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At least 919 records · Page 51
All sourcesNASA NTRSDOE OSTI
DOE OSTI2016

Characterization of 3D Cirrus Cloud and Radiation Fields Using ARS/AIRS/MODIS data and its Application to Climate Model

During the report period, we have made the following research accomplishments. First, we performed analysis for a number of MODIS scenes comprising of heavy dust events and ice clouds, covering regions of frequent dust outbreaks in East Asia, Middle East, and West Africa, as well as areas associated with long-range dust transports over the Equatorial Tropical Atlantic Ocean. These scenes contain both dust/aerosols and clouds. We collected suitable aerosol/ice-cloud data, correlated ice cloud and aerosol parameters by means of statistical analysis, and interpreted resulting correlation trends based on the physical principles governing cloud microphysics. Aerosol and cloud optical depths and cloud effective particle size inferred from MODIS for selected domains were analyzed from which the parameters including dust aerosol number concentration, ice cloud water path, and ice particle number concentration were subsequently derived. We illustrated that the Twomey (solar albedo) effect can be statistically quantified based on the slope of best-fit straight lines in the correlation study. Analysis of aerosol and cloud retrieval products revealed that for all cases, the region with a larger dust aerosol optical depth is always characterized by a smaller cloud particle size, consistent with the Twomey hypothesis for aerosol-cloud interactions. Second, we developed mean correlation curves with uncertainties associated with small ice-crystal concentration observations for the mean effective ice crystal size (De) and ice water content (IWC) by dividing the atmosphere into three characteristic regions: Tropics cirrus, Midlatitude cirrus, including a temperature classification to improve correlation, and Arctic ice clouds. We illustrated that De has a high correlation with IWC based on theoretical consideration and analysis of thousands of observed ice crystal data obtained from a number of ARM-DOE field campaigns and other experiments. The correlation has the form: ln(De) = a + b ln(IWC) + c ((ln(IWC))2, where a, b, and c are fitting coefficients and are functions of three regions. We demonstrated that this correlation can be effectively incorporated in GCMs and climate models that predict IWC - a significant advance in ice microphysics parameterization for interactive cloud-radiation analysis and feedback. Substantial July mean differences are shown in the OLR (W/m2) and precipitation (mm/day) patterns between UCLA GCM simulations based on Des determined from the De-IWC correlations and the control run using a fixed ice crystal size. Third, in order to improve the computation of spectral radiative transfer processes in the WRF model, we developed a consistent and efficient radiation scheme that can better resolve the spectral bands, determine the cloud optical properties, and provide more reliable and accurate radiative heating fields. In the newly developed radiation module, we have implemented in WRF a modified and improved version referred to as the Fu-Liou-Gu scheme, which includes a combination of delta-four-stream and delta-two-stream approximations for solar and IR flux calculations, respectively. This combination has been proven to be computationally efficient and at the same time to produce a high degree of accuracy. The incorporation of nongray gaseous absorption in multiple scattering atmospheres was based on the correlated k-distribution method. The solar and IR spectra are divided into 6 and 12 bands, respectively, according to the location of absorption bands of H2O, CO2, O3, CH4, N2O, and CFCs. We further included absorption by the water vapor continuum and a number of minor absorbers in the solar spectrum leading to an additional absorption of solar flux in a clear atmosphere on the order of 1-3 W/m2. Additionally, we incorporated the ice microphysics parameterization that includes an interactive mean effective ice crystal size in association with radiation parameterizations. The Fu-Liou-Gu scheme is an ideal tool for the simulation of radiative transfer and ice microphysics within the domain of WRF. It is particularly useful for studying direct and indirect aerosol radiative effects associated with ice cloud formation. The newly implemented radiation module has been demonstrated to work well in WRF and can be effectively used for studies related to cirrus cloud formation and evolution as well as aerosol-cloud-radiation interactions. With the newly implemented radiation scheme, the simulations of cloud cover and ice water path have been improved for cirrus clouds, with a more consistent comparison with the corresponding MODIS observations, especially for optically thin cirrus with an improvement of about 20% in the simulated mean ice water path.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1993

Coherent Doppler lidar signal covariance including wind shear and wind turbulence

The performance of coherent Doppler lidar is determined by the statistics of the coherent Doppler signal. The derivation and calculation of the covariance of the Doppler lidar signal is presented for random atmospheric wind fields with wind shear. The random component is described by a Kolmogorov turbulence spectrum. The signal parameters are clarified for a general coherent Doppler lidar system. There are two distinct physical regimes: one where the transmitted pulse determines the signal statistics and the other where the wind field dominates the signal statistics. The Doppler shift of the signal is identified in terms of the wind field and system parameters.

