Experimental test of the diffusion approximation for multiply scattered sound

Evaluation of stiffness loss of reinforced concrete beams using the diffuse ultrasound method

Flexural cracks are common in reinforced concrete (RC) beams. At service loads, the tensile stresses induced by the bending moments cause beam sections to crack, leading to loss of stiffness and a consequent increase in beam deflections. Serviceability limit states in RC beam design include maximum deflection and maximum crack widths. Cracks affect the propagation of ultrasound by disrupting its travel path, which leads to a strongly scattering of the ultrasonic waves. As a result, there is a delay in the arrival of the ultrasonic energy flux, which can be observed by the increasing formation of coda waves. This resultant incoherent wavefield can be approximated by the diffuse ultrasound method. The diffuse ultrasound method can better describe the cracking effects over a larger region of the RC element compared to the ultrasonic pulse velocity, the most used ultrasound parameter in concrete applications. Changes in the diffuse ultrasound parameters (diffusivity, dissipation and ATME) can be related to the extent of cracking in a RC element. The objective of this research was to apply the diffuse ultrasound method to evaluate the stiffness loss due to flexural cracking of RC beams. Beams with different longitudinal flexural reinforcement ratios were cast and submitted to a bending test. The deflection at mid-span, and thus beam stiffness, was monitored during the test. Ultrasound transducers were installed in the central region of the beams with ultrasound readings performed during the tests in order to acquire the waveforms at various loading stages. For each waveform, the diffuse ultrasound parameters were recovered using a time-frequency analysis. The behavior of the diffuse parameters with increasing progressive damage caused by flexural cracking was analyzed and correlated to the stiffness loss of the beams. As a result, it was observed that diffusivity and ATME were the most sensitive parameters to identify the onset of cracking and also were seen to be related to beam stiffness variation at early cracking stage. When correlated with stiffness loss values up to 70%, diffusivity and ATME presented high mean correlation coefficients, allowing to conclude that it is possible to estimate the stiffness loss through the diffuse ultrasound parameters in the interval following the beginning of the cracking process.

Bubbly flow velocity measurement in multiple scattering regime

We propose a technique to measure the velocity of a bubble cloud based on the coda correlation. The method is founded on successive recordings of multiple scattered waves from a bubble cloud. Our model predicts the dependence between the correlation coefficient of these coda waves and the velocity of the bubble cloud under diffusion approximation. The Acoustic experiments are validated by simultaneous optical measurements in a water tank, with a good agreement between the acoustical and the optical methods (relative difference smaller than 7%). This technique can be transposed to any particle flow velocity problems involving multiple scattering effects in acoustics.

Effect of an interstitial fluid on the dynamics of three-dimensional granular media

The propagation of mechanical energy in granular materials has been intensively studied in recent years given the wide range of fields that have processes related to this phenomena, from geology to impact mitigation and protection of buildings and structures. In this paper, we experimentally explore the effect of an interstitial fluid on the dynamics of the propagation of a mechanical pulse in a granular packing under controlled confinement pressure. The experimental results reveal the occurrence of an elastohydrodynamic mechanism at the scale of the contacts between wet particles. We describe our results in terms of an effective medium theory, including the presence of the viscous fluid. Finally, we study the nonlinear weakening of the granular packing as a function of the amplitude of the pulses. Our observations demonstrate that the softening of the material can be impeded by adjusting the viscosity of the interstitial fluid above a threshold at which the elastohydrodynamic interaction overcomes the elastic repulsion due to the confinement.

Multiple scattering of an ultrasonic shock wave in bubbly media

This experimental study deals with the propagation of an ultrasonic shock wave in a random heterogeneous medium, constituted of identical 75μm radius bubbles, trapped in a yield-stress fluid. The fundamental frequency of the incident wave (in the MHz range) was much larger than the resonance frequency of bubbles (38kHz). A well-expanded coda, resulting from the multiple scattering of the incident shock wave through the heterogeneous medium, was experimentally measured in transmission. Despite the significant amplitude of the shock wave (90kPa), no sign of nonlinear response of the bubbles was detected. Both the coherent and incoherent fields were successfully described by linear theories. Using a shock wave presents the advantage of characterizing the medium over a large frequency range (1.5-15MHz).

