it to propel themselves in some coherent way in a viscous fluid and/or over a dissipative substrate. In this last case, or whenever hydrodynamic effects can be neglected, physicists speak of "dry active matter" [3]. Systems as diverse as animal flocks [4,5,6], human crowds [7,8], subcellular proteins [9], bacterial colonies [10], and driven granular matter [11,12,13] have been described in this framework.
In the context of dry active matter, there is now some consensus in the physics community that minimal models such as the celebrated Vicsek model [14,15] play a crucial role, since they stand as simple representatives of universality classes which have started to emerge from a combination of numerical and theoretical results: for instance, many different microscopic (particle) models have been shown to exhibit the same collective properties as the Vicsek model, and the continuous equation proposed by Toner and Tu [16] is widely believed to account for its collective properties. Such hydrodynamic theories formulated at the mesoscopic level (stochastic PDEs) are the natural framework to characterize and define universality classes.
In early approaches these mesoscopic theories have been built on the principle of including all that is not explicitly forbidden, retaining all leading terms (in a gradient expansion sense) allowed by symmetries and conservation laws [16,17]. This grants access to the general structure of these equations and has been successful in describing relevant features of active matter systems such as their anomalously large number density fluctuations [16,12,18,13]. Despite the attractions of a gradient expansion, it typically contains many transport coefficients of unknown dependence on microscopic control parameters and hydrodynamic fields such as local density. Moreover, the dependence of the noise terms on the dynamical fields in such equations remains arbitrary, and frequently neglected, whereas it could have profound consequences for important phenomena such as spontaneous segregation, clustering and interface dynamics.
Ideally, thus, one would be able to derive well-behaved mesoscopic theories using a systematic procedure starting from a given microscopic model. Kinetic-theory-like approaches [19,20,21,22,23] go one step towards this goal, by allowing one to compute hydrodynamic transport coefficients and nonlinear terms. One of the most successful versions is arguably the "Boltzmann-Ginzburg-Landau" (BGL) framework recently put forward by some of us [24,25], where, in the spirit of weakly nonlinear analysis, one performs well-controlled expansions in the vicinity of ordering transitions. Kinetic approaches alone thus yield good deterministic "mean-field" equations but one still need to "reintroduce" fluctuations in order to get bona fide mesoscopic descriptions.
In this work, we show how this complete program can be achieved for the case of active nematics, i.e. systems where particles are energized individually but not really self-propelled, moving along the axis of the nematic degree of freedom they carry, with equal probability forward or back. (Think of shaken apolar rods aligning by inelastic collisions [12].) Starting from the Vicsek-style model for active nematics introduced in [26], we formulate a version of the BGL scheme mentioned above adapted to problems dominated by diffusion, derive the corresponding hydrodynamic equations, and study their homogeneous and inhomogeneous solutions. In a last section, we show how these equations can be complemented by appropriate noise terms using a direct coarse-graining approach.

Microscopic dynamics
We consider the microscopic model for active nematics of [26] in two space dimensions. This Vicsek-style model can be thought of as a minimal model for a single layer of vibrated granular rods [12] although it does not deal explicitly with any volume exclusion forces. Here, rather, pointwise particles j = 1, . . . , N are characterized by their position x t j and an axial direction θ t j ∈ [−π/2, π/2]. They interact synchronously with all neighboring particles situated within distance r 0 in a characteristic driven-overdamped dynamics implemented at discrete timesteps ∆t: where V j is the neighborhood of particle j, d 0 < r 0 is the elementary displacement, n t j ≡ cos θ t j , sin θ t j T is the nematic director, and ψ and κ are two white noises: the random angle ψ t j , familiar of Vicsek-style models, is drawn from a symmetric distributionP η (ψ) of variance η 2 , and the zero average bimodal noise κ t j = ±1 determines the actual orientation of motion. Both noises are delta correlated, namely κ t j κ t k ∼ ψ t j ψ t k ∼ δ t t δ j k . In the following, we adopt the convention [nn] αβ ≡n αnβ and label coordinates by greek indices, α, β, . . . = 1, 2, summing over repeated indices.

