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This chapter deals with the theoretical aspects of multifluid magnetohydrodynamics problems. In addition to the coupling between hydrodynamics and electromagnetics examined in Chapter 2, the high nonlinearity that now needs to be addressed is geometrical in nature. It results from the presence of one (or many) free interface(s) separating the fluids, assumed non-miscible. The mathematical analysis is substantially more intricate, and a long list of simple, however open, problems can be drawn up. Throughout the chapter, the equation that plays a crucial role is the equation of the conservation of mass. Owing to an argument based on the theory of renormalized solutions, a global-in-time existence result of weak solution is proved. The long-time behaviour of sufficiently regular solutions is also investigated.

This chapter draws on material from all preceding chapters to give an overview of the field of coherent X-ray imaging. It opens with an introduction to the operator theory of imaging, this being the language in which the remainder of the chapter is couched. It then discusses Talbot self imaging (Talbot effect), diffraction-free beams, in-line holography, off-axis holography, Fourier holography, Zernike phase contrast, differential interference contrast, analyzer-based phase contrast, propagation-based phase contrast, phase retrieval (Gerchberg-Saxton algorithm, transport-of-intensity equation, one-dimensional phase retrieval), interferometry (Bonse-Hart interferometer, Young interferometer, intensity interferometer), and virtual optics. Throughout, experimental examples are drawn from the contemporary research literature.

The time-dependent Ginzburg-Landau equation is phenomenologically introduced in relation to the order parameter relaxation in quasi-stationary approximation. This is applied to the fluctuation transport relative to superconducting transition. A general formula for the direct contribution of fluctuation Cooper pairs to the conductivity tensor of a superconductor with the arbitrary spectrum in the a.c. field and in the presence of external constant magnetic field is obtained. This is used to analyse fluctuation conductivity of the samples of arbitrary dimensionality, a.c., magnetoconductivity, and Hall conductivity of a layered superconductor nanotube. The transport equation of Boltzmann type for the fluctuation Cooper pair distribution function is derived.

This chapter presents simple, illustrative descriptions of collective dynamics in plasmas—in particular, in fully-ionized plasmas. These descriptions entail choices of collective plasma models, and the simplest models of plasma dynamics which describe collective modes spanning regimes from LF and long-wavelengths to HF and short-wavelengths. The chapter also addresses reduced hydrodynamic model equations appropriate for the first regime with emphasis on the Braginskii transport equations and the magnetohydrodynamics (MHD) model equations. The topics covered here are critical to students in two dimensions: The first is important in evaluating the validity of a result that follows from a reduced hydrodynamic description. The second is important to appreciate and understand the quasi-steady state, classical properties of a magnetically-confined plasma, and to compare these to experiments in current magnetic confinement fusion (MCF) plasmas exhibiting “anomalous” transport. Additionally, it is also important to be able to appreciate and evaluate the time and space scales of such collisional transport.

This chapter discusses conductivity in metals and semiconductor heterostructures. It aims to exhibit the similarities to nuclear transport problems. Semi-classical transport equations are used and the relaxation time approximation is exploited for the collision term. For metals, several processes are possible whose temperature dependence is discussed, namely electron-electron and electron-phonon collisions and scattering at impurities. The extent to which such features can be taken over to mesoscopic systems, such as quantum wires, is examined. For the two-dimensional electron gas it is shown that the electron transport may behave ballistically and that the conductivity is quantized. The resistance becomes finite because of contacts with leads. This makes up an essential difference from the nuclear case, where no external heat bath exists.

An example of a random process is the propagation of a particle (or photons, or acoustic wave) in a turbid medium, where particles undergo multiple scattering by randomly distributed scatterers in the medium. The kinetic equation governing particle propagation is the classic Boltzmann transport equation, which is also called the radiative transfer equation in the case of light propagation. The search for an analytical solution of the time-dependent elastic Boltzmann transport equation has lasted for many years. This chapter considers the problem of the classic elastic Boltzmann transport equation based on cumulant expansion. An analytical expression for cumulants of the spatial distribution of particles at any angle and time, exact up to an arbitrarily high order, is derived in an infinite uniform scattering medium. Up to the second order, a Gaussian approximation of the distribution function for the Boltzmann transport equation is obtained, with exact average center and exact half-width with time.

This chapter describes the main experimental techniques used to measure the drift velocity in superfluid ⁴He at low temperature. The experimental results are then presented by showing the contributions to the ion drag due to the different elementary excitations of the superfluid. The theoretical description of the processes of ion scattering off phonons, rotons, and ³He atomic impurities is also presented, and the theoretical predictions are compared with experimental results. The use of the formalism of the Boltzmann transport equation to predict how the drag force on an ion in the superfluid is determined by the different scattering mechanisms is discussed.

