Session L41: Pattern Formation, Nonlinear Dynamics, Computational Fluid Dynamics

A Thermodynamic Model for Behavioral Intelligence

Recent advances in cosmology and computer science have hinted at a potentially deep connection between intelligence and thermodynamics. Here we attempt to elucidate that connection by showing that a generalization of entropic forces can induce archetypically intelligent behaviors in a variety of classical mechanical systems. These results suggest a simple, but general, thermodynamic model for behavioral intelligence.   

Data collapse and critical dynamics in neuronal avalanche data

The tasks of information processing, computation, and response to stimuli require neural computation to be remarkably flexible and diverse. To optimally satisfy the demands of neural computation, neuronal networks have been hypothesized to operate near a non-equilibrium critical point. In spite of their importance for neural dynamics, experimental evidence for critical dynamics has been primarily limited to power law statistics that can also emerge from non-critical mechanisms. By tracking the firing of large numbers of synaptically connected cortical neurons and comparing the resulting data to the predictions of critical phenomena, we show that cortical tissues in vitro can function near criticality. Among the most striking predictions of critical dynamics is that the mean temporal profiles of avalanches of widely varying durations are quantitatively described by a single universal scaling function (data collapse). We show for the first time that this prediction is confirmed in neuronal networks. We also show that the data have three additional features predicted by critical phenomena: approximate power law distributions of avalanche sizes and durations, samples in subcritical and supercritical phases, and scaling laws between anomalous exponents.   

Self Organized Criticality as a new paradigm of sleep regulation

Humans and animals often exhibit brief awakenings from sleep (arousals), which are traditionally viewed as random disruptions of sleep caused by external stimuli or pathologic perturbations. However, our recent findings show that arousals exhibit complex temporal organization and scale-invariant behavior, characterized by a power-law probability distribution for their durations, while sleep stage durations exhibit exponential behavior. The co-existence of both scale-invariant and exponential processes generated by a single regulatory mechanism has not been observed in physiological systems until now. Such co-existence resembles the dynamical features of non-equilibrium systems exhibiting self-organized criticality (SOC). Our empirical analysis and modeling approaches based on modern concepts from statistical physics indicate that arousals are an integral part of sleep regulation and may be necessary to maintain and regulate healthy sleep by releasing accumulated excitations in the regulatory neuronal networks, following a SOC-type temporal organization.   

Frost nucleation, growth and propagation on a hydrophobic surface

We report experimental results on the condensation of water vapor on a substrate (-9~$^{\circ}$C) at supercooled conditions. The resulting breath figure grows until the liquid to solid phase transition takes place. The frost seeds start to grow by deposition at the expense of neighboring supercooled water drops that evaporate. Sometimes the propagation (due to the growth of the ice) is faster than the evaporation of the drops, hence they transit to the solid state via a percolation mechanism. In this work [1], we analyze the growth of supercooled condensed drops (first stage), the growth of the ice crystals and the evolution of the supercooled water drops (intermediate and late stages). We also consider the liquid - solid front propagation (growth of the frost figure).\\[4pt] [1] J. Guadarrama-Cetina, A. Mongruel, W. Gonz\'alez-Vi\~nas, D. Beysens. In preparation   

Pattern formation in oscillatory fluid flows

Rigid spherical particles in oscillating fluid flows form interesting structures as a result of fluid mediated interactions. Here we show that two spheres under horizontal vibration align themselves at right angles to the oscillation and sit with a gap between them, which scales in a non-classical way with the boundary layer thickness. The details of this behavior have been investigated through experiments and simulations. We then look at a collection of spherical particles, which form chains perpendicular to the direction of oscillation. Comparing experiments and simulations we study the stages of evolution from a dispersed initial configuration to an ordered chain structure. We investigate the details of the interactions and find that the nonlinear hydrodynamic effect of steady streaming is the driving force.   

