Boltzmann power laws

In this paper we propose a model based on the Boltzmann distribution as a mechanism for generating power laws, Boltzmann Power Laws (BPL). Some of these power laws are studied and compared to popular power laws such as ‘1/f’ noise and self-organized criticality (SOC). We will show how, in some cases, these BPLs reproduce behaviors similar to the finite size scaling (FSS) scenario, which is typical of SOC.     

Random Renormalization Group Operators Applied to Stochastic Dynamics

Let X(t) be a fixed point the renormalization group operator (RGO), R _p,r X(t)=X(rt)/r ^p . Scaling laws for the probability density, mean first passage times, finite-size Lyapunov exponents of such fixed points are reviewed in anticipation of more general results. A generalized RGO, $\mathcal{R}_{P,n}$ where P is a random variable, is introduced. Scaling laws associated with these random RGOs (RRGOs) are demonstrated numerically and applied to subdiffusion in bacterial cytoplasm and a process modeling the transition from subdiffusion to classical diffusion. The scaling laws for the RRGO are not simple power laws, but are a weighted average of power laws. The weighting used in the scaling laws can be determined adaptively via Bayes?? theorem.     

Potentials of divergent equations and new additional conservation laws for two-dimensional steady gas flows

An algorithm is constructed that compares a new divergent equation (the additional conservation law) to each of two divergent equations. Both the starting divergent equations themselves and their potentials participate in the additional conservation law, and both the first and second participate symmetrically. A characteristic feature of such additional conservation laws is that not only are the functions of gas-dynamic parameters and their derivatives taken along streamlines but so are the integrals, i.e., the functionals, and their derivatives participate in them. All these facts reveal the physical sense of the topical conservation laws constructed. A comparison with the asymmetric conservation laws constructed previously by the author (Doklady Physics, 2002) is performed. As an example, the relation that connects four additional laws comparable by dimensionality is constructed as an example. This is the conservation law of the momentum and its three analogs. Two laws are asymmetric (from Doklady Physics, 2002), while two others are constructed in this study.     

Statistical limit in a completely integrable system with deterministic initial conditions

The asymptotic behavior of the solutions of the KdV equation in the classical limit with an oscillating nonperiodic initial function u ₀(x) prescribed on the entire x axis is investigated. For such an initial condition, nonlinear oscillations, which become stochastic in the asymptotic limit t→∞, develop in the system. The complete system of conservation laws is formulated in the integral form, and it is demonstrated that this system is equivalent to the spectral density of the discrete levels of the initial problem. The scattering problem is studied for the Schrödinger equation with the initial potential ?u ₀(x), and it is shown that the scattering phase is a uniformly distributed random quantity. A modified method is developed for solving the inverse scattering problem by constructing the maximizer for an N-soliton solution with random initial phases. A one-to-one relation is established between the spectrum of the discrete levels of the initial state of the system and the spectrum established in phase space. It is shown that when the system passes into the stochastic state, all KdV integral conservation laws are satisfied. The first three laws are satisfied exactly, while the remaining laws are satisfied in the WKB approximation, i.e., to within the square of a small dispersion parameter. The concept of a quasisoliton, playing in the stochastic state of the system the role of a standard soliton in the dynamical limit, is introduced. A method is developed for determining the probability density f(u), which is calculated for a specific initial problem. Physically, the problem studied describes a developed one-dimensional turbulent state in dispersion hydrodynamics.     

A Cartesian grid embedded boundary method for hyperbolic conservation laws

We present a second-order Godunov algorithm to solve time-dependent hyperbolic systems of conservation laws on irregular domains. Our approach is based on a formally consistent discretization of the conservation laws on a finite-volume grid obtained from intersecting the domain with a Cartesian grid. We address the small-cell stability problem associated with such methods by hybridizing our conservative discretization with a stable, nonconservative discretization at irregular control volumes, and redistributing the difference in the mass increments to nearby cells in a way that preserves stability and local conservation. The resulting method is second-order accurate in L¹ for smooth problems, and is robust in the presence of large-amplitude discontinuities intersecting the irregular boundary.     

Emergence of the Second Law out of Reversible Dynamics

If one demystifies entropy the second law of thermodynamics comes out as an emergent property entirely based on the simple dynamic mechanical laws that govern the motion and energies of system parts on a micro-scale. The emergence of the second law is illustrated in this paper through the development of a new, very simple and highly efficient technique to compare time-averaged energies in isolated conservative linear large scale dynamical systems. Entropy is replaced by a notion that is much more transparent and more or less dual called ectropy. Ectropy has been introduced before but we further modify the notion of ectropy such that the unit in which it is expressed becomes the unit of energy. The second law of thermodynamics in terms of ectropy states that ectropy decreases with time on a large enough time-scale and has an absolute minimum equal to zero. Zero ectropy corresponds to energy equipartition. Basically we show that by enlarging the dimension of an isolated conservative linear dynamical system and the dimension of the system parts over which we consider time-averaged energy partition, the tendency towards equipartition increases while equipartition is achieved in the limit. This illustrates that the second law is an emergent property of these systems. Finally from our large scale linear dynamic model we clarify Loschmidt’s paradox concerning the irreversible behavior of ectropy obtained from the reversible dynamic laws that govern motion and energy at the micro-scale.     

