Scaling and spontaneous symmetry restoring of topological defect dynamics in liquid crystal.

Topological defects-locations of local mismatch of order-are a universal concept playing important roles in diverse systems studied in physics and beyond, including the universe, various condensed matter systems, and recently, even life phenomena. Among these, liquid crystal has been a platform for studying topological defects via visualization, yet it has been a challenge to resolve three-dimensional structures of dynamically evolving singular topological defects. Here, we report a direct confocal observation of nematic liquid crystalline defect lines, called disclinations, relaxing from an electrically driven turbulent state. We focus in particular on reconnections, characteristic of such line defects. We find a scaling law for in-plane reconnection events, by which the distance between reconnecting disclinations decreases by the square root of time to the reconnection. Moreover, we show that apparently asymmetric dynamics of reconnecting disclinations is actually symmetric in a comoving frame, in marked contrast to the two-dimensional counterpart whose asymmetry is established. We argue, with experimental supports, that this is because of energetically favorable symmetric twist configurations that disclinations take spontaneously, thanks to the topology that allows for rotation of the winding axis. Our work illustrates a general mechanism of such spontaneous symmetry restoring that may apply beyond liquid crystal, which can take place if topologically distinct asymmetric defects in lower dimensions become homeomorphic in higher dimensions and if the symmetric intermediate is energetically favorable.

Active colloid with externally induced periodic bipolar motility and its cooperative motion.

Active matter physics has been developed with various types of self-propelled particles, including those with polar and bipolar motility and beyond. However, the bipolar motions experimentally realized so far have been either random along the axis or periodic at intrinsic frequencies. Here we report another kind of bipolar active particles, whose periodic bipolar self-propulsion is set externally at a controllable frequency. We used Quincke rollers-dielectric particles suspended in a conducting liquid driven by an electric field-under an AC electric field instead of the usually used DC field. Reciprocating motion of a single particle at the external frequency was observed experimentally and characterized theoretically as stable periodic motion. Experimentally, we observed not only the reciprocating motion but also non-trivial active Brownian particle (ABP)-like persistent motion in a long time scale. This resulted in a Lorentzian spectrum around zero frequency, which is not accounted for by a simple extension of the conventional model of Quincke rollers to the AC field. It was found that ABP-like motion can be reproduced by considering the top-bottom asymmetry in the experimental system. Moreover, we found a rotational diffusion coefficient much larger than the thermal one, as also reported in previous experiments, which may have resulted from roughness of the electrode surface. We also found self-organized formation of small clusters, such as doublets and triplets, and characterized cooperative motion of particles therein. The AC Quincke rollers reported here may serve as a model experimental system of bipolar active matter, which appears to deserve further investigations.

Tilt-induced polar order and topological defects in growing bacterial populations.

Rod-shaped bacteria, such as Escherichia coli, commonly live forming mounded colonies. They initially grow two-dimensionally on a surface and finally achieve three-dimensional growth. While it was recently reported that three-dimensional growth is promoted by topological defects of winding number +1/2 in populations of motile bacteria, how cellular alignment plays a role in nonmotile cases is largely unknown. Here, we investigate the relevance of topological defects in colony formation processes of nonmotile E. coli populations, and found that both ±1/2 topological defects contribute to the three-dimensional growth. Analyzing the cell flow in the bottom layer of the colony, we observe that +1/2 defects attract cells and -1/2 defects repel cells, in agreement with previous studies on motile cells, in the initial stage of the colony growth. However, later, cells gradually flow toward -1/2 defects as well, exhibiting a sharp contrast to the existing knowledge. By investigating three-dimensional cell orientations by confocal microscopy, we find that vertical tilting of cells is promoted near the defects. Crucially, this leads to the emergence of a polar order in the otherwise nematic two-dimensional cell orientation. We extend the theory of active nematics by incorporating this polar order and the vertical tilting, which successfully explains the influx toward -1/2 defects in terms of a polarity-induced force. Our work reveals that three-dimensional cell orientations may result in qualitative changes in properties of active nematics, especially those of topological defects, which may be generically relevant in active matter systems driven by cellular growth instead of self-propulsion.