Impact of Strong Scattering Resonances on Ballistic and Diffusive Wave Transport

The strong impact of scattering resonances on all the key transport parameters of classical waves in disordered media is demonstrated through ultrasonic experiments on monodisperse emulsions. Through accurate measurements of both ballistic and diffusive transport over a wide range of frequencies, we show that the group velocity is large near sharp resonances, whereas the energy velocity (as well as the diffusion coefficient) is significantly slowed down by resonant scattering delay. Excellent agreement between theory and experiment is found, elucidating the effects of resonant scattering on wave transport in both acoustics and optics.

Radiative transfer of acoustic waves in continuous complex media: Beyond the Helmholtz equation

Heterogeneity can be accounted for by a random potential in the wave equation. For acoustic waves in a fluid with fluctuations of both density and compressibility (as well as for electromagnetic waves in a medium with fluctuation of both permittivity and permeability) the random potential entails a scalar and an operator contribution. For simplicity, the latter is usually overlooked in multiple scattering theory: whatever the type of waves, this simplification amounts to considering the Helmholtz equation with a sound speed c depending on position r. In this work, a radiative transfer equation is derived from the wave equation, in order to study energy transport through a multiple scattering medium. In particular, the influence of the operator term on various transport parameters is studied, based on the diagrammatic approach of multiple scattering. Analytical results are obtained for fundamental quantities of transport theory such as the transport mean-free path ℓ^{*}, scattering phase function f, and anisotropy factor g. Discarding the operator term in the wave equation is shown to have a significant impact on f and g, yet limited to the low-frequency regime, i.e., when the correlation length of the disorder ℓ_{c} is smaller than or comparable to the wavelength λ. More surprisingly, discarding the operator part has a significant impact on the transport mean-free path ℓ^{*} whatever the frequency regime. When the scalar and operator terms have identical amplitudes, the discrepancy on the transport mean-free path is around 300% in the low-frequency regime, and still above 30% for ℓ_{c}/λ=10^{3} no matter how weak fluctuations of the disorder are. Analytical results are supported by numerical simulations of the wave equation and Monte Carlo simulations.

Creep of sound paths in consolidated granular material detected through coda wave interferometry

The time evolution of the contact force structure of a consolidated granular material subjected to a constant stress is monitored using the coda wave interferometry method. In addition, the nature of the aging and rejuvenation processes are investigated. These processes are interpreted in terms of affine and nonaffine structural path deformations. During the later stages of creep, the rearrangements of subgrains are so small that they only produce affine deformations in the contact paths, without any significant changes in the structural configuration. As a result, the strain path distribution follows the macroscopic strain. Conversely, in the presence of ultrasonic perturbations, the nonaffine grain buckling mechanism dominates, producing relatively drastic changes in the structural configuration accompanied by path deformations of the order of the grain size. This plastic mechanism induces material rejuvenation that is observed macroscopically as an ultrasonically accelerated creep.

Mechanical impulse propagation in a three-dimensional packing of spheres confined at constant pressure

Mechanical impulse propagation in granular media depends strongly on the imposed confinement conditions. In this work, the propagation of sound in a granular packing contained by flexible walls that enable confinement under hydrostatic pressure conditions is investigated. This configuration also allows the form of the input impulse to be controlled by means of an instrumented impact pendulum. The main characteristics of mechanical wave propagation are analyzed, and it is found that the wave speed as function of the wave amplitude of the propagating pulse obeys the predictions of the Hertz contact law. Upon increasing the confinement pressure, a continuous transition from nonlinear to linear propagation is observed. Our results show that in the low-confinement regime, the attenuation increases with an increasing impulse amplitude for nonlinear pulses, whereas it is a weak function of the confinement pressure for linear waves.