This chapter explores the evolution of an ensemble of electrons under stimulus, classically and quantum-mechanically. The classical Liouville description is derived, and then reformed to the quantum Liouville equation. The differences between the classical and the quantum-mechanical description are discussed, emphasizing the uncertainty-induced fuzziness in the quantum description. The Fokker-Planck equation is introduced to describe the evolution of ensembles and fluctuations in it that comprise the noise. The Liouville description makes it possible to write the Boltzmann transport equation with scattering. Limits of validity of the relaxation time approximation are discussed for the various scattering possibilities. From this description, conservation equations are derived, and drift and diffusion discussed as an approximation. Brownian motion arising in fast-and-slow events and response are related to the drift and diffusion and to the Langevin and Fokker-Planck equations as probabilistic evolution. This leads to a discussion of Markov processes and the Kolmogorov equation.

This chapter discusses the studies of the electron bubble mobility in normal liquid ³He. The high temperature measurements are extended into the milliKelvin range down to the superfluid transition at approximately 2.7 mK. At such low temperatures, the mean free path of the quasiparticles of the liquid becomes larger than the size of the electron bubble, and the drag on them must be calculated in the Knudsen limit. A quantum modification of the Boltzmann transport equation has been used to describe theoretically the experimental results by taking into account the requirements of the Pauli exlusion principle for fermion scattering. It is shown that the ion recoil must be taken into account by means of the van Hove scattering functions.

This chapter solves the underdetermined, elliptic equation ∂ⱼQsuperscript jl = Usuperscript l and Qsuperscript jl = Qsuperscript lj (Equation 1069) in order to eliminate the error term in the parametrix. For the proof of the Main Lemma, estimates for Q and the material derivative as well as its spatial derivatives are derived. The chapter finds a solution to Equation (1069) with good transport properties by solving it via a Transport equation obtained by commuting the divergence operator with the material derivative. It concludes by showing the solutions, spatial derivative estimates, and material derivative estimates for the Transport-Elliptic equation, as well as cutting off the solution to the Transport-Elliptic equation.

This chapter illustrates how the foundations of the fluid description are rooted in statistical mechanics and in kinetic theory. This approach, which is appropriate for those systems composed of a very large number of free particles and extending over a length-scale much larger than the inter-particles separation, is first presented in the Newtonian framework and then extended to the relativistic regime. A number of fundamental conceptual steps are taken and treated in detail: the introduction of a distribution function that depends on the positions and on the four-momentum of the constituent particles, the definition of the energy–momentum tensor as the second moment of the distribution function, the discussion of the relativistic Maxwell–Boltzmann equation with the corresponding H-theorem and transport equations. Finally, equations of state are described for all possible cases of relativistic or non-relativistic, degenerate or non-degenerate fluids.

This chapter investigates the proof for Theorems M2 and M3. It relies on the decay of q to prove the decay estimates for α‎ and α‎. More precisely, the chapter relies on the results of Theorem M1 to prove Theorem M2 and M3. In Theorem M1, decay estimates are derived for q defined with respect to the global frame constructed in Proposition 3.26. To recover α‎ from q, the chapter derives a transport equation for α‎ where q is on the RHS. It then derives as Teukolsky–Starobinsky identity a parabolic equation for α‎. The chapter also improves bootstrap assumptions for α‎.

The random motion of particles in a turbid medium, due to multiple elastic scattering, obeys the classic Boltzmann transport equation. This chapter shows how the center position and the diffusion coefficients of an incident collimated beam into an infinite uniform turbid medium are derived using an elementary analysis of the random walk of photons in a turbid medium. Light propagation in a multiple scattering (turbid) medium such as the atmosphere, colloidal suspensions, and biological tissue is commonly treated by the theory of radiative transfer. The basic equation of radiative transfer is the elastic Boltzmann equation, a nonseparable integro-differential equation of first order for which an exact closed form solution is not known except for the case for isotropic scatterers. Solutions are often based on truncation of the spherical harmonics expansion of the photon distribution function or resort to numerical calculation including Monte Carlo simulations. Macroscopic and microscopic statistics in the direction space are also discussed, along with the generalised Poisson distribution.

