# Dynamics, order and fluctuations in active nematics: numerical and theoretical studies

Abstract (Summary)

In this thesis we studied theoretically and numerically dynamics, order and fluctuations
in two dimensional active matter with specific reference to the nematic
phase in collections of self-driven particles. The aim is to study the ways in which a
nonequilibrium steady state with nematic order differs from a thermal equilibrium
system of the same spatial symmetry. The models we study are closely related to
“flocking” [1], as well as to equations written down to describe the interaction of
molecular motors and filaments in a living cell [2, 3] and granular nematics [4].
We look at (i) orientational and density fluctuations in the ordered phase, (ii) the
way in which density fluctuations evolve in a nematic background, and finally (iii)
the coarsening of nematic order and the density field starting from a statistically
homogeneous and isotropic initial state. Our work establishes several striking differences
between active nematics and their thermal equilibrium counterparts.
We studied two-dimensional nonequilibrium active nematics. Two-dimensional
nonequilibrium nematic steady states, as found in agitated granular-rod monolayers
or films of orientable amoeboid cells, were predicted [5] to have giant number
fluctuations, with the standard deviation proportional to the mean. We studied
this problem more closely, asking in particular whether the active nematic
steady state is intrinsically phase-separated. Our work has close analogy to the
work of Das and Barma [6] on particles sliding downhill on fluctuating surfaces,
so we looked at a model in which particles were advected passively by the brokensymmetry
modes of a nematic, via a rule proposed in [5]. We found that an
initially homogeneous distribution of particles on a well-ordered nematic background
clumped spontaneously, with domains growing as t1/2, and an apparently
finite phase-separation order parameter in the limit of large system size. The density
correlation function shows a cusp, indicating that Porod’s Law does not hold
here and that the phase-separation is fluctuation-dominated [7].
Dynamics of active particles can be implemented either through microscopic rules
as in [8, 9] or in a long-wavelength phenomenological approach as in [5] It is important
to understand how the two methods are related. The purely phenomenological
approach introduces the simplest possible (and generally additive) noise consistent
with conservation laws and symmetries. Deriving the long-wavelength equation
by explicit coarse-graining of the microscopic rule will in general give additive and
multiplicative noise terms, as seen in e.g., in [10]. We carry out such a derivation
and obtain coupled fluctuating hydrodynamic equations for the orientational
order parameter (polar as well as apolar) and density fields. The nonequilibrium
“curvature-induced” current term postulated on symmetry grounds in [5] emerges
naturally from this approach. In addition, we find a multiplicative contribution
to the noise whose presence should be of importance during coarsening [11].
We studied nonequilibrium phenomena in detail by solving stochastic partial differential
equations for apolar objects as obtained from microscopic rules in [8].
As a result of “curvature-induced” currents, the growth of nematic order from an
initially isotropic, homogeneous state is shown to be accompanied by a remarkable
clumping of the number density around topological defects. The consequent
coarsening of both density and nematic order are characterised by cusps in the
short-distance behaviour of the correlation functions, a breakdown of Porod’s Law.
We identify the origins of this breakdown; in particular, the nature of the noise
terms in the equations of motion is shown to play a key role [12].
Lastly we studied an active nematic steady-state, in two space dimensions, keeping
track of only the orientational order parameter, and not the density. We apply
the Dynamic Renormalization Group to the equations of motion of the order parameter.
Our aim is to check whether certain characteristic nonlinearities entering
these equations lead to singular renormalizations of the director stiffness coefficients,
which would stabilize true long-range order in a two-dimensional active
nematic, unlike in its thermal equilibrium counterpart. The nonlinearities are related
to those in [13] but free of a constraint that applies at thermal equilibrium.
We explore, in particular, the intriguing but ultimately deceptive similarity between
a limiting case of our model and the fluctuating Burgers/KPZ equation. By
contrast with that case, we find that the nonlinearities are marginally irrelevant.
This implies in particular that 2-d active nematics too have only quasi-long-range
order [14].
[1] J. Toner, Y. Tu and S. Ramaswamy, Ann. Phys. 318, 170 (2005).
[2] T. B. Liverpool et al., Phys. Rev. Lett. 90, 138102 (2003).
[3] H. Y. Lee and M. Kardar, Phys. Rev. E 64, 056113 (2001).
[4] V. Narayan, S. Ramaswamy, and N. Menon, Science 317, 105 (2007).
[5] S. Ramaswamy, R.A. Simha, and J. Toner, Europhys. Lett. 62, (2003) 196.
[6] D. Das and M. Barma, Phys. Rev. Lett. 85, 1602 (2000).
[7] S. Mishra and S. Ramaswamy, Phys. Rev. Lett. 97, 090602 (2006).
[8] H. Chat´e et al., Phys. Rev. Lett. 96, 180602 (2006).
[9] T. Vicsek et al., Phys. Rev. Lett. 75, 1226 (1995).
[10] D.S. Dean, J. Phys. A: Math. Gen. 29 (1996) L613-617.
[11] S. Mishra et al. in preparation.
[12] S. Mishra, F. Ginelli, H. Chat´e, S. Puri and S. Ramaswamy (to be submitted).
[13] D. Nelson et al., Phys. Rev. B 16 (1977) 2191.
[14] S. Mishra and S. Ramaswamy J. Stat. Mech. (2010) P02003.
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Bibliographical Information:

Advisor:Sriram Ramaswamy

School:Indian Institute of Science

School Location:India

Source Type:Doctoral Dissertation

Keywords:active nematics, nonequilibrium statistical mechanics, self-propelled particles

ISBN:

Date of Publication:10/14/2008