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Rayleigh–Bénard convection, i.e. the flow of a fluid between two parallel plates that is driven by a temperature gradient, is an idealised set-up to study thermal convection. Of special interest are the statistics of the turbulent temperature field, which we are investigating and comparing for three different geometries, namely convection with periodic horizontal boundary conditions in three and two dimensions as well as convection in a cylindrical vessel, in order to determine the similarities and differences. To this end, we derive an exact evolution equation for the temperature probability density function. Unclosed terms are expressed as conditional averages of velocities and heat diffusion, which are estimated from direct numerical simulations. This framework lets us identify the average behaviour of a fluid particle by revealing the mean evolution of a fluid with different temperatures in different parts of the convection cell. We connect the statistics to the dynamics of Rayleigh–Bénard convection, giving deeper insights into the temperature statistics and transport mechanisms. We find that the average behaviour is described by closed cycles in phase space that reconstruct the typical Rayleigh–Bénard cycle of fluid heating up at the bottom, rising up to the top plate, cooling down and falling again. The detailed behaviour shows subtle differences between the three cases.

We investigate the unfolding pathway of a cytosine-rich DNA structure via molecular dynamics simulations. By the study of the essential dynamics, we are able to identify a hidden complexity in the description of the dynamics in terms of the first two eigenvectors which are used as collective variables. This complexity can be mainly explained by non-Gaussian fluctuations due to contributions arising from the disregarded set of eigenvectors. We introduce the local non-Gaussian parameter as a tool for the detection of hidden complexities. The usage of this parameter allows a fast and reliable investigation for the determination of the important minimal number of eigenvectors which is needed for a sufficient description of molecular unfolding motion.

In glaucoma surgery, fibrotic processes occur, leading to impairment of liquid outflow. Activated fibroblasts are responsible for postoperative scarring. The transforming growth factor-β (TGF-β) pathway plays a key role in fibroblast function, differentiation and proliferation. The aim of this study was the characterization of the fibrotic potential of two subtypes of primary human ocular fibroblasts and the attempt to inhibit fibrotic processes specifically, without impairing cell viability. For fibrosis inhibition we focused on the small molecule pirfenidone, which has been shown to prevent pulmonary fibrosis by the decrease of the expression of TGF-β1, TGF-β2 and TGF-β3 cytokines. For in vitro examinations, isolated human primary fibroblasts from Tenon capsule and human intraconal orbital fat tissues were used. These fibroblast subpopulations were analyzed in terms of the expression of matrix components responsible for postoperative scarring. We concentrated on the expression of collagen I, III, VI and fibronectin. Additionally, we analyzed the expression of α-smooth muscle actin, which serves as a marker for fibrosis and indicates transformation of fibroblasts into myofibroblasts. Gene expression was analyzed by rtPCR and synthesized proteins were examined by immunofluorescence and Western blot methods. Proliferation of fibroblasts under different culture conditions was assessed using BrdU assay. TGF-β1 induced a significant increase of cell proliferation in both cell types. Also the expression of some fibrotic markers was elevated. In contrast, pirfenidone decreased cell proliferation and matrix synthesis in both fibroblast subpopulations. Pirfenidone slightly attenuated TGF-β1 induced expression of fibronectin and α-smooth muscle actin in fibroblast cultures, without impairing cell viability. To summarize, manipulation of the TGF-β signaling pathway by pirfenidone represents a specific antifibrotic approach with no toxic side effects in two human orbital fibroblast subtypes. We presume that pirfenidone is a promising candidate for the treatment of fibrosis following glaucoma surgery.

We are interested in the stability of the localized stationary solutions of a three-component reaction-diffusion system with one activator and two inhibitors. We show that depending on control parameters, solutions in form of moving and breathing localized structures can be observed in the vicinity of the codimension-two bifurcation point. We analyze this situation performing multiple scale perturbation expansion in the vicinity of the bifurcation point and derive a set of order parameter equations, explicitly describing the dynamics of the single localized structure. Numerical simulations are carried out, showing good agreement with the analytical predictions.

This review summarizes the work conducted in the last decade on the fabrication of mesostructured patterns, which have lateral dimensions within the nano- and microscales, over a wafer-scaled size by means of dynamic self-assembly using Langmuir-Blodgett (LB) transfer or dip-coating. First, strategies to form mesostructures from a homogeneous Langmuir monolayer with controlled shape, size, and patterns alignment will be presented, followed by a detailed theoretical explanation of the pattern formation. In addition, the patterning of nanocrystals and other chemicals with LB transfer or other dynamic processes, such as dip-coating, will be summarized.