Light transmittance in turbid media slabs used to model biological tissues; scaling laws

Near infrared (NIR) light propagation through turbid media, such as some polymers or milky solutions, is dominated by multiple scattering. Common equations for light transmittance in slabs of these turbid media include summations of infinite order. These preclude to obtain simple expressions from which information such as scale laws, absorption and scattering coefficient, etc. can be retrieved.In the present work we show, starting from the transmittances for semi-infinite media, that the cumbersome expressions from the theory can be written in a much more compact way, thus allowing, for example, to relate some magnitudes which are, in principle, not easily compared. Moreover, it is shown that it is simple to find out the mentioned scale laws, to fit the shape of the distribution of time of flight of photons and to obtain the momenta, 〈t^k〉 or, eventually 〈l^k〉, being l=(c/n)t, which can be useful for recovering the optical properties of the medium. This is of great interest for the case of biological media because it allows to retrieve simultaneously both, the absorption and scattering coefficient.     

Localization of acoustic vibrational modes in a chain-type lattice

The problem of weak localization of acoustic phonon modes in a nonideal chain-type crystal lattice is studied. An expression is obtained for the diffusion coefficient tensor D. The role of the back coherent scattering processes is investigated. It is established that on account of such processes a substantial renormalization of the diffusion coefficient D can occur in the relatively low-frequency range, where the dispersion laws for phonon modes exhibit quasi-one-dimensional properties.     

Remarks on the electromagnetic waves in a medium with negative ɛ and µ

The question whether main electrodynamics laws accept the existence of materials with negative ? and µ is considered. An obstacle to this is the negativity of the wave energy density W in the theory of such media. It is shown that an attempt to change W sign only worsens the situation since it leads to the conflict of signs in the continuity equation.     

A local discontinuous Galerkin method for directly solving Hamilton–Jacobi equations

In this paper we propose a new local discontinuous Galerkin method to directly solve Hamilton–Jacobi equations. The scheme is a natural extension of the monotone scheme. For the linear case with constant coefficients, the method is equivalent to the discontinuous Galerkin method for conservation laws. Thus, stability and error analysis are obtained under the framework of conservation laws. For both convex and noneconvex Hamiltonian, optimal (k + 1)th order of accuracy for smooth solutions are obtained with piecewise kth order polynomial approximations. The scheme is numerically tested on a variety of one and two dimensional problems. The method works well to capture sharp corners (discontinuous derivatives) and have the solution converges to the viscosity solution.     

Gauge covariant formulation of the generating operator. 2. Systems on homogeneous spaces

A gauge covariant approach to the operator Λ, generating the n-wave type equations on homogeneous spaces is proposed. The operator Λ̃ for the gauge equivalent equations is explicitly constructed. The main results (such as conservation laws, hierarchies of hamiltonian structures, etc.) for the n-wave type equations and their gauge equivalent ones are formulated in terms of Λ and Λ̃ respectively.     

Hydrodynamic Limit of a B.G.K. Like Model on Domains with Boundaries and Analysis of Kinetic Boundary Conditions for Scalar Multidimensional Conservation Laws

In this paper we study the hydrodynamic limit of a B.G.K. like kinetic model on domains with boundaries via BV _loc theory. We obtain as a consequence existence results for scalar multidimensional conservation laws with kinetic boundary conditions. We require that the initial and boundary data satisfy the optimal assumptions that they all belong to L ¹∩L ^∞ with the additional regularity assumptions that the initial data are in BV _loc . We also extend our hydrodynamic limit analysis to the case of a generalized kinetic model to account for forces effects and we obtain as a consequence the existence theory for conservation laws with source terms and kinetic boundary conditions.     

General linear response formula for non integrable systems obeying the Vlasov equation

Long-range interacting N-particle systems get trapped into long-living out-of-equilibrium stationary states called quasi-stationary states (QSS). We study here the response to a small external perturbation when such systems are settled into a QSS. In the N → ∞ limit the system is described by the Vlasov equation and QSS are mapped into stable stationary solutions of such equation. We consider this problem in the context of a model that has recently attracted considerable attention, the Hamiltonian mean field (HMF) model. For such a model, stationary inhomogeneous and homogeneous states determine an integrable dynamics in the mean-field effective potential and an action-angle transformation allows one to derive an exact linear response formula. However, such a result would be of limited interest if restricted to the integrable case. In this paper, we show how to derive a general linear response formula which does not use integrability as a requirement. The presence of conservation laws (mass, energy, momentum, etc.) and of further Casimir invariants can be imposed a posteriori. We perform an analysis of the infinite time asymptotics of the response formula for a specific observable, the magnetization in the HMF model, as a result of the application of an external magnetic field, for two stationary stable distributions: the Boltzmann-Gibbs equilibrium distribution and the Fermi-Dirac one. When compared with numerical simulations the predictions of the theory are very good away from the transition energy from inhomogeneous to homogeneous states.     

