Active Brownian Motion Ronojoy Adhikari The Institute Of Mathematical Sciences ⌘ ⌘0 = (1 + 5 2 + 5.2 2) D = µkBT Better than anyone before or after him, he knew how to ... make use of statistical fluctuations.

Fluctuation, dissipation and momentum conservation ⌦n = µRT nm · Fm + µRR nm · Tm + q 2kBTµRT nm · ⇣T m + q 2kBTµRR nm · ⇣R m Vn = µT T nm · Fm + µT R nm · Tm + q 2kBTµT T nm · ⇠T m + q 2kBTµT R nm · ⇠R m mobility matrices many-body dissipation Wiener processes many-body fluctuation

Fluctuation, dissipation and momentum conservation ⌦n = µRT nm · Fm + µRR nm · Tm + q 2kBTµRT nm · ⇣T m + q 2kBTµRR nm · ⇣R m Vn = µT T nm · Fm + µT R nm · Tm + q 2kBTµT T nm · ⇠T m + q 2kBTµT R nm · ⇠R m mobility matrices many-body dissipation Wiener processes many-body fluctuation - mobility matrices encode momentum conservation - mobility matrices are symmetric (Onsager) - mobility matrices are positive definite (dissipation) - “square roots” are Cholesky factors - Gibbs distribution is stationary for gradient flows (FDT)

Fluctuation, dissipation and momentum conservation ⌦n = µRT nm · Fm + µRR nm · Tm + q 2kBTµRT nm · ⇣T m + q 2kBTµRR nm · ⇣R m Vn = µT T nm · Fm + µT R nm · Tm + q 2kBTµT T nm · ⇠T m + q 2kBTµT R nm · ⇠R m mobility matrices many-body dissipation Wiener processes many-body fluctuation - mobility matrices encode momentum conservation - mobility matrices are symmetric (Onsager) - mobility matrices are positive definite (dissipation) - “square roots” are Cholesky factors - Gibbs distribution is stationary for gradient flows (FDT) How do we extend this paradigm to suspensions of active particles ?

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M ˙ V = I · n dS + Fe I ˙ ⌦ = I r ⇥ · n dS + Te

neglect inertia, set external forces and torques to zero : M ˙ V = I · n dS + Fe I ˙ ⌦ = I r ⇥ · n dS + Te

neglect inertia, set external forces and torques to zero : M ˙ V = I · n dS + Fe I ˙ ⌦ = I r ⇥ · n dS + Te 0 = I · n dS 0 = I r ⇥ · n

neglect inertia, set external forces and torques to zero : M ˙ V = I · n dS + Fe I ˙ ⌦ = I r ⇥ · n dS + Te 0 = I · n dS 0 = I r ⇥ · n = pI + ⌘(rv + rvT )

neglect inertia, set external forces and torques to zero : M ˙ V = I · n dS + Fe I ˙ ⌦ = I r ⇥ · n dS + Te 0 = I · n dS 0 = I r ⇥ · n active particle : non-trivial flow in the absence of forces and torques = pI + ⌘(rv + rvT )

neglect inertia, set external forces and torques to zero : M ˙ V = I · n dS + Fe I ˙ ⌦ = I r ⇥ · n dS + Te 0 = I · n dS 0 = I r ⇥ · n active particle : non-trivial flow in the absence of forces and torques = pI + ⌘(rv + rvT )

neglect inertia, set external forces and torques to zero : M ˙ V = I · n dS + Fe I ˙ ⌦ = I r ⇥ · n dS + Te 0 = I · n dS 0 = I r ⇥ · n active particle : non-trivial flow in the absence of forces and torques = pI + ⌘(rv + rvT ) r · = rp + ⌘r2v = 0 r · v = 0

neglect inertia, set external forces and torques to zero : M ˙ V = I · n dS + Fe I ˙ ⌦ = I r ⇥ · n dS + Te 0 = I · n dS 0 = I r ⇥ · n active particle : non-trivial flow in the absence of forces and torques = pI + ⌘(rv + rvT ) r · = rp + ⌘r2v = 0 r · v = 0 boundary conditions

neglect inertia, set external forces and torques to zero : M ˙ V = I · n dS + Fe I ˙ ⌦ = I r ⇥ · n dS + Te 0 = I · n dS 0 = I r ⇥ · n active particle : non-trivial flow in the absence of forces and torques = pI + ⌘(rv + rvT ) r · = rp + ⌘r2v = 0 r · v = 0 boundary conditions v = Vn + ⌦n ⇥ (r Rn) + va n , r 2 Sn