Initial perturbation matters: Implications of geometry-dependent universal Kardar-Parisi-Zhang statistics for spatiotemporal chaos.

Infinitesimal perturbations in various systems showing spatiotemporal chaos (STC) evolve following the power laws of the Kardar-Parisi-Zhang (KPZ) universality class. While universal properties beyond the power-law exponents, such as distributions and correlations and their geometry dependence, are established for random growth and related KPZ systems, the validity of these findings to deterministic chaotic perturbations is unknown. Here, we fill this gap between stochastic KPZ systems and deterministic STC perturbations by conducting extensive simulations of a prototypical STC system, namely, the logistic coupled map lattice. We show that the perturbation interfaces, defined by the logarithm of the modulus of the perturbation vector components, exhibit the universal, geometry-dependent statistical laws of the KPZ class despite the deterministic nature of STC. We demonstrate that KPZ statistics for three established geometries arise for different initial profiles of the perturbation, namely, point (local), uniform, and "pseudo-stationary" initial perturbations, the last being the statistically stationary state of KPZ interfaces given independently of the Lyapunov vector. This geometry dependence lasts until the KPZ correlation length becomes comparable to the system size. Thereafter, perturbation vectors converge to the unique Lyapunov vector, showing characteristic meandering, coalescence, and annihilation of borders of piece-wise regions that remain different from the Lyapunov vector. Our work implies that the KPZ universality for stochastic systems generally characterizes deterministic STC perturbations, providing new insights for STC, such as the universal dependence on initial perturbation and beyond.

Phase-ordering kinetics in the Allen-Cahn (Model A) class: Universal aspects elucidated by electrically induced transition in liquid crystals.

The two-dimensional (2D) Ising model is the statistical physics textbook example for phase transitions and their kinetics. Quenched through the Curie point with Glauber rates, the late-time description of the ferromagnetic domain coarsening finds its place at the scalar sector of the Allen-Cahn (or Model A) class, which encompasses phase-ordering kinetics endowed with a nonconserved order parameter. Resisting exact results sought for theoreticians since Lifshitz's first account in 1962, the central quantities of 2D Model A-most scaling exponents and correlation functions-remain known up to approximate theories whose disparate outcomes urge experimental assessment. Here we perform such assessment based on a comprehensive study of the coarsening of 2D twisted nematic liquid crystals whose kinetics is induced by a superfast electrical switching from a spatiotemporally chaotic (disordered) state to a two-phase concurrent, equilibrium one. Tracking the dynamics via optical microscopy, we first show the sharp evidence of well-established Model A aspects, such as the dynamic exponent z=2 and the dynamic scaling hypothesis, to then move forward. We confirm the Bray-Humayun theory for Porod's regime describing intradomain length scales of the two-point spatial correlators and show that their nontrivial decay beyond the Porod's scale can be captured in a free-from-parameter fashion by Gaussian theories, namely the Ohta-Jasnow-Kawasaki (OJK) and Mazenko theories. Regarding time-related statistics, we corroborate the aging hypothesis in Model A systems, which includes the collapse of two-time correlators into a master curve whose format is, actually, best accounted for by a solution of the local scaling invariance theory: the same solution that fits the 2D nonconserved Ising model correlator along with the Fisher-Huse conjecture. We also suggest the true value for the local persistence exponent in Model A class, in disfavor of the exact outcome for the diffusion and OJK equations. Finally, we observe a fractal morphology for persistence clusters and extract their universal dimension. Given its accuracy and possibilities, this experimental setup may work as a prototype to address further universality issues in the realm of nonequilibrium systems.

Direct Evidence for Universal Statistics of Stationary Kardar-Parisi-Zhang Interfaces.