Dynamic effects induced by renormalization in anisotropic pattern forming systems

The dynamics of patterns in large two-dimensional domains remains a challenge in nonequilibrium phenomena. Often it is addressed through mild extensions of one-dimensional equations. We show that full two-dimensional generalizations of the latter can lead to unexpected dynamic behavior. As an example we consider the anisotropic Kuramoto-Sivashinsky equation, which is a generic model of anisotropic pattern forming systems and has been derived in different instances of thin film dynamics. A rotation of a ripple pattern by 90$^{\circ}$ occurs in the system evolution when nonlinearities are strongly suppressed along one direction. This effect originates in nonlinear parameter renormalization at different rates in the two system dimensions, showing a dynamic interplay between scale invariance and wavelength selection. Potential experimental realizations of this phenomenon are identified. A. Keller, M. Nicoli, S. Facsko, and R. Cuerno, Phys. Rev. E 84, 015202(R) (2011).   

Self-Sustained Front Propagation in Disordered Flow

We generate propagative fronts resulting from a balance between molecular diffusion and non-linear chemical reaction. These fronts behave as solitary waves with a constant velocity and a stationary concentration profile. The interaction between this self-sustained system and a disordered flow leads to complex structures formation. We have performed experiments of the front propagation over a wide range of stochastic flow rates, in porous media. We have determined the structure and the velocity distribution measured along the front. The concentration profile displays salient spatial features such as scaling laws and pattern formation.   

Sidebranching in the Dendritic Crystal Growth of Ammonium Chloride

We report measurements of the dendritic crystal growth of NH$_4$Cl from supersaturated aqueous solution at small supersaturations. Sidebranch growth in this regime is challenging to model well, and the origin of the sidebranches is not fully understood. The early detection of sidebranches requires measurements of small deviations from the smooth steady state shape, but that shape is not well known at the intermediate distances relevant for sidebranch measurements. One model is that sidebranches result from the selective amplification of microscopic noise. We compare measurements of the sidebranch envelope with predictions of the noise-induced sidebranching model of Gonz\'alez-Cinca, Ram\'irez-Piscina, Casademunt, and Hern\'andez-Machado [Phys Rev. E, 63, 051602 (2001)]. We find that the measured amplitude is somewhat larger than predicted, and the shape of the sidebranch envelope is also different. A second model is that sidebranches result from small oscillations of the tip. We have observed no such oscillations, but very small ones can not be ruled out. No measurement of the tip region can be completely free of contamination from early sidebranches, so it can be challenging to distinguish between an oscillating tip and a smooth tip with sidebranches starting nearby.   

Burning invariant manifolds in spatially disordered advection-reaction-diffusion

We introduce burning invariant manifolds (BIMs) which act as barriers to front propagation, similar to the role played by invariant manifolds as barriers to passive transport in two-dimensional flows. We present experimental studies of BIMs in a spatially disordered, time-independent flow. We generate the flow with a magnetohydrodynamic technique that uses a DC current and a disordered pattern of permanent magnets. The velocity field is determined from this flow using particle tracking velocimetry, and reaction fronts are produced using the Ferroin-catalyzed Belousov-Zhabotinsky (BZ) chemical reaction. We use the experimental velocity field and a three-dimensional set of ODEs to predict from theory the location and orientation of BIMs. These predicted BIMs are found to match up well with the propagation barriers observed experimentally in the same flow using the BZ reaction. We explore the nature of BIMs as one-sided barriers, in contrast to invariant manifolds that act as barriers for passive transport in all directions. We also explore the role of projection singularities in the theory and how these singularities affect front behavior.   

Chaotic advection of immiscible fluids

We consider a system of two immiscible fluids advected by a chaotic flow field. A nonequilibrium steady state arises from the competition between the coarsening of the immiscible fluids and the domain bursting caused by the chaotic flow. It has been established that the average domain size in this steady state scales as a inverse power of the Lyapunov exponent. We examine the issue of local structure and look for correlations between the local domain size and the finite-time Lyapunov exponent (FTLE) field. For a variety of chaotic flows, we consistently find the domains to be smallest in regions where the FTLE field is maximal. This raises the possibility of making universal predictions of steady-state characteristics based on Lyapunov analysis of the flow field.   

