Nanobubbles, gas-filled entities that possess a size of less than , have notably emerged as critical components within various domains such as environmental science and medicine, primarily due to their remarkable stability and distinctive diffusion characteristics that differentiate them from larger bubbles. This comprehensive study meticulously explores the diffusion rates and stability of hydrogen and oxygen when encapsulated in oxyhydrogen nanobubbles, specifically within aqueous environments, utilizing the Chapman-Enskog theory, which is a well-established framework for understanding gas diffusion processes, alongside a modified Arrhenius equation to effectively evaluate activation volume and stability parameters. The findings from this investigation reveal that the diffusion processes of both gasses experience a significant reduction in rate due to the influence of high internal pressure, as well as the molecular interactions occurring with water, whereby hydrogen is observed to exhibit a marginally accelerated diffusion rate in comparison to oxygen. It is crucial to understand that elements like surface charges, zeta potential, and the idea of activation volume significantly influence the enduring stability of nanobubbles, particularly in specific environmental scenarios that could boost their longevity. These significant findings pose a challenge to classical diffusion models, thereby suggesting a promising potential for the sustained application of nanobubbles in various fields such as water treatment processes and therapeutic medical interventions. Future research endeavors should prioritize the refinement of theoretical models while also aiming to experimentally validate the mechanisms of stability that have been observed across a range of diverse applications to further substantiate these findings.

We study in this work a dilute granular gas immersed in a thermal bath made of smaller particles with nonnegligible masses as compared with the granular ones. The kinetic theory for this system is developed and described by an Enskog--Fokker--Planck equation for the one-particle velocity distribution function. Granular particles are assumed to have inelastic and hard interactions, losing energy in collisions as accounted by a constant coefficient of normal restitution. The interaction with the thermal bath is based on a nonlinear drag force plus a white-noise stochastic force. To get explicit results of the temperature aging and steady states, Maxwellian and first Sonine approximations are developed. The latter takes into account the coupling of the excess kurtosis with the temperature. Theoretical predictions are compared with direct simulation Monte Carlo and event-driven molecular dynamics simulations. While good results for the granular temperature are obtained from the Maxwellian approximation, a much better agreement, especially as inelasticity and drag nonlinearity increase, is found when using the first Sonine approximation. The latter approximation is, additionally, crucial to account for memory effects like Mpemba and Kovacs-like ones.

Secondary forests (SF) are important carbon sinks, removing CO₂ from the atmosphere through the photosynthesis process and storing photosynthates in their aboveground live biomass (AGB). This process occurring at large-scales partially counteracts C emissions from land-use change, playing, hence, an important role in the global carbon cycle. The absorption rates of carbon in these forests depend on forest physiology, controlled by environmental and climatic conditions as well as on the past land use, which is rarely considered for retrieving AGB from remotely sensed data. In this context, the main goal of this study is to evaluate the potential of full polarimetric ALOS-2 PALSAR-2 data for estimating AGB by taking into account the past-land use of SF areas in the Brazilian Amazon. We surveyed a chronosequence of 42 SF plots (20 ha) near the Tapajós National Forest in Pará state to quantifying AGB growth rates. We explored the full polarimetric data testing three regression models including non-linear (NL), multiple linear regressions models (MLR), and the semi-empirical extended water cloud model (EWCM). The results showed that the intensity of previous use has affected the structure of SF by reducing the AGB accumulation and being noticeable by several polarimetric attributes. The combination of multiple prediction variables with MLR improved the AGB estimation by 70% comparing amongst other models (R² adj. = 0.51; RMSE = 13.2 Mg ha^-1) bias = 2.1 ± 37.9 Mg ha^-1. The error propagation of the MLR model was estimated to be 15%.

Following Dirac’s notation in Quantum Mechanics (QM), we propose the Probability Bracket Notation (PBN), by defining a probability-bra (P-bra), P-ket, P-bracket, P-identity, etc…Using the PBN, many formulas, such as normalizations and expectations in systems of one or more random variables, can now be written in abstract basis-independent expressions, easy to expand by inserting a proper P-identity. The time evolution of homogeneous Markov processes can also be formatted in such a way. Our system P-kets are identified with probability vectors and our P-bra system is comparable with Doi’s state function or Peliti’s standard bra. In the Heisenberg picture of the PBN, a random variable becomes a stochastic process, and the Chapman-Kolmogorov equations are obtained by inserting a time-dependent P-identity. Also, some QM expressions in Dirac notation are naturally transformed to probability expressions in PBN by a special Wick rotation. Potential Applications show the usefulness of the PBN beyond the constrained domain and range of Hermitian operators on Hilbert Spaces in QM all the way to IT.