Active and Driven Soft Matter Lecture 3: Self-Propelled Hard Rods

Transcription

1 A Tutorial: From Langevin equation to Active and Driven Soft Matter Lecture 3: Self-Propelled Hard Rods M. Cristina Marchetti Syracuse University Boulder School 2009

2 A Tutorial: From Langevin equation to Lecture 1: Table of Contents 1 The Model Active Hard Rod Nematic Plan and 2 A Tutorial: From Langevin equation to Langevin dynamics From Langevin to Fokker-Planck dynamics Low density limit & Smoluchowski equation Summary and Plan 3 Langevin dynamics of SP Rods 4 Fields Equations 5 Homogeneous States and Enhanced Nematic Order Stability and Novel Properties of Bulk States 6

3 A Tutorial: From Langevin equation to Active Hard Rod Nematic Active Hard Rod Nematic Plan and 2d hard rods Excluded volume interaction Overdamped dynamics Non-thermal white noise hard rod nematic + self-propulsion v 0 along long axis v 0 1. Start with Langevin dynamics of coupled orientational and translational degrees of freedom plus hard core collisions SP yields anisotropic enhancement of ˆk momentum transferred in a hard core 1 collision v 1 v2 2. Derive continuum theory 2

4 A Tutorial: From Langevin equation to Plan and Active Hard Rod Nematic Plan and The lecture illustrates how to use the tools of statistical physics to derive hydrodynamics from microscopic dynamics for a minimal model of a self-propelled system. Inspired by Vicsek model of SP point particles that align with neighbors according to prescribed rules in the presence of noise. This rule-based model orders in polar (moving) states below a critical value of the noise. Hard rods order in nematic states due to steric effects: can excluded volume interactions, plus self-propulsion, yield a polar state? Result: no homogeneous polar state, but other nonequilibrium effects: SP enhances nematic order SP enhances longitudinal diffusion ("persistent" random walk) SP yields propagating sound-like waves in the isotropic state SP destabilizes the nematic state SP+boundary effects can yield a polar state

5 A Tutorial: From Langevin equation to Langevin dynamics Langevin dynamics From Langevin to Fokker-Planck dynamics Low density limit & Smoluchowski equation Summary and Plan Spherical particle of radius a and mass m, in one dimension m dv dt = ζv + η(t) ζ = 6πηa friction noise is uncorrelated in η(t) = 0 time and Gaussian: η(t)η(t ) = 2 δ(t t ) Noise strength In equilibrium is determined by requiring lim t < [v(t)] 2 >=< v 2 > eq = k BT m = ζk BT m 2 Mean square displacement is diffusive < [ x(t)] 2 >= 2k [ BT t m ( 1 e ζt/m)] 2k BT t = 2Dt ζ ζ ζ

6 A Tutorial: From Langevin equation to Fokker-Plank equation Langevin dynamics From Langevin to Fokker-Planck dynamics Low density limit & Smoluchowski equation Summary and Plan Many-particle systems In this case it is convenient to work with phase-space distribution functions: ˆfN (x 1, p 1, x 2, p 2,..., x N, p N, t) ˆf N (x N, p N, t) These are useful when dealing with both Hamiltonian dynamics and Langevin (stochastic) dynamics. First step: Transform the Langevin equation into a Fokker-Plank equation for the noise-average distribution function f 1 (x, p, t) =< ˆf 1 (x, p, t) >

7 A Tutorial: From Langevin equation to Fokker-Plank equation - 2 Langevin dynamics From Langevin to Fokker-Planck dynamics Low density limit & Smoluchowski equation Summary and Plan Compact notation dv dt dx = p/m dt = ζv du dx + η(t) dx dt = V + η(t) «x X = p «p/m V = ζv U «0 η = η Conservation law for probability distribution ( dx ˆf (X, t) = 1 tˆf + X X ˆf ) t = 0 ( ) ( ) ( ) tˆf + X Vˆf + X ηˆf = 0 tˆf + Lˆf + X ηˆf = 0 t ˆf (X, t) = e Lt f (X, 0) ds e L(t s) 0 X η(s)ˆf (x, s)

