Statistical Mechanics of Active Matter Umberto Marini Bettolo Marconi University of Camerino, Italy Naples, 24 May,2017 Umberto Marini Bettolo Marconi (2017) Statistical Mechanics of Active Matter 2017 1 / 19
Outline Theoretical models of self-propelled particles: Langevin-like approach. Configurational properties of Steady state (Kramers Smoluchowski) Velocity distribution Active Pressure and mobility reduction Umberto Marini Bettolo Marconi (2017) Statistical Mechanics of Active Matter 2017 2 / 19
Cells alternate periods of smooth forward swimming (runs) with abrupt reorientations (tumbles). The direction of motion fluctuates, but on a short-time scale a persistence to move in the current direction is observed. Umberto Marini Bettolo Marconi (2017) Statistical Mechanics of Active Matter 2017 3 / 19 Non-equilibrium properties Active systems are able to convert energy from the environment into directed or persistent motion either by metabolic processes or by chemical reactions. Every particle has its own driving, in contrast to boundary driven systems, like a strip of metal heated at one end.
Models with Langevin dynamics Active Brownian model: particles are subject to constant drag (the active force) γv 0 e i in modulus and the angle associated with e i performs rotational diffusion D r = 1/τ. ẋ i (t) = v 0 e i (t) iu γ, θ i (t) = D r η θ i (t) Gaussian colored noise model: particles are subject to γu i fluctuating drag force having Gaussian distribution ẋ i (t) = u i (t) iu γ, u i (t) = 1 τ u i(t) + D1/2 ξ i (t) τ The correspondence between models is obtained by requiring the same free diffusion D = v 2 0 τ Umberto Marini Bettolo Marconi (2017) Statistical Mechanics of Active Matter 2017 4 / 19
Overdamped limit: Unified color approximation Exponentially correlated velocity, u i (t), is an Ornstein-Uhlenbeck process dx i dt du i dt = u i (t) 1 γ U x i = u i τ + D1/2 ξ i (t) τ u i (t)u j (t D ) = δ /τ ij τ e t t. Eliminating the active force γu i (t) and neglecting the acceleration term, we obtain an eq. of driven-diffusive type with space dependent friction matrix τ d 2 x i dt 2 + j ( δ ij + τ γ 2 U x i x j ) dxi dt = 1 U + D 1/2 ξ i (t). γ x i Unified Colored Noise Approximation is analogous to Kramers Smoluchowski reduction. It is exact in the limits τ 0 and τ. Umberto Marini Bettolo Marconi (2017) Statistical Mechanics of Active Matter 2017 5 / 19
Boltzmann-like distribution With zero current the probability distribution at temperature T = Dγ is P(x 1,..., x N ) = 1 Z exp [ U T τ 2T ( U i x i ) 2 + ln det( δ ij + τ γ ] 2 U ) x i x j (1) 7 6 5 4 =0.01 =0.3 =0.9 Density profile Effective mobility 1.5 1.0 1/ 0.5 3 2 0.0 0 0.5 1 1.5 2 2.5 3 3.5 4 x 1 0 0 0.5 1 1.5 2 2.5 3 3.5 4 x Umberto Marini Bettolo Marconi (2017) Statistical Mechanics of Active Matter 2017 6 / 19
Steady state Why Boltzmann-like? Detailed balance (microscopic reversibility)? In equilibrium systems, energy is exchanged reversibly with environment and P(x) determined by the potential. In non-equilibrium systems, energy is exchanged irreversibly with environment in general no one-to-one correspondence between potential and P(x). Fluctuating forces originate from the active nature of the system, such as fluctuations of the propulsion, and not from thermal noise. Effective forces can emerge as a consequence of microscopic breaking of detailed balance. Non equilibrium force is an attraction between self-propelled particles causing them to cluster. Umberto Marini Bettolo Marconi (2017) Statistical Mechanics of Active Matter 2017 7 / 19
Velocity distribution Particles subject to equilibrium noise have steady state distribution { P eq (x, v) exp v 2 + U(x) } 2T Instead, with time-correlated non-equilibrium noise { P neq (x, v) exp v 2 + U eff (x) } 2T eff (x) T eff (x) = T 1 + τ γ U (x) Integrating out the velocity we obtain the UCNA steady state pdf: [ P ucna (x) = exp U T τ ( ) ] du 2 + ln det(1 + τ 2γT dx γ U (x)) Velocity correlations determine a non local effective force. Umberto Marini Bettolo Marconi (2017) Statistical Mechanics of Active Matter 2017 8 / 19
Steady state of Gaussian colored model Effective temperature is not uniform, i.e. velocity variance is space dependent. Velocity distribution is only approximately Gaussian. Within the UCNA approximation detailed balance holds and entropy production vanishes. The Gaussian velocity pdf does not solve exactly the Fokker-Planck equation. The approximate Gaussian solution is not current-free. Umberto Marini Bettolo Marconi (2017) Statistical Mechanics of Active Matter 2017 9 / 19
