Coupled normal heat and matter transport in a simple model system.

We introduce the first simple mechanical system that shows fully realistic transport behavior while still being exactly solvable at the level of equilibrium statistical mechanics. The system is a Lorentz gas with fixed freely rotating circular scatterers which scatter point particles via perfectly rough collisions. Upon imposing either a temperature gradient and/or a chemical potential gradient, a stationary state is attained for which local thermal equilibrium holds. Transport in this system is normal in the sense that the transport coefficients which characterize the flow of heat and matter are finite in the thermodynamic limit. Moreover, the two flows are nontrivially coupled, satisfying Onsager's reciprocity relations.     

Generation of aperiodic tilings with fivefold symmetry by the method of intersecting decagons and diffraction from finite size tilings

A method for generating aperiodic tilings with five fold symmetry is discussed here. Basic patterns formed within decagons can be used to fill two dimensional space, by matching such suitable patterns. It appears to be possible to generate perfect tilings without retracing already established coordinates imposing conditions at the initial stages of generating them. Various possible ways to generate tilings, when perfectness is not required, are discussed. The calculated diffraction patterns for some representative finite size tilings are shown. There are subtle differences in the intensities of peaks in the diffraction patterns corresponding to different finite size tilings constructed using intersecting decagons. These effects persist for a larger number of scatterers in weak peaks than in strong peaks. They are unaffected by an introduction of systematic disorder. These effects could be termed as the finite size boundary effects. There are also small shifts in the peak positions owing to the finite size effects. The possibility of formation of large approximate square cells in large tilings is shown.     

Free Path Lengths in Quasicrystals

Previous studies of kinetic transport in the Lorentz gas have been limited to cases where the scatterers are distributed at random (e.g., at the points of a spatial Poisson process) or at the vertices of a Euclidean lattice. In the present paper we investigate quasicrystalline scatterer configurations, which are non-periodic, yet strongly correlated. A famous example is the vertex set of a Penrose tiling. Our main result proves the existence of a limit distribution for the free path length, which answers a question of Wennberg. The limit distribution is characterised by a certain random variable on the space of higher dimensional lattices, and is distinctly different from the exponential distribution observed for random scatterer configurations. The key ingredients in the proofs are equidistribution theorems on homogeneous spaces, which follow from Ratner’s measure classification.     

Transport mean free path for magneto-transverse light diffusion: an alternative approach

This paper presents a derivation of the transport mean free path for magneto-transverse light diffusion, ℓ*_⊥, in an arbitrary random mixture of Faraday-active and non-Faraday-active Mie scatterers. This derivation is based on the standard radiative transfer equation. The expression of the transport mean free path obtained previously from the Bethe-Salpeter equation, for the case where only Faraday-active scatterers are present, is recovered. This simpler formulation can include the case of homogeneous mixtures of Faraday-active and non-Faraday-active scatterers.     

Transport Properties of a Modified Lorentz Gas

We present a detailed study of the first simple mechanical system that shows fully realistic transport behavior while still being exactly solvable at the level of equilibrium statistical mechanics. The system under consideration is a Lorentz gas with fixed freely-rotating circular scatterers interacting with point particles via perfectly rough collisions. Upon imposing a temperature and/or a chemical potential gradient, a stationary state is attained for which local thermal equilibrium holds for low values of the imposed gradients. Transport in this system is normal, in the sense that the transport coefficients which characterize the flow of heat and matter are finite in the thermodynamic limit. Moreover, the two flows are non-trivially coupled, satisfying Onsager's reciprocity relations to within numerical accuracy as well as the Green–Kubo relations. We further show numerically that an applied electric field causes the same currents as the corresponding chemical potential gradient in first order of the applied field. Puzzling discrepancies in higher order effects (Joule heating) are also observed. Finally, the role of entropy production in this purely Hamiltonian system is shortly discussed.     

Multiple scattering of acoustic waves by a half-space of distributed discrete scatterers with modified T-matrix approach

In studying the multiple scattering of acoustic waves by a half-space of distributed discrete scatterers, the quasicrystalline approximation(QCA) approach together with the hole correction (HC) or the pair distribution functions (PDF) have been used extensively, in which a system of simultaneous equations must be solved to determine the effective propagation constant and the expansion coefficients of the coherent exciting field. In this paper, we analyse the same problem under Foldy's approximation (EFA) by using the so-called modified T-matrix approach (MTMA) which was first proposed by Twersky. Two equations in a considerably simple and clear form are obtained for determining the effective propagation constant and the amplitude of the coherent transmitted field, as the scatterers are identical spheres. The numerical results in the low-frequency limit are also discussed in brief.     