8 A Tutorial: From Langevin equation to Fokker-Plank equation - 3 Langevin dynamics From Langevin to Fokker-Planck dynamics Low density limit & Smoluchowski equation Summary and Plan Use properties of Gaussian noise to carry out averages t < ˆf > + X V < ˆf > + X < η(t)e Lt f (X, 0) > Z t X < η(t) 0 ds e L(t s) X η(s)ˆf (X, s) > = 0 t f = p m xf p [ U (x) ζp/m]f + 2 pf Fokker-Plank eq. easily generalized to many interacting particles dp α = ζv α X xα V (x α x β ) + η α(t) dt β Z t f 1 (1, t) = v 1 x1 f 1 (1) + ζ p1 v 1 f 1 (1) + p 2 1 f 1 (1) + p1 d2 x1 V (x 12 )f 2 (1, 2, t)

9 A Tutorial: From Langevin equation to Smoluchowski equation Langevin dynamics From Langevin to Fokker-Planck dynamics Low density limit & Smoluchowski equation Summary and Plan 1 One obtains a hierarchy of Fokker-Planck equations for f 1 (1), f 2 (1, 2), f 3 (1, 2, 3),... To proceed we need a closure ansatz. Low density (neglect correlations) f 2 (1, 2, t) f 1 (1, t)f 1 (2, t) 2 It is instructive to solve the FP equation by taking moments c(x, t) = R dp f (x, p, t) J(x, t) = R dp (p/m)f (x, p, t) concentration of particles density current Eqs. for the moments obtained by integrating the FP equation. t c(x, t) = x J(x, t) t J(x 1 ) = ζj(x 1 ) m ζ x 1 c(x 1) R dx 2 [ x1 V (x 12 )]c(x 1, t)c(x 2, t) For t >> ζ 1, we eliminate J to obtain a Smoluchowski eq. for c t c(x 1, t) = D 2 x 1 c(x 1, t) + 1 ζ x 1 x 2 [ x1 V (x 12 )]c(x 1, t)c(x 2, t)

10 A Tutorial: From Langevin equation to Langevin dynamics From Langevin to Fokker-Planck dynamics Low density limit & Smoluchowski equation Summary and Plan Due to the interaction with the substrate, momentum is not conserved. The only conserved field is the concentration of particles c(x, t). This is the only hydrodynamic field. To obtain a hydrodynamic equation form the Smoluchowski equation we recall that we are interested in large scales. Assuming the pair potential has a finite range R 0, we consider spatial variation of c(x, t) on length scales x >> R 0 and expand in gradients t c(x 1, t) = D x 2 1 c(x 1, t) + 1 Z ζ x 1 V (x )[ x c(x 1 + x, t)]c(x 1, t) x = D x 2 1 c(x 1, t) + 1 Z ζ x 1 V (x )[ x1 c(x 1, t) + x 2 x x 1 c(x 1, t) +...]c(x 1, t) The result is the expected diffusion equation, with a microscopic expression for D ren which is renormalized by interactions t c(x, t) = x [D ren x c(x, t)] D ren 2 x c(x, t)

11 A Tutorial: From Langevin equation to Summary of Tutorial and Plan Langevin dynamics From Langevin to Fokker-Planck dynamics Low density limit & Smoluchowski equation Summary and Plan Microscopic Langevin dynamics of interacting particles Approximations: noise average; low density: f 2 (1, 2) f 1 (1)f 1 (2) Fokker Planck equation Overdamped limit: t >> 1/ζ Smoluchowski equation [ t c(x 1, t) = x1 D x1 c(x 1, t) 1 ] F (x 12 )c(x 2, t)c(x 1, t ζ x 2 Pair interaction F (x 12 ): steric repulsion SP rods short-range active interactions cross-linkers in motor-filaments mixtures medium-mediated hydrodynamic interactions swimmers Smoluchowski Hydrodynamic equations

