Spatial Structures and Giant Number Fluctuations in Models of Active Matter

Abstract

The large scale fluctuations of the ordered state in active matter systems are usually characterised by studying the "giant number fluctuations" of particles in any finite volume, as compared to the expectations from the central limit theorem. However, in ordering systems, the fluctuations in density ordering are often captured through their structure functions deviating from Porod law. In this paper we study the relationship between giant number fluctuations and structure functions, for different models of active matter as well as other non-equilibrium systems. A unified picture emerges, with different models falling in four distinct classes depending on the nature of their structure functions. For one class, we show that experimentalists may find Porod law violation, by measuring subleading corrections to the number fluctuations. [4][5][6]. In these, the "activity" refers to conscious decision making or internally generated cellular thrusts in the biological systems, or impulses from a vibrating plate for the granular systems. The combination of activity and interaction can lead to macroscopic order [7][8][9][10][11][12]. However, these systems are far from equilibrium and the usual notions of equilibrium phase transitions come under unexpected challenges [13,14]. In particular, macroscopic order, and large scale fluctuations reminiscent of critical equilibrium systems, coexist.Depending on their dynamics and symmetries, different active matter systems exhibit macroscopic polar, nematic, and/or density order. Polar ordering has been demonstrated in point polar particle (PP) models [7,9], experiments with granular disks [6], and continuum theories [13,14]. For polar rods (PR), continuum theories rule out macroscopic polar ordering [15], and experiment on mobile bacteria [2] and simulations of models of polar rods are in agreement [11,16]. Apolar rods (AR) or active nematics have been studied experimentally [4], in hydrodynamic theories [14,17], and in simulation [10] and exhibit nematic and density ordering.The density fluctuations in the ordered state has been characterised by the number fluctuations σ 2 l = n 2 l − n 2 l of particles in a finite box of linear size l, where n is the particle number. In active matter systems, σ 2 l ∼ n α with α > 1, indicating "giant" number fluctuations (GNF) in comparison to what is expected from the central limit theorem. The exponent α has been used to infer the long range correlations in the system. In two dimensions, for the PP [6, 9, 14], and PR [2, 11] systems, it is now known that α = 1.6, and for AR systems [4, 14, 17] α = 2.0.Consider now an active matter system relaxing to its ordered state from an initial disordered state. As the density order grows with time t, there is an increasing macroscopic length scale L(t) over which there is enhanced clustering. Information about the spatial structures in such a coarsening system can be obtained by studying the spatial density-density correlation function C(r, t) = ρ(0, t)ρ(r, t) where ρ(r, t) is the local density at poin...

Mammalian cortical bone in tension is non-Haversian

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Low-noise phase of a two-dimensional active nematic system

“…In Ref. [62], the relation between the structure factor and the scaling of number fluctuations is addressed numerically in detail. The constraints imposed by rotational symmetry of the model force the scaling of the giant number fluctuations to be modified from the linearized prediction,…”

Active particles in heterogeneous media display new physics

“…It has been predicted that in spatially homogeneous systems, the SPPs in the ordered phase -far away from the band regime -exhibit GNF [28]. This prediction has been observed in simulations of SPP particles, where a value of β ∼ 0.8 was found [38,40]. Here, we are interested in knowing the impact of an heterogeneous environment on the number fluctuations.…”