Timescales and lengthscales
We consider low density systems in which particles, at a given time, are either noninteracting or involved in a binary interaction. In this dilute limit we can neglect interactions between more than two particles. We also treat interactions as collision-like events, with the mean intercollision time where ρ 0 is the global particle density and τ d is shortest microscopic timescale of the dynamics, associated to the inversion of the rods direction of motion τ d ∼ ∆t. This inter-collision time is much larger than the collision timescale For driven granular rods, τ d may be thought of as the inverse of the shaking frequency, and for typical parameters it is much smaller than both the collision (τ coll ) and the mean intercollision (τ free ) timescales; at low enough densities τ d τ coll τ free . Note that the timescales (4)-(3) are different from the ones characteristic of ballistic dynamics [20].
To develop a kinetic approach we consider a mesoscopic timescale τ B such that τ coll τ B τ free . As a consequence, we will treat the inversion of the direction of motion as a noisy term through Itô stochastic calculus [27]. We also consider a mesoscopic coarse-graining lengthscale B which, while being much smaller than the system size L, is larger than the microscopic scales, such as the step-size d 0 , the mean interparticle distance ρ −1/2 0 and the interaction range r 0 . To summarize, in a dilute system one has and where L is the system size and we have made explicit the condition that the typical coarse-graining lengthscale B is such that many particles are contained in a box of linear size B , that is ρ 0 2 B 1.

Master equation
We now write down a Boltzmann-like master equation in terms of the single particle probability distribution f (x, θ, t), with − π 2 < θ ≤ π 2 , evolving over the timescale ∆t ≈ τ B . The minimal spatial resolution is such that many particles are contained in a spatial volume d 2 x centered around the position x. Moreover, we consider a dilute system, so that interactions (collisions) between particles are sufficiently rare to justify i) binary interactions (as explained above, particles then either self-diffuse or experience noisy binary, collision-like interactions), ii) decorrelation of the orientation between successive binary collisions of the same pair of particles, that is We first omit collisions and angular diffusion, only considering Eq. (2) to get where we have considered that a particle moves along one of the two orientations ofn with equal probability. On the mesoscopic timescale τ B τ d ∼ ∆t, Itô calculus [27] to second order gives where is the microscopic diffusion parameter.
To account for angular diffusion and binary collisions, the appropriate integrals need to be added to the right hand side of Eq. (8), The diffusion integral describes self-diffusion which takes place at a rate λ = 1/τ d where we used the simplified notation f (θ) ≡ f (x, θ, t), δ π is a generalized Dirac delta imposing that the argument is equal to zero modulo π and P (ζ) is a symmetric noise distribution of variance σ 2 , corresponding to the effective noise arising at the timescale τ B from the sum of the microscopic stochastic contributions to angular dynamics. Binary collisions are described by where, for the sake of simplicity, we have used the same noise distribution P (ζ) as in the self-diffusion integral, and the out-coming angle Ψ from deterministic binary collisions is, for − π 2 < θ 1 , θ 2 ≤ π 2 , Note that the role of the function h(θ) is to ensure that Ψ(θ 1 , θ 2 ) is π-periodic with respect to θ 1 and θ 2 independently. The collision kernel K(θ 1 , θ 2 ), i.e. the number of collisions per unit time and volume, is calculated as follows. Consider two particles with nematic axisn(θ) andn(θ ) located in the volume d 2 x centered around position x. In the reference frame of the first particle the second one diffuses either along the |n(θ) −n(θ )| or the |n(θ) +n(θ )| nematic axis. In unit time, taking into account the characteristic timescales τ d and step-size d 0 of its motion, it sweeps a surface (its cross section, which is conserved going back to the lab reference frame) equal to where we have introduced the microscopic collision parameter Note that K(θ, θ ) ≡K(θ − θ ) is an even function of the difference (θ − θ ), and fulfills the nematic symmetry, being invariant under rotation of either angle by π. Before proceeding to derive hydrodynamic equations, we simplify all notations by rescaling timet = λt = t/τ d and spacex = √ 2 d 0 x. As in [24,25] we also set the collision surface S = 2r 0 d 0 to 1 by a global rescaling of the one-particle probability density f , without loss of generality. This amounts to set λ 0 = 1, D 0 = 1 and 2α 0 = 1, so that, dropping the tildes, our Boltzmann-like master equation now depends only on the global density ρ 0 and the noise intensity σ.

