non-Gaussian Theory Research Articles

An amended approximation of the non-Gaussian probability distribution function

Network models of rubber elasticity are based on the conformational entropy of an idealized chain and mostly motivated by the non-Gaussian statistical theory by Kuhn and Grün. However, the non-Gaussian probability distribution function cannot be expressed in a closed form and requires an approximation. All such approximations applied in the literature demonstrate pronounced inaccuracies in comparison to the analytical solution. The ideal choice of the approximation function depends on a variety of factors, such as the chain parameters or the desired application of the approximation (the probability distribution function itself, the corresponding entropic energy, or the force developed by the chain). In addition, when making a choice regarding the best approximation for a given application, the applied error measure plays a significant role since the approximation that grants, for example, the best maximal relative error is not necessarily the same that provides the best mean absolute error. In the literature, this application-specific evaluation of available approximations is commonly disregarded. In this paper, we evaluate previously proposed approximations on the application-specific basis and develop an approach to derive a family of approximations for the free energy of a polymer chain in a broader range of the number of its chain segments. The analytical method based on the Padé technique delivers an approximation of the non-Gaussian probability distribution function that can be easily tailored depending on the desired application. The proposed approach is capable to provide much stronger predictions in comparison to the Kuhn and Grün model in a wide range of chain segment numbers.

К моделированию динамической эволюции Вселенной под действием энтропийной силы, связанной с модифицированной энтропией Шарма-Миттала

Using the Verlind formalism, the paper considers several scenarios of the evolution of the Friedman-Robertson-Walker Universe, which arise in the framework of entropic cosmology based on the formulated new modification of the Sharma-Mittal entropy. The research, carried out in the framework of non-Gaussian statistical theory, uses several entropies associated with the surface of the horizon of the Universe due to the holographic information stored there. A set of new generalized Friedmann equations is obtained, in which, instead of the cosmological constant, control forces appear based on the Bekenstein-Hawking, Tsallis-Chirto and Barrow entropies, as well as modified Sharma-Mittal and Renyi entropies containing additional nonextensity parameters. The proposed approach, associated with the use of probabilistic nonextensive aspects of the Hubble horizon of the surface of the Universe, meets all the basic requirements for thermodynamic modeling of the dynamic behavior of outer space without involving the concept of dark energy.

HIGH ORDER NONLINEAR WAVE INTERACTIONS FROM DEEP TO FINITE WATER DEPTH WITH BOTTOM TOPOGRAPHY CHANGE

In recent years, freak wave/rouge wave has become an important problem in science and engineering. Modulational instability is considered to be an important factor leading to freak wave in the wave evolution of deep water, and Janssen (2003) defined Benjamin-Feir index (BFI) to reflect it. Mori and Janssen (2006) gave the occurrence probability of freak waves based on a weakly non-Gaussian theory, and distribution of wave height is determined by skewness and kurtosis of surface elevation to a considerable extent in deep water. According to observational record, freak wave has not only been found in deep water in the ocean, but also been observed in shallow water and coastal areas. In the process of water wave entering continental shelf, water depth is changing with mild slope after a long distance propagation. This study focus on investigating how water depth affect skewness and kurtosis in the high order nonlinear wave evolution from deep water to finite water depth in two-dimension.Recorded Presentation from the vICCE (YouTube Link): https://youtu.be/a8LiJvXWRrw

Improved approximations of non-Gaussian probability, force, and energy of a single polymer chain.

The mechanical behavior of polymers such as their probability density function (PDF), strain energy, and entropic force, has long been described by the non-Gaussian statistical model. Non-Gaussian models are often approximated by the Kuhn-Grün (KG) distribution function, which is derived from the first-order approximation of the complex Rayleigh's exact Fourier integral distribution. The KG function is widely accepted in polymer physics, where the non-Gaussian theory is often used to describe energy of the chains with various flexibility ratios. However, the KG function is shown to be relevant only for long chains and becomes extremely inaccurate for chains with fewer than 40 segments. In comparison to the KG model, other approximations of the non-Gaussian statistical model are often less accurate, and those with higher accuracy are usually too complex to be implemented in large-scale simulations. Here a new accurate approximation of the non-Gaussian PDF, entropic force, and strain energy of a single chain subsequently is developed to describe the mechanics of a polymer chain. With a similar level of complexity, the presented approximations of non-Gaussian PDF, strain energy, and entropic force are at least 10 times more accurate than KG approximations and thus are an excellent alternative option to be used in micromechanical constitutive models.