The nonequilibrium steady state of the one-dimensional (1D) Kardar-Parisi-Zhang (KPZ) universality class has been studied in-depth by exact solutions, yet no direct experimental evidence of its characteristic statistical properties has been reported so far. This is arguably because, for an infinitely large system, infinitely long time is needed to reach such a stationary state and also to converge to the predicted universal behavior. Here we circumvent this problem in the experimental system of growing liquid-crystal turbulence, by generating an initial condition that possesses a long-range property expected for the KPZ stationary state. The resulting interface fluctuations clearly show characteristic properties of the 1D stationary KPZ interfaces, including the convergence to the Baik-Rains distribution. We also identify finite-time corrections to the KPZ scaling laws, which turn out to play a major role in the direct test of the stationary KPZ interfaces. This paves the way to explore unsolved properties of the stationary KPZ interfaces experimentally, making possible connections to nonlinear fluctuating hydrodynamics and quantum spin chains as recent studies unveiled relation to the stationary KPZ.

Kardar-Parisi-Zhang Interfaces with Curved Initial Shapes and Variational Formula.

We study fluctuations of interfaces in the Kardar-Parisi-Zhang (KPZ) universality class with curved initial conditions. By simulations of a cluster growth model and experiments with liquid-crystal turbulence, we determine the universal scaling functions that describe the height distribution and the spatial correlation of the interfaces growing outward from a ring. The scaling functions, controlled by a single dimensionless time parameter, show crossover from the statistical properties of the flat interfaces to those of the circular interfaces. Moreover, employing the KPZ variational formula to describe the case of the ring initial condition, we find that the formula, which we numerically evaluate, reproduces the numerical and experimental results precisely without adjustable parameters. This demonstrates that precise numerical evaluation of the variational formula is possible at all, and underlines the practical importance of the formula, which is able to predict the one-point distribution of KPZ interfaces for general initial conditions.

Lane formation and critical coarsening in a model of bacterial competition.

We study competition of two nonmotile bacterial strains in a three-dimensional channel numerically and analyze how their configuration evolves in space and time. We construct a lattice model that takes into account self-replication, mutation, and killing of bacteria. When mutation is not significant, the two strains segregate and form stripe patterns along the channel. The formed lanes are gradually rearranged, with increasing length scales in the two-dimensional cross-sectional plane. We characterize it in terms of coarsening and phase ordering in statistical physics. In particular, for the simple model without mutation and killing, we find logarithmically slow coarsening, which is characteristic of the two-dimensional voter model. With mutation and killing, we find a phase transition from a monopolistic phase, in which lanes are formed and coarsened until the system is eventually dominated by one of the two strains, to an equally mixed and disordered phase without lane structure. Critical behavior at the transition point is also studied and compared with the generalized voter class and the Ising class. These results are accounted for by continuum equations, obtained by applying a mean-field approximation along the channel axis. Our findings indicate relevance of critical coarsening of two-dimensional systems in the problem of bacterial competition within anisotropic three-dimensional geometry.

When fast and slow interfaces grow together: Connection to the half-space problem of the Kardar-Parisi-Zhang class.

We study height fluctuations of interfaces in the (1+1)-dimensional Kardar-Parisi-Zhang (KPZ) class, growing at different speeds in the left half and the right half of space. Carrying out simulations of the discrete polynuclear growth model with two different growth rates, combined with the standard setting for the droplet, flat, and stationary geometries, we find that the fluctuation properties at and near the boundary are described by the KPZ half-space problem developed in the theoretical literature. In particular, in the droplet case, the distribution at the boundary is given by the largest-eigenvalue distribution of random matrices in the Gaussian symplectic ensemble, often called the GSE Tracy-Widom distribution. We also characterize crossover from the full-space statistics to the half-space one, which arises when the difference between the two growth speeds is small.

Measuring Lyapunov exponents of large chaotic systems with global coupling by time series analysis.