Hydrodynamic description
In two spatial dimensions, hydrodynamic fields can be obtained by expanding the single particle probability density f in Fourier series of its angular variable θ ∈ [−π/2, π/2] §: The number density and the density-weighted nematic tensor field w ≡ ρQ are then given by and Note that when Imf 1 = 0 the nematic field is aligned either along the Injecting the Fourier expansion (16) in the master equation (10), one gets, after some lengthy calculations detailed in Appendix A, the infinite hierarchy: 21) § These k-modes are equivalent to even harmonics if one would define particles orientation in [−π, π] in spite of the symmetry under rotations by π (with odd ones being zero by symmetry).
whereP k is the Fourier transform of the noise distribution P (ζ) (namely,P k = ∞ −∞ dζP (ζ)e i2kζ ) and and we have introduced the following "complex" operators The equation at order k = 0 is thus expressed in the simple form and is nothing but the continuity equation for diffusive active matter with local anisotropy characterized byf 1 . Eq. (21) possesses a trivial, isotropic and homogeneous solution: We are interested in a nematically ordered homogeneous solution which could eventually arise following some instability of the isotropic solution above. In analogy to the scaling ansatz used for polar particles [20,25], the interaction term in Eq. (21) suggests a simple scaling ansatz to close the infinite hierarchy of equations onf k (x, t): Near an instability threshold with continuous onset, Fourier coefficients should scale asf k (x, t) ∼ |k| where is a small parameter characterizing the distance to threshold. Moreover, the curvature induced current (last term of (23)) also induces an order variation in the density field, ρ(x, t) − ρ 0 ∼ . Then, assuming spatial derivatives to be of order , the request that all terms in Eq. (23) are of the same order also fixes the diffusive structure of the scaling of time and spatial gradients: Using the above scaling ansatz, we proceed by discarding all terms appearing in (21) of order higher than 3 . For k = 1, 2 we get: and where the coefficients are Eq. (25) shows that at this orderf 2 is enslaved tof 1 (given that a 2 > 0) and, further, where the coefficient a 2 is evaluated at the mean density ρ 0 , since the δρ = ρ − ρ 0 corrections are of higher order. By substituting Eq. (25) into (24) one finally gets, where we have introduced the transport coefficients Note that the coefficient ξ is only a function of the average density ρ 0 , as space and time dependent corrections are of order 4 . Note also that the coefficients µ and ξ are exactly the same as those found for the nematic field equation of nematically-aligning polar particles [25] Eqs. (23) and (31) can be expressed in tensorial notation. To this aim, we introduce the linear differential operator Γ, such that Γ 11 = −Γ 22 ≡ ∂ 1 ∂ 1 − ∂ 2 ∂ 2 and Γ 12 = Γ 21 ≡ 2∂ 1 ∂ 2 , and the Frobenius inner product A : B = A αβ B αβ (note that w : w = ||w|| 2 and Γ : w = 2∂ α ∂ β w αβ ). After some manipulation of the terms and the use of Eqs. (19,20), we obtain the hydrodynamic equations for the density and nematic field Although the tensorial notation might be more familiar to some readers, it is in fact easier here to continue manipulating the complex fieldf 1 and the complex operators Note that in [25], the equations obtained are not entirely correct: (i) there is a sign error and a misplaced factor π in the expression of ξ; (ii) the term ν 4 ∇ 2 f 2 should read ν 4 ∆f 2 , where ∆ is the Laplacian. In addition, let us emphasize that the Fourier coefficientsP k have a different definition in [25], due to the absence of global nematic symmetry:P k here corresponds toP 2k in [25], leading to (only apparent) differences. defined above. Moreover, in the following we drop the "ˆ" superscript to ease notations. Eqs. (35) and (36) are also derived from an apolar Vicsek-style model in [29].
The parameter-free character of the Laplacian term in (36) means, consistent with our expansion in , that the nematic phase of our system will be characterized by a single Frank constant [28]. The nonlinearities studied in [30] are therefore also absent to this order. The last term in Eq. (23) (or Eq. (35)), i.e. 1 2 Re (∇ 2 f 1 ) (or 1 2 (Γ : w)), is a curvature induced current which couples the density and the nematic field. While its existence was first deduced from general principles [17], here we have computed it directly from microscopic dynamics. Our calculations also give an exact expression for the corresponding transport coefficient, which is equal to the diffusive one (in Eq. (23) or Eq. (35)), here set to 1/2 by our rescaling. In Appendix B, we show explicitly that this curvature-induced current originates from the coupling of orientation with motility.
We note finally that Eqs. (35,36) are similar to those found by Baskaran and Marchetti [22] but simpler, largely due to our simpler starting point.

