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

by Mishra, Shradha , PhD

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.