Velocity distribution in active particles systems, Supplemental Material

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Velocity distribution in active particles systems, Supplemental Material Umberto Marini Bettolo Marconi, 1 Claudio Maggi, 2 Nicoletta Gnan, 2 Matteo Paoluzzi, 3 and Roberto Di Leonardo 2 1 Scuola di Scienze e Tecnologie, Università di Camerino, Via Madonna delle Carceri, 62032, Camerino, INFN Perugia, Italy 2 Universita Sapienza, Rome, Italy 3 Syracuse University, NY, USA (Dated: February 19, 2016 A simple derivation of the stationary velocity distribution of a collection of active particles subject to non equilibrium colored noise is provided in this note. Moreover simulation results for the 2- dimensional case are reported. I. DERIVATION OF THE STATIONARY VELOCITY DISTRIBUTION The eective dynamics for space coordinates of an assembly of active spheres 1,2 is ṙ i (t = 1 γ F i(r 1,..., r N + u i (t (1 where the velocities u i evolve according to the law: u i (t = 1 u i(t + D1/2 a η i (t (2 The force F i = i U acting on the i-th particle is conservative and associated to the potential U(r 1,..., r N, γ is the drag coecient, whereas the stochastic vectors η i (t are Gaussian and Marovian processes distributed with zero mean and moments η i (tη j (t = 2δ ij δ(t t. where d is the spatial dimensionality. The coecient D a due to the activity is related to the correlation of the Ornstein-Uhlenbec process u i (t via u i (tu j (t = d D a δ ij exp( t t /. where d is the spatial dimension. To simplify the notation we switch from r i to an array x i as done in previous publications: ẋ i (t = 1 γ F i(x 1,..., x N + u i (t (3 Dierentiate again ẍ i (t = 1 γ F i x ẋ + u i (t (4 Let us introduce the variable v i and recast eq.(5 as: ẍ i (t = 1 F i ẋ 1 γ x [ẋ i F i γ ] + D1/2 a η i (5 ẋ i = v i (6 v i = 1 F i v 1 γ x [v i F i γ ] + D1/2 a η i (7

2 A. One particle Before presenting the multidimensional result, we digress to illustrate the inetic method of solution in a simple one-dimensional case. We begin with a single particle in one dimension and drop the index i. We dierentiate eq. (1 with respect to time and introduce the velocity variable v = ẋ so that instead of the original system (1 and (2 we have: We obtain the Kramers equation for the phase-space distribution f(x, ẋ; t: v = 1 (1 df γ dx v + 1 γ F + D1/2 a η ẋ = v (8 f t + v f x U f γ v = 1 v ( D v + Γ(xv f (9 with Γ(x = 1 + γ 2 U(x. By multiplying and integrating over v eq. (9 and considering only time independent solutions f 0 (x, v one obtains: dvv 2 f 0 (x, v + U dvf 0 (x, v = 1 x γ Γ(x dvvf 0 (x, v (10 Such an integro-dierential equation can be solved by the following ( ansatz f 0 (x, v = Π(v x P(x where Π is the Gaussian velocity distribution : Π(v x = ( 2πD 1/2 Γ(x exp 2D Γ(xv2 whose width depends on the particle position and the average velocity v vanishes. After substituting the factorization f 0 into (9 and evaluating the velocity variance dvv 2 Π(v (x = D a 1 (11 Γ(x we arrive at the following dierential equation determining the steady state coordinate distribution P(x : d dx [Γ 1 (xp(x] + U(x P(x = 0 (12 Dγ which is identical to the dierential equation determining the stationary coordinate distribution in the unied color noise approximation (UCNA: with P(x = 1 Z 1 exp( H(x T s (13 H(x = U(x + 2γ (U(x x 2 T s ln[1 + γ d 2 U(x dx 2 ] (14 and T s = D a γ and Z 1 a normalization constant dened as: ( Z 1 = dx exp H(x. (15 T s The method can be easily extended to the multidimensional case. II. A KINETIC DERIVATION OF THE STEADY DISTRIBUTION, VELOCITY CORRELATIONS FOR MANY PARTICLE SYSTEMS We now generalize the inetic argument above illustrated to a many particle system and write the following multidimensional Kramers equation 3 describing the evolution of the phase-space distribution f N (x 1,..., x N ;, v 1..., v N ; t f N t + i v i f N x i + i F i f N γ v i = 1 i ( Da v i v i + Γ i v f N (16