Frehlich, R. G.↗
DOE OSTI2019

Collaborative Research: Experimental and Theoretical Study of the Plasma Physics of Antihydrogen Generation and Trapping (Final Technical Report)

The charge-parity-time (CPT) invariance principle and the weak equivalence principle (WEP) undergird our current understanding of Nature. CPT, which is based on only Lorentz invariance, locality, and spin-statistics, is basic to local relativistic quantum field theory (QFT) models of physical laws. Testing established principles like these is a long-standing and honorable tradition in physics. Beginning with Newton’s Laws, many firmly established laws have been superseded, often to the great surprise of contemporaneous physicists. Certainly, CPT rests on firmer grounds than the earlier widely believed, but erroneous principles, of parity (P) and charge-parity (CP) invariance, but this does not necessarily make CPT inviolable. Tests of CPT confront the entire framework of QFT, and are quite distinct from most other “beyond the Standard Model” searches, which look for new physics within the QFT framework. With the failure, so far, to find new physics at the LHC or in dark matter searches, the motivation to test QFT itself is urgent. The revolutionary consequences of CPT and WEP violation would be so great that these principles are under intense scrutiny in many different systems. Here we focus on tests of CPT and WEP that employ antimatter, specifically antihydrogen. Such tests are difficult to realize experimentally, but constitute some of the simplest and most profound matter-antimatter comparisons that can be performed. Antimatter particles have been investigated since the prediction and subsequent discovery of the positron in the 1930s. That early discovery motivated the search for the antiproton, culminating in its creation and detection in Berkeley two decades later. As far as we know today, particles and their antiparticles share identical properties except for the sign of their charge. Antihydrogen is the simplest antimatter system—it is the antiparticle of the simplest atom, hydrogen. Differences between the spectra of the two are forbidden by CPT. The weak equivalence principle (WEP), demands that matter and antimatter accelerate identically in a gravitational field. WEP has never been subjected to a direct, free fall test using antimatter. Because of the weakness of the gravita tional force, direct antimatter WEP tests must employ a neutral particle, of which antihydrogen is the simplest to obtain. There are reasons to suspect that something is amiss with our understanding of antimatter. All generally accepted theories of the Big Bang predict that roughly equal amounts of matter and antimatter should have been created. Instead, very little antimatter has been observed in the Universe. This baryogenesis problem is arguably the most fundamental issue with the theory of the Big Bang. Its resolution may come from the discovery of enhanced CP violation, but could also come from CPT or WEP violation.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2016

Uncertainty quantification of effective nuclear interactions

We give a brief review on the development of phenomenological NN interactions and the corresponding quanti cation of statistical uncertainties. We look into the uncertainty of effective interactions broadly used in mean eld calculations through the Skyrme parameters and effective eld theory counter-terms by estimating both statistical and systematic uncertainties stemming from the NN interaction. We also comment on the role played by different tting strategies on the light of recent developments.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
NASA NTRS2024

A Physics-Based Antarctic Melt Detection Technique: Combining Advanced Microwave Scanning Radiometer 2, Radiative-Transfer Modeling, and Firn Modeling

Surface melt on ice shelves has been linked to hydrofracture and subsequent ice shelf breakup. Since the 1990s, scientists have been using microwave radiometers to detect melt on ice shelves and ice sheets by applying various statistical thresholding techniques to identify significant increases in brightness temperature that are associated with melt. In this study, instead of using a fixed threshold, we force the Snow Radiative Transfer model (SMRT) with outputs from the Community Firn Model (CFM) to create a dynamic, physics-based threshold for melt. In the process, we also combine our method with statistical thresholding techniques and produce microwave grain size information in the process. We run this “Hybrid Method” across the Larsen C ice shelf as well as thirteen sites on the Antarctic Ice Sheet. Melt and non-melt days from the Hybrid Method and three statistical thresholding techniques match with surface energy balance within 94±1%; the effect of melt on the passive microwave is mostly binary and thus largely detectable by statistical thresholding techniques as well as physics-based techniques. Rather than always replacing statistical thresholding techniques with the Hybrid Method, we recommend using the Hybrid Method in studies where melt volume or grain size is of interest. In this study, we show that the Hybrid Method can be used to (a) model dry snow brightness temperatures of Antarctic snow and (b) derive a measure of grain size; therefore, it is an important step forwards towards using firn and radiative transfer modeling to quantify melt rather than to simply detect melt days.

