Universality in active fluids - Chiu Fan Lee Department of Bioengineering, Imperial College London, UK - Department of ...
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Theoretical Physics, Warwick University, Feb 28, 2019 Universality in active fluids Chiu Fan Lee Department of Bioengineering, Imperial College London, UK
Biophysics across scales Amyloid fibrils RNA granules Active matter 200nm Brangwynne [Hyman Lab] https://www.youtube.com/watch?v=Ei mBzUSmak8 L Hong, CFL & YJ CA Weber, D Zwicker, F Jülicher & Huang (2017) CFL (2019) Statistical Mechanics and Physics of Active Emulsions. Kinetics of Amyloid Rep. Prog. Phys. Fibrillation. E- print: arxiv:1609.01569
Biophysics across scales Amyloid fibrils RNA granules Active matter 200nm Brangwynne [Hyman Lab] https://www.youtube.com/watch?v=Ei mBzUSmak8 L Hong, CFL & YJ CA Weber, D Zwicker, F Jülicher & Huang (2017) CFL (2019) Statistical Mechanics and Physics of Active Emulsions. Kinetics of Amyloid Rep. Prog. Phys. Fibrillation. E- print: arxiv:1609.01569.
Plan • Active fluids: the Vicsek model as an example • Symmetries + conservation laws -> hydrodynamic equations of motion • Universal behaviour • Active fluids with contact inhibition • Moving incompressible active fluids • Summary & Outlook
ACTIVE FLUIDS: THE VICSEK MODEL AS AN EXAMPLE
The Vicsek model - algorithm Figure from F. Ginelli, arXiv:1511.01451 T Vicsek, A Czirók, E Ben-Jacob, I Cohen & O Shochet (1995) Novel Type of Phase Transition in a System of Self-Driven Particles Phys. Rev. Lett. 75 1226–9
The Vicsek model – observations Homogeneous disordered (HD) Homogeneous ordered (HO) Noise level T Vicsek, A Czirók, E Ben-Jacob, I Cohen & O Shochet (1995) Novel Type of Phase Transition in a System of Self-Driven Particles Phys. Rev. Lett. 75 1226–9
What’s the big deal? 1. A continuous symmetry (rotational symmetry) seems to be broken in 2D! – A violation of the Mermin-Wagner-Hohenberg theorem at thermal equilibrium 2. Non-equilibrium dynamics is a crucial ingredient 3. It could be a new state of matter
What’s the big deal? 1. A continuous symmetry (rotational symmetry) seems to be broken in 2D! – A violation of the Mermin-Wagner-Hohenberg theorem at thermal equilibrium 2. Non-equilibrium dynamics is a crucial ingredient 3. It could be a new state of matter
SYMMETRIES + CONSERVATION LAWS -> HYDRODYNAMIC EQUATIONS OF MOTION
Hydrodynamic variables and conservation laws? • Hydrodynamic variables: density and momentum Ԧ • Conservation law: Mass conservation
Symmetries • What is the force F? • Starting with symmetries: – Temporal invariance: F does not depend on time – Translational invariance: F does not depend on position r – Rotational invariance: F does not depend on a particular direction – Chiral (parity) invariance: F is not right-handed or left handed
Hydrodynamics EOM Gaussian noise terms
Problems • What are the coefficients? • How can we say anything meaningful without knowing their numerical values?
UNIVERSAL BEHAVIOUR
Universal behaviour • Universal properties do not depend on model parameters • What are universal properties? 1. Nature of phase transitions 2. Critical exponents at the critical transitions 3. Scaling exponents in the ordered phase in which a continuous symmetry is spontaneously broken D Belitz, TR Kirkpatrick & T Vojta (2005) Rev. Mod. Phys.