3 with the non dimensional friction matrix Γ i dened as Γ i = δ i + γ 2 U x i x. (17 The probability conservation and the momentum balance equations are straightforwardly obtained by projection, i.e. by integrating eq.(16 over the dn dimensional velocity space after multiplying by 1 and v i, respectively: J i ({x i }, t t + P N ({x i }, t t + i J i ({x i }, t x i = 0 (18 p i ({x i }, t F i({x i }, t P N ({x i }, t = 1 Γ i ({x i }, tj ({x i }, t (19 x γ where P N ({x i }, t is the marginalized distribution giving the distribution of positions of the particles regardless their velocities: P N ({x i }, t = d N v f N ({x i }, ({v i }, t, (20 while the current J i ({x i }, t is the N-dimensional vector: J i ({x i }, t = d N v v i f N ({x i }, ({v i }, t. (21 Finally, p ij ({x i }, t is the N N dimensional generalized inetic pressure tensor p ij ({x i }, t = d N v v i v j f N ({x i }, ({v i }, t (22 Equations (18-(19 are an exact consequence of (16, but require the nowledge of p ij which can be obtained by continuing the projection procedure in velocity space to higher order in v i and closing the hierarchy by a suitable truncation ansatz, such as the introduction of phenomenological transport coecients. On the other hand, if we limit ourselves to study the stationary regime the approach is fruitful and leads to a dierent interpretation of the MUCNA equations. Let us rst, notice that the operator featuring in the right hand side of (16 posseses a null eigenvalue whose associated eigenfunction (the ground state is the multivariate Gaussian velocity distribution : Π(({v i } {x i } = ( N/2 ( det Γ exp v i Γ ij ({x}v j (23 2πD a 2D a In other words Π({v i } {x i } is the conditional probability distribution of velocities given the positions tae on the values {x i }. We now construct a time-independent trial phase-space distribution having a factorized form: ij f (K T ({x i }, ({v i } = Π(({v i } {x i } P st ({x i } (24 and corresponding to zero current J i (x 1,..., x N. For generic potentials the velocity moments of G are non constant in space, but depend on coordinates x i and dierent velocity components may be correlated and are given by the formula: v i v j = d N vv i v j Π({v i } {x i } = D a Γ 1 ij ({x i} (25 so that the "pressure " tensor reads p ij ({x i } = Da Γ 1 ij ({x i}p st ({x i }. Using eq. condition of vanishing currents, J i = 0, we have the following balance equations: D a (19 in conjunction with the p i ({x i } F i({x i } P st ({x i } = 0 (26 x D a γ Eliminating the pressure tensor using (25 we arrive at the dierential equation determining P st ({x i } : [Γ 1 i x ({x i}p st ({x i }] + 1 U({x i } P st ({x i } = 0 (27 D a γ x i

Finally, by multiplying by the matrix Γ equation (27 we arrive at an equation identical to the MUCNA equation (see ref 2 ( PN ( (D a + D t γ P N ln det Γ P N x n x n δ n + γ 2 U U(x1,..., x N = 0. (28 x n x x in the case D t = 0, so that we may identify the congurational part of the trial solution P st with P N derived by the MUCNA method 2 : P N (x 1,..., x N = 1 Z N { exp 1 [ U(x 1,..., x N + D a γ 2γ N ( U(x1,..., x N 2 ]} Da γ ln det Γ i x where Z N is a normalization constant. We turn now to the formula for the velocity covariance matrix for many particles. Using the Gaussianity of the distribution we can immediately write the averages of the velocity as < v i v j >= d N x v i v j P N ({x i } = D a 4 (29 d N xγ 1 ij ({x i}p N ({x i } (30 where stands for the double average over velocities and over space since the velocity distribution depends on the positions, unlie the equilibrium case. In the general case we must evaluate the matrix elements Γ ij. As shown in the appendix this can be done to rst order in /γ with the result for the velocity self-correlations: < v i αv i α >= D a d N rp st (r 1,..., r N [1 w αα (r i, r j ] (31 γ j i Finally, averaging over the whole system and introducing the positional pair correlation function g(r r one obtains 1 N N i=1 < vαv i α i >= D a (1 γ ρ drg 2 (rw αα (r (32 In analogy with granular gases we can dene the second moment of the velocity distribution to be the average inetic temperature, T of the system, via T = 1 N N i v iv i. III. APPROXIMATION FOR THE DETERMINANT AND VELOCITY CORRELATIONS The exact evaluation of the determinant Γ associated with the Hessian matrix is beyond the authors capabilities and we loo for approximations in order to evaluate the eective forces. We consider the associated determinant in the case of two spatial dimensions and vanishing external potential: [1 + γ wxx(r1, rj] wxy(r1, rj wxx(r1, r2... wxy(r1, rn γ γ γ wyx(r1, rj [1 + γ γ wyy(r1, rj] wyx(r1, r2... wyy(r1, rn γ γ wxx(r2, r1 wyx(r2, r1 [1 + γ γ γ j 2 wxx(r2, rj]... wxy(r2, rn γ γ wxy(rn, r1 γ wyy(rn, r1...... [1 + γ j N wyy(rn, rj] It is interesting to remar that the o-diagonal elements contain only one term, while the diagonal elements and their neighbors contain N terms. Thus in the limit of N we expect that the matrix becomes eectively diagonal. However, even with such a limit, we see that to order /γ the inverse matrix, which is directly related to velocity