Marissa E Dattler↗
DOE OSTI2024

Probabilistic neural networks for improved analyses with phenomenological R -matrix

Here we present a method for measurement analyses based on probabilistic deep neural networks that provide several advantages over conventional analyses with phenomenological models. These include predicting physical quantities directly from data, the rapid generation of statistically robust uncertainties, and the ability to bypass some parameters that may induce ambiguities and complications in data analysis. As deep learning methods make predictions through “black boxes,” the uncertainty quantification is typically challenging. We use a probabilistic framework that provides thorough uncertainty quantification and is straightforward to follow in practice. With the network architecture based on the Transformer, we demonstrate the current method for predicting nuclear resonance parameters from scattering data using the phenomenological R-matrix model.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2020

Quantitative metrics for evaluating thermonuclear design codes and physics models applied to the National Ignition Campaign

In this study, physics models and design codes for hot dense plasmas undergoing thermonuclear burn are evaluated objectively using statistical metrics that compare the difference between calculations and data relative to the experimental uncertainties. The analysis is applied to the National Ignition Campaign (NIC) because it is relevant, comprehensive, and well documented. The statistics confirm that a key process afflicting NIC performance is mix driven by hydrodynamic instabilities as approximated here using the KL model [G. Dimonte and R. Tipton, Phys. Fluids 18, 085101 (2006)]. New physics models are also presented for instability-driven magnetic fields [B. Srinivasan et al., Phys. Rev. Lett. 108, 165002 (2012)] and the Coulomb logarithm for electron–ion thermal relaxation [G. Dimonte and J. Daligault, Phys. Rev. Lett. 101, 135001 (2008)]. The plasma-generated magnetic fields improve the agreement between code and data in a statistically significant manner by reducing the electron thermal conduction in the hot-spot via the Hall term. The Coulomb logarithm is presented in a numerically practical form that incorporates recent theoretical advances. However, it does not improve the agreement between the code and data because the thermal relaxation is so fast in non-ignited NIC plasmas that the typical 20% error does not change the tight coupling between electrons and ions. Even with these improved physics models, the one-dimensional simulations presented here are not able to describe the high-convergence NIC implosions in a statistically acceptable manner. Of the heroic multi-dimensional simulations of Clark et al. [Phys. Plasmas 23, 056302 (2016)], only those in three-dimensions satisfy the statistical acceptance criterion.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2019

Python GST Implementation (PyGSTi) v. 0.9

PyGSTi is a Python package for modeling and characterizing noise (errors) in small quantum information processors. In addition to being a basic framework for describing quantum circuits and noise models, it implements mainstream quantum characterization, verification, and validation (QCVV) protocols such as Gate Set Tomography (GST), Randomized Benchmarking (RB), Robust Phase Estimation, and Idle Tomography. It also implements prototype protocols used for timeseries analysis and crosstalk detection, all of which have the goal of better understanding the noise found in existing as-built experimental devices. The central protocol of pyGSTi (from where it derives its name) is Gate Set Tomography. GST is a theory and protocol for simultaneously estimating the state preparation, gate operations, and measurement effects of a physical system of one or many quantum bits (qubits). These estimates are based entirely on the statistics of experimental measurements, and their interpretation and analysis can provide a detailed understanding of the types of errors/imperfections in the physical system. In this way, GST provides not only a means of certifying the “goodness” of qubits but also a means of debugging (i.e. improving) them. The other protocols follow this similar pattern in that they use statistical inference and analysis of experimental data to estimate one or more properties of the noise in a device.

