E. H. Hauge

Asymptotic time behavior of correlation functions. II. Kinetic and potential terms

On the basis of the mode-coupling theory we obtain the long-time behavior ∼t−d/2 for the kinetic, potential, and cross-terms in the Green-Kubo integrands, expressed completely in terms of transport coefficients and thermodynamic quantities. All two-mode amplitudes are explicitly evaluated in terms of measurable quantities such as specific heats, thermal expansion coefficients, etc.

Asymptotic time behavior of correlation functions. III. Local equilibrium and mode-coupling theory

The decay to equilibrium is discussed from a general point of view based on the assumed rapid approach to local equilibrium for well-chosen initial states. The assumption is applied to the problem of time correlation functions and it is shown that the mode-coupling formula describes the asymptotics of the so-called projected wavenumber-dependent correlation functions. The local equilibrium assumption thus provides a general basis for thet−3/2 behavior of correlation functions derived in previous papers in this series, as well as for the infinite series of correction termst−(2−Pn (n⩾ 2), withPn=2−n, and for the corresponding series of corrections of orderk3−Pn (n⩾1) to Navier-Stokes hydrodynamics.

From the Liouville Equation to the Generalized Boltzmann Equation for Magnetotransport in the 2D Lorentz Model

We consider a system of non-interacting charged particles moving in two dimensions among fixed hard scatterers, and acted upon by a perpendicular magnetic field. Recollisions between charged particles and scatterers are unavoidable in this case. We derive from the Liouville equation for this system a generalized Boltzmann equation with infinitely long memory, but which still is analytically solvable. This kinetic equation has been earlier written down from intuitive arguments.

Can a local repulsive potential trap an electron

We study the classical dynamics of a charged particle in two dimensions, under the influence of a perpendicular magnetic and an in-plane electric field. We prove the surprising fact that there is a finite region in phase space that corresponds to the otherwise drifting particle being trapped by a local repulsive potential. Our result is a direct consequence of KAM-theory and, in particular, of Moser’s theorem. We illustrate it by numerical phase portraits and by an analytic approximation to invariant curves. The detailed dynamics of electrons in two dimensions (2D), in electromagnetic fields and with localized scatterers, is important for several aspects of the quantum Hall effect. Classical dynamics is more relevant in this context than one might think, see the recent review by Trugman [1]. In addition, classical kinetic theory for such systems is surprisingly subtle [2], and requires for its foundation an understanding of the underlying dynamics. In this letter we focus on the classical dynamics of an “electron” (i.e., a particle with charge −e) confined to the xy−plane. Perpendicular to these two dimensions there is a constant magnetic field ~

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.

The effective interface potential for a superconducting layer

The effective interface potential is derived for a superconducting layer attached to a wall. The expression applies to the neighborhood of a continuous wetting or delocalization transition, which exists for type I superconductors with a negative extrapolation length. From this potential a number of features can be easily derived, such as the locus of the phase transition and the critical exponents. Whereas the order parameter exponent is universal, other exponents, like the susceptibility exponent, are not.

Bound states and metastability near a scatterer in crossed electromagnetic fields

The eigenvalue problem of an electron in the plane in the presence of a repulsive scatterer is studied. The electron is subject to a weak in-plane electric field and a magnetic field perpendicular to the plane. The associated magnetic length is much larger than the range of the scatterer. In this parameter region it is natural to follow Prange and treat the scatterer basically as a repulsive δ-function. However, the finite range of the scatterer is essential in that it provides the cutoffs necessary to make the problem mathematically well posed. We demonstrate that a true δ-function is unable to trap an electron in a finite electric field, no matter how small. At high Landau levels we find semi-quantitative agreement with recent classical results on electron trapping. With sharp cutoffs one bound state per Landau level is found for sufficiently weak electric fields. As the strength of the electric field is increased, the role of the bound state is taken over by a metastable wave packet which remains close to the scatterer for an exceedingly long time. This wave packet is explicitly constructed. With smooth cutoffs, all bound states become submerged in the continuum, and only long-lived wavepackets remain.

Magnetotransport in the 2D Lorentz model: linear and nonlinear effects of a weak electric field

We study the two-dimensional (2D) classical Lorentz model in a transverse magnetic and an in-plane electric field, in the regime where the dimensionless electric field is smaller than all other parameters. Since, therefore, the dimensionless density must be kept finite, we start from the Liouville equation and derive, by the multiple time scale method, the equations governing nonlinear as well as linear transport. The same diffusion tensor, formally rederived as the Kubo expression is, surprisingly, found to govern both regimes, albeit in different manner. Subsequently, explicit asymptotic results for the two components of the current density are calculated in the low-density regime.

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.

Is quantum tunnelling faster than the speed of light

The time taken for a single photon to tunnel through a potential energy barrier has been measured for the first time. Surprisingly, perhaps, the photon appears to tunnel through the barrier faster than an identical photon travelling through free space. However, closer scrutiny shows that this apparently superluminal velocity is not in conflict with the theory of relativity.