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A novel class of random media with designed correlation functions are introduced. When light waves are scattered by this kind of media, the spectral density distributions of the scattered field present concentric rings-like profiles. By adjusting different parameters of the media, the number and the size of concentric rings can be modulated properly. Besides, it is found that the spectral density function of the scattered field and the non-negative definiteness function of the media belong to the same kind of functions. Our results have potential applications in optical trapping and particle manipulation.

We develop the p-adic model of propagation of fluids (e.g., oil or water) in capillary networks in a porous random medium. The hierarchic structure of a system of capillaries is mathematically modeled by endowing trees of capillaries with the structure of an ultrametric space. Considerations are restricted to the case of idealized networks represented by homogeneous p-trees with p branches leaving each vertex, where p >1 is a prime number. Such trees are realized as the fields of p-adic numbers. We introduce and study an inhomogeneous Markov process describing the penetration of fluid into a porous random medium.

This paper presents an analytical model to investigate the effect of randomly distributed micro-organic dust particles on the flame propagation through the combustible mixture by considering the thermal resistance and temperature difference between the gas and the particles. The effect of thermal resistance on flame characteristics is intended by the dimensionless Biot number. It is concluded that the increase in Biot number leads to decrease in the burning velocity as well as flame temperature. Conclusively, flame characteristics for different Lewis numbers and particles radii are reported. Results show that both flame temperature and burning velocity increase with enhancement in the Lewis number. This research leads us to a deep understanding of random combustion of lycopodium particles and more practical prediction of particle combustion compared to the published theoretical studies.

The large deformation response of Ti-6Al-4V structures made through shaped metal deposition is heavily affected by the prior-beta grain structure induced by the manufacturing process. Based on more than 25 tension and shear experiments on individual prior-beta grains, the statistical distribution of the isotropic hardening response of prior-beta grains is identified. In close analogy with Crystal Plasticity Finite Element (CPFE) analysis, Voronoi tesselation is used to generate finite element models of additively-manufactured structures containing multiple prior-beta grains. For each prior-beta grain, a different hardening curve is assigned using a random pull from the truncated Gaussian distribution of Hockett-Sherby type of hardening curves. Similarly, the parameters of the Hosford-Coulomb fracture initiation model are assigned. The comparison of the CPFE simulation results with the experimental results on tension specimens containing multiple prior-beta grains, shows good qualitative and quantitative agreement with regards to the force-displacement response, the surface strain fields, and fracture occurrences. The results demonstrate that the failure of additively-manufactured Ti-6Al-4V components through ductile fracture requires a heterogeneous random medium modeling approach. While the present study focuses on strength and hardening variations, it is noted that further improvements are expected when considering the known texture mismatch between prior-beta grains along with detailed models of the prior-beta grain boundaries.

We study the directed polymer in random media with an attractive line defect at the center of a three-dimensional substrate in d = 3 + 1 . It is found that a small defect strength of ϵ ϵ c does not make the polymer localized, whereas the defect strength of ϵ > ϵ c restricts the polymer fluctuation to a finite region, suggesting that the critical strength ϵ c is nonzero but finite. A localization transition appears at ϵ c , where various critical exponents characterizing the polymer dynamics are also calculated.

We continue the study of the homogenization of coercive non-convex Hamilton–Jacobi equations in random media identifying two general classes of Hamiltonians with very distinct behavior. For the first class there is no homogenization in a particular environment while for the second homogenization takes place in environments with finite range dependence. Motivated by the recent counter-example of Ziliotto, who constructed a coercive but non-convex Hamilton–Jacobi equation with stationary ergodic random potential field for which homogenization does not hold, we show that same happens for coercive Hamiltonians which have a strict saddle-point, a very local property. We also identify, based on the recent work of Armstrong and Cardaliaguet on the homogenization of positively homogeneous random Hamiltonians in environments with finite range dependence, a new general class Hamiltonians, namely equations with uniformly strictly star-shaped sub-level sets, which homogenize.

We use the superposition T-matrix method to compare the far-field scattering matrices generated by spheroidal and spherical volumes of discrete random medium having the same volume and populated by identical spherical particles. Our results fully confirm the robustness of the previously identified coherent and diffuse scattering regimes and associated optical phenomena exhibited by spherical particulate volumes and support their explanation in terms of the interference phenomenon coupled with the order-of-scattering expansion of the far-field Foldy equations. We also show that increasing nonsphericity of particulate volumes causes discernible (albeit less pronounced) optical effects in forward and backscattering directions and explain them in terms of the same interference/multiple-scattering phenomenon.