Homogeneous solutions
From now on, we use for P (ζ) a centered Gaussian distribution of variance σ 2 , in which caseP k = e −2k 2 σ 2 . The linear stability with respect to homogeneous perturbations of the disordered solution ρ(x, t) = ρ 0 ,f 1 (x, t) = 0 is given by the sign of µ(ρ 0 ) which yields the basic transition line Note that in the dilute limit ρ 0 1, where the equations have been derived, one has σ t ∼ √ ρ 0 .
For σ < σ t , µ > 0, and the homogeneous nematically ordered solution exists and is stable w.r.t. homogeneous perturbations. The critical line is shown in Fig. 1a (black solid line). Note that for σ < σ t , all transport coefficients (32-34) are positive. This will be useful in the rest of the paper.

Linear stability analysis
We now study the linear stability of the above homogeneous solutions w.r.t. to arbitrary perturbations. Linearizing Eqs. (23) and (31) around a homogeneous solution, f 1 = f 1,0 + δf 1 and ρ = ρ 0 + δρ, one has where µ 0 ≡ µ(ρ 0 ) and µ is the derivative of µ w.r.t. ρ. We then introduce the real and imaginary parts of the order parameter perturbation, δf 1 = δf , and express the spatial dependence of all perturbation fields in Fourier space, with a wavevector q = (q x , q y ), by introducing the ansatz 1,q e st+iqr , δf The stability of the stationary solution f 1,0 is then ruled by the real part of the growth rate s.

Stability of the disordered isotropic solution
We first study the stability of the disordered solution f 1,0 = 0, in the case µ 0 < 0. Substituting Eqs.
s δf where q 2 = q 2 x + q 2 y . All directions of the wavevector q being equivalent, we choose for simplicity q x = q and q y = 0. From Eq. (43), one then sees that the component δf In the disordered state µ 0 < 0, so that β 1 and β 2 are positive and one always has Re(s) < 0. Therefore, the homogeneous disordered solution is stable w.r.t. to all perturbations if µ 0 < 0, i.e. σ > σ t .

Stability of the ordered solution
To study the stability of the anisotropic ordered solution, it is convenient to choose a reference frame in which order is along one of the axes: This solution is aligned along x, if f 1,0 is positive, or along y if negative. For simplicity we will concentrate further on the case f 1,0 ≥ 0, i.e., on the nematic solution aligned along the x axis. The real part δf s δf 1,q . We performed a full numerical stability analysis of these equations. The results are presented in Fig. 1. The transition to the homogeneous solution is given by the line σ t . This solution is unstable to finite wavelength transversal perturbations of angle |θ| > π 4 between the lines σ t and σ s (dotted purple line in Fig. 1), but is stable deeper in the ordered phase.
Two remarks are in order. First, the angle of the most unstable mode is here always perfectly π 2 . It is thus possible to obtain the "restabilization" line σ s analytically as shown below. Second, there is no spurious instability at low noise and/or high density (although we have found that such an instability appears if the truncation of the equations is made to the fourth order).
To obtain the analytic expression of the line σ s , we write the wavevector in terms of its modulus q and its angle θ q , so that q 2 x − q 2 y = q 2 cos 2θ q and 2q x q y = q 2 sin 2θ q . We can then analyze Eqs. (46) in the longitudinal and perpendicular wavedirections θ q = 0 , ± π 2 , where the imaginary perturbation δf (I) 1,q decouples from the other two. The latter is stable towards long wavelength perturbations, since the corresponding eigenvalue s = −q 2 /2 is negative. The stability towards density and real perturbations depends on a 2 × 2 matrix which yields the quadratic eigenvalue equation whose solutions are The sign ± in front of the µ f 1,0 term in Eq. (47) corresponds to the case θ q = 0 (positive sign) and θ q = π 2 (negative sign) respectively. Note that µ is strictly positive, as typical for all active matter system with metric interactions, where the interaction rate grows with local density. Also µ 0 is positive and of order 2 (see Eq. (45)). It it thus easy to see that in the case of large q, [s] ≤ 0. For small values of q, we perform an expansion to order q 2 of the largest growth rate s + , obtained by taking the positive sign in front of the square root in Eq. (48), leading to We can then conclude that for longitudinal perturbations (θ q = 0, negative sign in front of µ ), the homogenous solution is stable confirming the results of numerical analysis.
In the case of transversal perturbations (θ q = ±π/2), the stability condition is given by meaning that close to the instability threshold of the disordered solution, when µ 0 is positive but small, the state of homogeneous order is unstable with respect to long wavelength perturbations. This instability was first identified in a kinetic-equation analysis by Shi and Ma [23]. Note that condition (50) is valid up to the third order in (or, equivalently, in the order parameter ||w||). It yields the stability line where µ 2 = µ(ρ = 0), ξ 1 = (1/ξ) and ξ 2 = (1/ξ(ρ 0 = 0)). We do not provide here the explicit analytical expression for σ s because this requires solving a sixth order polynomial.
We remark that the near-threshold instability discussed above is rather generic and appears in "dry" active matter systems with metric interactions, as opposed to systems with metric-free ones, where the interaction rate is density-independent, and µ = 0 [31,24,32]. In this case (topological active nematics), stability would be enforced by the positive higher order corrections µ 0 q 2 which dominates arbitrarily close to threshold.