Composition law of cardinal ordering permutations

In this paper the theorems that determine composition laws both cardinal ordering permutations and their inverses are proven. So, the relative position of points in an hs-periodic orbit is completely known as well as which order those points are visited, no matter how the hs-periodic orbit emerges, be it through a period doubling cascade (s=2ⁿ) of as the h-periodic orbit or a primary window (like saddle-node bifurcation cascade with h=2ⁿ) or a secondary window (the birth of an s-periodic window inside the h-periodic one). Certainly, period doubling cascade orbits are particular cases with h=2 and s=2ⁿ. Both composition laws are also shown in algorithmic way for easy use.     

Randomness, evenness, and Rényi’s index

This paper links together the notion of entropy and the notion of inequality indices—the former is applied in Statistical Physics to measure randomness, and the latter is applied in Economics to measure evenness. We explore the profound similarities between these diametric notions, construct a mathematical transformation between them, and show how randomness can be used to measure evenness-and vice versa. In particular, we devise and study Rényi’s index—a randomness-based measure of evenness with special properties. Rényi’s index is established as an effectual gauge of statistical heterogeneity in the context of general probability laws defined on the positive half-line.     

Wave theory of light leading to the generalized vectorial laws of reflection and refraction

This paper deals with the application of Huygens’ wave theory of light for the derivation of the generalized vectorial laws of reflection and refraction discovered by the author in 2005. The long-running literature falls short of such a theoretical proof of the generalized vectorial laws of reflection and refraction on the basis of wave theory. As such the present work is novel and original. At the same time it also enhances the theoretical foundation of the discovery of the generalized vectorial laws of reflection and refraction there by proving the efficiency and increasing the range of applicability of the wave theory of light as well.     

Self-organized criticality in a block lattice model of the brittle crust

An earthquake model is introduced, in which the brittle crust is treated as a two-dimensional system of many blocks divided by faults, and the mechanical behavior of the faults is described by the Burridge-Knopoff stick-slip model. The coherent system naturally evolves into a self-organized critical state. Some universal scaling laws of seismicity, such as the Gutenberg-Richter law with the b value in agreement with the observational result and the fractal feature of fault patterns, are reproduced. Some ambiguity in simple cellular automata models is also solved.     

Modélisation de séchoirs à tapis. Utilisation des réseaux de neurones

Modelling of convective layer dryers. Using neural networks. Dryer modelling is considered in this paper. A dryer scale approach is implemented in order to write the classical differential equations through parameters such as the heat transfer coefficient or drying kinetics. The behaviour of the dryers is described by a non-linear system which integrates these equations in a transfer network using the finite difference method. The finite difference method is easy to implement, but appears to be too slow for dryer designing. So, in the second part of the study, neural networks are used to model drying process in steady state. When applying neural networks method to the design of dryers, one of the main problems is to find necessary and sufficient inputs so that the neural networks can learn transfers laws. To reduce the problem, each output is defined by a single neural network and non-dimensional numbers are used. The following step deals with the determination of the number of neurones and the minimization of output error for each efficiency (change of training points). Then, neural networks are used to simulate different configurations of dryers. Results are compared with the finite difference method and an industrial application is studied in the last chapter.     

Autowaves in an active two-wire line with exponential current-voltage characteristics

Nonlinear wave processes in two-wire lines containing an active element with an exponential current-voltage characteristic (CVC) similar to that of a p-n junction are investigated. These lines are models of systems that are encountered in various physical and biological applications, such as biological membranes and semiconductor devices. It is shown that such systems may operate in different modes each of which has different dispersion and dissipation properties and, as a consequence, is described by autowave processes of different types. The behavior of a system in all basic modes is analyzed. For each mode, exact solutions to relevant equations are found and their differential conservation laws and intrinsic symmetries are investigated. One of common properties of such equations is the presence of a special superposition principle that describes the discrete structure of excitations in a line that consist of individual elementary excitations. It is shown that autopulses may be generated in such systems.     

Uncertainty quantification for systems of conservation laws

Uncertainty quantification through stochastic spectral methods has been recently applied to several kinds of non-linear stochastic PDEs. In this paper, we introduce a formalism based on kinetic theory to tackle uncertain hyperbolic systems of conservation laws with Polynomial Chaos (PC) methods. The idea is to introduce a new variable, the entropic variable, in bijection with our vector of unknowns, which we develop on the polynomial basis: by performing a Galerkin projection, we obtain a deterministic system of conservation laws. We state several properties of this deterministic system in the case of a general uncertain system of conservation laws. We then apply the method to the case of the inviscid Burgers’ equation with random initial conditions and we present some preliminary results for the Euler system. We systematically compare results from our new approach to results from the stochastic Galerkin method. In the vicinity of discontinuities, the new method bounds the oscillations due to Gibbs phenomenon to a certain range through the entropy of the system without the use of any adaptative random space discretizations. It is found to be more precise than the stochastic Galerkin method for smooth cases but above all for discontinuous cases.