Fractional calculus is entering the field of nonlinear optics to describe unconventional regimes, as disorder biological media and soft-matter. Here we investigate spatiotemporal modulational instability (MI) in a fractional nonlinear Schrodinger equation. We derive the MI gain spectrum in terms of the Levy indexes and a varying number of spatial dimensions. We show theoretically and numerically that the Levy indexes affect fastest growth frequencies and MI bandwidth and gain. Our results unveil a very rich scenario that may occur in the propagation of ultrashort pulses in random media and metamaterials, and may sustain novel kinds of propagation invariant optical bullets.

Uniform regularity for second order elliptic equations in a highly heterogeneous random medium is concerned. The medium is separated by a random ensemble of simply closed interfaces into a connected sub-region with high conductivity and a disconnected subset with low conductivity. The elliptic equations, whose diffusion coefficients depend on the conductivity, have fast diffusion in the connected sub-region and slow diffusion in the disconnected subset. Without a stationary–ergodic assumption, a uniform Holder estimate in ω , ϵ , λ for the elliptic solutions is derived, where ω is a realization of the random ensemble, ϵ ∈ ( 0 , 1 ] is the length scale of the interfaces, and λ 2 ∈ ( 0 , 1 ] is the conductivity ratio of the disconnected subset to the connected sub-region. Results show that if external sources are small enough in the disconnected subset, the uniform Holder estimate in ω , ϵ , λ holds in the whole domain. If not, it holds only in the connected sub-region. Meanwhile, the elliptic solutions change rapidly in the disconnected subset.

The current paper is devoted to the investigation of wave propagation phenomenon in reaction-diffusion equations with ignition type nonlinearity in time heterogeneous and random media. It is proven that such equations in time heterogeneous media admit transition fronts or generalized traveling wave solutions with time dependent profiles and that such equations in time random media admit generalized traveling wave solutions with random profiles. Important properties of generalized traveling wave solutions, including the boundedness of propagation speeds and the uniform decaying estimates of the propagation fronts, are also obtained.

Preface to a special issue of the International Journal of Solids and Structures. This is not a research article.; International audience; This volume gathers contributions presented at the International Workshop on Physics and Mechanics of Random Structures: From Morphology to Material Properties held on the Ile d’Oléron (Atlantic coast, France), June 17–22, 2018 (Willot and Forest, 2018; Oleron, 2019). This exceptional workshop was organized on the occasion of the (official) retirement of Prof. Dominique Jeulin from his position at Mines ParisTech, where he has been working for more than 30 years to the development of mathematical morphology and physics and mechanics of random media.

The T-matrix method is one of the most versatile and efficient direct computer solvers of the macroscopic Maxwell equations and is widely used for the computation of electromagnetic scattering by single and composite particles, discrete random media, and particles in the vicinity of an interface separating two half-spaces with different refractive indices. This paper is the seventh update to the comprehensive thematic database of peer-reviewed T-matrix publications initiated by us in 2004 and includes relevant publications that have appeared since 2013. It also lists a number of earlier publications overlooked previously.

The possibility for different media to generate scattered field with identical spectral coherence is discussed. It is shown that two random media, with different characters of correlation function, may generate scattered field with identical spectral coherence property. An example of light waves on scattering from Gaussian-Schell model media is discussed, and a condition for identical spectral coherence of the far-zone scattered field is obtained.

Driven by sintering kinetics, stochastically distributed voids exist inevitably in nano‑silver sintered joints to affect their electrical, mechanical and thermal properties, reliability, etc. In this paper, a holistic approach including both void morphology characterization and reconstruction stages is developed for the reconstruction of Stochastically Distributed Void Morphologies (SDVMs) in nano‑silver sintered joints. A set of dispersion parameters, including area A, distance l, angle α, and aspect ratio δ, are firstly proposed for quantitatively characterizing the voids dispersion features of the SDVMs. Then on the basis of the characterization results, a reconstruction approach is developed based on the random medium theory and similarity comparison algorithm. The applicability and validity of the proposed voids dispersion characterization and SDVM reconstruction methods are verified through numerical and experimental case studies, respectively. And effects on the dispersion features of the SDVMs by such as autocorrelation function parameters and the sintering pressure are also investigated via the comparative discussions on the cumulative distribution function (cdf) curves of all dispersion parameters.