An example of the effect of irreversible microscopic dynamics on the macroscopic behavior of a large system

The original Kac ring model is examined for the situation in which some of the scatterers are non-time-reversal invariant. It is shown that the system tends to absolute equilibrium (no fluctuations) even in the limit of very small density of these anomalous scatterers.     

Free Path Lengths in Quasi Crystals

The Lorentz gas is a model for a cloud of point particles (electrons) in a distribution of scatterers in space. The scatterers are often assumed to be spherical with a fixed diameter d, and the point particles move with constant velocity between the scatterers, and are specularly reflected when hitting a scatterer. There is no interaction between point particles. An interesting question concerns the distribution of free path lengths, i.e. the distance a point particle moves between the scattering events, and how this distribution scales with scatterer diameter, scatterer density and the distribution of the scatterers. It is by now well known that in the so-called Boltzmann–Grad limit, a Poisson distribution of scatterers leads to an exponential distribution of free path lengths, whereas if the scatterer distribution is periodic, the free path length distribution asymptotically behaves as a power law.     

The Lorentz process converges to a random flight process

The Lorentz process is the stochastic process defined by a particle moving, according to Newton's law of motion, through static scatterers distributed according to some probability measure in space. We consider the Boltzmann-Grad limit: The density of scatterers increases to infinity and at the same time the diameter of the scatterers decreases to zero in such a way that the mean free path of the particle is kept constant. We show that the Lorentz process converges in the weak*-topology of regular Borel measures on the paths space to some stochastic process. The limit process is Markovian if and only if the rescaled density of scatterers converges in probability to its mean. In that case the limit process is a (spatially inhomogeneous) random flight process.On leave of absence of Fachbereich Physik der Universität München. Work supported by a DFG fellowship     

Coherent acoustic wave propagation in media with pair-correlated spheres

Propagation of plane compressional waves in a non-viscous fluid with a dense distribution of identical spherical scatterers is investigated. The analysis is based on the multiple scattering approach proposed by Fikioris and Waterman, and is generalized to include arbitrary choice of the pair-correlation functions used to represent the distribution of the scatterers. A closed form solution for the effective wavenumber as a function of the concentration of pair-correlated finite-size spheres is derived up to the second order. In the limit of uncorrelated point-scatterers, this solution is identical to that obtained by Lloyd and Berry. Different pair-correlation functions are exemplified and compared, and the resulting differences discussed.     

Transport mean free path for magneto-transverse light diffusion: an alternative approach

Abstract

This paper presents a derivation of the transport mean free path for magneto-transverse light diffusion, ?*_⊥, in an arbitrary random mixture of Faraday-active and non-Faraday-active Mie scatterers. This derivation is based on the standard radiative transfer equation. The expression of the transport mean free path obtained previously from the Bethe–Salpeter equation, for the case where only Faraday-active scatterers are present, is recovered. This simpler formulation can include the case of homogeneous mixtures of Faraday-active and non-Faraday-active scatterers.     

Heat conduction in caricature models of the Lorentz gas

Heat transport coefficients are calculated for various random walks with internal states (the Markov partition of the Sinai billiard connects these walks with the Lorentz gas among a periodic configuration of scatterers). Models with reflecting or absorbing barriers and also those without or with local thermal equilibrium are investigated. The method is unified and is based on the Keldysh expansion of the resolvent of a matrix polynomial.     