We investigate the single-point probability density function of the velocity in three-dimensional stationary and decaying homogeneous isotropic turbulence. To this end, we apply the statistical framework of the Lundgren–Monin–Novikov hierarchy combined with conditional averaging, identifying the quantities that determine the shape of the probability density function. In this framework, the conditional averages of the rate of energy dissipation, the velocity diffusion and the pressure gradient with respect to velocity play a key role. Direct numerical simulations of the Navier–Stokes equation are used to complement the theoretical results and assess deviations from Gaussianity.

The paper is devoted to the construction of the superstatistical description for nonequilibrium Markovian systems. It is based on Kirchhoff’s diagram technique and the assumption on the system under consideration to possess a wide variety of cycles with vanishing probability fluxes. The latter feature enables us to introduce equivalence classes called channels within which detailed balance holds individually. Then stationary probability as well as flux distributions are represented as some sums over the channels. The latter construction actually forms the superstatistical description, which, however, deals with a certain superposition of equilibrium subsystems rather than a formal expansion of the nonequilibrium steady state distribution into terms of the Boltzmann type.

We present a numerical investigation of two-dimensional decaying turbulence in the Lagrangian framework. Focusing on single particle statistics, we investigate Lagrangian trajectories in a freely evolving turbulent velocity field. The dynamical evolution of the tracer particles is strongly dominated by the emergence and evolution of coherent structures. For a statistical analysis we focus on the Lagrangian acceleration as a central quantity. For more geometrical aspects we investigate the curvature along the trajectories. We find strong signatures for the self-similar universal behavior.

In this work a method is developed for analyzing time series of periodically driven stochastic systems involving time-delayed feedback. The proposed data-analysis method yields dynamical models in terms of stochastic delay differential equations. On the basis of these dynamical models differential effects of driving forces and time-delayed feedback forces can be identified.

Obwohl Methoden zur Inaktivierung schadlicher Mikroorganismen durch Hitze sowohl bei der Behandlung von Lebensmitteln als auch im pharmazeutisch-medizinischen Bereich seit langer Zeit erfolgreich angewendet werden, sind prazise mathematische Modellierungen derartiger Prozesse erst in neuester Zeit in Entwicklung begriffen. Dies gilt insbesondere fur moglichst „schonende” Verfahren, bei denen Temperaturen und zeitliche Einwirkung gering gehalten werden sollen (Pasteurisierung, Thermisierung). Hier spielt eine quantitative Kontrolle auch uber die Phasen der Erwarmung und Abkuhlung eine Rolle. Beschrieben wird ein solches hinreichend flexibles Modell fur derartige nicht-isotherme Inaktivierungsvorgange. Es wird ein expliziter Algorithmus angegeben, der vom Anwender mittels Microsoft Excel™ implementiert werden kann.

Small scale velocity statistics are measured in different turbulent flows with Reynolds numbers up to 106. The stochastic features of the turbulent signals are investigated by means of Markov processes for the velocity increments evolving along the cascade. The analysis of the validity of the Markovian properties yields to a new small scale coherence length. This coherence length can be seen as analogue to the mean free path length of a Brownian motion as pointed out in [A. Einstein, Ann. Phys. 17 (1905) 549]. For length scales larger than this coherence length the complexity of turbulence can be treated as a Markov process. We present experimental evidence that this Markov–Einstein coherence length scales with Re 1 / 2 and show that it is closely related to the Taylor micro-scale and not to the Kolmogorov dissipation length.

We obtain the exact solution for the Burgers equation with a time dependent forcing, which depends linearly on the spatial coordinate. For the case of a stochastic time dependence an exact expression for the joint probability distribution for the velocity fields at multiple spatial points is obtained. A connection with stretched vortices in hydrodynamic flows is discussed., Comment: 10 pages

Complex, self-regulating systems such as the human heart must process inputs with a broad range of characteristics to generate physiological data and time series. Many of these physiological time series seem to be highly chaotic, represent nonstationary data, and fluctuate in an irregular and complex manner. One hypothesis is that the seemingly chaotic structure of physiological time series arises from external and intrinsic perturbations that push the system away from a homeostatic set point. An alternative hypothesis is that the fluctuations are due, at least in part, to the system's underlying dynamics. In this review, we describe new computational approaches - based on new theoretical concepts - for analyzing physiological time series. We'll show that the application of these methods could potentially lead to a novel diagnostic tool for distinguishing healthy individuals from those with congestive heart failure (CHF)

