Understanding the ideal glass transition: Lessons from an equilibrium study of hard disks in a channel

We use an exact transfer-matrix approach to compute the equilibrium properties of a system of hard disks of diameter $\sigma$ confined to a two-dimensional channel of width $1.95\,\sigma$ at constant longitudinal applied force. At this channel width, which is sufficient for next-nearest-neighbor disks to interact, the system is known to have a great many jammed states. Our calculations show that the longitudinal force (pressure) extrapolates to infinity at a well-defined packing fraction $\phi_K$ that is less than the maximum possible $\phi_{\rm max}$, the latter corresponding to a buckled crystal. In this quasi-one-dimensional problem there is no question of there being any \emph{real} divergence of the pressure at $\phi_K$. We give arguments that this avoided phase transition is a structural feature -- the remnant in our narrow channel system of the hexatic to crystal transition -- but that it has the phenomenology of the (avoided) ideal glass transition. We identify a length scale $\tilde{\xi}_3$ as our equivalent of the penetration length for amorphous order: In the channel system, it reaches a maximum value of around $15\,\sigma$ at $\phi_K$, which is larger than the penetration lengths that have been reported for three dimensional systems. It is argued that the $\alpha$-relaxation time would appear on extrapolation to diverge in a Vogel-Fulcher manner as the packing fraction approaches $\phi_K$.Comment: 17 pages, 16 figure

Absence of hyperuniformity in amorphous hard-sphere packings of nonvanishing complexity

We relate the structure factor $S(\mathbf{k} \to \mathbf{0})$ in a system of jammed hard spheres of number density $\rho$ to its complexity per particle $\Sigma(\rho)$ by the formula $S(\mathbf{k} \to \mathbf{0})=-1/ [\rho^2\Sigma''(\rho)+2\rho\Sigma'(\rho)]$. We have verified this formula for the case of jammed disks in a narrow channel, for which it is possible to find $\Sigma(\rho)$ and $S(\mathbf{k})$ analytically. Hyperuniformity, which is the vanishing of $S(\mathbf{k} \to \mathbf{0})$, will therefore not occur if the complexity is nonzero. An example is given of a jammed state of hard disks in a narrow channel which is hyperuniform when generated by dynamical rules that produce a non-extensive complexity.Comment: 5 pages, 3 figure

Electromagnetic Wave Transmission Through a Subwavelength Nano-hole in a Two-dimensional Plasmonic Layer

An integral equation is formulated to describe electromagnetic wave transmission through a sub-wavelength nano-hole in a thin plasmonic sheet in terms of the dyadic Green's function for the associated Helmholtz problem. Taking the subwavelength radius of the nano-hole to be the smallest length of the system, we have obtained an exact solution of the integral equation for the dyadic Green's function analytically and in closed form. This dyadic Green's function is then employed in the numerical analysis of electromagnetic wave transmission through the nano-hole for normal incidence of the incoming wave train. The electromagnetic transmission involves two distinct contributions, one emanating from the nano-hole and the other is directly transmitted through the thin plasmonic layer itself (which would not occur in the case of a perfect metal screen). The transmitted radiation exhibits interference fringes in the vicinity of the nano-hole, and they tend to flatten as a function of increasing lateral separation from the hole, reaching the uniform value of transmission through the sheet alone at large separations.Comment: 14 pages, 24 individual figures organized in 9 captioned group

Transition state theory and the dynamics of hard disks

The dynamics of two and five disk systems confined in a square has been studied using molecular dynamics simulations and compared with the predictions of transition state theory. We determine the partition functions Z and Z^\ddagger of transition state theory using a procedure first used by Salsburg and Wood for the pressure. Our simulations show this procedure and transition state theory are in excellent agreement with the simulations. A generalization of the transition state theory to the case of a large number of disks N is made and shown to be in full agreement with simulations of disks moving in a narrow channel. The same procedure for hard spheres in three dimensions leads to the Vogel-Fulcher-Tammann formula for their alpha relaxation time.Comment: 1 new author, new simulations and figures, less speculation. Now 6 pages, 6 figures, 1 animation. Animation may be viewed at http://www.theory.physics.manchester.ac.uk/~godfrey/supplement/activated_dynamics2.htm

Carrier localization mechanisms in InGaN/GaN quantum wells

Localization lengths of the electrons and holes in InGaN/GaN quantum wells have been calculated using numerical solutions of the effective mass Schr\"odinger equation. We have treated the distribution of indium atoms as random and found that the resultant fluctuations in alloy concentration can localize the carriers. By using a locally varying indium concentration function we have calculated the contribution to the potential energy of the carriers from band gap fluctuations, the deformation potential and the spontaneous and piezoelectric fields. We have considered the effect of well width fluctuations and found that these contribute to electron localization, but not to hole localization. We also simulate low temperature photoluminescence spectra and find good agreement with experiment.Comment: 7 pages, 7 figure

Influence of reheating on the trispectrum and its scale dependence

We study the evolution of the non-linear curvature perturbation during perturbative reheating, and hence how observables evolve to their final values which we may compare against observations. Our study includes the evolution of the two trispectrum parameters, \gnl and \taunl, as well as the scale dependence of both \fnl and \taunl. In general the evolution is significant and must be taken into account, which means that models of multifield inflation cannot be compared to observations without specifying how the subsequent reheating takes place. If the trispectrum is large at the end of inflation, it normally remains large at the end of reheating. In the classes of models we study, it is very hard to generate \taunl\gg\fnl^2, regardless of the decay rates of the fields. Similarly, for the classes of models in which \gnl\simeq\taunl during slow--roll inflation, we find the relation typically remains valid during reheating. Therefore it is possible to observationally test such classes of models without specifying the parameters of reheating, even though the individual observables are sensitive to the details of reheating. It is hard to generate an observably large \gnl however. The runnings, \nfnl and \ntaunl, tend to satisfy a consistency relation \ntaunl=(3/2)\nfnl, but are in general too small to be observed for the class of models considered regardless of reheating timescale

The Effect of a Magnetic Field on the Acoustoelectric current in a Narrow Channel

The effect of a perpendicular magnetic field on the quantized current induced by a surface acoustic wave in a quasi-1D channel is studied. The channel has been produced experimentally in a GaAs heterostructure by shallow etching techniques and by the application of a negative gate voltage to Schottky split gates. Commensurability oscillations of the quantized current in this constriction have been observed in the interval of current between quantized plateaus. The results can be understood in terms of a moving quantum dot with the electron in the dot tunneling into the adjacent two-dimensional region. The goal is to explain qualitatively the mechanism for the steplike nature of the acoustoelectric current as a function of gate voltage and the oscillations when a magnetic field is applied. A transfer Hamiltonian formalism is employed.Comment: 5 pages, 2 figure