Importance Sampling Variance Reduction for the Fokker-Planck Rarefied Gas Particle Method

Models and methods that are able to accurately and efficiently predict the flows of low-speed rarefied gases are in high demand, due to the increasing ability to manufacture devices at micro and nano scales. One such model and method is a Fokker-Planck approximation to the Boltzmann equation, which can be solved numerically by a stochastic particle method. The stochastic nature of this method leads to noisy estimates of the thermodynamic quantities one wishes to sample when the signal is small in comparison to the thermal velocity of the gas. Recently, Gorji et al have proposed a method which is able to greatly reduce the variance of the estimators, by creating a correlated stochastic process which acts as a control variate for the noisy estimates. However, there are potential difficulties involved when the geometry of the problem is complex, as the method requires the density to be solved for independently. Importance sampling is a variance reduction technique that has already been shown to successfully reduce the noise in direct simulation Monte Carlo calculations. In this paper we propose an importance sampling method for the Fokker-Planck stochastic particle scheme. The method requires minimal change to the original algorithm, and dramatically reduces the variance of the estimates. We test the importance sampling scheme on a homogeneous relaxation, planar Couette flow and a lid-driven-cavity flow, and find that our method is able to greatly reduce the noise of estimated quantities. Significantly, we find that as the characteristic speed of the flow decreases, the variance of the noisy estimators becomes independent of the characteristic speed

GMC Collisions As Triggers of Star Formation. IV. The Role of Ambipolar Diffusion

We investigate the role of ambipolar diffusion (AD) in collisions between magnetized giant molecular clouds (GMCs), which may be an important mechanism for triggering star cluster formation. Three dimensional simulations of GMC collisions are performed using a version of the Enzo magnetohydrodynamics code that has been extended to include AD. The resistivities are calculated using the 31-species chemical model of Wu et al. (2015). We find that in the weak-field, $10\:{\rm \mu G}$ case, AD has only a modest effect on the dynamical evolution during the collision. However, for the stronger-field, $30\:{\rm \mu G}$ case involving near-critical clouds, AD results in formation of dense cores in regions where collapse is otherwise inhibited. The overall efficiency of formation of cores with $n_{\rm H}\geq10^{6}\:{\rm cm}^{-3}$ in these simulations is increases from about 0.2% to 2% once AD is included, comparable to observed values in star-forming GMCs. The gas around these cores typically has relatively slow infall at speeds that are a modest fraction of the free-fall speed.Comment: 15 pages, 15 figures, Accepted to Ap

GMC Collisions As Triggers of Star Formation. IV. The Role of Ambipolar Diffusion

We investigate the role of ambipolar diffusion (AD) in collisions between magnetized giant molecular clouds (GMCs), which may be an important mechanism for triggering star cluster formation. Three dimensional simulations of GMC collisions are performed using a version of the Enzo magnetohydrodynamics code that has been extended to include AD. The resistivities are calculated using the 31-species chemical model of Wu et al. (2015). We find that in the weak-field, $10\:{\rm \mu G}$ case, AD has only a modest effect on the dynamical evolution during the collision. However, for the stronger-field, $30\:{\rm \mu G}$ case involving near-critical clouds, AD results in formation of dense cores in regions where collapse is otherwise inhibited. The overall efficiency of formation of cores with $n_{\rm H}\geq10^{6}\:{\rm cm}^{-3}$ in these simulations is increases from about 0.2% to 2% once AD is included, comparable to observed values in star-forming GMCs. The gas around these cores typically has relatively slow infall at speeds that are a modest fraction of the free-fall speed.Comment: 15 pages, 15 figures, Accepted to Ap

Model waveform accuracy standards for gravitational wave data analysis

Model waveforms are used in gravitational wave data analysis to detect and then to measure the properties of a source by matching the model waveforms to the signal from a detector. This paper derives accuracy standards for model waveforms which are sufficient to ensure that these data analysis applications are capable of extracting the full scientific content of the data, but without demanding excessive accuracy that would place undue burdens on the model waveform simulation community. These accuracy standards are intended primarily for broadband model waveforms produced by numerical simulations, but the standards are quite general and apply equally to such waveforms produced by analytical or hybrid analytical-numerical methods

On the complementarity of galaxy clustering with cosmic shear and flux magnification