In this paper we present the vector radiative transfer theory for a discrete random medium illuminated by a Gaussian beam. The analysis is based on a plane wave spectrum representation for a Gaussian beam and uses an approach developed previously for a discrete random medium illuminated by a plane electromagnetic wave. Specifically, we establish an integral representation for the coherent field, define an approximate coherent field that satisfies the differential equation fulfilled by the coherent field corresponding to a plane electromagnetic wave and matches the Gaussian beam at the interface of the particulate medium, and finally, derive the vector radiative transfer equation. For weakly focused Gaussian beams, the resulting equation is the traditional radiative transfer equation.

We present an algorithm and FORTRAN codes to compute the opposition effects in the reflection of light from a semi-infinite discrete random medium at normal incidence to the boundary of the medium. It is assumed that the medium is sparse enough that the waves propagating between the scatterers are spherical. In this case, the reflection matrix is determined only by contributions of the incoherent (diffuse) and coherent components. When calculating the coherent component, the contribution of the doubly scattered radiation to the reflection matrix is rigorously taken into account, while the contributions of the higher orders are calculated approximately. To be more specific, the multiply scattered radiation coming to some point of the medium “from above” is calculated exactly, but the radiation coming “from below”, approximately. Under this supposition, the solution of the system of integral equations is reduced to that of the system of linear algebraic equations. The matrix of this system is calculated with the recurrent relation, which radically speeds up the computations as compared to the direct procedure. This allows the opposition effect characteristics to be computed rapidly enough so that the codes may be used in interpretation of the remotely measured intensity and polarization of light reflected by different media to estimate, at least at a qualitative level, their properties.

When lidar is used to detect the microphysical properties of clouds and aerosols, it is a spot when the laser reaches the clouds. This is because the lidar has a certain divergence angle, and the location of the lidar is at a certain distance from the clouds. In this paper, a Monte Carlo method for simulating multiple scattering in clouds by lidar is presented, and the influence of the size and shape of the spot on the backscattering of polarized light in clouds is analyzed. From the calculation results, it can be seen that the spot will increase the depolarization ratio of lidar, and the larger of the size of the spot, the larger the depolarization ratio. In addition, depolarization ratio increases with the increase of optical thickness and field of view angle. The methods and conclusions in this paper are of great significance for long-distance detection of discrete random media by laser.

International audience; This paper analyzes the dynamic behavior of a randomly-heterogeneous continuum model of a ballasted railway track through the numerical construction of dispersion curves along the axis of the track. At intermediate frequencies (50-100 Hz), the dispersion curves display a strong decrease of the group velocity. At higher frequencies (above 100 Hz), the wave are strongly localized and cannot propagate along the track. This behavior is coherent with experimental observations reported in the literature on ballasted railway tracks and other granular media. Additional simulations are performed to prove that the localization is not a consequence of the geometry of the track or the impedance mismatch between the ballast and the underlying soil, but is indeed Anderson localization due to the heterogeneity of the ballast.

We experimentally show that the non-separability of polarization and orbital angular momentum present in a light beam remains preserved under scattering through a random medium like rotating ground glass. We verify this by measuring the degree of polarization and observing the intensity distribution of the beam when projected to different polarization states, before as well as after the scattering. We extend our study to the non-maximally non-separable states also.

Light transport and the physical phenomena related to light propagation in random media are very intriguing, they also provide scope for new paradigms of device functionality, most of which remain unexplored. Here we demonstrate, experimentally and by simulation, a novel kind of asymmetric light transmission (diffusion) in a stack of random media (SRM) with graded transport mean free path. The structure is studied in terms of transmission, of photons propagated through and photons generated within the SRM. It is observed that the SRM exhibits asymmetric transmission property with a transmission contrast of 0.25. In addition, it is shown that the SRM works as a perfect optical low-pass filter with a well-defined cutoff wavelength at 580 nm. Further, the photons generated within the SRM found to exhibit functionality similar to an optical diode with a transmission contrast of 0.62. The basis of this functionality is explained in terms of wavelength dependent photon randomization and the graded transport mean free path of SRM. MOE (Min. of Education, S’pore) Accepted version

The expressions for the correlation of intensity fluctuations in the far-zone that occurs in stochastic electromagnetic beams scattered by a random medium are derived within the validity of the first-order Born approximation. Some numerical results are presented to illustrate the influences of different source parameters and scatterer parameters on the normalized correlation of intensity fluctuations of the far-zone scattered field.

