The usefulness of higher-order constitutive relations for describing the Knudsen layer

The Knudsen layer is an important rarefaction phenomenon in gas flows in and around microdevices. Its accurate and efficient modeling is of critical importance in the design of such systems and in predicting their performance. In this paper we investigate the potential that higher-order continuum equations may have to model the Knudsen layer, and compare their predictions to high-accuracy DSMC (direct simulation Monte Carlo) data, as well as a standard result from kinetic theory. We find that, for a benchmark case, the most common higher-order continuum equation sets (Grad's 13 moment, Burnett, and super-Burnett equations) cannot capture the Knudsen layer. Variants of these equation families have, however, been proposed and some of them can qualitatively describe the Knudsen layer structure. To make quantitative comparisons, we obtain additional boundary conditions (needed for unique solutions to the higher-order equations) from kinetic theory. However, we find the quantitative agreement with kinetic theory and DSMC data is only slight

Capturing the Knudsen layer in continuum-fluid models of non-equilibrium gas flows

In hypersonic aerodynamics and microflow device design, the momentum and energy fluxes to solid surfaces are often of critical importance. However, these depend on the characteristics of the Knudsen layer - the region of local non-equilibrium existing up to one or two molecular mean free paths from the wall in any gas flow near a surface. While the Knudsen layer has been investigated extensively using kinetic theory, the ability to capture it within a continuum-fluid formulation (in conjunction with slip boundary conditions) suitable for current computational fluid dynamics toolboxes would offer distinct and practical computational advantages

Models for local ohmic quantum dissipation

We construct model master equations for local quantum dissipation. The master equations are in the form of Lindblad generators, with imposed constraints that the dissipations be strictly linear (i.e. ohmic), isotropic and translationally invariant. A particular form for is chosen to satisfy the constraints. The resulting master equations are given in both the Schr\"odinger and Heisenberg forms. We obtain fluctuation-dissipation relations, and discuss the relaxation of average kinetic energy to effective thermal equilibrium values. We compare our results to the Dekker and the Caldeira-Leggett master equations. These master equations allow a more general approach to quantum dissipation and the dynamics of quantum coherence to account for the nontrivial system-environment coupling in a local environment.Comment: 19 pages, REVTEX, PSU/TH/12

Exact solution of Riemann--Hilbert problem for a correlation function of the XY spin chain

A correlation function of the XY spin chain is studied at zero temperature. This is called the Emptiness Formation Probability (EFP) and is expressed by the Fredholm determinant in the thermodynamic limit. We formulate the associated Riemann--Hilbert problem and solve it exactly. The EFP is shown to decay in Gaussian.Comment: 7 pages, to be published in J. Phys. Soc. Jp

A comparative study of no-time-counter and majorant collision frequency numerical schemes in DSMC

The direct simulation Monte Carlo (DSMC) method is a stochastic approach to solve the Boltzmann equation and is built on various numerical schemes for transport, collision and sampling. This work aims to compare and contrast two popular O(N) DSMC collision schemes - no-time-counter (NTC) and majorant collision frequency (MCF) - with the goal of identifying the fundamental differences. MCF and NTC schemes are used in DSMC simulations of a spatially homogeneous equilibrium gas to study convergence with respect to various collision parameters. While the MCF scheme forces the reproduction of the exponential distribution of time between collisions, the NTC scheme requires larger number of simulators per cell to reproduce this Poisson process. The two collision schemes are also applied to the spatially homogeneous relaxation from an isotropic non-Maxwellian given by the Bobylev exact solution to the Boltzmann equation. While the two schemes produce identical results at large times, the initial relaxation shows some differences during the first few timesteps

Master-equations for the study of decoherence

Different structures of master-equation used for the description of decoherence of a microsystem interacting through collisions with a surrounding environment are considered and compared. These results are connected to the general expression of the generator of a quantum dynamical semigroup in presence of translation invariance recently found by Holevo.Comment: 10 pages, latex, no figures, to appear in Int. J. Theor. Phy

Quantum open systems and turbulence

We show that the problem of non conservation of energy found in the spontaneous localization model developed by Ghirardi, Rimini and Weber is very similar to the inconsistency between the stochastic models for turbulence and the Navier-Stokes equation. This sort of analogy may be useful in the development of both areas.Comment: to appear in Physical Review

Comparison of Kinetic Models for Gas Damping of Moving Microbeams

Numerical investigations of the gas flow structure and the gas-damping force on moving and heated microbeams are carried out using the Navier-Stokes equations with first-order velocity-slip and temperature-jump boundary conditions (the NSSJ method) and two kinetic numerical techniques: the particle-based direct simulation Monte Carlo (DSMC) method, and a deterministic discrete-ordinate solution of the ellipsoidal statistical (ES) kinetic model equation. The gas-damping coefficients on a moving microbeam for quasi-static isothermal conditions are estimated by the three numerical methods for Kn = 0.1-1.0. The NSSJ simulations tend to overestimate the gas-damping coefficient for Knudsen numbers larger than 0.1, whereas the DSMC and ES kinetic approaches are in good agreement for the slip and transitional flow regimes. The flow structure and the Knudsen force are calculated using the ES kinetic model for a heated microbeam over a wide range of Knudsen numbers. The Knudsen force peaks in the transitional regime (Kn ≈ 2), and the numerically predicted variation of the force with Knudsen number is consistent with experimentally observed displacements of the heated microbeam

Decoherence in a Talbot Lau interferometer: the influence of molecular scattering

We study the interference of C70 fullerenes in a Talbot-Lau interferometer with a large separation between the diffraction gratings. This permits the observation of recurrences of the interference contrast both as a function of the de Broglie wavelength and in dependence of the interaction with background gases. We observe an exponential decrease of the fringe visibility with increasing background pressure and find good quantitative agreement with the predictions of decoherence theory. From this we extrapolate the limits of matter wave interferometry and conclude that the influence of collisional decoherence may be well under control in future experiments with proteins and even larger objects.Comment: 8 pages, 5 figure

Two Derivations of the Master Equation of Quantum Brownian Motion

Central to many discussion of decoherence is a master equation for the reduced density matrix of a massive particle experiencing scattering from its surrounding environment, such as that of Joos and Zeh. Such master equations enjoy a close relationship with spontaneous localization models, like the GRW model. This aim of this paper is to present two derivations of the master equation. The first derivation is a pedagogical model designed to illustrate the origins of the master equation as simply as possible, focusing on physical principles and without the complications of S-matrix theory. This derivation may serve as a useful tutorial example for students attempting to learn this subject area. The second is the opposite: a very general derivation using non-relativistic many body field theory. It reduces to the equation of the type given by Joos and Zeh in the one-particle sector, but correcting certain numerical factors which have recently become significant in connection with experimental tests of decoherence. This master equation also emphasizes the role of local number density as the ``preferred basis'' for decoherence in this model.Comment: 19 pages, RevTe