Order-disorder critical transition (?) • At the mean-field level, our EOM are Homogeneous ordered (HO) Homogeneous disordered (HD) α α
Beyond mean-field: linear stability α • At the linear level, our EOM are ρ • Around the moving phase, the system is unstable if >0 where E Bertin, M Droz & G Grégoire (2006) PRE
Beyond mean-field: linear stability Unstable region Homogeneous Homogeneous disordered (HD) ordered (HO) Density ρ
Movie 1: Homogeneous disordered (HO) phase Colour wheel used to determine direction of velocity D Nesbitt, G Pruessner & CFL (2019) Uncovering novel phase transitions in dry polar active fluids using a lattice Boltzmann method. E-print: arXiv:1902.00530
Movie 2: Banding (B) regime Colour wheel used to determine direction of velocity D Nesbitt, G Pruessner & CFL (2019) Uncovering novel phase transitions in dry polar active fluids using a lattice Boltzmann method. E-print: arXiv:1902.00530
Movie 3: Homogeneous ordered (HO) phase Colour wheel used to determine direction of velocity D Nesbitt, G Pruessner & CFL (2019) Uncovering novel phase transitions in dry polar active fluids using a lattice Boltzmann method. E-print: arXiv:1902.00530
Beyond mean-field: linear stability Unstable region Homogeneous Homogeneous disordered (HD) ordered (HO) Density ρ
Phase separation pre-empts critical transition Two-phase coexistence Homogeneous Homogeneous disordered (HD) ordered (HO) Density ρ
So critical transition in active fluids is impossible !?
ACTIVE FLUIDS WITH CONTACT INHIBITION
Linear stability again α • At the linear level, our EOM are ρ • Around the moving phase, the system is unstable if >0 where
Linear stability again α • At the linear level, our EOM are ρ • Around the moving phase, the system is unstable if >0 where D Nesbitt, G Pruessner & CFL (2019) E-print: arXiv:1902.00530
Two instability regions Banding ? HD HO HD Density ρ
Contact inhibition -> Reverse banding Colour wheel used to determine direction of velocity S Schnyder et al. (2017) Collective motion of cells crawling on a substrate: roles of cell shape and contact inhibition. Sci. Rep. D Nesbitt, G Pruessner & CFL (2019) E-print: arXiv:1902.00530
Novel phase diagram Ԧ D Nesbitt, G Pruessner & CFL E-print: arXiv:1902.00530
Active fluids with contact inhibition • Universal nature of transitions – 4 first-order transitions – 1 critical transition achieved by fine tuning 2 parameters • Unknown critical exponents
MOVING INCOMPRESSIBLE ACTIVE FLUIDS
EOM of incompressible active fluids • From symmetry considerations again Ԧ = − + Ԧ − Ԧ ∙ Ԧ − + 2 Ԧ − 2 Ԧ + 4 Ԧ + 2 2 Ԧ + ⋯ Novel non-equilibrium universality class L Chen, J Toner & CFL (2015) NJP Parameter a
Ordered phase of 2D incompressible active fluids EOM: 1. RG transformation Ordered phase: 2. Use streaming function ℎ to enforce Incompressibility: 3. Equal-time (static) correlations in 2D active fluids are mapped onto the 2D smectic model: L Chen, CFL & J Toner (2016) Nature. Comm.
Kardar-Parisi-Zhang universality class y Incompressible flock (living) Smectic layers (equilibrium) Kardar-Parisi-Zhang surface growth Time (nonequilibrium) 2D Smectics = (1+1) KPZ [Golubović & Wang (1992) PRL] x L Chen, CFL & J Toner (2016) Nature. Comm.
Summary Active fluids with contact Incompressible active fluids inhibition Ԧ L Chen, J Toner & CFL (2015) NJP D Nesbitt, G Pruessner & CFL (2019) L Chen, CFL & J Toner (2016) Nature Comm. E-print: arXiv:1902.00530 L Chen, CFL & J Toner (2018) NJP
Outlook Biological physics is the new condensed matter physics! CFL & JD Wurtz (2019) Novel physics arising from phase transitions in biology JPD: Appl. Phys. 52, 023001
The Team at Imperial College Ben Jean David David Andrea Alice Partridge Wurtz Nesbitt Cairoli Spenlauhauer Collaborators • Gunnar Pruessner (Imperial College) Thank you for • Leiming Chen (China University of Mining and Technology) • John Toner (Oregon) your attention!
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