Inhomogeneous solution
We now show how a spatially-inhomogeneous stationary "band" solution to our hydrodynamic equations can be found. First we remark that our equation for the nematic field Eq. (31) is formally the same as that derived in [25] for polar particles with nematic alignment when the polar field is set to zero, as it is imposed here by the complete nematic symmetry of our system. We thus expect an ordered band solution made of two fronts connecting a linearly stable homogeneous disordered state (ρ = ρ gas < ρ t ) and a linearly stable homogeneous ordered state (ρ = ρ band > ρ s ) (see Fig. 1). Following [25], we rewrite with ρ t = (1 −P 1 )/µ , suppose that the nematic field is aligned along one of the axes and varies only along y. In other words: Eqs. (35) then becomes which can be integrated to give where A and ρ gas are integration constants. Furthermore, to keep the fields finite for |y| → ∞, one has A = 0. By substituting Eqs. (54) and (55) into Eq. (31) one gets We multiply equation (56) by ∂ y f 1 and integrate it once to obtain Separating the variables we obtain Integration of this equation under the condition lim y→±∞ f 1 (y)=0 gives after simplifications where a = 1 − 9ξ 2µ (ρ t − ρ gas ). We still need to obtain the value of ρ gas which is fixed by the condition L ρ (y) dy = ρ 0 L, where L is the length of the box. In the integral on the l.h.s we can neglect the exponentially decaying tails and integrate instead on the infinite domain. Furthermore, in the limit L → ∞ we can neglect the exponentially weak dependence of ρ gas on L everywhere except the a term. We then obtain Substituting it back into Eq. (59) we get, under the assumption L → ∞: and we finally obtain the ordered solution density with, as expected, ρ band > ρ t > ρ gas , which guarantees the stability of both the ordered and disordered parts of the solution. Note that since f band 1 > 0 the nematic order is parallel to the x direction (i.e. along the band orientation). This is the opposite of what happens in the Vicsek model, where bands extend transversally with respect to their polarization [15].
We can introduce the band fraction Ω which indicates the fraction of the box occupied by the band. If we suppose that the front width is negligible (once again justified in the limit L → ∞), this band fraction is determined by the equation Substituting inside the values of ρ gas and ρ band , we obtain The condition 0 < Ω < 1 gives us the lower σ min and upper σ max limits of the existence of bands. As found for polar particles aligning nematically, these limits of existence of the band solution extend beyond the region of linear instability of the homogeneous ordered solution (given by σ ∈ [σ s , σ t ], see Fig. 1). In Fig. 1, we provide a graphical illustration of the shape and properties of the band solution. An important problem left for future work is the linear stability analysis of the band solution in two space dimensions. This is all the more important as the unpublished work of Shi and Ma [23] suggests the existence of some instability mechanism.