Prediction of Soil Formation as a Function of Age Using the Percolation Theory Approach

Recent modeling and comparison with field results showed that soil formation by chemical weathering, from bedrock or unconsolidated material, is limited largely by solute transport. Chemical weathering rates are proportional to solute velocities. Nonreactive solute transport described by non-Gaussian transport theory appears compatible with soil formation rates. This change in understanding opens new possibilities for predicting soil production and depth across orders of magnitude of time scales. Percolation theory for modeling the evolution of soil depth and production was applied to new and published data for alpine and Mediterranean soils. The first goal was to check whether the empirical data conform to the theory. Secondly we analyzed discrepancies between theory and observation to find out if the theory is incomplete, if modifications of existing experimental procedures are needed and what parameters might be estimated improperly. Not all input parameters required for current theoretical formulations (particle size, erosion and infiltration rates) are collected routinely in the field; thus, theory must address how to find these quantities from existing climate and soil data repositories, which implicitly introduces some uncertainties. Existing results for soil texture, typically reported at relevant field sites, had to be transformed to results for a median particle size, d50, a specific theoretical input parameter. The modeling tracked reasonably well the evolution of the alpine and Mediterranean soils. For the Alpine sites we found, however, that we consistently overestimated soil depths by approximately 45%. Particularly during early soil formation, chemical weathering is more severely limited by reaction kinetics than by solute transport. The kinetic limitation of mineral weathering can affect the system until 1kyr to a maximum of 10kyr of soil evolution. Thereafter, solute transport seems dominant. The trend and scatter of soil depth evolution is well captured, particularly for Mediterranean soils. We assume that some neglected processes, such as bioturbation, tree throw, and land use change contributed to local reorganization of the soil and thus to some differences to the model. Nonetheless, the model is able to generate soil depth and confirms decreasing production rates with age. A steady state for soils is not reached before about 100 kyr.

Edwards field theory for glasses and granular matter

A minimal description of the inherent states of amorphous solids is presented. Using field theory, applicable when a system is probed at long length scales, it is shown that athermal amorphous solids have long-range correlations in their stresses, as recently observed in supercooled liquids, colloids, and granular matter. Explicit predictions for the correlators are presented, in both 2D and 3D, in excellent agreement with simulation data on supercooled liquids. It is shown that when applied to solids with strictly repulsive interactions, the simplest, na\"ive theory leads to a paradox. This paradox is resolved, and it is shown that a nontrivial, non-Gaussian theory is necessary for such materials. Modifications to the correlators are shown, at the saddle-point level. In all cases, `equations of state' relating fluctuations to imposed stresses are derived, as well as field equations that fix the spatial structure of stresses in arbitrary geometries. A new holographic quantity in 3D amorphous systems is identified.

Electrode redox reactions with polarizable molecules.

A theory of redox reactions involving electron transfer between a metal electrode and a polarizable molecule in solution is formulated. Both the existence of molecular polarizability and its ability to change due to electron transfer distinguish this problem from classical theories of interfacial electrochemistry. When the polarizability is different between the oxidized and reduced states, the statistics of thermal fluctuations driving the reactant over the activation barrier becomes non-Gaussian. The problem of electron transfer is formulated as crossing of two non-parabolic free energy surfaces. An analytical solution for these free energy surfaces is provided and the activation barrier of electrode electron transfer is given in terms of two reorganization energies corresponding to the oxidized and reduced states of the molecule in solution. The new non-Gaussian theory is, therefore, based on two theory parameters in contrast to one-parameter Marcus formulation for electrode reactions. The theory, which is consistent with the Nernst equation, predicts asymmetry between the cathodic and anodic branches of the electrode current. They show different slopes at small electrode overpotentials and become curved at larger overpotentials. However, the curvature of the Tafel plot is reduced compared to the Marcus-Hush model and approaches the empirical Butler-Volmer form with different transfer coefficients for the anodic and cathodic currents.

On the determination of the crystal-vapor surface free energy, and why a Gaussian expression can be accurate for a system far from Gaussian

The crystal-vapor surface free energy γ is an important physical parameter governing physical processes, such as wetting and adhesion. We explore exact and approximate routes to calculate γ based on cleaving an intact crystal into non-interacting sub-systems with crystal-vapor interfaces. We do this by turning off the interactions, ΔV, between the sub-systems. Using the soft-core scheme for turning off ΔV, we find that the free energy varies smoothly with the coupling parameter λ, and a single thermodynamic integration yields the exact γ. We generate another exact method, and a cumulant expansion for γ by expressing the surface free energy in terms of an average of e(-βΔV) in the intact crystal. The second cumulant, or Gaussian approximation for γ is surprisingly accurate in most situations, even though we find that the underlying probability distribution for ΔV is clearly not Gaussian. We account for this fact by developing a non-Gaussian theory for γ and find that the difference between the non-Gaussian and Gaussian expressions for γ consist of terms that are negligible in many situations. Exact and approximate methods are applied to the (111) surface of a Lennard-Jones crystal and are also tested for more complex molecular solids, the surface of octane and nonadecane. Alkane surfaces were chosen for study because their crystal-vapor surface free energy has been of particular interest for understanding surface freezing in these systems.

