Subgeometric ergodicity and β-mixing

It is well known that stationary geometrically ergodic Markov chains are $\beta$ -mixing (absolutely regular) with geometrically decaying mixing coefficients. Furthermore, for initial distributions other than the stationary one, geometric ergodicity implies $\beta$ -mixing under suitable moment assumptions. In this note we show that similar results hold also for subgeometrically ergodic Markov chains. In particular, for both stationary and other initial distributions, subgeometric ergodicity implies $\beta$ -mixing with subgeometrically decaying mixing coefficients. Although this result is simple, it should prove very useful in obtaining rates of mixing in situations where geometric ergodicity cannot be established. To illustrate our results we derive new subgeometric ergodicity and $\beta$ -mixing results for the self-exciting threshold autoregressive model.

Renormalization and mixing of the Gluino-Glue operator on the lattice

We study the mixing of the Gluino-Glue operator in $$\mathcal{N}=1$$ N = 1 Supersymmetric Yang–Mills theory (SYM), both in dimensional regularization and on the lattice. We calculate its renormalization, which is not merely multiplicative, due to the fact that this operator can mix with non-gauge invariant operators of equal or, on the lattice, lower dimension. These operators carry the same quantum numbers under Lorentz transformations and global gauge transformations, and they have the same ghost number. We compute the one-loop quantum correction for the relevant two-point and three-point Green’s functions of the Gluino-Glue operator. This allows us to determine renormalization factors of the operator in the $${\overline{\mathrm{MS}}}$$ MS ¯ scheme, as well as the mixing coefficients for the other operators. To this end our computations are performed using dimensional and lattice regularizations. We employ a standard discretization where gluinos are defined on lattice sites and gluons reside on the links of the lattice; the discretization is based on Wilson’s formulation of non-supersymmetric gauge theories with clover improvement. The number of colors, $$N_c$$ N c , the gauge parameter, $$\beta $$ β , and the clover coefficient, $$c_{\mathrm{SW}}$$ c SW , are left as free parameters.

Atmospheric phenomena affecting the methane lifetime on Mars

<p>We investigate a range of atmospheric phenomena concerning their potential to address the Martian methane lifetime discrepancy. This refers to the over-estimate of the modelled lifetimes compared to observations by a factor of up to six hundred. We apply a newly developed atmospheric photochemical model where we vary in a Monte Carlo approach the chemical rate and Eddy mixing coefficients within their current uncertainties. We also investigate the effect of air shower events due to galactic cosmic rays and solar cosmic rays. Our results suggest that the current uncertainty in chemical rates and transport together with seasonal changes in the water column can likely account for up to a factor of about thirty in the Mars methane lifetime discrepancy whereas the air shower effects are likely to be of secondary importance.</p>

3D1D hydro-nucleosynthesis simulations – I. Advective–reactive post-processing method and its application to H ingestion into He-shell flash convection in rapidly accreting white dwarfs

ABSTRACT We present two mixing models for post-processing of 3D hydrodynamic simulations applied to convective–reactive i-process nucleosynthesis in a rapidly accreting white dwarf (RAWD) with [Fe/H] = −2.6, in which H is ingested into a convective He shell. A 1D advective two-stream model adopts physically motivated radial and horizontal mixing coefficients constrained by 3D hydrodynamic simulations. A simpler approach uses diffusion coefficients calculated from the same simulations. All 3D simulations include the energy feedback of the 12C(p, γ)13N reaction from the H entrainment. Global oscillations of shell H ingestion in two of the RAWD simulations cause bursts of entrainment of H and non-radial hydrodynamic feedback. With the same nuclear network as in the 3D simulations, the 1D advective two-stream model reproduces the rate and location of the H burning within the He shell closely matching the 3D simulation predictions, as well as qualitatively displaying the asymmetry of the XH profiles between the upstream and downstream. With a full i-process network the advective mixing model captures the difference in the n-capture nucleosynthesis in the upstream and downstream. For example, 89Kr and 90Kr with half-lives of $3.18\,\,\mathrm{\mathrm{min}}$ and $32.3\,\,\mathrm{\mathrm{s}}$ differ by a factor 2–10 in the two streams. In this particular application the diffusion approach provides globally the same abundance distribution as the advective two-stream mixing model. The resulting i-process yields are in excellent agreement with observations of the exemplary CEMP-r/s star CS31062-050.

