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Register a new userOn-lattice Vicsek model in confined geometries
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The Vicsek model (Vicsek et al. 1995) is a very popular minimalist model to study active matter with a number of applications to biological systems at different length scales. With its off-lattice implementation and the periodic boundary conditions, it aims at the analysis of bulk behaviour of a limited number of particles. To expand the applicability of the model to further biological systems, we introduce an on-lattice implementation and analyse its behaviour for three different geometries with reflective boundary conditions. For sufficiently fine lattices, the model behaviour does not differ between off-lattice and on-lattice implementation. The reflective boundary conditions introduce an alignment of the particles with the boundary for low levels of noise. Numerical sensitivity analysis of the swarming behaviour results in a detailed characterisation of the Vicsek model for confined geometries with reflective boundary conditions. In a channel geometry, the boundary alignment causes swarms to move along the channel. In a box, the edges act as swarm traps and the trapping shows a discontinuous noise dependence. In a disk geometry, an ordered rotational state arises. This state is well described by a novel order parameter. These results provide the basis for applications of the Vicsek model to biological questions involving large particle numbers in confined environments.
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The well-known Vicsek model describes the dynamics of a flock of self-propelled particles (SPPs). Surprisingly, there is no direct measure of the chaotic behavior of such systems. Here, we discuss the dynamical phase transition present in Vicsek systems in light of the largest Lyapunov exponent (LLE), which is numerically computed by following the dynamical evolution in tangent space for up to one million SPPs. As discontinuities in the neighbor weighting factor hinder the computations, we propose a smooth form of the Vicsek model. We find that there is chaotic behavior in the disordered phase, which supports the claim that the LLE can be useful as an indicator of phase transitions even for this out-of-equilibrium system.
We investigate relations between spatial properties of the free energy and the radius of Gaussian curvature of the underlying curved lattice geometries. For this purpose we derive recurrence relations for the analysis of the free energy normalized per lattice site of various multistate spin models in the thermal equilibrium on distinct non-Euclidean surface lattices of the infinite sizes. Whereas the free energy is calculated numerically by means of the Corner Transfer Matrix Renormalization Group algorithm, the radius of curvature has an analytic expression. Two tasks are considered in this work. First, we search for such a lattice geometry, which minimizes the free energy per site. We conjecture that the only Euclidean flat geometry results in the minimal free energy per site regardless of the spin model. Second, the relations among the free energy, the radius of curvature, and the phase transition temperatures are analyzed. We found out that both the free energy and the phase transition temperature inherit the structure of the lattice geometry and asymptotically approach the profile of the Gaussian radius of curvature. This achievement opens new perspectives in the AdS-CFT correspondence theories.
Magnetic properties of the transverse-field Ising model on curved (hyperbolic) lattices are studied by a tensor product variational formulation that we have generalized for this purpose. First, we identify the quantum phase transition for each hyperbolic lattice by calculating the magnetization. We study the entanglement entropy at the phase transition in order to analyze the correlations of various subsystems located at the center with the rest of the lattice. We confirm that the entanglement entropy satisfies the area law at the phase transition for fixed coordination number, i.e., it scales linearly with the increasing size of the subsystems. On the other hand, the entanglement entropy decreases as power-law with respect to the increasing coordination number.
Collective behavior, both in real biological systems as well as in theoretical models, often displays a rich combination of different kinds of order. A clear-cut and unique definition of phase based on the standard concept of order parameter may therefore be complicated, and made even trickier by the lack of thermodynamic equilibrium. Compression-based entropies have been proved useful in recent years in describing the different phases of out-of-equilibrium systems. Here, we investigate the performance of a compression-based entropy, namely the Computable Information Density (CID), within the Vicsek model of collective motion. Our entropy is defined through a crude coarse-graining of the particle positions, in which the key role of velocities in the model only enters indirectly through the velocity-density coupling. We discover that such entropy is a valid tool in distinguishing the various noise regimes, including the crossover between an aligned and misaligned phase of the velocities, despite the fact that velocities are not used by this entropy. Furthermore, we unveil the subtle role of the time coordinate, unexplored in previous studies on the CID: a new encoding recipe, where space and time locality are both preserved on the same ground, is demonstrated to reduce the CID. Such an improvement is particularly significant when working with partial and/or corrupted data, as it is often the case in real biological experiments.
Motivated by recent experiments on the rod-like virus bacteriophage fd, confined to circular and annular domains, we present a theoretical study of structural transitions in these geometries. Using the continuum theory of nematic liquid crystals, we examine the competition between bulk elasticity and surface anchoring, mediated by the formation of topological defects. We show analytically that bulk defects are unstable with respect to defects sitting at the boundary. Moreover, in case of an annulus, whose topology does not require the presence of topological defects, under weak anchoring conditions we find that nematic textures with boundary defects are stable compared to the defect free configurations. Thus our simple approach, with no fitting parameters, suggests a possible symmetry breaking mechanism responsible for the formation of one-, two- and three-fold textures under annular confinement.
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