Measurements of ultrasonic diffusivity and transport speed from coda waves in a resonant multiple scattering medium

Frequency-resolved experimental measurements of ultrasonic diffusivity in the MHz range are presented. The samples under study are two-dimensional random arrangements of parallel steel rods immersed in water and exhibit high-order multiple scattering. Their physical characteristics, particularly the density and pair-correlation functions of the scatterers, are well controlled. These synthetic samples are used as phantoms for actual inhomogeneous materials. The resonant nature of the scatterers has a strong effect on diffusivity, which is shown to vary significantly with frequency. This may affect the result of broadband measurements of apparent diffusivity, which can be expected to depend on time and sample thickness, whereas diffusivity is intrinsically an intensive parameter. Moreover, the transport speed is shown to vary drastically with frequency, sometimes by more than 50%, due to a very narrow resonance that slows down transport. Interestingly, this sharp resonance could only be revealed by experiments performed with coda waves, and not with ballistic or coherent waves whose frequency resolution is intrinsically limited from an experimental point of view.

Sound pulse broadening in stressed granular media

The pulse broadening and decay of coherent sound waves propagating in disordered granular media are investigated. We find that the pulse width of these compressional waves is broadened when the disorder is increased by mixing the beads made of different materials. To identify the responsible mechanism for the pulse broadening, we also perform the acoustic attenuation measurement by spectral analysis and the numerical simulation of pulsed sound wave propagation along one-dimensional disordered elastic chains. The qualitative agreement between experiment and simulation reveals a dominant mechanism by scattering attenuation at the high-frequency range, which is consistent with theoretical models of sound wave scattering in strongly random media via a correlation length.

Recurrent scattering and memory effect at the Anderson localization transition

We report on ultrasonic measurements of the propagation operator in a strongly scattering mesoglass. The backscattered field is shown to display a deterministic spatial coherence due to a remarkably large memory effect induced by long recurrent trajectories. Investigation of the recurrent scattering contribution directly yields the probability for a wave to come back close to its starting spot. The decay of this quantity with time is shown to change dramatically near the Anderson localization transition. The singular value decomposition of the propagation operator reveals the dominance of very intense recurrent scattering paths near the mobility edge.

Dynamic acoustoelastic testing of weakly pre-loaded unconsolidated water-saturated glass beads

Dynamic acoustoelastic testing is applied to weakly pre-loaded unconsolidated water-saturated glass beads. The gravitational acceleration produces, on the probed beads, a static stress of order 130 Pa, thus the granular medium is close to the jamming transition. A low-frequency (LF) acoustic wave gently disturbs the medium, inducing successively slight expansion and compaction of the granular packing expected to modulate the number of contacts between beads. Ultrasound (US) pulses are emitted simultaneously to dynamically detect the induced modification of the granular skeleton. US propagation velocity and attenuation both increase when the LF pressure increases. The quadratic nonlinear elastic parameter β, related to the pressure dependence of US propagation velocity, was measured in the range 60-530 if water-saturated glass beads are considered as an effective medium. A dynamic modification of US scattering induced by beads is proposed to modulate US attenuation. Complex hysteretic behaviors and tension-compression asymmetry are also observed and analyzed by time-domain and spectral analyses. Furthermore acoustic nonlinearities are measured in cases of quasi-static and dynamic variations of the LF wave amplitude, providing quantitatively similar acoustic nonlinearities but qualitatively different.

Transverse confinement of waves in three-dimensional random media

We study the transmission of a tightly focused beam through a thick slab of three-dimensional disordered medium in the Anderson localized regime. We show that the transverse profile of the transmitted beam exhibits clear signatures of Anderson localization and that its mean square width provides a direct measure of the localization length. For a short incident pulse, the width is independent of absorption.