Impacts of Stochastic Mixing in Idealized Convection-Permitting Simulations of Squall Lines

This study investigates impacts of altering subgrid-scale mixing in “convection-permitting” kilometer-scale horizontal-grid-spacing (Δh) simulations by applying either constant or stochastic multiplicative factors to the horizontal mixing coefficients within the Weather Research and Forecasting Model. In quasi-idealized 1-km Δh simulations of two observationally based squall-line cases, constant enhanced mixing produces larger updraft cores that are more dilute at upper levels, weakens the cold pool, rear-inflow jet, and front-to-rear flow of the squall line, and degrades the model’s effective resolution. Reducing mixing by a constant multiplicative factor has the opposite effect on all metrics. Completely turning off parameterized horizontal mixing produces bulk updraft statistics and squall-line mesoscale structure closest to an LES “benchmark” among all 1-km simulations, although the updraft cores are too undilute. The stochastic mixing scheme, which applies a multiplicative factor to the mixing coefficients that varies stochastically in time and space, is employed at 0.5-, 1-, and 2-km Δh. It generally reduces midlevel vertical velocities and enhances upper-level vertical velocities compared to simulations using the standard mixing scheme, with more substantial impacts at 1- and 2-km Δh compared to 0.5-km Δh. The stochastic scheme also increases updraft dilution to better agree with the LES for one case, but has less impact on the other case. Stochastic mixing acts to weaken the cold pool but without a significant impact on squall-line propagation. It also does not affect the model’s overall effective resolution unlike applying constant multiplicative factors to the mixing coefficients.

SUBGEOMETRICALLY ERGODIC AUTOREGRESSIONS

In this paper, we discuss how the notion of subgeometric ergodicity in Markov chain theory can be exploited to study stationarity and ergodicity of nonlinear time series models. Subgeometric ergodicity means that the transition probability measures converge to the stationary measure at a rate slower than geometric. Specifically, we consider suitably defined higher-order nonlinear autoregressions that behave similarly to a unit root process for large values of the observed series but we place almost no restrictions on their dynamics for moderate values of the observed series. Results on the subgeometric ergodicity of nonlinear autoregressions have previously appeared only in the first-order case. We provide an extension to the higher-order case and show that the autoregressions we consider are, under appropriate conditions, subgeometrically ergodic. As useful implications, we also obtain stationarity and $\beta $ -mixing with subgeometrically decaying mixing coefficients.

Exploring CP-violation, via heavy neutrino oscillations, in rare B meson decays at Belle II

In this article we study the rare B-meson decay via two on-shell almost-degenerate Majorana Heavy Neutrinos, into two charged leptons and two pseudoscalar mesons ($$B^{\pm } \rightarrow D^0 \ell ^{\pm }_1 \ell ^{\pm }_2 \pi ^{\mp }$$ B ± → D 0 ℓ 1 ± ℓ 2 ± π ∓ ). We consider the scenario where the heavy neutrino masses are $$\sim 2$$ ∼ 2 GeV and the heavy-light mixing coefficients are $$|B_{\ell N}|^2 \sim 10^{-5}$$ | B ℓ N | 2 ∼ 10 - 5 , and evaluate the possibility to measure the CP-asymmetry at Belle II. We present some realistic conditions under which the asymmetry could be detected.

Mixing characterization in different helically coiled configurations by laser-induced fluorescence

Flow Mixing of two miscible liquids has been characterized experimentally in three different helically coiled reactor configurations of two different lengths in the laminar flow regime at Re = 50…1000. A straight helical coil, a coiled flow inverter, and a new coiled flow reverser have been built, each in a 3-turn and a 6-turn configuration. Laser-induced fluorescence of resorufin has been used to visualize and quantify mixing in cross-sections throughout the reactors. A mixing coefficient is derived from the fluorescence images to allow for a quantitative measure and comparison of the six configurations. It becomes obvious from these experimental results, that an early flow redirection in the helical configuration is beneficial to mixing. The 3-turn reactors achieve nearly the same mixing coefficients as the 6-turn reactors with the double length. This can be explained by the stabilizing effect of the Dean vortices in the helix, which develop during the first two turns. After that, the liquid is trapped inside the vortices and further mixing is inhibited. Accordingly, the coiled flow inverter and coiled flow reverser configurations lead to much higher mixing coefficients than the straight helical coil. The results of these measurements are now used for validation of numerical simulations, which reproduce the geometrical and flow conditions of the experiments. Some exemplary results of these calculations are also shown in this article. Graphic abstract Mass fractions of tracer fluid at Re = 500 in the six examined helix configurations.