Langevin formulation
Being based on a master equation, the derivation we have discussed in the previous sections leads to a set of deterministic PDEs. This is a standard approach in equilibrium statistical physics, where the microscopic fluctuations are integrated out in the coarse graining process implicit in the definition of a mesoscopic cell size B . Fluctuations, when needed, can be eventually introduced as an additive, delta correlated stochastic term as in Ref. [16]. However, the presence of large density fluctuations [17] suggests that fluctuations may not be faithfully accounted for by some additive noise term. The precise nature of noise correlations at the mesoscopic level cannot be safely overlooked in non-equilibrium systems, as it is known that stochastic terms multiplicative in the relevant fields can radically alter the universality class of mesoscopic theories [33].
In this section, we perform a direct coarse-graining of the microscopic dynamics in order to compute the (multiplicative) stochastic terms which emerge at the mesoscopic level. We however restrict the computation to the stochastic terms emerging from the collisionless dynamics. For real-space coarse-graining, we make use of a smooth, isotropic, normalized (to one) filter g s (r) decaying exponentially or faster for r > s, e.g., a Gaussian of width s. The fluctuating coarse-grained density and nematic order field are then defined as and where we have introduced the microscopic traceless tensor

Density field fluctuations
The correlations of density field fluctuations can be derived by generalizing an approach first outlined by Dean [34] for Brownian particles. As mentioned above, we use the collisionless dynamics. We are interested in the time evolution of the density field (66), which is given by where ∆x t i = x t+∆t i − x t i . Expanding up to second order in powers of ∆x t i according to Itô calculus [27] and by virtue of Eq. (2) one has where and Note that derivatives are taken w.r.t. the argument of the function g s , and not w.r.t. x. The second order term T 0 yields the deterministic part of the density dynamics. By Eqs. (66,67) and the definition of the microscopic nematic tensorQ [Eq. (68)] one easily gets that is, the right hand side of the diffusion Eq. (35). The first-order term T 1 gives rise to the (zero average) stochastic term we are interested in. At this stage, T 1 is not a simple function of the mesoscopic fields; however, following Ref. [34] it is possible to show that its two point correlation can be recast as a function of ρ and w. Averaging over the random numbers κ t i , we have, in the limit s → 0, Using Eq. (68), one then finds, approximating the filter g s by a Dirac delta in the limit s → 0, We can rewrite the noise term T 1 in the stochastically equivalent (i.e., with the same correlations on the mesoscopic scale) form where h is a Gaussian, zero-average vectorial noise, delta-correlated in time with correlations Such a noise term can finally be expressed in the more convenient form where the Gaussian noiseh has correlations independent from the hydrodynamic fields and the tensor K is implicitly defined from the relation K · K = (ρ/2)I + w (with I being the identity matrix). In the limit of small w considered here, we can expand K to first order in w, yielding The divergence term ∇· appearing in T 1 reflects global density conservation, while the proportionality of noise variance to number density can be interpreted as a consequence of the central limit theorem. Adding up the two contributions, one finally gets