Detection of damage in concrete using diffuse ultrasound

This letter demonstrates the potential for using diffuse ultrasound measurements to detect damage in concrete. Two different solutions to the diffusion equation, an infinite three-dimensional (3D) volume model that neglects geometric boundaries and a finite 3D cuboid model, are used for the required curve fitting procedure to determine the influence of geometric boundaries on the solution. The measurements consider two types of microcrack damage in concrete, alkali-silica reaction and thermal damage, and show that the measured diffusivity parameter is related to the amount of damage in each specimen.

A statistical approach to direct density of states measurements in disordered systems

A statistical method for measuring the modal density of elastic waves through direct mode counting in strongly scattering disordered systems is presented. To illustrate this approach, the results of ultrasonic experiments in a highly porous sintered glass bead network are reported. This method is shown to yield a reliable and robust measurement of the density of states, enabling mode-counting techniques to be applied to increasingly complex systems, where modal overlap and sensitivity to experimental conditions have previously hampered definitive results.

Ultrasound propagation in wet and airless non-consolidated granular materials

This paper deals with an experimental description of the acoustic behaviour of non-consolidated granular materials submitted to static force. The aim of this work is to investigate the effect of a small amount of an interstitial fluid on the acoustic propagation. Measurements of the velocity and of the transmission of the coherent wave are performed for different values of the applied force. It is shown that the behaviour of the speed of the ultrasonic coherent wave according to pressure have a slope close to the one of the Hertz-Mindlin's model in the case of a dry medium. When a small amount of a low viscosity fluid is added in a mono-disperse granular medium, the speed of the ultrasonic wave increases according to the power 1/6 to the force applied following the Hertz-Mindlin law (v approximately P(1/6)). Moreover, measurements of the velocity and of the transmission of the ultrasonic wave are strongly dependent on the nature of the interstitial fluid. In order to quantify its effect on the propagation, measurements are performed using various fluids having different characteristics. In a first step, silicon oils of different viscosities (from 50 x 10(-3) to 10 Pa s) are used, showing that with increasing viscosity, the wave velocity no longer varies according to the power law 1/6. The transmission coefficient also increases with the viscosity, showing a better propagation of the wave through the medium. Then, measurements are done in the vacuum allowing a comparison with ultrasonic propagation in presence of an interstitial fluid. This experiment shows a strong increase of the transmission coefficient while velocity remains the same as in the dry case. The study of scattered waves in vacuum shows also a significant increase in amplitude and duration of these typical waves. Then, different saturating inert gases are added to the medium showing that the propagation of the scattered wave is not influenced by their different characteristics.

Synthetic aperture diffraction tomography for three-dimensional imaging

Tomography of complex three-dimensional objects with ultrasound or microwave has been a long-standing goal since the introduction of these technologies after World War II. While current state-of-the-art systems can provide high-resolution images of cylindrical objects characterized by a two-dimensional structure, the three-dimensional case remains an open challenge owing to current limitations of sensor technology and computer power. Here, this problem is addressed by means of a synthetic aperture technique that, while using hardware technology developed for two-dimensional problems, accounts for the complexity of three-dimensional scattering and leads to high-resolution three-dimensional reconstructions. In this paper, we present the theoretical formulation of this new approach and illustrate it by means of a numerical example.

Random matrix theory applied to acoustic backscattering and imaging in complex media

The singular values distribution of the propagation operator in a random medium is investigated in a backscattering configuration. Experiments are carried out with pulsed ultrasonic waves around 3 MHz, using an array of transducers. Coherent backscattering and field correlations are taken into account. Interestingly, the distribution of singular values shows a dramatically different behavior in the single and multiple-scattering regimes. Based on a matrix separation of single and multiple-scattered waves, an experimental illustration of imaging through a highly scattering slab is presented.