12 SP Rods The Model A Tutorial: From Langevin equation to Langevin dynamics of SP Rods The algebra is quite a bit more involved for a number of reasons: 1 translational and rotational degrees of freedom are coupled v α t = v 0 ˆν α ζ(ˆν α) v α P β T (α, β) vα + ηα (t) ω α t = ζ R ω α P β T (α, β) ωα + ηr α (t) where η αi (t)η βj (t ) = 2k B T a ζ ij (ˆν α)δ ij δ (t t ) D E ηα R (t) ηβ R (t ) = 2k B T a (ζ R /I) δ αβ δ (t t ), I = l 2 /12 ζ ij (ˆν α) = ζ ˆν αi ˆν αj + ζ (δ ij ˆν αi ˆν αj ) 2 hard core interactions must be treated with care to handle properly the instantaneous momentum transfer 3 coupling of self-propulsion and collisional dynamics yields angular correlations

13 A Tutorial: From Langevin equation to Langevin dynamics of SP Rods The Smoluchowski equation for c(r, ˆν, t) is given by t c + v 0 c = D R 2 θc + (D + D S ) 2 c + D 2 c (Iζ R ) 1 θ (τ ex + τ SP ) ζ 1 (F ex + F SP ) = ˆν D S = v0 2/ζ enhancement of = ˆν(ˆν ) longitudinal diffusion Torques and forces exchanged upon collision as the sum of Onsager excluded volume terms and contributions from self-propulsion: τ ex = θ V ex V F ex = V ex (1) = k B T ac(1, t) R R ξ ex 12 ˆν ˆν 2 1 ˆν 2 c (r 1 + ξ 12, ˆν 2, t) ξ 12 = ξ 1 ξ 2 «Z Z! FSP = v τ 0 2 b k SP 2,ˆk ẑ (ξ 1 b [ẑ (ˆν 1 ˆν 2 )] 2 k) s 1,s 2 Θ( ˆν 12 bk)c(1, t)c(2, t)

14 A Tutorial: From Langevin equation to SP terms in Smoluchowski Eq. Langevin dynamics of SP Rods Convective term describes mass flux along the rod s long axis. Longitudinal diffusion enhanced by self-propulsion: D D + v 2 0 /ζ. Longitudinal diffusion of SP rod as persistent random walk with bias v 0 towards steps along the rod s long axis. The SP contributions to force and torque describe, within mean-field, the additional anisotropic linear and angular momentum transfers during the collision of two SP rods. D E pcoll Mean-field Onsager: v th t τ kb T a k BT a coll l/ k B T a l D E pcoll SP rods: t v 0 ˆν 1 ˆν 2 SP l/v 0 ˆν 1 ˆν 2 v 0 2 ˆν 1 ˆν 2 2

15 A Tutorial: From Langevin equation to Fields Fields Equations Conserved density: ρ(r, t) = c(r, ˆν, t) ˆν Order parameter fields: Polarization vector: P(r, t) = ˆν c(r, ˆν, t) ˆν Polar order p -p fish, bacteria, motor-filaments p Nematic alignment tensor: Q ij (r, t) = ˆν (ˆν i ˆν j 1 2 δ ij)c(r, ˆν, t) Nematic order n -n melanocytes, granular rods n

16 A Tutorial: From Langevin equation to Equations Fields Equations t ρ + v 0 P = D ρ 2 ρ + D Q : ρq t P = D R P + λp Q v 0 Q λ 1 (P )P λ 2 P 2 λ 3 P( P) v 0 2 ρ + D bend 2 P + (D splay D benb ) ( P) t Q = 4D R 1 ρ «Q v 0 [ P] ST λ 4 [P Q] ST λ 5 [Q P] ST λ 6 [ P] ST ρ IN [T] ST ij = 1 2 (T ij + T ji 1 2 δ ij T kk ) + D Q 4 ( 1 2 1)ρ + D Q 2 Q