Fluctuation theory of Rashba Fermi gases: Gaussian and beyond

Fermi gases with generalized Rashba spin orbit coupling inducedby a synthetic gauge field have the potential of realizing many interesting states such as rashbon condensates and topological phases. Here we develop a fluctuation theory of such systems and demonstrate that beyond-Gaussian effects are essential to capture the physics of such systems. We obtain their phase diagram by constructing an approximate non-Gaussian theory. We conclusively establish that spin-orbit coupling can enhance the exponentially small transition temperature ($T_c$) of a weakly attracting superfluid to the order of Fermi temperature, paving a pathway towards high $T_c$ superfluids.

A simple and accurate approximation of the inverse Langevin function

The inverse Langevin function cannot be represented in an explicit form and requires an approximation by a series, a non-rational or a rational function as for example by a Pade approximation. In the current paper, an analytical method based on the Pade technique and the multiple point interpolation is presented for the inverse Langevin function. Thus, a new simple and accurate approximation of the inverse Langevin function is obtained. It might be advantageous, for example, for non-Gaussian statistical theory of rubber elasticity where the inverse Langevin function plays an important role.

On a molecular statistical basis for Ogden's model of rubber elasticity

In this paper, a link is established between the statistical theory of long chain molecules and Ogden's phenomenological model of rubber elasticity. It has been shown by several authors in the past that many invariant-based phenomenological models for rubber-like materials are related to the classical statistical theories. The essential means to reach this reconciliation were methods to account for a non-affine deformation of polymer chains in the network, appropriate techniques to calculate their averaged response, and an approximation of the inverse Langevin function appearing in the non-Gaussian statistical theory. It is shown in this paper that the very same approach, if appropriately implemented, allows to express the strain-energy function of Ogden's material in terms of physical constants characterising the polymer chain and network, together with few additional parameters that account for the non-affine deformation of the polymer chains. Particularly, it is shown that Ogden's model can be represented as a non-affine non-Gaussian 3-chain model with topological constraints.

Elastic Behavior of Polymer Chains

The elastic behavior of the polymer chain was investigated in a three-dimensional off-lattice model. We sample more than 109 conformations of each kind of polymer chain by using a Monte Carlo algorithm, then analyze them with the non-Gaussian theory of rubberlike elasticity, and end with a statistical study. Through observing the effect of the chain flexibility and the stretching ratio on the mean-square end-to-end distance, the average energy, the average Helmholtz free energy, the elastic force, the contribution of energy to the elastic force, and the entropy contribution to elastic force of the polymer chain, we find that a rigid polymer chain is much easier to stretch than a flexible polymer chain. Also, a rigid polymer chain will become difficult to stretch only at a quite high stretching ratio because of the effect of the entropy contribution. These results of our simulation calculation may explain some of the macroscopic phenomena of polymer and biomacromolecular elasticity.

A statistical geoacoustic inversion scheme based on a modified radial basis functions neural network

This paper addresses the task of recovering the geoacoustic parameters of a shallow-water environment using measurements of the acoustic field due to a known source and a neural network based inversion process. First, a novel efficient "observable" of the acoustic signal is proposed, which represents the signal in accordance with the recoverable parameters. Motivated by recent studies in non-Gaussian statistical theory, the observable is defined as a set of estimated model parameters of the alpha-stable distributions, which fit the marginal statistics of the wavelet subband coefficients, obtained after the transformation of the original signal via a one-dimensional wavelet decomposition. Following the modeling process to extract the observables as features, a radial basis functions neural network is employed to approximate the vector function that takes as input the observables and gives as output the corresponding set of environmental parameters. The performance of the proposed approach in recovering the sound speed and density in the substrate of a typical shallow-water environment is evaluated using a database of synthetic acoustic signals, generated by means of a normal-mode acoustic propagation algorithm.

Generalized Concept of the Reinforcement of Elastomers. Part 1: Carbon Black Reinforcement of Rubbers

Abstract The author had a detailed discussion about the new interface model and the concepts proposed by the author concerning the carbon black reinforcement of rubbers, with additional experiments and calculations, and came to the final conclusion. Fundamentally, the changeable and deformable characters of the SH layer in the bound rubber produce the typical phenomena in the carbon black-filled rubbers. The super-networks are constructed through the process of molecular sliding and orientation in the SH layer under large extension, which generate the stress upturn and the great tensile strength in the filled rubber. Both the large scale of molecular slippage and the very large stress increase take place simultaneously in the SH layer. The former occurs for regulating the molecular length between carbon particles and the latter results from the limited extensibility of molecules between carbon particles. Thus, the stress-strain relation of the carbon-filled rubber is governed by the distance between adjacent carbon particles, whose role is similar to the molecular length between crosslinks in the non-Gaussian theory. The very large hysteresis loop observed in the carbon black-filled rubber results from the buckling of the super-network in unloading, whose contribution becomes 40% when the carbon content is 60 phr. Very recently, the above model and the concepts have been verified gradually by other researchers as well.

