2-fold and 3-fold mixing: why 3-dot-type counterexamples are impossible in one dimension

V.A. Rohlin asked in 1949 whether 2-fold mixing implies 3-fold mixing for a stationary process indexed by Z, and the question remains open today. In 1978, F. Ledrappier exhibited a counterexample to the 2-fold mixing implies 3-fold mixing problem, the so-called "3-dot system", but in the context of stationary random fields indexed by ZxZ. In this work, we first present an attempt to adapt Ledrappier's construction to the one-dimensional case, which finally leads to a stationary process which is 2-fold but not 3-fold mixing conditionally to the sigma-algebra generated by some factor process. Then, using arguments coming from the theory of joinings, we will give some strong obstacles proving that Ledrappier's counterexample can not be fully adapted to one-dimensional stationary processes

Melt-Mixing by Novel Pitched-Tip Kneading Disks in a Co-Rotating Twin-Screw Extruder

Melt-mixing in twin-screw extruders is a key process in the development of polymer composites. Quantifying the mixing performance of kneading elements based on their internal physical processes is a challenging problem. We discuss melt-mixing by novel kneading elements called "pitched-tip kneading disk (ptKD)". The disk-stagger angle and tip angle are the main geometric parameters of the ptKDs. We investigated four typical arrangements of the ptKDs, which are forward and backward disk-staggers combined with forward and backward tips. Numerical simulations under a certain feed rate and screw revolution speed were performed, and the mixing process was investigated using Lagrangian statistics. It was found that the four types had different mixing characteristics, and their mixing processes were explained by the coupling effect of drag flow with the disk staggering and pitched-tip and pressure flows, which are controlled by operational conditions. The use of a pitched-tip effectively to controls the balance of the pressurization and mixing ability

Development of a trench cutting re-mixing deep wall method model test device

The trench cutting re-mixing deep wall (TRD) is a new type of underground waterproof curtain. Mixing uniformity is the key index affecting the efficiency and quality of this method. However, because of many influencing factors, existing theories cannot be used to express the relationship between various factors and mixing uniformity. By analyzing the cutting and mixing process of the TRD method, the main factors affecting the uniformity of the mixing were obtained. A model test device was designed and manufactured, based on Buckingham's pi theorem. The validity of the model test device was verified through a comparative analysis of model and field test results. The model test device was demonstrated to be able to simulate the mixing process of the TRD method. The results provide guidance for promotion and better application of the TRD method

Is turbulent mixing a self convolution process ?

Experimental results for the evolution of the probability distribution function (PDF) of a scalar mixed by a turbulence flow in a channel are presented. The sequence of PDF from an initial skewed distribution to a sharp Gaussian is found to be non universal. The route toward homogeneization depends on the ratio between the cross sections of the dye injector and the channel. In link with this observation, advantages, shortcomings and applicability of models for the PDF evolution based on a self-convolution mechanisms are discussed.Comment: 4 page

Comparison of Swendsen-Wang and Heat-Bath Dynamics

We prove that the spectral gap of the Swendsen-Wang process for the Potts model on graphs with bounded degree is bounded from below by some constant times the spectral gap of any single-spin dynamics. This implies rapid mixing of the Swendsen-Wang process for the two-dimensional Potts model at all temperatures above the critical one, as well as rapid mixing at the critical temperature for the Ising model. After this we introduce a modified version of the Swendsen-Wang algorithm for planar graphs and prove rapid mixing for the two-dimensional Potts models at all non-critical temperatures.Comment: 22 pages, 1 figur

Short-range dependent processes subordinated to the Gaussian may not be strong mixing

There are all kinds of weak dependence. For example, strong mixing. Short-range dependence (SRD) is also a form of weak dependence. It occurs in the context of processes that are subordinated to the Gaussian. Is a SRD process strong mixing if the underlying Gaussian process is long-range dependent? We show that this is not necessarily the case.Comment: 3 page

Chaos, Determinacy and Fractals in Active-Sterile Neutrino Oscillations in the Early Universe

The possibility of light sterile neutrinos allows for the resonant production of lepton number in the early universe through matter-affected neutrino mixing. For a given a mixing of the active and sterile neutrino states it has been found that the lepton number generation process is chaotic and strongly oscillatory. We undertake a new study of this process' sensitivity to initial conditions through the quantum rate equations. We confirm the chaoticity of the process in this solution, and moreover find that the resultant lepton number and the sign of the asymmetry produces a fractal in the parameter space of mass, mixing angle and initial baryon number. This has implications for future searches for sterile neutrinos, where arbitrary high sensitivity could not be determinate in forecasting the lepton number of the universe.Comment: 6 pages, 3 figure

Discrete Element Study Mixing in an Industrial Sized Mixer

The mixing quality as function of the operating time is one major parameter to be considered for the design of industrial sized mixers. Different mixer types with different operating principles can be found for different special tasks, and the question about the proper quantity that quantifies mixing is still open.\ud \ud Computer simulations based on the Discrete Element Method (DEM) provide a close,\ud detailed look inside the mixing device and process and thus a better understanding of the particle flow in the mixer. Therefore such simulations can be used for an improvement of the mixer design or operating conditions.\ud \ud DEM simulations allow the “online”- and “inline”-measurement of the mixing quality over time. But in mixers with a complex design it is not only interesting at which time a certain mixing quality is reached. It is also interesting to analyse which part of the mixing (either location in the device or process conditions) are of special importance due to a strong effect on the mixing process.\ud \ud Therefore, a time- and space-dependent analysis was developed, using several approaches for mixing-quality that can be found in literature. More explicitly, the particle numbers, the number of contacts between different particle-types, and the generalized mean mixing index (GMMI) have been examined. All have their regimes of reasonable use that will be discussed - the most promising approaches will be compared