Despite the prominent importance of the Lyapunov exponents for characterizing chaos, it still remains a challenge to measure them for large experimental systems, mainly because of the lack of recurrences in time series analysis. Here, we develop a method to overcome this difficulty, valid for highly symmetric systems such as systems with global coupling, for which the dimensionality of recurrence analysis can be reduced drastically. We test our method numerically with two globally coupled systems, namely, logistic maps and limit-cycle oscillators with global coupling. The evaluated exponent values are successfully compared with the true ones obtained by the standard numerical method. We also describe a few techniques to improve the accuracy of the proposed method.

Kardar-Parisi-Zhang Interfaces with Inward Growth.

We study the (1+1)-dimensional Kardar-Parisi-Zhang (KPZ) interfaces growing inward from ring-shaped initial conditions, experimentally and numerically, using growth of a turbulent state in liquid-crystal electroconvection and an off-lattice Eden model, respectively. To realize the ring initial condition experimentally, we introduce a holography-based technique that allows us to design the initial condition arbitrarily. Then, we find that fluctuation properties of ingrowing circular interfaces are distinct from those for the curved or circular KPZ subclass and, instead, are characterized by the flat subclass. More precisely, we find an asymptotic approach to the Tracy-Widom distribution for the Gaussian orthogonal ensemble and the Airy_{1} spatial correlation, as long as time is much shorter than the characteristic time determined by the initial curvature. Near this characteristic time, deviation from the flat KPZ subclass is found, which can be explained in terms of the correlation length and the circumference. Our results indicate that the sign of the initial curvature has a crucial role in determining the universal distribution and correlation functions of the KPZ class.

Memory and Universality in Interface Growth.

Recently, very robust universal properties have been shown to arise in one-dimensional growth processes with local stochastic rules, leading to the Kardar-Parisi-Zhang (KPZ) universality class. Yet it has remained essentially unknown how fluctuations in these systems correlate at different times. Here, we derive quantitative predictions for the universal form of the two-time aging dynamics of growing interfaces and we show from first principles the breaking of ergodicity that the KPZ time evolution exhibits. We provide corroborating experimental observations on a turbulent liquid crystal system, as well as a numerical simulation of the Eden model, and we demonstrate the universality of our predictions. These results may give insight into memory effects in a broader class of far-from-equilibrium systems.

Crossover from growing to stationary interfaces in the Kardar-Parisi-Zhang class.

This Letter reports on how the interfaces in the (1+1)-dimensional Kardar-Parisi-Zhang (KPZ) class undergo, in the course of time, a transition from the flat, growing regime to the stationary one. Simulations of the polynuclear growth model and experiments on turbulent liquid crystal reveal universal functions of the KPZ class governing this transition, which connect the distribution and correlation functions for the growing and stationary regimes. This in particular shows how interfaces realized in experiments and simulations actually approach the stationary regime, which is never attained unless a stationary interface is artificially given as an initial condition.

Hyperbolic decoupling of tangent space and effective dimension of dissipative systems.

We show, using covariant Lyapunov vectors, that the tangent space of spatially extended dissipative systems is split into two hyperbolically decoupled subspaces: one comprising a finite number of frequently entangled "physical" modes, which carry the physically relevant information of the trajectory, and a residual set of strongly decaying "spurious" modes. The decoupling of the physical and spurious subspaces is defined by the absence of tangencies between them and found to take place generally; we find evidence in partial differential equations in one and two spatial dimensions and even in lattices of coupled maps or oscillators. We conjecture that the physical modes may constitute a local linear description of the inertial manifold at any point in the global attractor.

Extensive and subextensive chaos in globally coupled dynamical systems.

Using a combination of analytical and numerical techniques, we show that chaos in globally coupled identical dynamical systems, whether dissipative or Hamiltonian, is both extensive and subextensive: their spectrum of Lyapunov exponents is asymptotically flat (thus extensive) at the value λ(0) given by a single unit forced by the mean field, but sandwiched between subextensive bands containing typically O(logN) exponents whose values vary as λ≃λ(∞)+c/logN with λ(∞)≠λ(0).

