Asymptotic Derivation and Numerical Investigation of Time-Dependent Simplified Pn Equations

The steady-state simplified Pn (SPn) approximations to the linear Boltzmann equation have been proven to be asymptotically higher-order corrections to the diffusion equation in certain physical systems. In this paper, we present an asymptotic analysis for the time-dependent simplified Pn equations up to n = 3. Additionally, SPn equations of arbitrary order are derived in an ad hoc way. The resulting SPn equations are hyperbolic and differ from those investigated in a previous work by some of the authors. In two space dimensions, numerical calculations for the Pn and SPn equations are performed. We simulate neutron distributions of a moving rod and present results for a benchmark problem, known as the checkerboard problem. The SPn equations are demonstrated to yield significantly more accurate results than diffusion approximations. In addition, for sufficiently low values of n, they are shown to be more efficient than Pn models of comparable cost.Comment: 32 pages, 7 figure

Simple models suffice for the single dot quantum shuttle

A quantum shuttle is an archetypical nanoelectromechanical device, where the mechanical degree of freedom is quantized. Using a full-scale numerical solution of the generalized master equation describing the shuttle, we have recently shown [Novotn\'{y} {\it et al.}, Phys. Rev. Lett. {\bf 92}, 248302 (2004)] that for certain limits of the shuttle parameters one can distinguish three distinct charge transport mechanisms: (i) an incoherent tunneling regime, (ii) a shuttling regime, where the charge transport is synchronous with the mechanical motion, and (iii) a coexistence regime, where the device switches between the tunneling and shuttling regimes. While a study of the cross-over between these three regimes requires the full numerics, we show here that by identifying the appropriate time-scales it is possible to derive vastly simpler equations for each of the three regimes. The simplified equations allow a clear physical interpretation, are easily solved, and are in good agreement with the full numerics in their respective domains of validity.Comment: 23 pages, 14 figures, invited paper for the Focus issue of the New Journal of Physics on Nano-electromechanical system

Quantum Continuum Mechanics Made Simple

In this paper we further explore and develop the quantum continuum mechanics (CM) of [Tao \emph{et al}, PRL{\bf 103},086401] with the aim of making it simpler to use in practice. Our simplifications relate to the non-interacting part of the CM equations, and primarily refer to practical implementations in which the groundstate stress tensor is approximated by its Kohn-Sham version. We use the simplified approach to directly prove the exactness of CM for one-electron systems via an orthonormal formulation. This proof sheds light on certain physical considerations contained in the CM theory and their implication on CM-based approximations. The one-electron proof then motivates an approximation to the CM (exact under certain conditions) expanded on the wavefunctions of the Kohn-Sham (KS) equations. Particular attention is paid to the relationships between transitions from occupied to unoccupied KS orbitals and their approximations under the CM. We also demonstrate the simplified CM semi-analytically on an example system

Nonlinear effects in resonant layers in solar and space plasmas

The present paper reviews recent advances in the theory of nonlinear driven magnetohydrodynamic (MHD) waves in slow and Alfven resonant layers. Simple estimations show that in the vicinity of resonant positions the amplitude of variables can grow over the threshold where linear descriptions are valid. Using the method of matched asymptotic expansions, governing equations of dynamics inside the dissipative layer and jump conditions across the dissipative layers are derived. These relations are essential when studying the efficiency of resonant absorption. Nonlinearity in dissipative layers can generate new effects, such as mean flows, which can have serious implications on the stability and efficiency of the resonance

Density and current response functions in strongly disordered electron systems: Diffusion, electrical conductivity and Einstein relation

We study consequences of gauge invariance and charge conservation of an electron gas in a strong random potential perturbed by a weak electromagnetic field. We use quantum equations of motion and Ward identities for one- and two-particle averaged Green functions to establish exact relations between density and current response functions. In particular we find precise conditions under which we can extract the current-current correlation function from the density-density correlation function and vice versa. We use these results in two different ways to extend validity of a formula associating the density response function with the electrical conductivity from semiclassical equilibrium to quantum nonequilibrium systems. Finally we introduce quantum diffusion via a response relating the current with the negative gradient of the charge density. With the aid of this response function we derive a quantum version of the Einstein relation and prove the existence of the diffusion pole in the zero-temperature electron-hole correlation function with the the long-range spatial fluctuations controlled by the static diffusion constant.Comment: 16 pages, REVTeX4, 6 EPS figure

StaRMAP - A second order staggered grid method for spherical harmonics moment equations of radiative transfer

We present a simple method to solve spherical harmonics moment systems, such as the the time-dependent $P_N$ and $SP_N$ equations, of radiative transfer. The method, which works for arbitrary moment order $N$, makes use of the specific coupling between the moments in the $P_N$ equations. This coupling naturally induces staggered grids in space and time, which in turn give rise to a canonical, second-order accurate finite difference scheme. While the scheme does not possess TVD or realizability limiters, its simplicity allows for a very efficient implementation in Matlab. We present several test cases, some of which demonstrate that the code solves problems with ten million degrees of freedom in space, angle, and time within a few seconds. The code for the numerical scheme, called StaRMAP (Staggered grid Radiation Moment Approximation), along with files for all presented test cases, can be downloaded so that all results can be reproduced by the reader.Comment: 28 pages, 7 figures; StaRMAP code available at http://www.math.temple.edu/~seibold/research/starma

Lindblad dynamics of the quantum spherical model

The purely relaxational non-equilibrium dynamics of the quantum spherical model as described through a Lindblad equation is analysed. It is shown that the phenomenological requirements of reproducing the exact quantum equilibrium state as stationary solution and the associated classical Langevin equation in the classical limit $g\to 0$ fix the form of the Lindblad dissipators, up to an overall time-scale. In the semi-classical limit, the models' behaviour become effectively the one of the classical analogue, with a dynamical exponent $z=2$, and an effective temperature $T_{\rm eff}$, renormalised by the quantum coupling $g$. A distinctive behaviour is found for a quantum quench, at zero temperature, deep into the ordered phase $g\ll g_c(d)$, for $d>1$ dimensions. Only for $d=2$ dimensions, a simple scaling behaviour holds true, with a dynamical exponent $z=1$, while for dimensions $d\ne 2$, logarithmic corrections to scaling arise. The spin-spin correlator, the growing length scale and the time-dependent susceptibility show the existence of several logarithmically different length scales.Comment: 61 pages, 14 figure