17 A Tutorial: From Langevin equation to Homogeneous States Homogeneous States and Enhanced Nematic Order Stability and Novel Properties of Bulk States Neglecting all gradients, the equations are given by Bulk states: t ρ = 0 t P = D R P + λp Q Isotropic State: ρ = ρ 0, P = Q = 0 No bulk uniform polar state: P = 0 Nematic state P = 0, Q 0 for ρ > ρ IN SP enhances nematic order: ρ IN = ρ N 1+ v2 0 5k B T t Q = 4D R [1 ρ/ρ IN (v 0 )] Q with ρ N = 3/(πl 2 )

18 A Tutorial: From Langevin equation to Homogeneous States and Enhanced Nematic Order Stability and Novel Properties of Bulk States Enhanced Nematic order in Simulations of Actin Motility Assay - +

19 A Tutorial: From Langevin equation to Homogeneous States and Enhanced Nematic Order Stability and Novel Properties of Bulk States Hydrodynamic Modes Stability of Bulk States We examine the dynamics of the fluctuations of the hydrodynamic fields about their mean values and analyze the hydrodynamic modes of the system. ρ = ρ Isotropic state (neglect Q) 0 + δρ P = 0 + δp Linearized equations: Fourier modes: t δρ = D ρ 2 δρ + v 0 ρ 0 δp t δp = D R δp + (v 0 /2ρ 0 ) δρ δρ(r, t) = k δρ k(t)e ik r δp(r, t) = k δp k(t)e ik r z ± = 1 2 (D R + D ρ k 2 ) ± 1 2 (D R D ρ k 2 ) 2 2v 2 0 k 2 δρ k (t) e z(k)t δp k (t) e z(k)t

20 A Tutorial: From Langevin equation to Sound Waves in Isotropic State Homogeneous States and Enhanced Nematic Order Stability and Novel Properties of Bulk States Propagating density waves in isotropic state of overdamped SP rods t = D 2 + v 0 0 P tp = DRP + v0 0 fluctuations: = 0 +, v 0 P = P P Above a critical v 0 the system supports sound-like waves in a range of wavevectors v 0c v 0c D R Ramaswamy & Mazenko, 1982: fluid on frictional substrate --> movie of fluidized rods by Durian s lab

21 A Tutorial: From Langevin equation to Instability of Nematic State Homogeneous States and Enhanced Nematic Order Stability and Novel Properties of Bulk States The nematic state is unstable for v 0 > v c (φ, S), as shown in the figure for S = 1 (red line) and S < 1 (dashed blue line). The instability arises from a subtle interplay of splay and bend deformations. cosφ = ˆn 0 ˆk φ = 0 pure bend φ = π/2 pure splay

22 A Tutorial: From Langevin equation to Homogeneous States and Enhanced Nematic Order Stability and Novel Properties of Bulk States Enhanced polar order near boundaries SP can enhance the size of correlated polar regions, as seen in simulations [F. Peruani, A. Deutsch and M. Bär, Phys. Rev. E 74, (2006).]. Polarization profile across channel (P x (±L/2) = P 0 ) P x (y) = P 0 cosh(y/δ)/ cosh(l/2δ) δ = l/2 5/2 + v0 2/k BT δ(v 0 = 0) l

23 A Tutorial: From Langevin equation to A minimal model of interacting SP hard rods exhibits several novel nonequilibrium phenomena at large scales No bulk polar state enhancement of nematic order enhancement of longitudinal diffusion sound waves in isotropic state enhanced polarization correlations We will consider in the last lecture a richer model that incorporates the role of fluid flow. References 1 R. Zwanzig, Nonequilibrium Statistical Mechanics (Oxford University Press, 2001), Chapters 1 and 2. 2 A. Baskaran and M. C. Marchetti, of Self-Propelled Hard Rods, Phys. Rev. E 77, (2008). 3 A. Baskaran and M. C. Marchetti, Enhanced Diffusion and Ordering of Self-Propelled Rods, Phys. Rev. Lett. 101, (2008).