In this paper, we motivate the use of galaxy clustering measurements using photometric redshift information, including a contribution from flux magnification, as a probe of cosmology. We present cosmological forecasts when clustering data alone is used, and when clustering is combined with a cosmic shear analysis. We consider two types of clustering analysis: firstly, clustering with only redshift auto-correlations in tomographic redshift bins; secondly, using all available redshift bin correlations. Finally, we consider how inferred cosmological parameters may be biased using each analysis when flux magnification is neglected. Results are presented for a Stage III ground-based survey, and a Stage IV space-based survey modelled with photometric redshift errors, and values for the slope of the luminosity function inferred from CFHTLenS catalogues. We find that combining clustering information with shear can improve constraints on cosmological parameters, giving an improvement to a Dark Energy Task Force-like figure of merit by a factor of 1.33 when only auto-correlations in redshift are used for the clustering analysis, rising to 1.52 when cross-correlations in redshift are also included. The addition of galaxy-galaxy lensing gives further improvement, with increases in figure of merit by a factor of 2.82 and 3.7 for each type of clustering analysis respectively. The presence of flux magnification in a clustering analysis does not significantly affect the precision of cosmological constraints when combined with cosmic shear and galaxy-galaxy lensing. However if magnification is neglected, inferred cosmological parameter values are biased, with biases in some cosmological parameters larger than statistical errors. (Abridged)Comment: Accepted by MNRAS, 18 pages, 12 Figures, 3 Table

Toward quantum processing in molecules: A THz-bandwidth coherent memory for light

The unusual features of quantum mechanics are enabling the development of technologies not possible with classical physics. These devices utilize nonclassical phenomena in the states of atoms, ions, and solid-state media as the basis for many prototypes. Here we investigate molecular states as a distinct alternative. We demonstrate a memory for light based on storing photons in the vibrations of hydrogen molecules. The THz-bandwidth molecular memory is used to store 100-fs pulses for durations up to 1ns, enabling 10,000 operational time bins. The results demonstrate the promise of molecules for constructing compact ultrafast quantum photonic technologies.Comment: 5 pages, 3 figures, 1 tabl

Topological Phases in the Plaquette Random-Cluster Model and Potts Lattice Gauge Theory

The $i$-dimensional plaquette random-cluster model on a finite cubical complex is the random complex of $i$-plaquettes with each configuration having probability proportional to p^{\text{# of plaquettes}}(1-p)^{\text{# of complementary plaquettes}}q^{\mathbf{ b}_{i-1}}, where $q\geq 1$ is a real parameter and $\mathbf{b}_{i-1}$ denotes the rank of the $(i-1)$-homology group with coefficients in a specified coefficient field. When $q$ is prime and the coefficient field is $\mathbb{F}_q$, this model is coupled with the $(i-1)$-dimensional $q$-state Potts lattice gauge theory. We prove that the probability that an $(i-1)$-cycle in $\mathbb{Z}^d$ is null-homologous in the plaquette random-cluster model equals the expectation of the corresponding generalized Wilson loop variable. This provides the first rigorous justification for a claim of Aizenman, Chayes, Chayes, Fr\"olich, and Russo that there is an exact relationship between Wilson loop variables and the event that a loop is bounded by a surface in an interacting system of plaquettes. We also prove that the $i$-dimensional plaquette random-cluster model on the $2i$-dimensional torus exhibits a sharp phase transition at the self-dual point $p_{\mathrm{sd}} \mathrel{\vcenter{:}}= \frac{\sqrt{q}}{1+\sqrt{q}}$ in the sense of homological percolation. This implies a qualitative change in the generalized Swendsen--Wang dynamics from local to non-local behavior.Comment: Minor change

A Sharp Deconfinement Transition for Potts Lattice Gauge Theory in Codimension Two

We prove that Wilson loop expectations in $(d-2)$-dimensional $q$-state Potts lattice gauge theory on $\mathbb{Z}^d$ undergo a sharp phase transition dual to that of the Potts model. This is a consequence of a more general theorem on the asymptotic probability that the boundary of a $(d-1)$-dimensional box is null-homologous in the $(d-1)$-dimensional plaquette random-cluster model. Our proof is unconditional for Ising lattice gauge theory, but relies on a regularity conjecture for the random-cluster model in slabs when $q>2.$ As another special case, we generalize a theorem of Aizenman, Chayes, Chayes, Fr\"olich, and Russo for $2$-dimensional Bernoulli percolation on $\mathbb{Z}^3$ to $(d-1)$-dimensional Bernoulli plaquette percolation on $\mathbb{Z}^d.