110

# Active Brownian Motion

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

February 08, 2015

## Transcript

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

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

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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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

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".'