Active Brownian Motion

Active Brownian Motion

Lecture at the Statistical Physics meeting at the Indian Institute of Science

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Ronojoy Adhikari

February 08, 2015
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  1. 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.
  2. 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
  3. 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)
  4. 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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  7. M ˙ V = I · n dS + Fe

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

    M ˙ V = I · n dS + Fe I ˙ ⌦ = I r ⇥ · n dS + Te
  9. 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
  10. 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 )
  11. 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 )
  12. 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 )
  13. 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
  14. 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
  15. 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
  16. 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
  17. 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
  18. E. Coli Paramecium

  19. larger faster ⇠ µm ⇠ 100µm ⇠ cm ⇠ m

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

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

    E. Coli Paramecium Produce flow in the absence of body forces and torques
  22. The main theoretical result

  23. 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
  24. Irreducible expansions of active flow rotation 0 = I ·

    n dS 0 = I r ⇥ · n translation + ...
  25. 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)
  26. 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)
  27. 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)
  28. 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)
  29. Experiment Theory J. S. Guasto, K. A. Johnson, J. P.

    Gollub PRL 105 (2010) S. Ghose, RA PRL 112 (2014)
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  37. The main numerical result

  38. Moran and Posner, JFM 680 (2011) Pandey, Kumar, RA, arxiv:1408.0433

    Non-equilibrium steady states in suspensions of active rods
  39. 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
  40. Extensile active rods - pair dynamics

  41. Extensile active rods - pair dynamics

  42. Contractile active rods - pair dynamics

  43. Contractile active rods - pair dynamics

  44. Extensile suspension : phase aggregated steady state

  45. Extensile suspension : phase aggregated steady state

  46. Relaxation to steady state : mean cluster size

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

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

    concentration
  49. Contractile suspension : micro-structured steady states clusters form at high

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

    concentration activity destabilizes smectic order
  51. Low entropy states maintained by energy dissipation Palacci et al,

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

    Science 339 (2013) extensile flow contractile flow
  53. '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".'