A theoretical calculation of tunneling spectroscopy of an electron waveguide with or without a scatterer

A theoretical study on the tunneling spectroscopy of an electron waveguide recently observed by Eugster and del Alamo is presented. A narrow electron waveguide coupled with another much wider one by a thin barrier between them is taken as a theoretical model for the leaky electron waveguide implemented by Eugster et al., and the transport properties of electrons are studied comprehensively through the wavefunction of the system. The results demonstrate that the conductance for the current tunneling out the barrier oscillates strongly with the width of the narrow electron waveguide, in line with its conductance steps. The theory is in good agreement with the experiments and confirm that the oscillations of the tunneling current can be considered as a spectroscopy of the 1D DOS (one dimensional electron density of states) in the electron waveguide as proposed by Eugster et al. In order to study the effects of scatterers on the transport properties of the leaky electron waveguide, a δ-function is used to simulate the scattering potential The results show that the presence of even a single scatterer located in the waveguide will lead to obvious distortion of the shape of conductance steps, and will greatly influence the oscillations of the tunneling current observed in clean waveguides. However the effects of scatterers located outside the tunneling barrier on either the conductance steps or the oscillations of the tunneling current are negligible.     

Spatial fluctuations of the chemical potential in case of nearly coherent transport along an ordered chain

The Landauer-Büttiker approach is used to describe electron transport along a chain of scatterers which allow elastic as well as inelastic processes. The inelastic scattering takes place via side branches coupling the chain to electron reservoirs which serve as a heat bath. For small inelastic coupling of the scatterers to the heat bath strong interference effects lead to spatial fluctuations of the charge density. The corresponding oscillations of the chemical potential are discussed in view of phase-sensitive experiments measuring the four-probe resistance.     

Hot Scatterers and Tracers for the Transfer of Heat in Collisional Dynamics

We introduce stochastic models for the transport of heat in systems described by local collisional dynamics. The dynamics consists of tracer particles moving through an array of hot scatterers describing the effect of heat baths at fixed temperatures. Those models have the structure of Markov renewal processes. We study their ergodic properties in details and provide a useful formula for the cumulant generating function of the time integrated energy current. We observe that out of thermal equilibrium, the generating function is not analytic. When the set of temperatures of the scatterers is fixed by the condition that in average no energy is exchanged between the scatterers and the system, different behaviours may arise. When the tracer particles are allowed to travel freely through the whole array of scatterers, the temperature profile is linear. If the particles are locked in between scatterers, the temperature profile becomes nonlinear. In both cases, the thermal conductivity is interpreted as a frequency of collision between tracers and scatterers.     

Measurement of scattering rate and minimum conductivity in graphene

The conductivity of graphene samples with various levels of disorder is investigated for a set of specimens with mobility in the range of 1-20x10(3) cm2/V sec. Comparing the experimental data with the theoretical transport calculations based on charged impurity scattering, we estimate that the impurity concentration in the samples varies from 2-15x10(11) cm(-2). In the low carrier density limit, the conductivity exhibits values in the range of 2-12e2/h, which can be related to the residual density induced by the inhomogeneous charge distribution in the samples. The shape of the conductivity curves indicates that high mobility samples contain some short-range disorder whereas low mobility samples are dominated by long-range scatterers.     

Low energy density of states in several disordered systems

Disordered systems of arbitrary dimension consisting of randomly distributed scatterers are studied in the low energy limit (Roger Waxler, Ph.D. thesis, Columbia University). We evaluate the density of electronic energy levels for long range, attractive, and electric dipole scatterers.     

Concentric layered Hermite scatterers

The long wavelength limit of scattering from spheres has a rich history in optics, electromagnetics, and acoustics. Recently it was shown that a common integral kernel pertains to formulations of weak spherical scatterers in both acoustics and electromagnetic regimes. Furthermore, the relationship between backscattered amplitude and wavenumber k was shown to follow power laws higher than the Rayleigh scattering $k^2$ power law, when the inhomogeneity had a material composition that conformed to a Gaussian weighted Hermite polynomial. Although this class of scatterers, called Hermite scatterers, are plausible, it may be simpler to manufacture scatterers with a core surrounded by one or more layers. In this case the inhomogeneous material property conforms to a piecewise continuous constant function. We demonstrate that the necessary and sufficient conditions for supra-Rayleigh scattering power laws in this case can be stated simply by considering moments of the inhomogeneous function and its spatial transform. This development opens an additional path for construction of, and use of scatterers with unique power law behavior.     

Quantum transport of massless Dirac fermions

Motivated by recent graphene transport experiments, we undertake a numerical study of the conductivity of disordered two-dimensional massless Dirac fermions. Our results reveal distinct differences between the cases of short-range and Coulomb randomly distributed scatterers. We speculate that this behavior is related to the Boltzmann transport theory prediction of dirty-limit behavior for Coulomb scatterers.