We introduce a generalization of Obukhov's model [A.M. Obukhov, Adv. Geophys. 6, 113 (1959)] for the description of the joint position-velocity statistics of a single fluid particle in fully developed turbulence. In the presented model the velocity is assumed to undergo a continuous time random walk. This takes into account long time correlations. As a consequence the evolution equation for the joint position-velocity probability distribution is a Fokker-Planck equation with a fractional time derivative. We determine the solution of this equation in the form of an integral transform and derive a relation for arbitrary single time moments. Analytical solutions for the joint probability distribution and its moments are given., Comment: 10 pages

A data driven characterization of time-delayed stochastic systems is proposed in terms of linear delay differential equations and two drift parameters. It is shown how these parameters determine the states of such systems with respect to generalized phase diagrams. This approach allows for a comparison of systems with different parameters as exemplified for two motor control tasks: tracking and force production. © 2005 Elsevier B.V. All rights reserved.

We consider nonextensive systems that are related to the nonextensive entropy proposed by Tsallis and can be described by means of the nonlinear porous medium equation and the nonlinear Fokker–Planck equation proposed by Plastino and Plastino. We show how to determine the degree of nonextensivity of these systems from experimental data. Both transient and stationary cases are addressed.

Time series data from stochastic processes described by the Langevin equation are analyzed. Analysis is based on estimation of the deterministic and random terms of the Langevin equation from data. The terms are presented as fields and inspected visually. Forming a model of the process, the terms are also used to reconstruct the deterministic and stochastic process trajectories. The deterministic term is approximated by an analytical ansatz. The equations obtained by the approximation are used to generate deterministic process trajectories and to study their linear stability. Influence of measurement noise on the estimates is also discussed.

A method for analysis of stochastic processes is presented which makes feasible the extraction of deterministic and random components of process dynamics directly from data. The method is applied to time series from metal cutting. Three regimes of turning are treated: (a) chatter-free cutting, (b) cutting accompanied by a strong, and (c) by a weak chatter. It is shown that the transition from chatter-free cutting to chatter exhibits some properties of the Hopf bifurcation from a stable fixed point to a stable limit cycle. Other possible applications of the method are mentioned.

We present a stochastic analysis of a data set consisting of 1.25 × 107 samples of the local velocity measured in the turbulent region of a round free jet. We find evidence that the statistics of the longitudinal velocity increment v(r) can be described as a Markov process. This new approach to characterize small-scale turbulence leads to a Fokker–Planck equation for the r-evolution of the probability density function (p.d.f.) of v(r). This equation for p(v, r) is completely determined by two coefficients D1(v, r) and D2(v, r) (drift and diffusion coefficient, respectively). It is shown how these coefficients can be estimated directly from the experimental data without using any assumptions or models for the underlying stochastic process. The solutions of the resulting Fokker–Planck equation are compared with experimentally determined probability density functions. It is shown that the Fokker–Planck equation describes the measured p.d.f.(s) correctly, including intermittency effects. Furthermore, knowledge of the Fokker–Planck equation also allows the joint probability density of N increments on N different scales p(v1, r1, …, vN, rN) to be determined.

We consider the joint probability functions f(v3,L3;v2,L2;v1,L1) for the velocity increments v(L) across scale L of a turbulent field measured in the helium gas jet experiment of Chabaud et al. (Chabaud B. et al., Phys. Rev. Lett., 73 (1994) 3227). We show that the conditional probability distribution p(v3,L3|v2,L2;v1,L1), L1 > L2 > L3, becomes independent of v1 and L1 provided L1 − L2 > Lmar, where Lmar is comparable to the crossover scale from the inertial to the viscous subrange. This indicates that the N-point probability distributions f(vN,LN;...v2,L2;v1,L1) may be considered as a stochastic process exhibiting Markovian properties.