Temperature-dependent diffusing acoustic wave spectroscopy with resonant scatterers

The influence of a slight temperature change on the correlation of multiply scattered acoustic waves is studied, and experimental results are discussed. The technique presented here, similar to diffusing-acoustic-wave spectroscopy, is based on the sensitivity of a multiply scattering medium to a slight change. Ultrasonic waves around 3 MHz are transmitted through a sample made of steel rods in water and recorded by an array of transducers at different temperatures. The cross correlations between highly scattered signals are computed. The main effect of the temperature change is a simple dilation of the times of arrival, due to a change of the sound velocity in water. But the scatterers also play a role in the progressive decorrelation of wave forms. An analysis resolved in both time and frequency shows that at some particular frequencies, the resonant behavior of the scatterers is responsible for a significantly larger decorrelation. Interestingly, the experimental results allow one to detect the presence of a small resonance that was not detected earlier on the same scatterers with classical measurement of the scattering mean free path. A simple model is proposed to interpret the experimental results.

Ultrasonic imaging of highly scattering media from local measurements of the diffusion constant: separation of coherent and incoherent intensities

As classical imaging fails with diffusive media, one way to image a multiple-scattering medium is to achieve local measurements of the dynamic transport properties of a wave undergoing diffusion. This paper presents a method to obtain local measurements of the diffusion constant D in a multiple-scattering medium. The experimental setup consists in an array of programmable transducers placed in front of the multiple-scattering medium to be imaged. By achieving Gaussian beamforming both at emission and reception, an array of virtual sources and receivers located in the near field is constructed. The time evolution of the incoherent component of the intensity backscattered on this virtual array is shown to represent directly the growth of the diffusive halo as sqrt[Dt]. A matrix treatment is proposed to separate the incoherent intensity from the coherent backscattering peak. Once the incoherent contribution is isolated, a local measurement of the diffusion constant is possible. The technique is applied to image the long-scale variations of D in a random-scattering sample made of two parts with a different concentration of cylindrical scatterers. This experimental result is obtained with ultrasonic waves around 3 MHz. It illustrates the possibility of imaging diffusive media from local measurements of the diffusion constant, based on coherent Gaussian beamforming and a matrix "antisymmetrization," which creates a virtual antireciprocity.

The wavelet response as a multiscale characterization of scattering processes at granular interfaces

We perform a multiscale analysis of the backscattering properties of a complex interface between water and a layer of randomly arranged glass beads with diameter D=1 mm. An acoustical experiment is done to record the wavelet response of the interface in a large frequency range from lambda/D=0.3 to lambda/D=15. The wavelet response is a physical analog of the mathematical wavelet transform which possesses nice properties to detect and characterize abrupt changes in signals. The experimental wavelet response allows to identify five frequency domains corresponding to different backscattering properties of the complex interface. This puts quantitative limits to the validity domains of the models used to represent the interface and which are flat elastic, flat visco-elastic, rough random half-space with multiple scattering, and rough elastic from long to short wavelengths respectively. A physical explanation based on Mie scattering theory is proposed to explain the origin of the five frequency domains identified in the wavelet response.

Wave transport in two-dimensional random media: the ballistic to diffusive transition and the extrapolation length

By using a first-principles approach based on the Bethe-Salpeter equation, we study the behavior of wave propagation through a two-dimensional random slab as a function of thickness, L , in the region where L is much smaller than the localization length. A general two-dimensional vertex function for the ladder diagrams is derived from the Ward identity. We calculate both the static and the time-resolved transmitted intensities as functions of L/l , where l is the mean free path. When L is comparable to l , we study the ballistic to diffusive transition. A sharp crossover is observed when L(c) approximately = 6l, which is significantly larger than the crossover thickness of L(c) approximately = 3l found in three dimensions. When L>>l , we obtain the extrapolation length in two dimensions, i.e., z2De approximately = 0.82l, which is noticeably larger than the previously used value of z2De = pi/4 obtained by an analytical approach.