In micromechanical modelling, covariograms are often used as a statistical analysis to estimate the features or morphological Representative Elementary Volumes (REV), from representative pictures of the initial microstructure (i.e. static covariogram). The aim of this article is to present an extension of the method to the time evolution of the microstructure, through the idea of dynamic covariogram built up of successive pictures of the microstructure along time. The possible enhanced evolution of covariogram parameters is analyzed first from a general point of view. It is then illustrated on the cavitation damage occurring during decompression in rubbers exposed to diffusing gas.

This paper presents a multiple length-scale asymptotic analysis for transport problems in 1-D diffusive random media. This analysis shows that the Levermore-Pomraning (LP) equations can be adjusted in order to achieve the correct asymptotic behavior. This adjustment appears in the form of a rescaling of the Markov transition functions by a factor $\eta$, which can be chosen in a simple way. Numerical results are given that (i) validate the theoretical predictions; and (ii) show that the adjusted LP equations greatly outperform the standard LP model for this class of transport problems., Comment: V.2.: We have added around 7 pages of new text; three new sets of numerical results; 6 new figures; and two new references [13, 17]. Current version has 18 pages, 10 figures. The changes correspond to an expansion of Sections 3 and 4. Section 3 now has more detailed and complete analysis; Section 4 has new numerical results

Divergence-form operators with stationary random coefficients homogenize over large scales. We investigate the effect of certain perturbations of the medium on the homogenized coefficients. The perturbations that we consider are rare at the local level, but when occurring, have an effect of the same order of magnitude as the initial medium itself. The main result of the paper is a first-order expansion of the homogenized coefficients, as a function of the perturbation parameter., 32 pages. V3: revised introduction, some corrections and clarifications

Clarifies misrepresentations of existing iterative methods in Feng et al.Demonstrates the efficiency of the ITAM method for random media simulation.Addresses compatibility and generality of random media simulation methods. The recent work of Feng et al. (2014) presents a new approach for generation of 2-phase random media using a translation-based stochastic field. The method is useful, elegant, and efficient. The intention of this discussion is to clarify some misrepresentations of prior work as they are presented in terms of the efficiency of iterative methods and the Iterative Translation Approximation Method (ITAM) of Shields et al. (2011) specifically. The ITAM approach is far more efficient than presented and represents a negligibly small increase in expense over the proposed approach. Some issues pertaining to compatibility and generalization of translation fields are also discussed.

The reconstruction of microstructure of biological and synthetic materials is of great importance for theoretical and practical application. The obtaining of this purpose through limited statistical information is a challenge at which some pioneering researchers are focusing their efforts. For the first time, a reconstruction method was used to rebuild bread microstructure by using the information of lineal-path distribution function (LPF). The method was a powerful tool to reconstruct bread microstructure; a perfect match between lineal-path function of reference and reconstructed images were obtained. The reconstructed images progressively improve, compared to the reference, increasing the number of LPFs used during reconstruction. The use of LPFs in different direction did not allowed to obtain a perfect match between the bread and the reconstructed images because, LPF did not contain sufficient microstructure information however, the essential features of bread were captured proving the possibility to reconstruct food microstructure.

We consider scattering of a pulse propagating through a three-dimensional random media and study the shape of the pulse in the parabolic approximation. We show that, similarly to the one-dimensional O’Doherty-Anstey theory, the pulse undergoes a deterministic broadening. Its amplitude decays only algebraically and not exponentially in time, due to the signal low/midrange frequency component. We also argue that the parabolic approximation captures the front evolution (but not the signal away from the front) correctly even in the fully threedimensional situation.

It is shown that the correlation function of scattering potential of random medium can be determined by utilizing a monochromatic Gauss vortex beam radiated from a planar source. The determination can be achieved provided that the scattering angle between the spatially drifted dark spots in scattered field and the initial one is obtained. The principles for the determination as well as the applicable scopes for the proposed method are demonstrated.