On Kurtosis and Occurrence Probability of Freak Waves

Abstract Based on a weakly non-Gaussian theory, the occurrence probability of freak waves is formulated in terms of the number of waves in a time series and the surface elevation kurtosis. Finite kurtosis gives rise to a significant enhancement of freak wave generation in comparison with the linear narrowbanded wave theory. For a fixed number of waves, the estimated amplification ratio of freak wave occurrence due to the deviation from the Gaussian theory is 50%–300%. The results of the theory are compared with laboratory and field data.

Asymptotic freedom, non-Gaussian perturbation theory, and the application of renormalization group theory to isotropic turbulence

A model equation with an extended range of asymptotic freedom is used as the basis of a non-Gaussian perturbation expansion in powers of the control parameter of the conditional average. Re-expansion in the local Reynolds number allows the systematic rederivation of an earlier, heuristic theory [W. D. McComb and A. G. Watt, Phys. Rev. Lett. 65, 3281 (1990)].

Mechanical behavior of amorphous polymers in shear

Based on the non-equilibrium thermodynamic theory, a new thermo-viscoelastic constitutive model for an incompressible material is proposed. This model can be considered as a kind of generalization of the non-Gaussian network theory in rubber elasticity to include the viscous and the thermal effects. A set of second rank tensorial internal variables was introduced, and in order to adequately describe the evolution of these internal variables, a new expression of the Helmholtz free energy was suggested. The mechanical behavior of the thermo-viscoelastic material under simple shear deformation was studied, and the “viscous dissipation induced” anisotropy due to the change of orientation distribution of molecular chains was examined. Influences of strain rate and thermal softening produced by the viscous dissipation on the shear stress were also discussed. Finally, the model predictions were compared with the experimental results performed by G' Sell et al., thus the validity of the proposed model is verified.

Void growth and cavitation in nonlinear viscoelastic solids

This paper discusses the growth of a pre-existing void in a nonlinear viscoelastic material subjected to remote hydrostatic tensions with different loading rates. The constitutive relation of this viscoelastic material is the one recently proposed by the present authors, which may be considered as a generalization of the non-Gaussian statistical theory in rubber elasticity. As the first order approximation, the above constitutive relation can be reduced to the “neo-Hookean” type viscoelastic one. Investigations of the influences of the material viscosity and the loading rate on the void growth, or on the cavitation are carried out. It is found that: (1) for generalized “inverse Langevin approximation” nonlinear viscoelastic materials, the cavitation limit does not exist, but there is a certain (remote) stress level at which the void will grow rapidly; (2) for generalized “Gaussian statistics” (neo-Hookean type) viscoelastic materials, the cavitation limit exists, and is an increasing function of the loading rate. The present discussions may be of importance in understanding the material failure process under high triaxial stress.

Wavelet scale analysisof bivariate time series ii:statistical properties for linear processes

Scaling characteristics of stochastic processes can be examined using wavelet cross-co- variances. For jointly stationary but generally non-Gaussian linear processes, the asymptotic properties of the resulting wavelet cross-covariance estimator are derived. The linear processes are assumed to have only a square-summable weight sequence, so that the class of processes includes long-memory processes. The variance of the estimator can in each case be expressed as a spectrum value at zero frequency, and can hence be readily estimated. A simulation experiment is reported which demonstrates the utility of this approach. A comparison of estimated standard deviations of wavelet autocovar-iances and cross-covariances for Beaufort Sea albedo/temperature data under both Gaussian and non-Gaussian assumptions, illustrates the necessity of developing the non-Gaussian theory.

Hysteresis and time-dependent constitutive modelling of filled vulcanised rubber

A constitutive model for the description of the time-dependent behaviour as well as the hysteresis behaviour of filled vulcanised rubber materials is presented. The time dependent behaviour for long-duration loading situations is described by extending Tobolsky's network alteration theory to the large deformation range. In contrast with previous Works. The proposed model employs the non-Gaussian network theory as the point of departure. A chain-rupture function and a chain-growth function govern the long-term time-dependent behaviour. The general reaction laws being applied for these functions are applicable for long-duration situations, while possible flexibility changes can appropriately he described. In order to be able to account or the Mullins effect. the time-dependent model is combined with Simo's damage theory. The damage parameter is related to the previous maximum value of the deviatoric part of the Helmholtz free energy. Model parameter characterisation is performed for a modified SI-rubber by means of a true stress controlled measuring method. The constitutive model is implemented into a FEM program, in order to be able to consider rubber products of arbitrary shape and with arbitrary loading situations.