Codalike multiple scattering of elastic waves in dense granular media

We study the multiple scattering of short-wavelength ultrasound through the force networks in dry and wet glass bead packings under stress. Over long distance scales, the diffusion approximation is shown to describe adequately the transport of elastic waves dominated by shear waves. The recovered transport mean path reveals a short-range correlation of the force chains. Also we observe the drastic effect of wetting liquids on the energy dissipation in the granular medium. The relevance of these experimental findings for the seismological applications is discussed.

Influence of ballistics to diffusion transition in primary wave propagation on parametric antenna operation in granular media

A model of parametric transmitting antenna in granular media is developed, which takes into account velocity dispersion, frequency-dependent absorption, and frequency-dependent scattering of acoustic waves in granular media. The latter process may induce a transition from the ballistics to the diffusion regime of pump (primary) high-frequency wave propagation with increasing frequency. The conditions under which the transition from ballistics to diffusion manifests itself in the change of the demodulated (rectified) low-frequency acoustic pulse profile are established. It is demonstrated that parametric low-frequency radiation contains information on both absorption and scattering of high-frequency acoustic waves.

Wave transport through thin slabs of random media with internal reflection: Ballistic to diffusive transition

The static and dynamic properties of wave transport through thin slabs of random media in the presence of internal reflection are investigated by performing first-principles calculations. These results are compared with results from time-independent and time-dependent diffusion equations, respectively, where the effects due to internal reflection are incorporated into an average extrapolation length in the boundary conditions. For the static properties, we find an abrupt transition from ballistic to diffusive behavior when sample thickness is about three mean free paths, i.e., L approximately 3l. The diffusion approximation is valid when L>3l, independent of the amount of internal reflection. For the dynamic properties, both the peak arrival time at short times and the diffusion constant at long times of the transmitted pulse indicate that there is a region of anomalous diffusion when 3l<L<L(c). The diffusion constant in this region increases with decreasing L. It also increases with the amount of internal reflection. The physical origin of the existence of such an anomalous region is the resonance-induced wave focusing effect. Due to the presence of internal reflection, the wave energy tends to concentrate in the forward direction at output boundary. It makes direction randomization difficult in the scattered waves. A similar wave focusing effect has been found in resonant tunneling systems of electrons in the presence of elastic scattering. The diffusion approximation is valid when L>L(c). The value of L(c) is about ten times the average extrapolation length, i.e., L approximately 10z(e), where z(e) is a fast increasing function of the amount of internal reflection.

Diffusing acoustic wave spectroscopy

We have developed a technique in ultrasonic correlation spectroscopy called diffusing acoustic wave spectroscopy (DAWS). In this technique, the motion of the scatterers (e.g., particles or inclusions) is determined from the temporal fluctuations of multiply scattered sound. In DAWS, the propagation of multiply scattered sound is described using the diffusion approximation, which allows the autocorrelation function of the temporal field fluctuations to be related to the dynamics of the multiply scattering medium. The expressions relating the temporal field autocorrelation function to the motion of the scatterers are derived, focusing on the types of correlated motions that are most likely to be encountered in acoustic measurements. The power of this technique is illustrated with ultrasonic data on fluidized suspensions of particles, where DAWS provides a sensitive measure of the local relative velocity and strain rate of the suspended particles over a wide range of time and length scales. In addition, when combined with the measurements of the rms velocity of the particles using dynamic sound scattering, we show that DAWS can be used to determine the spatial extent of the correlations in the particle velocities, thus indirectly measuring the particle velocity correlation function. Potential applications of diffusing acoustic wave spectroscopy are quite far reaching, ranging from the ultrasonic nondestructive evaluation of the dynamics of inhomogeneous materials to geophysical studies of mesoscopic phenomena in seismology.

Stress transmission through textured granular packings

We propose a theoretical approach aimed to describe the stress transmission through granular pilings. According to our framework, the stress transmission obeys a diffusive process in random isotropic packings, while it follows a local convection-diffusion equation in textured packings characterized by an anisotropic distribution of the contact orientations. In the latter case, the direction of the convection depends on the angular distribution function of the contact orientations. Our theoretical approach agrees with both experimental and numerical recent evidences, and moreover, succeeds in capturing the role of the pile preparing history.