In this paper, based on the generalized coated coherent potential approximation method, we derive the mathematical formulae, for the extended effective medium theory, to investigate the optical properties of disordered media composed of core–shell cylinders. The effective indices of such media are obtained in the long-wavelength limit and in the Mie-scattering region. Moreover, we use this method to study optical properties of random media composed of core–shell cylinders with the core layer consisting of epsilon-less-than-one material.

The correspondence principle is widely used to estimate the effective behaviour of non ageing linear viscoelastic composites, as it allows to take advantage of homogenization approaches originally developed in the elastic framework. However, this principle is no longer valid when the phases exhibit ageing. The approach proposed in this paper overcomes this limitation. Operating in the time domain, it allows to use homogenization of random media on composites made up of isotropic ageing linear viscoelastic phases. Several homogenization schemes are revisited to take into account information on morphology, namely the dilute, Mori Tanaka and self-consistent schemes with spherical shapes. This approach is validated against results obtained from either analytical or numerical computations. It is found to be efficient on nowadays computers: the effective relaxation tensor of a matrix-inclusions composite can be obtained in less than one second, whereas numerical FEM homogenization takes more than twelve minutes on a very coarse mesh.

This work is a combined analytical and numerical study of the extreme values of the thermal and elastic fields occurring in a propellant composite material, a granular medium containing dense self-assembled spheroidal grains embedded in a matrix. First, a 3D microtomography image of a representative sample is segmented using morphological operators. Second, the local temperature and heat flux in the quasi-static thermal response is computed numerically, making use of the segmented microstructure and of experimental values of the heat conductivity in each phase. Such fields are readily derived by means of a Fast Fourier Transform method. Emphasis is put on the maximum values of the local fields: even at low contrast of properties between the grain and the matrix, the heat flux patterns is made of hot-spot zones located in-between grains that are close to each other with preferential directions. Third, the local extrema of the fields are investigated in the context of linear elasticity. Finally, analytical approximations are examined at low and high contrast of properties, on a simple hard-core model of cylinders, which is tantamount to a 2D granular microstructure.

A procedure to reconstruct two phase porous media, given the porosity and the two point correlation function of such media is described. The random media are modelled as a discrete valued random field Z(x?), which takes value 1 in regions of pores and 0 in regions of solid phase. The field Z(x?) is obtained by applying a non-linear filter - Nataf's transformation - to a correlated Gaussian random field Y(x?). The two point correlation function RYY of the Gaussian field Y is related to the two point correlation function RZZ of the field Z and can be calculated by expanding the bivariate Gaussian probability in terms of Hermite polynomials. The correlation function of the Gaussian field is decomposed into eigenfunctions and eigenvalues required by the Karhunen-L�eve expansion. The eigenfunctions and eigenvalues are used to generate as many samples of the Gaussian field as required and the discrete field corresponding to each such sample can be obtained by applying the non-linear filter mentioned above. The method was tested by generating a large number of samples of one and two dimensional Debye random media using different porosities and different correlation lengths and the statistics of the ensemble was found to agree favourably with the input data. Also one and two dimensional 'chess board' patterns were reconstructed to see how well the geometry is reconstructed. The one dimensional case was reconstructed very accurately, whereas the two dimensional case, though not very satisfactory, indicates that the method captures some of the essentials of the geometry. The method also has the advantage that it gives an analytical framework for the porous media in terms of the random fields. These random fields could be used for further studies related to porous media. � 2013 Elsevier Ltd.

A methodology is presented for bounding the higher L p norms, 2 ⩽ p ⩽ ∞ , of the local strain inside random media. We present optimal lower bounds that are given in terms of the applied loading and volume fractions for random two phase composites. These bounds provide a means to measure load transfer across length scales relating the excursions of the local fields to applied loads. These results deliver tight upper bounds on the macroscopic strength domains for statistically defined heterogeneous media.