Pair distribution function and Effective pair potential Back to UCNA In 3 dimensions there are contributions from normal and tangential fluctuations of the velocity. g(r) = exp U(r) τ γ [(U (r)] 2 + T ln[(1 + 2 τ γ U (r))(1 + 2 τ U (r) γ r ) 2 ] T Effective pair potential U eff (r) = T ln g(r) There is an attraction out of repulsion as τ γ increases. Attraction results from enhancement of friction due to interactions. Approximation still captures important aspects of the non-equilibrium solution. Umberto Marini Bettolo Marconi (2017) Statistical Mechanics of Active Matter 2017 10 / 19
Umberto Marini Bettolo Marconi (2017) Statistical Mechanics of Active Matter 2017 11 / 19
Pressure of an active fluid (Brady-Takatori) Pressure excess is the sum of active (swim ) and passive pressure Through their self-motion, all active matter systems generate a unique swim pressure that is entirely athermal in origin. An active body would swim away in space unless confined by boundaries- this confinement pressure is the swim pressure p total = p thermal + p active direct interaction + p For pure white noise (= thermal fluctuations) : p active = 0. Hydrostatic balance condition leads to definition of a mechanical Pressure (Force Virial) Mobility decreases in densely populated regions or near boundaries. Pressure decreases with reduced mobility of particles leading to MIPS. Does exist a thermodynamic relation between probability distribution and pressure? p =? T ln Z V Themodynamic pressure and mechanical pressure are equivalent? Umberto Marini Bettolo Marconi (2017) Statistical Mechanics of Active Matter 2017 12 / 19
Active Pressure is also a tensor In spherical geometry we obtain the analog of the Laplace equation d dr p N(r) + 2 r (p N(r) p T (r)) = ρ(r)u ext(r). The difference between normal and tangential components integrated over the interface defines a surface tension ρ(r) p N (r) = T 1 + τ γ u ext(r), p T (r) = T ρ(r) 1 + τ γ u ext(r)/r tangential and normal pressure 4.5 4 3.5 3 2.5 2 1.5 1 =0.1 =0.3 =0.1 =0.3 Pressure profiles 0.5 0 6 8 10 12 14 r Umberto Marini Bettolo Marconi (2017) Statistical Mechanics of Active Matter 2017 13 / 19
Velocity covariance for many particles Umberto Marini Bettolo Marconi (2017) Statistical Mechanics of Active Matter 2017 14 / 19
Umberto Marini Bettolo Marconi (2017) Statistical Mechanics of Active Matter 2017 15 / 19
Phase space distribution and its velocity moments Kramers equation describing evolution of phase-space distribution f N t + i v i f N x i + i F i f N γτ v i = 1 τ i v i ( D τ v i + k M ik v k )f N Probability conservation for the marginalised distribution of positions: P N ({x i }, t) t + i J i ({x i }, t) x i = 0 (2) Momentum balance for the current J i d N v v i f N ({x i }, {v i }, t). J i ({x i }, t) t + k Π ik ({x i }, t) x k F i γτ P N({x i }, t) = 1 τ M ik J k k Generalized Pressure tensor : Π ij ({x i } d N v v i v j f N ({x i }, {v i }) Umberto Marini Bettolo Marconi (2017) Statistical Mechanics of Active Matter 2017 16 / 19
Steady state and zero currents J i = 0 Gaussian trial distribution f N ({x i }, ({v i }) = G({v i } {x i }) P trial ({x i }) τ ( G(({v i } {x i }) = ( 2πD )N/2 det M exp τ ) v i M ij (x)v j 2D By substituting in the momentum eq. we get the UCNA equation for P trial. Π ij ({x i } = v i v j P trial ({x i }). ij v i v j = d N vv i v j G({v i } {x i }) = D τ M 1 ij ({x i }) D τ [1 τ γ w αα (r i, r j )] j i Averaging over the whole system and introducing the positional pair correlation function g(r r ) one obtains 1 N N < v i v i >= D (1 τ τ γ ρ i=1 ) drg 2 (r)w αα (r) Umberto Marini Bettolo Marconi (2017) Statistical Mechanics of Active Matter 2017 17 / 19
Future directions Explore stationary regime in the presence of steady currents. Go beyond the Unified color approximation. Non Gaussian effects. Collaborations with C.Maggi, M. Paoluzzi, A.Puglisi, A. Sarracino, R. Di Leonardo, N.Gnan Umberto Marini Bettolo Marconi (2017) Statistical Mechanics of Active Matter 2017 18 / 19
Figure : (a) Stationary probability distribution p(x, v) in the case of a double-well potential and persistence time τ = 0.7. (b) Position probability distribution n(x). (c) Temperature profiles v 2 x and θ(x). (d) Local heat flow and local entropy production (full and dashed lines respectively) for persistence time τ = 0.7. Both quantities are negative in the potential wells since there the particle transfers heat to the bath, whereas the opposite occurs in the peak region. Umberto Marini Bettolo Marconi (2017) Statistical Mechanics of Active Matter 2017 19 / 19