5 correlator v αi v βj (see (25, reads: [1 γ wxx(r1, rj] wxy(r1, rj wxx(r1, r2... wxy(r1, rn γ γ γ wyx(r1, rj [1 γ γ wyy(r1, rj] wyx(r1, r2... wyy(r1, rn γ γ wxx(r2, r1 wyx(r2, r1 D v [1 γ γ γ j 2 wxx(r2, rj]... wxy(r2, rn γ αiv βj wxy(rn, r1 γ γ wyy(rn, r1...... [1 γ j N wyy(rn, rj] We, now, assume translational invariance and perform the average over the spatial distribution P N (r 1,..., r N, and introduce the pair correlation as P 2 (r 1, r 2 = ρ2 N(N 1 g(r 1 r 2. The sought average is: D < v αiv βj >= 1 ρ γ xx(rg(r ρ γ xy(rg(r ρ Nγ xx(rg(r... ρ Nγ xy(rg(r ρ γ yx(rg(r 1 ρ γ yy(rg(r ρ Nγ yx(r... ρ Nγ yy(rg(r ρ rw Nγ xx(rg(r ρ Nγ yx(rg(r 1 ρ γ xx(rg(r... ρ Nγ xy(rg(r One can see that in the limit of large N the o-diagonal terms become neglegible because they carry factors 1/N and one obtains a bloc diagonal matrix whose blocs are 2 2 matrices: ( < vxv x > < v xv y > = D ( 1 ρ γ xx(rg(r ρ γ xy(rg(r < v yv x > < v yv y > yx(rg(r 1 ρ γ yy(rg(r If the system is rotationally invariant < v x v x >=< v y v y > and < v x v y >=< v y v x >= 0. ρ γ A. Evaluation of the ensemble average of the determinant to linear order in /γ and velocity variance Notice that to linear order /γ the ensemble average of the determinant is: dr 1... dr N P N (r 1,..., r N det Γ(r 1,..., r N ([1 + γ ρ xx (rg(r][1 + γ ρ yy (rg(r] N (33 dr 1... dr N P N (r 1,..., r N det Γ(r 1,..., r N (1 + γ ρ dr[w xx (r + w yy (r]g(r N (34 To order /γ: dr 1... dr N P N (r 1,..., r N ln det Γ N ln (1+ γ ρ ( D dr[w xx (r+w yy (r]g(r N ln 1 ( D +N ln < v x v x > 1 < v y v y > (35 IV. 2D SIMULATION RESULTS We simulate a 2d system driven by GCN and composed by N = 1000 particles with periodic boundary conditions. The particles interact via the purely repulsive pair potential ϕ(r 1 r 2 = r 1 r 2 12. We x = 1 and we explore several values of density ρ = N/L 2 and diusivity D. The results are reported in Fig. 1. Fig. 1(a shows the normalized velocity variance as a function of density for dierent values of D. It is seen that, as in the 1d case, the quantity ẋ 2 /(D/ is a decreasing function of ρ. However in the 1d case ẋ 2 /(D/ is also a decreasing function of D (at xed ρ, while in 2d a non-monotonic behavior in D can be observed at high ρ. Fig. 1(b shows the amplitude of the Fourier-transformed density uctuations at low q (q (20σ 1 which shows a very similar behaviour to the 1d case. Fig. 1(c and (d show two snapshots at high D and intermediate densities where evident clustering of the particles is observed but not a full phase separation. This is the case also upon further increasing to very high values as shown in Fig. 2.

6 Figure 1: (a Velocity variance as a function of density at dierent values of D. (b The colormap represents the values of the structure factor at low q for several values of the density and the diusivity. (c and (d Snapshot of the system at dierent values of D and ρ, an evident clustering is observed. 1 C. Maggi, U. M. B. Marconi, N. Gnan, and R. Di Leonardo, Scientic Reports 5 10742 (2015. 2 U. M. B. Marconi and C. Maggi, Soft matter 11 8768-8781 (2015. 3 H. Risen, Foer-Planc Equation (Springer, 1984.

Figure 2: Snapshot of the system at high for two dierent values of D and ρ, an evident clustering is observed without a full phase separation. 7

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