This paper comprises numerical and theoretical studies of spatiotemporal patterns in advection-reaction-diffusion systems in which the chemical species interact with the hydrodynamic fluid. Due to the interplay between the two, we obtained the spiral defect chaos in the activator-inhibitor-type model. We formulated the generalized Swift-Hohenberg-type model for this system. Then the evolution of fractal boundaries due to the effect of the strong nonlinearity at the interface of the two chemical species is studied numerically. The purpose of the present paper is to point out that spiral defect chaos, observed in model equations of the extended Swift-Hohenberg equation for low Prandtl number convection, may actually be obtained also in certain advection-reaction-diffusion systems.

We study defect motion in a rotating convection cell. We present numerical results of a generalized Swift-Hohenberg equation, which provides a model description of the vertically averaged three-dimensional (3-D) hydrodynamic equations. Our model includes non-Boussinesq effects. It also accounts for the effects of the Coriolis force induced by a rotation of the fluid layer around its vertical symmetry axis. We show that even a slow rotation well below the Kuppers-Lortz instability causes defect motion perpendicular to the convective rolls. We derive an analytic estimation of that motion.

We derive the order parameter equations for higher instabilities in synergetic systems with continuous symmetry. For the case of a multidimensional Lie-group we show that the state vector can be decomposed into a drift along the group orbit defined by the unstable state vector and a motion transvers to the group orbit.

We propose a novel method to analyze spatiotemporal signals emerging from synergetic systems. By this approach we are able to reconstruct the spatial modes, as well as their dynamic interaction close to instabilities. Our method is an extension of the principal component analysis to the case of nonlinear self-organizing systems. We demonstrate our method by an example of a codimension one instability, apply the algorithm to a simulated Benard instability and present a generalization to bifurcations with several order parameters.

We derive the order parameter equation which describes the evolution of spatio-temporal patterns close to the Benard instability in a rotating large aspect ratio system for high Prandtl number fluids. Since this order parameter equation contains rather complicated nonlinear terms we present a model equation which can be obtained from the order parameter equation by suitable simplification of the nonlinearity. For this model equation we calculate the family of roll solutions and investigate their stability with respect to long scale instabilities and examine the onset of the Kuppers-Lortz instability. Then we present spatiotemporal patterns which are obtained from a numerical evaluation of the model equation.

By symmetry arguments we present the phase diffusion equation for the description of spatially modulated roll patterns of isotropic nonequilibrium systems lacking reflectional symmetry such as high Prandtl number Rayleigh-Benard convection in a fluid layer rotating about a vertical axis.

A statistical analysis of the two-point vorticity statistics in the inverse energy cascade of two-dimensional turbulence is presented in terms of probability density functions (PDFs). Evolution equations for the PDFs are derived in the framework of the Lundgren–Monin–Novikov hierarchy, and the unclosed terms are studied with the help of direct numerical simulations (DNS). Furthermore, the unclosed terms are evaluated in a Gaussian approximation and compared to the DNS results. It turns out that the statistical equations can be interpreted in terms of the dynamics of screened vortices. The two-point statistics is related to the dynamics of two point vortices with screened velocity field, where an effective relative motion of the two point vortices originating from the turbulent surroundings is identified to be a major characteristics of the dynamics underlying the inverse cascade.

We report on spiral type convection patterns obtained from a numerical treatment of a generalized Swift-Hohenberg equation.

We derive the generalized Ginzburg-Landau equation for the case of an oscillatory instability of a spatially homogeneous state in systems whose geometry is characterized by two entirely different length scales. This evolution equation is applied to describe the spatio-temporal behaviour of the onset of convection in binary fluid mixtures in large aspect ratio systems. We obtain time periodic traveling wave motions, quasiperiodic fluid motions with two and more frequencies modulating the intensities of the traveling waves as well as chaotic temporal behaviour.

Traveling wave patterns formed at the oscillatory onset of convection in binary mixtures with free slip and permeable horizontal boundary conditions are theoretically derived by means of an order parameter equation for the case of large aspect ratio systems with circular and rectangular geometry. We obtain in addition to ‘Zipper states’ the so called ‘confined states’, which so far has been observed only experimentally. Our present study of traveling wave patterns in circular systems, which are the first theoretical investigations in this respect, exhibit a characteristic change of behaviour. Close to onset the waves travel in radial direction towards the sidewalls. For higher Rayleigh-numbers the waves are confined to the circular boundary traveling in azimuthal direction. The occurrence of this transition should be confirmed experimentally.

We derive the exact stationary probability distribution for the coupled system of Langevin equationsdtu=ɛu−u s,dts=−γs+d2+F(t).