Statistical State Dynamics: a new perspective on turbulence in shear flow

Traditionally, single realizations of the turbulent state have been the object of study in shear flow turbulence. When a statistical quantity was needed it was obtained from a spatial, temporal or ensemble average of sample realizations of the turbulence. However, there are important advantages to studying the dynamics of the statistical state (the SSD) directly. In highly chaotic systems statistical quantities are often the most useful and the advantage of obtaining these statistics directly from a state variable is obvious. Moreover, quantities such as the probability density function (pdf) are often difficult to obtain accurately by sampling state trajectories even if the pdf is stationary. In the event that the pdf is time dependent, solving directly for the pdf as a state variable is the only alternative. However, perhaps the greatest advantage of the SSD approach is conceptual: adopting this perspective reveals directly the essential cooperative mechanisms among the disparate spatial and temporal scales that underly the turbulent state. While these cooperative mechanisms have distinct manifestation in the dynamics of realizations of turbulence both these cooperative mechanisms and the phenomena associated with them are not amenable to analysis directly through study of realizations as they are through the study of the associated SSD. In this review a selection of example problems in the turbulence of planetary and laboratory flows is examined using recently developed SSD analysis methods in order to illustrate the utility of this approach to the study of turbulence in shear flow.Comment: 27 pages, 18 figures. To appear in the book "Zonal jets: Phenomenology, genesis, physics", Cambridge University Press, edited by B. Galperin and P. L. Rea

Structural Kinetic Modeling of Metabolic Networks

To develop and investigate detailed mathematical models of cellular metabolic processes is one of the primary challenges in systems biology. However, despite considerable advance in the topological analysis of metabolic networks, explicit kinetic modeling based on differential equations is still often severely hampered by inadequate knowledge of the enzyme-kinetic rate laws and their associated parameter values. Here we propose a method that aims to give a detailed and quantitative account of the dynamical capabilities of metabolic systems, without requiring any explicit information about the particular functional form of the rate equations. Our approach is based on constructing a local linear model at each point in parameter space, such that each element of the model is either directly experimentally accessible, or amenable to a straightforward biochemical interpretation. This ensemble of local linear models, encompassing all possible explicit kinetic models, then allows for a systematic statistical exploration of the comprehensive parameter space. The method is applied to two paradigmatic examples: The glycolytic pathway of yeast and a realistic-scale representation of the photosynthetic Calvin cycle.Comment: 14 pages, 8 figures (color

Distributed privacy-preserving network size computation: A system-identification based method

In this study, we propose an algorithm for computing the network size of communicating agents. The algorithm is distributed: a) it does not require a leader selection; b) it only requires local exchange of information, and; c) its design can be implemented using local information only, without any global information about the network. It is privacy-preserving, namely it does not require to propagate identifying labels. This algorithm is based on system identification, and more precisely on the identification of the order of a suitably-constructed discrete-time linear time-invariant system over some finite field. We provide a probabilistic guarantee for any randomly picked node to correctly compute the number of nodes in the network. Moreover, numerical implementation has been taken into account to make the algorithm applicable to networks of hundreds of nodes, and therefore make the algorithm applicable in real-world sensor or robotic networks. We finally illustrate our results in simulation and conclude the paper with discussions on how our technique differs from a previously-known strategy based on statistical inference.Comment: 52nd IEEE Conference on Decision and Control (CDC 2013) (2013

Spontaneous nucleation of structural defects in inhomogeneous ion chains

Structural defects in ion crystals can be formed during a linear quench of the transverse trapping frequency across the mechanical instability from a linear chain to the zigzag structure. The density of defects after the sweep can be conveniently described by the Kibble-Zurek mechanism. In particular, the number of kinks in the zigzag ordering can be derived from a time-dependent Ginzburg-Landau equation for the order parameter, here the zigzag transverse size, under the assumption that the ions are continuously laser cooled. In a linear Paul trap the transition becomes inhomogeneous, being the charge density larger in the center and more rarefied at the edges. During the linear quench the mechanical instability is first crossed in the center of the chain, and a front, at which the mechanical instability is crossed during the quench, is identified which propagates along the chain from the center to the edges. If the velocity of this front is smaller than the sound velocity, the dynamics becomes adiabatic even in the thermodynamic limit and no defect is produced. Otherwise, the nucleation of kinks is reduced with respect to the case in which the charges are homogeneously distributed, leading to a new scaling of the density of kinks with the quenching rate. The analytical predictions are verified numerically by integrating the Langevin equations of motion of the ions, in presence of a time-dependent transverse confinement. We argue that the non-equilibrium dynamics of an ion chain in a Paul trap constitutes an ideal scenario to test the inhomogeneous extension of the Kibble-Zurek mechanism, which lacks experimental evidence to date.Comment: 19 pages, 5 figure

Self-organized escape of oscillator chains in nonlinear potentials

We present the noise free escape of a chain of linearly interacting units from a metastable state over a cubic on-site potential barrier. The underlying dynamics is conservative and purely deterministic. The mutual interplay between nonlinearity and harmonic interactions causes an initially uniform lattice state to become unstable, leading to an energy redistribution with strong localization. As a result a spontaneously emerging localized mode grows into a critical nucleus. By surpassing this transition state, the nonlinear chain manages a self-organized, deterministic barrier crossing. Most strikingly, these noise-free, collective nonlinear escape events proceed generally by far faster than transitions assisted by thermal noise when the ratio between the average energy supplied per unit in the chain and the potential barrier energy assumes small values

Reaction diffusion processes on random and scale-free networks

We study the discrete Gierer-Meinhardt model of reaction-diffusion on three different types of networks: regular, random and scale-free. The model dynamics lead to the formation of stationary Turing patterns in the steady state in certain parameter regions. Some general features of the patterns are studied through numerical simulation. The results for the random and scale-free networks show a marked difference from those in the case of the regular network. The difference may be ascribed to the small world character of the first two types of networks.Comment: 8 pages, 7 figure

Structural defects in ion crystals by quenching the external potential: the inhomogeneous Kibble-Zurek mechanism

The non-equilibrium dynamics of an ion chain in a highly anisotropic trap is studied when the transverse trap frequency is quenched across the value at which the chain undergoes a continuous phase transition from a linear to a zigzag structure. Within Landau theory, an equation for the order parameter, corresponding to the transverse size of the zigzag structure, is determined when the vibrational motion is damped via laser cooling. The number of structural defects produced during a linear quench of the transverse trapping frequency is predicted and verified numerically. It is shown to obey the scaling predicted by the Kibble-Zurek mechanism, when extended to take into account the spatial inhomogeneities of the ion chain in a linear Paul trap.Comment: 5 pages, 3 figure