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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.

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.

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 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 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.

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.

This chapter discusses remote processes that influence electron transport and manifest themselves in a variety of properties of interest. Coulomb and phonon-based interactions have appeared in many discussions in the text. Coulomb interactions can be short range or long range, but phonons have been treated as a local effect. At the nanoscale, the remote aspects of these interactions can become significant. An off-equilibrium distribution of phonons, in the limit of low scattering, will lead to the breakdown of the local description of phonon-electron coupling. Phonons can drag electrons, and electrons can drag phonons. Soft phonons—high permittivity—can cause stronger electron-electron interactions. So, plasmon scattering can become significant. Remote phonon scattering too becomes important. These and other such changes are discussed, together with phonon drag’s consequences for the Seebeck effect, as illustrated through the coupled Boltzmann transport equation. The importance of the zT coefficient for characterizing thermoelectric capabilities is stressed.

The discussion of electron transport in Chapter 19 was based on the semi-classical Boltzmann transport equation. Other than implicitly assuming plane-wave-like electron states, as numbered via their wave vectors, k, the primary place where quantum mechanics entered the model was in the use of the Fermi–Dirac distribution function for filling these states, so as to systematically incorporate the Pauli exclusion principle. This chapter begins a quantum mechanical discussion of electrical transport with the relatively simple approach of Mattis and Bardeen. The approach is based on averaging a quantity involving a position-dependent correlation of the single particle wave function that enters the quantum mechanical expression for the current density. After introducing a simple assumption for mathematical form of this correlation, it recovers some results obtained in Chapter 19 that were based on the Boltzmann equation. In particular, it re-derives the Chambers expression for the conductivity.