Static and dynamic transport of light close to the Anderson localization transition

Anderson localization of light refers to an inhibition of wave transport in scattering media due to the interference of multiple scattered waves. We present wavelength dependent midinfrared optical transport measurements in slabs of randomly packed germanium (Ge) micron-sized particles, using a free electron laser as a tunable source of pulsed radiation. Because of their high refractive index and low absorption, Ge and similar semiconductors are excellent systems to study Anderson localization of light. To characterize the samples fully, we have employed several complementary optical techniques: total diffuse transmission, total diffuse reflection, coherent transmission, and time-resolved speckle interferometry. In this way we obtained the scattering (l(s)) and transport (l) mean free paths, the absorption coefficient (alpha), the diffusion constant (D), and the energy transport velocity (v(e)). These measurements have been made as a function of midinfrared wavelength, so that the scattering cross section and absorption coefficients can be varied in the same samples. We found that the Ge samples are close (kl(s) approximately 3) to the localization transition, but still above it. Our measurements of l(s) and l suggest that l is renormalized due to interference at the proximity of the localization transition. We also found that the diffusion constant is significantly reduced in samples thinner than approximately 7l.

Velocity fluctuations in fluidized suspensions probed by ultrasonic correlation spectroscopy

Velocity fluctuations in a fluidized suspension of particles are investigated using two new ultrasonic correlation spectroscopies: diffusing acoustic wave spectroscopy and dynamic sound scattering. These techniques probe both the local strain rate and rms velocity of the particles, providing important information about the spatial extent of velocity correlations. Our results demonstrate the power of these techniques to probe particle dynamics of fluidized suspensions, and suggest that the velocity correlations are essentially independent of Reynolds numbers for Re(p)<1.

Nonequilibrium statistical mechanics of drifting particles

This paper describes a method for obtaining nonequilibrium one-particle energy distributions of fermions or bosons. For the program to be carried out, particle transport should occur in the drifting mode in which the average velocity is much lower than the instantaneous velocity. Under this condition, the spectral current density has a drift-diffusion structure involving a mobility-diffusion relationship unrelated to statistics. When a local-equilibrium energy distribution is used, the linear response theory is recovered. Next, the particle-medium energy exchange is treated within a Fokker-Planck framework in order to obtain the nonequilibrium energy distribution; a nonlinear framework is used to account for the quantum-statistical correlations. Explicit formulas are obtained for homogeneous distributions at steady state. The rate of change of entropy is a simple generalization of the second law of thermodynamics. The positivity of the total entropy production stems from the positive definiteness of the diffusion tensors. Minimal entropy production is not necessarily achieved in the stationary state.

Wave transport in random media: the ballistic to diffusive transition

The character of wave transport through a strongly scattering medium, excited by a pulsed plane-wave source, is investigated as a function of sample thickness over the range from about one to 13 mean free paths. To examine the behavior theoretically, we perform a first-principles calculation of both the frequency correlation function of the transmitted field and the time-domain profile of the transmitted intensity. These quantities are investigated experimentally using an ultrasonic technique, which allows us to separate the ballistic and scattered components of the total transmitted field, and hence to measure the scattered component unambiguously in thin samples. For sample thicknesses greater than about four mean free paths, we find good agreement between our theory, the diffusion approximation, and our experimental data for both the frequency correlation function and the intensity time profile. In thinner samples, there are systematic differences between theory and experiment. To characterize the transition from ballistic to diffusive behavior in thin samples, we focus on the arrival time of the peak in the scattered component of the transmitted intensity; unexpectedly we find that the scattered peak arrival time exhibits an abrupt crossover between ballistic and diffusive behavior when the ratio of sample thickness to mean free path, L/l, is approximately equal to 3. Excellent agreement is obtained between our theory and experiment for this crossover behavior over the entire range of sample thicknesses investigated.