Nematic field fluctuations
We next discuss fluctuations of the nematic tensor. As seen from Eq. (67), w is a function of the 2N microscopic stochastic variables x t i and -through the microscopic nematic tensor (68)-θ t i , whose dynamics is given by Eqs. (1)- (2). According to Itô calculus, one has where Ω 0 is the deterministic part of the coarse-grained collisionless dynamics (which we do not write here explicitly), while Ω 1 and Ω 2 are two stochastic contributions, where ψ t i and κ t i are the microscopic noises and Note that in Ω 1 we have retained only the linear contribution in the microscopic noise ψ t i . We first focus on the stochastic terms Ω 1 . On coarse-graining scales, averaging over the microscopic noise ψ t i , correlations of Ω 1 are given by To evaluate this correlator, we determine the average value i g s (A ·Q t i )(A ·Q t i ) , in the framework of the deterministic dynamics studied in Sect. 2, namely (87) After some rather lengthy calculations, using the closure equations (30), one finds where we have introduced the tensor which plays the role of a unit tensor for the double contraction of symmetric traceless tensors, e.g., w αβ = J αβµν w µν . In order to characterize the noise Ω 1 , we introduce the following change of variables: whereΩ is a tensorial symmetric traceless white noise, such that with D = 2η 2 /τ d . The correlation of Ω 1 then reads By identification with Eq. (86), and using Eq. (88), one eventually finds for H Note that, in agreement with the central limit theorem, Ω 1 is (at least to first order in w) proportional to the square root of local density.
The second stochastic term Ω 2 , finally, can be treated similarly, but it would give rise to a conserved noise (due to the presence of ∇ terms) akin to the one discussed for the density equations, thus related to density fluctuations affecting the w = ρQ field.
We discard such conserved term as irrelevant (in the renormalization group sense) with respect to the non-conserved multiplicative noise Ω 1 .
In order to write down the complete Langevin equation, one also needs to evaluate the contribution of the deterministic part Ω 0 . However, expressing this contribution in terms of the fluctuating fields ρ and w turns out to be a very complicated task. One should also take into account collisions between particles, and not only the collisionless dynamics described by Ω 0 . Then some further approximations would be required to treat the non-linear part of the dynamics.
In addition, microscopic collisions could provide a further fluctuation source due to disorder below the coarse-graining scale. While we conjecture them to be irrelevant, we leave a final settlement of this difficult problem for future work, and use for the deterministic part of the dynamics the terms the hydrodynamic equation (36), derived from the Boltzmann approach.
We thus finally obtain the stochastic equation for the nematic field A few remarks are in order: first, our expressions of the noise amplitudes K and H (Eqs. (80) and (93)) suggest that the stochastic terms might be better expressed in terms of the field Q, rather than w = ρQ; second, Eqs. (81) and (94) are also derived from an apolar Vicsek-style model in [29]. In spite of the limitations listed above, the present approach already provides us with useful information on the statistics of the noise terms, which is seen to differ significantly from the white noise postulated on a phenomenological basis in previous works. On top of the overal ρ 1/2 dependency, our calculation reveals a non-trivial dependence of the correlation of the noise on the nematic order parameter [see Eqs. (80, 81, 92, 93)].

Conclusions
To summarize, using as a starting point the simple active nematics model of [26], we have demonstrated how one can derive in a systematic manner a continuous mesoscopic description: We formulated a version of the Boltzmann-Ginzburg-Landau approach put forward in [24,25] for this case where (anisotropic) diffusion dominates, deriving a simple hydrodynamic equation for the nematic ordering field -Eq. (36). We have then used a direct coarse-graining approach to endow the hydrodynamic equations with proper noise terms.
The next stage, left for future work, consists in studying the stochastic PDEs obtained. At the linear level, it is clear that in the long wavelength limit, standard results on giant density fluctuations [17] are recovered. However, the large amplitude of density fluctuations calls for a non-linear analysis (which turns out to be very difficult), where the density dependence of the noise derived in Sect. 5 may play an important role. Ideally, one should try to tackle this issue by applying methods from field theory and renormalization group analysis. In addition, we note that the multiplicative nature of the noise may also affect finite-wavelength properties, like coarsening behavior. The analysis of the stochastic PDEs can be done numerically, but some care must be taken when dealing with the multiplicative, conserved noise terms in (81).
Pending such attempts, some remarks and comments are already in order: like all previous cases studied before, the hydrodynamic equations found exhibit a domain of linear instability of the homogeneous ordered solution bordering the basic transition line σ t . This solution does become linearly stable deeper in the ordered phase (for σ below σ s ). Moreover, we have found that the long wavelength instability of the homogeneous ordered solution leads to a nonlinear, inhomogeneous band solution -see Eq. (62)-and that this band solution exists beyond the [σ s , σ t ] interval. These coexistence regions suggest, at the fluctuating level, discontinuous transitions. This seems to be at odds with the reported behavior of the original microscopic model: (i) the order/disorder transition has been reported to be of the Kosterlitz-Thouless type [26]; (ii) there is no trace, at the microscopic level, of the existence of a non-segregated, homogeneous phase; (iii) coming back to giant number fluctuations, we note that the standard calculation is made in the homogeneous ordered phase whereas the numerical evidence for them reported in [26] appears now to have been obtained in the inhomogeneous phase. All this calls for revisiting the simple particle-based model and, eventually, understanding its behaviour in the context of the stochastic continuum theory constructed here.