Frank A. Maao

THERE IS MORE TO BE LEARNED FROM THE LORENTZ MODEL

The classical Lorentz model for charged noninteracting point particles in a perpendicular magnetic field is reconsidered in 2D. We show that the standard Boltzmann equation is not valid for this model, even in the Grad limit. We construct a generalized Boltzmann equation which is, and solve the corresponding initial value problem exactly. By an independent calculation, we find the same solution by directly constructing the Green function from the dynamics of the model in the Grad limit. From this solution an expression for the diffusion tensor, valid for arbitrary short-range forces, is derived. For hard disks we calculate the diffusion tensor explicitly. Away from the Grad limit a percolation problem arises. We determine numerically the percolation threshold and the corresponding geometric critical exponents. The numerical evidence strongly suggests that this continuum percolation model is in the universality class of 2D lattice percolation. Although we have explicitly determined a number of limiting properties of the model, several intriguing open problems remain.

ACOUSTOELECTRIC EFFECTS IN QUANTUM CONSTRICTIONS

A dc current induced in a quantum constriction by a traveling acoustic wave (or by nonequilibrium ballistic phonons) is considered. We show that in many important situations the effect is originated from acoustically induced scattering between the propagating and reflecting states in the constriction. Two particular regimes corresponding to relatively high and low acoustic frequencies are discussed. In the first regime, the acoustoelectric effect in a smooth constriction can be understood by semiclassical considerations based on local conservation laws. For the low-frequency regime, we show that the acoustoconductance is closely related to the zero-field conductance. The qualitative considerations are confirmed by numerical calculations both for smooth and abrupt channels.

Liesegang pattern formation by gas diffusion in silica aerogels

Abstract Molecular gas diffusion of HCl and NH3 has been studied both in air and in aerogel. The two gases react to form NH4Cl which precipitates when a threshold concentration is reached. When the gases diffuse from opposite ends of a tube, the precipitate appears as a white ring whose position is determined by the ratio of the gas diffusion constants and follows Grahams law. When the diffusion process takes place in an aerogel, NH4Cl precipitates in sheets that span a cross-section of the gel. As the diffusion process continues, more bands are formed. The assembly of bands is called a Liesegang pattern. This phenomenon is known to occur in diffusion reaction systems. Numerical simulations based on the Ostwald nucleation theory qualitatively agree with experiments.

Band formation during gaseous diffusion in aerogels

We study experimentally how gaseous HCl and NH_3 diffuse from opposite sides of and react in silica aerogel rods with porosity of 92 % and average pore size of about 50 nm. The reaction leads to solid NH_4Cl, which is deposited in thin sheet-like structures. We present a numerical study of the phenomenon. Due to the difference in boundary conditions between this system and those usually studied, we find the sheet-like structures in the aerogel to differ significantly from older studies. The influence of random nucleation centers and inhomogeneities in the aerogel is studied numerically.

Criticality in the Integer Quantum Hall Effect

We review some elementary aspects of the critical properties of the series of metal-insulator transitions that constitute the integer quantum Hall effect. Numerical work has proven essential in charting out this phenomenon. Without being complete, we review network models that seem to capture the essentials of this critical phenomenon.

Photoelectric effects in a quantum point contact with an artificial impurity

I present exact numerical calculations on photon induced currents in a quantum point contact configuration with an artificial impurity potential. The presence of the impurity potential induces a net particle current in one direction only. The induced voltages for a corresponding open circuit are estimated. The calculations show that the effect is robust and easily within the experimental range.