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Register a new userChaotic Properties of Dilute Two and Three Dimensional Random Lorentz Gases I: Equilibrium Systems
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We compute the Lyapunov spectrum and the Kolmogorov-Sinai entropy for a moving particle placed in a dilute, random array of hard disk or hard sphere scatterers - i.e. the dilute Lorentz gas model. This is carried out in two ways: First we use simple kinetic theory arguments to compute the Lyapunov spectrum for both two and three dimensional systems. In order to provide a method that can easily be generalized to non-uniform systems we then use a method based upon extensions of the Lorentz-Boltzmann (LB) equation to include variables that characterize the chaotic behavior of the system. The extended LB equations depend upon the number of dimensions and on whether one is computing positive or negative Lyapunov exponents. In the latter case the extended LB equation is closely related to an anti-Lorentz-Boltzmann equation where the collision operator has the opposite sign from the ordinary LB equation. Finally we compare our results with computer simulations of Dellago and Posch and find very good agreement.
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We use the kinetic theory of gases to compute the Kolmogorov-Sinai entropy per particle for a dilute gas in equilibrium. For an equilibrium system, the KS entropy, h_KS is the sum of all of the positive Lyapunov exponents characterizing the chaotic behavior of the gas. We compute h_KS/N, where N is the number of particles in the gas. This quantity has a density expansion of the form h_KS/N = a u[-ln{tilde{n}} + b + O(tilde{n})], where u is the single-particle collision frequency and tilde{n} is the reduced number density of the gas. The theoretical values for the coefficients a and b are compared with the results of computer simulations, with excellent agreement for a, and less than satisfactory agreement for b. Possible reasons for this difference in b are discussed.
We focus here on the thermodynamic properties of adsorbates formed by two-species $A+B to oslash$ reactions on a one-dimensional infinite lattice with heterogeneous catalytic properties. In our model hard-core $A$ and $B$ particles undergo continuous exchanges with their reservoirs and react when dissimilar species appear at neighboring lattice sites in presence of a catalyst. The latter is modeled by supposing either that randomly chosen bonds in the lattice promote reactions (Model I) or that reactions are activated by randomly chosen lattice sites (Model II). In the case of annealed disorder in spatial distribution of a catalyst we calculate the pressure of the adsorbate by solving three-site (Model I) or four-site (Model II) recursions obeyed by the corresponding averaged grand-canonical partition functions. In the case of quenched disorder, we use two complementary approaches to find $textit{exact}$ expressions for the pressure. The first approach is based on direct combinatorial arguments. In the second approach, we frame the model in terms of random matrices; the pressure is then represented as an averaged logarithm of the trace of a product of random $3 times 3$ matrices -- either uncorrelated (Model I) or sequentially correlated (Model II).
We present an experimental and numerical study of missing-level statistics of chaotic three-dimensional microwave cavities. The nearest-neighbor spacing distribution, the spectral rigidity, and the power spectrum of level fluctuations were investigated. We show that the theoretical approach to a problem of incomplete spectra does not work well when the incompleteness of the spectra is caused by unresolved resonances. In such a case the fraction of missing levels can be evaluated by calculations based on random matrix theory.
Transport and mixing of scalar quantities in fluid flows is ubiquitous in industry and Nature. Turbulent flows promote efficient transport and mixing by their inherent randomness. Laminar flows lack such a natural mixing mechanism and efficient transport is far more challenging. However, laminar flow is essential to many problems and insight into its transport characteristics of great importance. Laminar transport, arguably, is best described by the Lagrangian fluid motion (`advection) and the geometry, topology and coherence of fluid trajectories. Efficient laminar transport being equivalent to `chaotic advection is a key finding of this approach. The Lagrangian framework enables systematic analysis and design of laminar flows. However, the gap between scientific insights into Lagrangian transport and technological applications is formidable primarily for two reasons. First, many studies concern two-dimensional (2D) flows yet the real world is three dimensional (3D). Second, Lagrangian transport is typically investigated for idealised flows yet practical relevance requires studies on realistic 3D flows. The present review aims to stimulate further development and utilisation of know-how on 3D Lagrangian transport and its dissemination to practice. To this end 3D practical flows are categorised into canonical problems. First, to expose the diversity of Lagrangian transport and create awareness of its broad relevance. Second, to enable knowledge transfer both within and between scientific disciplines. Third, to reconcile practical flows with fundamentals on Lagrangian transport and chaotic advection. This may be a first incentive to structurally integrate the `Lagrangian mindset into the analysis and design of 3D practical flows.
Motivated by electronic transport in graphene-like structures, we study the diffusion of a classical point particle in Fermi potentials situated on a triangular lattice. We call this system a soft Lorentz gas, as the hard disks in the conventional periodic Lorentz gas are replaced by soft repulsive scatterers. A thorough computational analysis yields both normal and anomalous (super) diffusion with an extreme sensitivity on model parameters. This is due to an intricate interplay between trapped and ballistic periodic orbits, whose existence is characterized by tongue-like structures in parameter space. These results hold even for small softness showing that diffusion in the paradigmatic hard Lorentz gas is not robust for realistic potentials, where we find an entirely different type of diffusion.
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