Hamiltonian traffic dynamics in microfluidic-loop networks

Recent microfluidic experiments revealed that large particles advected in a fluidic loop display long-range hydrodynamic interactions. However, the consequences of such couplings on the traffic dynamics in more complex networks remain poorly understood. In this letter, we focus on the transport of a finite number of particles in one-dimensional loop networks. By combining numerical, theoretical, and experimental efforts, we evidence that this collective process offers a unique example of Hamiltonian dynamics for hydrodynamically interacting particles. In addition, we show that the asymptotic trajectories are necessarily reciprocal despite the microscopic traffic rules explicitly break the time reversal symmetry. We exploit these two remarkable properties to account for the salient features of the effective three-particle interaction induced by the exploration of fluidic loops.   

Intensity statistics of branched flow

Branched flow is a universal phenomenon of particle and wave flows which are subjected to weak, correlated disorder. It has been observed on length scales ranging from a few micrometres, affecting the transport properties of semiconductor devices [1], up to several thousand kilometres, influencing sound propagation through the ocean [2]. It is also responsible for the appearance of large and hazardous freak waves and tsunamis [3]. While the statistics of the number of such branches has recently been calculated [4], the influence on the statistics of the intensity of the waves remains an open question [5]. Here, we show how the classical ray intensity impacts on the wave intensity statistics, and illuminate the role played by the decoherence of the wavefunction.\\ {[}1{]} Topinka et al., Nature 410, 183 (2001), Jura et al., Nat. Phys. 3, 841 (2007)\\ {[}2{]} Wolfson \& Tomsovich, J. Acous. Soc. Am. 109, 2693 (2001)\\ {[}3{]} Berry, Proc. R. Soc. A 463, 3055 (2007); Heller et al., J. Geophys. Res. 113, C09023 (2008)\\ {[}4{]} Metzger, Fleischmann and Geisel, Phys. Rev. Lett. 105, 020601 (2010)\\ {[}5{]} H\"ohmann et al., Phys. Rev. Lett. 104, 093901 (2010), Arecchi et al., Phys. Rev. Lett. 106, 153901 (2011), Ying et al., Nonlinearity 24, R67 (2011)   

Flow past a circular cylinder with momentum injection: Optimal control cylinder design

The primary aim of this work is to suppress vortex shedding behind a circular cylinder by placing two small rotating control cylinders very close to it and hence injecting momentum into the boundary layer. The position and circulation strengths of the control cylinders are the important aspects of our study. Solving the complete Navier-Stokes (NS) equations can be time consuming while identifying the position and circulation strength of the control cylinders. Instead, reduced order models (ROM) can be used to save the computational expenditure associated with solving the complete NS model. Physics-based approaches to reduced order modeling include many of the techniques for modeling and simplification commonly used in fluid dynamics analysis such as potential flow analysis, vortex methods etc. Each of these are approximations to the full NS equations and each can serve as effective ROMs under appropriate conditions. In the present study, we try to achieve potential flow behavior by optimum positioning of the control cylinders and hence potential flow analysis is carried out with different analytical methods like F\"oppl vortex model and conformal mapping techniques. For these optimum values, the analytical solution obtained is compared with the numerical viscous flow simulations.   

Simulating Tablet Dissolution in Complex Hydrodynamic Environment with Lattice-Boltzmann Method

Using the Lattice-Boltzmann method, we developed a 3D mesoscopic model to study the drug-dissolution process in a complex hydrodynamic environment involving spatially varying velocity and shear forces. The results showed turbulent flow in region above tablet, which was also obtained by visualization experiments. The dissolution profiles obtained by incorporating detailed kinetics showed good agreement with case studies from literature. The influence of the paddle speed and the size of the system were studied, and a multicomponent approach was also incorporated. Our results show how that the hydrodynamic environment would affect the dissolution process by changing the local concentration of components near the tablet and by the particle erosion under high fluid velocity. The code was also successfully parallelized so that the simulation of comparatively large system is now possible.