In this paper we study the asymptotic phase space energy distribution of solution of the Schr\"{o}dinger equation with a time-dependent random potential. The random potential is assumed to be with slowly decaying correlations. We show that the Wigner transform of a solution of the random Schr\"{o}dinger equation converges in probability to the solution of a radiative transfer equation. Moreover, we show that this radiative transfer equation with long-range coupling has a regularizing effect on its solutions. Finally, we give an approximation of this equation in term of a fractional Laplacian. The derivations of these results are based on an asymptotic analysis using perturbed-test-functions, martingale techniques, and probabilistic representations., Comment: 33 pages

This paper studies particle propagation in a one-dimensional inhomogeneous medium where the laws of motion are generated by chaotic and deterministic local maps. Assuming that the particle's initial location is random and uniformly distributed, this dynamical system can be reduced to a random walk in a one-dimensional inhomogeneous environment with a forbidden direction. Our main result is a local limit theorem which explains in detail why, in the long run, the random walk's probability mass function does not converge to a Gaussian density, although the corresponding limiting distribution over a coarser diffusive space scale is Gaussian., Comment: 26 pages, 2 figures. To appear in Stochastic Processes and their Applications

This paper deals with the evaluation of the strength domain for non-periodic masonry using a random media micromechanical approach. The generalized plane state formulation is used in order to more accurately describe the masonry behavior and failure criteria proposed in literature. Considering the masonry as a heterogeneous material with random microstructure, the elastic characteristics of the homogenized continuum and the strength domain are evaluated by using the hierarchy theory related to partitions with increasing dimensions through the application of natural and essential boundary conditions with proportionally growing values. An overall failure criterion based on the mean stress state of each phase is introduced. The proposed procedure is validated by comparison with the experimental results obtained with periodic masonry subjected to biaxial stress states recovering the main failure mechanisms. Then the approach is applied to an actual non-periodic masonry introducing peculiar algorithms in order to evaluate strength surfaces and to verify the convergence of the domains obtained through the application of natural and essential boundary conditions with the increasing size of portion dimensions at the mesoscale level.

Following Keller(Proc Symp Appl Math 1962;13:227:46), we classify all theoretical treatments of electromagnetic scattering by a morphologically complex object into first- principle (or "honest" in Keller s terminology) and phenomenological (or "dishonest") categories. This helps us identify, analyze, and dispel several profound misconceptions widespread in the discipline of electromagnetic scattering by solitary particles and discrete random media. Our goal is not to call for a complete renunciation of phenomenological approaches but rather to encourage a critical and careful evaluation of their actual origin, virtues, and limitations. In other words, we do not intend to deter creative thinking in terms of phenomenological short-cuts, but we do want to raise awareness when we stray (often for practical reasons) from the fundamentals. The main results and conclusions are illustrated by numerically-exact data based on direct numerical solutions of the macroscopic Maxwell equations.

The parabolic Anderson problem with a random potential obtained by attaching a long tailed potential around a randomly perturbed lattice is studied. The moment asymptotics of the total mass of the solution is derived.The results show that the total mass of the solution concentrates on a small set in the space of configuration., Comment: 18 pages

We present a method to determine the correlation function of scattering potentials of a random medium. We show that some information about the correlation function of scattering potentials of a random medium can be determined from the far-zone spectral density of the scattered field as the medium is illuminated with a pair of correlated scalar plane light waves. The method has been illustrated by considering an example of a centered symmetric quasi-homogeneous medium.

The concept of the representative volume element (RVE) for softening materials is revised in this contribution. It is demonstrated by means of numerical simulations that there exists a sample which is statistically representative for quasi-brittle materials with random microstructure like concrete. This finding is an important ingredient for homogenization-based multiscale modelling of softening materials.

Dimensionless conductance for light propagating through a random medium with amplification tends to diverge with an increase of gain. This raises questions on the applicability of the localization criteria based on this quantity. To circumvent this problem, we study the properties of the ratio between the transmission (conductance) and the energy stored in the random medium. We argue that the generalized conductance gG ¼ g �ð E0=EÞFconductance normalized by the energy buildup (ratio between energy stored in the medium with gain E to that in the passive system E0)—may be a convenient quantity on which a localization criterion can be built.

The radiative transfer (RT) approach is widely used in applications involving scattering from layered random media with rough interfaces. Although it has been successful in several applications in various disciplines it is well known that this approach involves certain approximations. In this paper these assumptions and approximations are reexamined. To enable this a statistical wave approach is employed to this problem and the governing equations for the first and second moments of the wave functions are derived. A transition is hence made to arrive at a system of equations corresponding to that of the RT approach. It is hence found that more conditions are implicitly involved in the RT approach than generally believed to be sufficient.