neglect inertia, set external forces and torques to zero : M ˙ V = I · n dS + Fe I ˙ ⌦ = I r ⇥ · n dS + Te 0 = I · n dS 0 = I r ⇥ · n active particle : non-trivial flow in the absence of forces and torques = pI + ⌘(rv + rvT ) r · = rp + ⌘r2v = 0 r · v = 0 boundary conditions can produce fluid flow without rigid body motion! v = Vn + ⌦n ⇥ (r Rn) + va n , r 2 Sn

neglect inertia, set external forces and torques to zero : M ˙ V = I · n dS + Fe I ˙ ⌦ = I r ⇥ · n dS + Te 0 = I · n dS 0 = I r ⇥ · n active particle : non-trivial flow in the absence of forces and torques = pI + ⌘(rv + rvT ) r · = rp + ⌘r2v = 0 r · v = 0 boundary conditions can produce fluid flow without rigid body motion! what systems can these describe ? v = Vn + ⌦n ⇥ (r Rn) + va n , r 2 Sn

E. Coli Paramecium

larger faster ⇠ µm ⇠ 100µm ⇠ cm ⇠ m E. Coli Paramecium

larger faster ⇠ µm ⇠ 100µm ⇠ cm ⇠ m E. Coli Paramecium

larger faster ⇠ µm ⇠ 100µm ⇠ cm ⇠ m E. Coli Paramecium Produce flow in the absence of body forces and torques

The main theoretical result

Propulsion matrices Singh, Ghose, RA, arxiv:1411.0278 Vn = µT T nm · Fm + µT R nm · Tm + q 2kBTµT T nm · ⇠T m + q 2kBTµT R nm · ⇠R m + ⇡T l nm · Vl+1 m ⌦n = µRT nm · Fm + µRR nm · Tm + q 2kBTµRT nm · ⇣T m + q 2kBTµRR nm · ⇣R m + ⇡Rl nm · Vl+1 m mobility matrices many-body dissipation Wiener processes many-body fluctuation propulsion matrices many-body activity - propulsion matrices encode momentum conservation - propulsion matrices produce ballistic motion - propulsion matrices are positive definite (dissipation) - Gibbs distribution is not stationary for gradient flows - energy from boundary condition is dissipated in fluid v = Vn + ⌦n ⇥ (r Rn) + va n , r 2 Sn Extension of Einstein’s theory of Brownian motion to active suspensions

Irreducible expansions of active flow rotation 0 = I · n dS 0 = I r ⇥ · n translation + ...

Chlamydomonas Reinhardtii Microswimming of C. Reinhardtii Flagella driven by dyenin motors J. S. Guasto, K. A. Johnson, J. P. Gollub, Physical Review Letters 105 (2010)

Chlamydomonas Reinhardtii Microswimming of C. Reinhardtii Flagella driven by dyenin motors J. S. Guasto, K. A. Johnson, J. P. Gollub, Physical Review Letters 105 (2010)

Chlamydomonas Reinhardtii Microswimming of C. Reinhardtii Flagella driven by dyenin motors J. S. Guasto, K. A. Johnson, J. P. Gollub, Physical Review Letters 105 (2010)

Chlamydomonas Reinhardtii Microswimming of C. Reinhardtii Flagella driven by dyenin motors Complex, time-dependent flow field created by activity J. S. Guasto, K. A. Johnson, J. P. Gollub, Physical Review Letters 105 (2010)

Experiment Theory J. S. Guasto, K. A. Johnson, J. P. Gollub PRL 105 (2010) S. Ghose, RA PRL 112 (2014)

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The main numerical result

Moran and Posner, JFM 680 (2011) Pandey, Kumar, RA, arxiv:1408.0433 Non-equilibrium steady states in suspensions of active rods

Apolar, active rods contractile flow || long axis extensile flow || long axis “contractile active rod” “extensile active rod” Recently synthesized by Roy group (IISER-K) Pandey, Kumar, RA, arxiv:1408.0433

Extensile active rods - pair dynamics

Extensile active rods - pair dynamics

Contractile active rods - pair dynamics

Contractile active rods - pair dynamics

Extensile suspension : phase aggregated steady state

Extensile suspension : phase aggregated steady state

Relaxation to steady state : mean cluster size

Contractile suspension : micro-structured steady states asters form at low concentration

Contractile suspension : micro-structured steady states asters form at low concentration

Contractile suspension : micro-structured steady states clusters form at high concentration activity destabilizes smectic order

Contractile suspension : micro-structured steady states clusters form at high concentration activity destabilizes smectic order

Low entropy states maintained by energy dissipation Palacci et al, Science 339 (2013)

Low entropy states maintained by energy dissipation Palacci et al, Science 339 (2013) extensile flow contractile flow

'I called on him [Brown] two or three times before the voyage of the Beagle (1831), and on one occasion he asked me to look through a microscope and describe what I saw. This I did, and believe now that it was the marvelous currents of protoplasm in some vegetable cell. I then asked him what I had seen; but he answered me, "That is my little secret".'