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Title:  Dynamics, Order And Fluctuations In Active Nematics : Numerical And Theoretical Studies 
Authors:  Mishra, Shradha 
Advisors:  Ramaswamy, Sriram 
Keywords:  Nematics Liquid Crystals Nematic Liquid Crystals Dynamic Renormalization Group (DRG) Nonequilibrium Statistical Mechanics Particle Dynamics Statistical Mechanics Active Nematic 
Submitted Date:  Oct2008 
Series/Report no.:  G22619 
Abstract:  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 selfdriven 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 twodimensional nonequilibrium active nematics. Twodimensional nonequilibrium nematic steady states, as found in agitated granularrod 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 phaseseparated. 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 wellordered nematic background clumped spontaneously, with domains growing as t1/2, and an apparently finite phaseseparation 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 phaseseparation is fluctuationdominated[7].
Dynamics of active particles can be implemented either through microscopic rules as in[8,9]or in a longwavelength 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 longwavelength equation by explicit coarsegraining 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 “curvatureinduced” 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 “curvatureinduced” 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 shortdistance 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 steadystate, 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 longrange order in a twodimensional 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/KPZequation. By contrast with that case, we find that the nonlinearities are marginally irrelevant. This implies in particular that 2dactive nematics too have only quasilongrange order[14]. 
URI:  http://hdl.handle.net/2005/832 
Appears in Collections:  Physics (physics)

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