Growing interfaces uncover universal fluctuations behind scale invariance.

Stochastic motion of a point - known as Brownian motion - has many successful applications in science, thanks to its scale invariance and consequent universal features such as Gaussian fluctuations. In contrast, the stochastic motion of a line, though it is also scale-invariant and arises in nature as various types of interface growth, is far less understood. The two major missing ingredients are: an experiment that allows a quantitative comparison with theory and an analytic solution of the Kardar-Parisi-Zhang (KPZ) equation, a prototypical equation for describing growing interfaces. Here we solve both problems, showing unprecedented universality beyond the scaling laws. We investigate growing interfaces of liquid-crystal turbulence and find not only universal scaling, but universal distributions of interface positions. They obey the largest-eigenvalue distributions of random matrices and depend on whether the interface is curved or flat, albeit universal in each case. Our exact solution of the KPZ equation provides theoretical explanations.

Chaos in the Hamiltonian mean-field model.

We study the dynamical properties of the canonical ordered phase of the Hamiltonian mean-field (HMF) model, in which N particles, globally coupled via pairwise attractive interactions, form a rotating cluster. Using a combination of numerical and analytical arguments, we first show that the largest Lyapunov exponent remains strictly positive in the infinite-size limit, converging to its asymptotic value with 1/lnN corrections. We then elucidate the scaling laws ruling the behavior of this asymptotic value in the critical region separating the ordered, clustered phase and the disordered phase present at high-energy densities. We also show that the full spectrum of Lyapunov exponents consists of a bulk component converging to the (zero) value taken by a test oscillator forced by the mean field, plus subextensive bands of O(ln N) exponents taking finite values. We finally investigate the robustness of these results by studying a "2D" extension of the HMF model where each particle is endowed with 4 degrees of freedom, thus allowing the emergence of chaos at the level of a single particle. Altogether, these results illustrate the subtle effects of global (or long-range) coupling and the importance of the order in which the infinite-time and infinite-size limits are taken: For an infinite-size HMF system represented by the Vlasov equation, no chaos is present, while chaos exists and subsists for any finite system size.

Universal fluctuations of growing interfaces: evidence in turbulent liquid crystals.

We investigate growing interfaces of topological-defect turbulence in the electroconvection of nematic liquid crystals. The interfaces exhibit self-affine roughening characterized by both spatial and temporal scaling laws of the Kardar-Parisi-Zhang theory in 1+1 dimensions. Moreover, we reveal that the distribution and the two-point correlation of the interface fluctuations are universal ones governed by the largest eigenvalue of random matrices. This provides quantitative experimental evidence of the universality prescribing detailed information of scale-invariant fluctuations.

Lyapunov analysis captures the collective dynamics of large chaotic systems.

We show, using generic globally coupled systems, that the collective dynamics of large chaotic systems is encoded in their Lyapunov spectra: most modes are typically localized on a few degrees of freedom, but some are delocalized, acting collectively on the trajectory. For globally coupled maps, we show, moreover, a quantitative correspondence between the collective modes and some of the so-called Perron-Frobenius dynamics. Our results imply that the conventional definition of extensivity must be changed as soon as collective dynamics sets in.

Hyperbolicity and the effective dimension of spatially extended dissipative systems.

Using covariant Lyapunov vectors, we reveal a split of the tangent space of standard models of one-dimensional dissipative spatiotemporal chaos: A finite extensive set of N dynamically entangled vectors with frequent common tangencies describes all of the physically relevant dynamics and is hyperbolically separated from possibly infinitely many isolated modes representing trivial, exponentially decaying perturbations. We argue that N can be interpreted as the number of effective degrees of freedom, which has to be taken into account in numerical integration and control issues.