A methodology for the analysis of local pore size in an arbitrary porous medium, based on the maximum sphere inscription method, has been developed. The method has been validated for regular cylindrical pores and used for the evaluation of pore size distribution on several random media, both computer-generated and scanned using X-ray microtomography. The local pore radii have then been used for the calculation of effective diffusion across the porous medium, whereby local Knudsen diffusion coefficients were employed depending on the local pore radius. This continuum-based approach was then compared with the results of Knudsen diffusion through the same porous media simulated by the molecular dynamics method. A difference of maximum 40% was obtained, which is deemed acceptable given the significantly faster computing speed and higher versatility of the continuum-based models.

Classical continuum theories are formulated based on the assumption of large scale separation. For scale-coupling problems involving uncertainties, novel multiscale methods are desired. In this study, by employing the generalized variational principles, a Green-function-based multiscale method is formulated to decompose a boundary value problem with random microstructure into a slow scale deterministic problem and a fast scale stochastic one. The slow scale problem corresponds to common engineering practices by smearing out fine-scale microstructures. The fast scale problem evaluates fluctuations due to random microstructures, which is important for scale-coupling systems and particularly failure problems. Two numerical examples are provided at the end.

We consider a random variable X satisfying almost-sure conditions involving G : = 〈 D X , − D L − 1 X 〉 where D X is X ’s Malliavin derivative and L − 1 is the pseudo-inverse of the generator of the Ornstein-Uhlenbeck semigroup. A lower- (resp. upper-) bound condition on G is proved to imply a Gaussian-type lower (resp. upper) bound on the tail P [ X > z ] . Bounds of other natures are also given. A key ingredient is the use of Stein’s lemma, including the explicit form of the solution of Stein’s equation relative to the function 1 x > z , and its relation to G . Another set of comparable results is established, without the use of Stein’s lemma, using instead a formula for the density of a random variable based on G , recently devised by the author and Ivan Nourdin. As an application, via a Mehler-type formula for G , we show that the Brownian polymer in a Gaussian environment, which is white-noise in time and positively correlated in space, has deviations of Gaussian type and a fluctuation exponent χ = 1 / 2 . We also show this exponent remains 1 / 2 after a non-linear transformation of the polymer’s Hamiltonian.

A tunneling mechanism of radiative transfer through a dielectric random medium is revealed applying technique of Dyson and Bethe–Salpeter equations for electromagnetic wave multiple scattering by medium inhomogeneities (scatterers) with near fields effects in scattered fields. The mechanism consists in existing inside of a random inhomogeneity a pair of virtually opposite decaying evanescent waves whose interference results in energy flux.

International audience; The overall mechanical response as well as strain and stress field statistics of an heterogeneous material made of two randomly distributed, linear elastic phases, are investigated numerically. The Boolean model of spheres is used to generate microstructures consisting of either porous or rigid inclusions, at any volume fraction of the phases. Stress and strain field integral ranges, or equivalently the representative volume element, are computed and linked to features of the field statistics, and to the microstructure geometry.

We describe an approximate method for the calculation of all characteristics of coherent backscattering for a homogeneous, semi-infinite particulate medium. The method allows one to transform a system of integral equations describing coherent backscattering exactly into a system of linear algebraic equations affording an efficient numerical solution. Comparisons of approximate theoretical results with experimental data as well as with benchmark numerical results for a medium composed of nonabsorbing Rayleigh scatterers have shown that the method can be expected to give a good accuracy.

For a set of two-dimensional passive random media that have the same randomness and different shapes, the effects of morphology on the time evolution of the power spectrum of the localized modes supported by the media are investigated. The results demonstrate that the evolving process of the spectrum, the lifetime of short-lived modes and the amount of long-lived modes are morphological-dependent, while the lifetime of long-lived modes is morphological-independent. The denser the medium is, the quicker the evolving process and the shorter the lifetime of the short-lived modes are. Single-mode operation is more possible and occurs more early for a denser medium, which is of practical importance for proposing a mode-selecting technique for random lasers.

In a two-dimensional random medium, when the transverse magnetic (TM) polarization state and the transverse electric (TE) polarization state share the same population inversion, there is a competition in the population inversion between the two states. Such competition is analyzed by using the finite difference time domain method to numerically solve Maxwell equations with a set of system parameters. The results indicate that TM state has a larger lasing threshold than TE state does so that TM state is strongly suppressed by TE state in the competition. The larger the size of the medium, the easier the lasing modes in TM state appears.