For the detection of charged particles many detector principles exploit the ionisation in sensing layers and the collection of the generated charges by electrical fields on electrodes, from where the signals can be deduced. In gases and liquids the charge carriers are electrons and ions, in semiconductors they are electrons and holes. To describe the ordered and unordered movement of the charge carriers in electric and magnetic fields the Boltzmann transport equation is introduced and approximate solutions are derived. On the basis of the transport equation drift and diffusion are discussed, first in general and then for applications to gases and semiconductors. It turns out that, at least for the simple approximations, the treatment for both media is very similar, for example also for the description of the movement in magnetic fields (Lorentz angle and Hall effect) or of the critical energy (Nernst-Townsend-Einstein relation).

This chapter discusses the statics and dynamics of particle ensemble evolution under multiple stimuli—electrical, magnetic and thermal, particularly (thermoelectromagnetic interaction)—by developing the evolution of the distribution function in a generalized form from its thermal equilibrium form. In the presence of electrical and magnetic fields, this shows the Hall effect, magnetoresistance, et cetera. Add thermal gradients, and one can elaborate additional consequences that can be calculated in terms of momentum relaxation times and the nature of impulse interaction, since momentum and energies carried by the ensemble are accounted for. So, parameters such as thermal conductivity due to the carriers can be determined, thermoelectric, thermomagnetic and thermoelectromagnetic interactions can be quantified and the Ettinghausen effect, the Nernst effect, the Righi-Leduc effect, the Ettinghausen-Nernst effect, the Seebeck effect, the Peltier effect and the Thompson coefficient understood. The dynamics also makes it possible to determine the frequency dependence of the phenomena.

Volatile organic chemicals (VOCs) are the most prevalent group of organic goundwater contaminants and originate primarily from industrial sources (Westerick et al., 1984). Many VOCs are aliphatics, of which a number are halogenated, and aromatics, which may or may not be chlorinated. As a class, these chemicals are highly volatile, many are relatively insoluble, and some have densities greater than that of water. The high volatility of these chemicals has supported a belief that fluxes of these materials from soil to the atmosphere arc so great that their persistence in soil is short-lived and the probability of groundwater contamination is small. However, groundwater monitoring data show that this is not the case and that many of these chemicals are posing potential threats to human health and the environment through contamination of water supplies (U.S. EPA, 1980). Even though VOCs are so widespread in the environment, our ability to adequately predict their transport and transformation in soil and the vadose zone is greatly lacking. This problem is further compounded by the fact that most sites are not contaminated with single compounds, but with mixtures of VOCs. As will be discussed later, the presence of other VOCs greatly complicates our ability to predict the behavior, both physical and biological, of a given chemical. Microbial communities play a pivotal role in the duration of contaminants from natural and managed environments and thus in reducing human exposure to toxins. A broad range of bioremediation approaches exist for contaminated soils and vadose material (Nelson et al., 1987, 1988; Harker and Kim 1990; Gibson and Sayler, 1992) and range from unmanaged to highly engineered systems. The unmanaged biodegradation of pollutants by indigenous microbial communities is increasingly becoming a remediation option in certain cases and is called “passive” or “intrinsic” bioremediation. Biostimulation usually involves additions of nutrients, electron acceptors, or cosubstrates to enhance the activity of indigenous microbial communities.

This chapter explores the kinetic theory of collisions and transport for fully-ionized plasmas. It first derives the Fokker–Planck collision equation analyzed in previous chapters by presenting them in two commonly used forms—the Landau form, and the form in terms of Rosenbluth potentials—and examines electrical conductivity through the Spitzer–Härm problem. The chapter then discusses the collisional transport theory, which entails slowly-varying (low-frequency and long-wavelength) hydrodynamics in a short mean-free-path limit. It shows that responses to drives such as the thermodynamics equilibrium entail the collisional transport coefficients of electrical conductivity, particle and heat diffusivities, and viscosity that enter into, and that specify the hydrodynamic transport equations. The kinetic theory analysis of arriving at the electrical conductivity of a fully-ionized plasma is carried out and used to appreciate some general aspects in the analysis and computation required for finding any transport coefficient.

This chapter derives estimates for quantities which are transported by the coarse scale flow and for their derivatives. It first considers the phase functions which satisfy the Transport equation, with the goal of choosing the lifespan parameter τ‎ sufficiently small so that all the phase functions which appear in the analysis can be guaranteed to remain nonstationary in the time interval, and so that the Stress equation can be solved. In order for these requirements to be met, τ‎ small enough is chosen so that the gradients of the phase functions do not depart significantly from their initial configurations. The chapter presents a proposition that bounds the separation of the phase gradients from their initial values in terms of b (b is less than or equal to 1, a form related to τ‎). Finally, it gathers estimates for relative velocity and relative acceleration.