Fine structure of distributions and central limit theorem in diffusive billiards

We investigate deterministic diffusion in periodic billiard models, in terms of the convergence of rescaled distributions to the limiting normal distribution required by the central limit theorem; this is stronger than the usual requirement that the mean square displacement grow asymptotically linearly in time. The main model studied is a chaotic Lorentz gas where the central limit theorem has been rigorously proved. We study one-dimensional position and displacement densities describing the time evolution of statistical ensembles in a channel geometry, using a more refined method than histograms. We find a pronounced oscillatory fine structure, and show that this has its origin in the geometry of the billiard domain. This fine structure prevents the rescaled densities from converging pointwise to gaussian densities; however, demodulating them by the fine structure gives new densities which seem to converge uniformly. We give an analytical estimate of the rate of convergence of the original distributions to the limiting normal distribution, based on the analysis of the fine structure, which agrees well with simulation results. We show that using a Maxwellian (gaussian) distribution of velocities in place of unit speed velocities does not affect the growth of the mean square displacement, but changes the limiting shape of the distributions to a non-gaussian one. Using the same methods, we give numerical evidence that a non-chaotic polygonal channel model also obeys the central limit theorem, but with a slower convergence rate.Comment: 16 pages, 19 figures. Accepted for publication in Physical Review E. Some higher quality figures at http://www.maths.warwick.ac.uk/~dsander

Welding, brazing, and soldering handbook

Handbook gives information on the selection and application of welding, brazing, and soldering techniques for joining various metals. Summary descriptions of processes, criteria for process selection, and advantages of different methods are given

Theoretical characterization of a model of aragonite crystal orientation in red abalone nacre

Nacre, commonly known as mother-of-pearl, is a remarkable biomineral that in red abalone consists of layers of 400-nm thick aragonite crystalline tablets confined by organic matrix sheets, with the $(001)$ crystal axes of the aragonite tablets oriented to within $\pm$ 12 degrees from the normal to the layer planes. Recent experiments demonstrate that this orientational order develops over a distance of tens of layers from the prismatic boundary at which nacre formation begins. Our previous simulations of a model in which the order develops because of differential tablet growth rates (oriented tablets growing faster than misoriented ones) yield patterns of tablets that agree qualitatively and quantitatively with the experimental measurements. This paper presents an analytical treatment of this model, focusing on how the dynamical development and eventual degree of order depend on model parameters. Dynamical equations for the probability distributions governing tablet orientations are introduced whose form can be determined from symmetry considerations and for which substantial analytic progress can be made. Numerical simulations are performed to relate the parameters used in the analytic theory to those in the microscopic growth model. The analytic theory demonstrates that the dynamical mechanism is able to achieve a much higher degree of order than naive estimates would indicate.Comment: 20 pages, 3 figure

Thermodynamics and Fractional Fokker-Planck Equations

The relaxation to equilibrium in many systems which show strange kinetics is described by fractional Fokker-Planck equations (FFPEs). These can be considered as phenomenological equations of linear nonequilibrium theory. We show that the FFPEs describe the system whose noise in equilibrium funfills the Nyquist theorem. Moreover, we show that for subdiffusive dynamics the solutions of the corresponding FFPEs are probability densities for all cases where the solutions of normal Fokker-Planck equation (with the same Fokker-Planck operator and with the same initial and boundary conditions) exist. The solutions of the FFPEs for superdiffusive dynamics are not always probability densities. This fact means only that the corresponding kinetic coefficients are incompatible with each other and with the initial conditions

Mesoscopic description of reactions under anomalous diffusion: A case study

Reaction-diffusion equations deliver a versatile tool for the description of reactions in inhomogeneous systems under the assumption that the characteristic reaction scales and the scales of the inhomogeneities in the reactant concentrations separate. In the present work we discuss the possibilities of a generalization of reaction-diffusion equations to the case of anomalous diffusion described by continuous-time random walks with decoupled step length and waiting time probability densities, the first being Gaussian or Levy, the second one being an exponential or a power-law lacking the first moment. We consider a special case of an irreversible or reversible A ->B conversion and show that only in the Markovian case of an exponential waiting time distribution the diffusion- and the reaction-term can be decoupled. In all other cases, the properties of the reaction affect the transport operator, so that the form of the corresponding reaction-anomalous diffusion equations does not closely follow the form of the usual reaction-diffusion equations

Optimization of the Asymptotic Property of Mutual Learning Involving an Integration Mechanism of Ensemble Learning

We propose an optimization method of mutual learning which converges into the identical state of optimum ensemble learning within the framework of on-line learning, and have analyzed its asymptotic property through the statistical mechanics method.The proposed model consists of two learning steps: two students independently learn from a teacher, and then the students learn from each other through the mutual learning. In mutual learning, students learn from each other and the generalization error is improved even if the teacher has not taken part in the mutual learning. However, in the case of different initial overlaps(direction cosine) between teacher and students, a student with a larger initial overlap tends to have a larger generalization error than that of before the mutual learning. To overcome this problem, our proposed optimization method of mutual learning optimizes the step sizes of two students to minimize the asymptotic property of the generalization error. Consequently, the optimized mutual learning converges to a generalization error identical to that of the optimal ensemble learning. In addition, we show the relationship between the optimum step size of the mutual learning and the integration mechanism of the ensemble learning.Comment: 13 pages, 3 figures, submitted to Journal of Physical Society of Japa

DNA bubble dynamics as a quantum Coulomb problem

We study the dynamics of denaturation bubbles in double-stranded DNA on the basis of the Poland-Scheraga model. We demonstrate that the associated Fokker-Planck equation is equivalent to a Coulomb problem. Below the melting temperature the bubble lifetime is associated with the continuum of scattering states of the repulsive Coulomb potential, at the melting temperature the Coulomb potential vanishes and the underlying first exit dynamics exhibits a long time power law tail, above the melting temperature, corresponding to an attractive Coulomb potential, the long time dynamics is controlled by the lowest bound state. Correlations and finite size effects are discussed.Comment: 4 pages, 3 figures, revte

Subdiffusion and weak ergodicity breaking in the presence of a reactive boundary

We derive the boundary condition for a subdiffusive particle interacting with a reactive boundary with finite reaction rate. Molecular crowding conditions, that are found to cause subdiffusion of larger molecules in biological cells, are shown to effect long-tailed distributions with identical exponent for both the unbinding times from the boundary to the bulk and the rebinding times from the bulk. This causes a weak ergodicity breaking: typically, an individual particle either stays bound or remains in the bulk for very long times. We discuss why this may be beneficial for in vivo gene regulation by DNA-binding proteins, whose typical concentrations are nanomolarComment: 4 pages, 1 figure, REVTeX4, accepted to Phys Rev Lett, some typos correcte

Transform of Riccati equation of constant coefficients through fractional procedure

We use a particular fractional generalization of the ordinary differential equations that we apply to the Riccati equation of constant coefficients. By this means the latter is transformed into a modified Riccati equation with the free term expressed as a power of the independent variable which is of the same order as the order of the applied fractional derivative. We provide the solutions of the modified equation and employ the results for the case of the cosmological Riccati equation of FRW barotropic cosmologies that has been recently introduced by FaraoniComment: 7 pages, 2 figure