Nielsen, Erik↗
DOE OSTI2021

Data Science Meets Physical Organic Chemistry

At the heart of synthetic chemistry is the holy grail of predictable catalyst design. In particular, researchers involved in reaction development in asymmetric catalysis have pursued a variety of strategies toward this goal. This is driven by both the pragmatic need to achieve high selectivities and the inability to readily identify why a certain catalyst is effective for a given reaction. While empiricism and intuition have dominated the field of asymmetric catalysis since its inception, enantioselectivity offers a mechanistically rich platform to interrogate catalyst-structure response patterns that explain the performance of a particular catalyst or substrate. In the early stages of an asymmetric reaction development campaign, the overarching mechanism of the reaction, catalyst speciation, the turnover limiting step, and many other details are unknown or posited based on related reactions. Considering the unclear details leading to a successful reaction, initial enantioselectivity data are often used to intuitively guide the ultimate direction of optimization. However, if the conditions of the Curtin-Hammett principle are satisfied, then measured enantioselectivity can be directly connected to the ensemble of diastereomeric transition states (TSs) that lead to the enantiomeric products, and the associated free energy difference between competing TSs (ΔΔ G ‡ = - RT ln[( S )/( R )], where ( S ) and ( R ) represent the concentrations of the enantiomeric products). We, and others, speculated that this important piece of information can be leveraged to guide reaction optimization in a quantitative way. Although traditional linear free energy relationships (LFERs), such as Hammett plots, have been used to illuminate important mechanistic features, we sought to develop data science derived tools to expand the power of LFERs in order to describe complex reactions frequently encountered in modern asymmetric catalysis. Specifically, we investigated whether enantioselectivity data from a reaction can be quantitatively connected to the attributes of reaction components, such as catalyst and substrate structural features, to harness data for asymmetric catalyst design. In this context, we developed a workflow to relate computationally derived features of reaction components to enantioselectivity using data science tools. The mathematical representation of molecules can incorporate many aspects of a transformation, such as molecular features from substrate, product, catalyst, and proposed transition states. Statistical models relating these features to reaction outputs can be used for various tasks, such as performance prediction of untested molecules. Perhaps most importantly, statistical models can guide the generation of mechanistic hypotheses that are embedded within complex patterns of reaction responses. Overall, merging traditional physical organic experiments with statistical modeling techniques creates a feedback loop that enables both evaluation of multiple mechanistic hypotheses and future catalyst design. In this Account, we highlight the evolution and application of this approach in the context of a collaborative program based on chiral phosphoric acid catalysts (CPAs) in asymmetric catalysis.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2018

The Enskog Equation for Confined Elastic Hard Spheres

A kinetic equation for a system of elastic hard spheres or disks confined by a hard wall of arbitrary shape is derived. It is a generalization of the modified Enskog equation in which the effects of the confinement are taken into account and it is supposed to be valid up to moderate densities. From the equation, balance equations for the hydrodynamic fields are derived, identifying the collisional transfer contributions to the pressure tensor and heat flux. A Lyapunov functional, H[f], is identified. For any solution of the kinetic equation, H decays monotonically in time until the system reaches the inhomogeneous equilibrium distribution, that is a Maxwellian distribution with a density field consistent with equilibrium statistical mechanics.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1978

Statistical models of summer rainshowers derived from fine-scale radar observations

A statistical modeling of rain cell characteristics is presented; the modeling is based on high-resolution radar measurements of the cell structure. The analysis of the fine-scale three-dimensional structure of summer rainshowers in the mid-Atlantic region leads to statistical descriptions of the rain cells in terms of a variety of physical cell parameters. Core reflectivity profiles, contour area, and altitude extent of the cells have been determined along with the frequency of occurrence for various storm classes and categories. The statistical descriptions of the rain cells for the mid-Atlantic region were compared to descriptions for other areas, and the similarities and differences are described. Simplified models of rain cells based on the statistical descriptions are developed for the different rain categories as a function of frequency of occurrence.

Konrad, T. G.↗
DOE OSTI2024

Perspectives for Measuring Neutrino Cross-Sections at Short Baseline Detectors (ICARUS and DUNE-PRISM-like Detector) and Their Implications for Precision Physics

Neutrino physics is rapidly advancing with current and future experiments promising high-statistics data, enabling unprecedented accuracy in measuring Standard Model (SM) parameters and providing crucial insights into neutrino-nucleus interactions. The ICARUS experiment, utilizing LAr TPC technology, is pivotal in this growth. As the far detector of the SBN program at Fermilab, ICARUS is set to investigate the neutrino anomalies observed in LSND and MiniBooNE and prepare for future long-baseline experiments like DUNE. Positioned along the BNB and 5.75°off-axis from the NuMI beamline, ICARUS benefits from numerous neutrino interactions, allowing for precise neutrino-argon nucleus cross-section measurements, which will benchmark future DUNE measurements. This work presents the prospects and progress in measuring the cross-section of the NuMI νµ CC inclusive channel in the ICARUS experiment. High-statistics data will also enable precise determinations of fundamental parameters like the weak mixing angle and the neutrino charge radius (NCR), crucial for testing the SM and exploring potential new physics. This work includes studies of future near detectors, such as DUNE-PRISM, which are sensitive to radiative corrections in neutrino-electron scattering. The DUNE-PRISM configuration allows for the analysis of different neutrino energy spectra, promising accurate NCR measurements with controlled systematic uncertainties.

Moreno-Granados, Guadalupe↗
DOE OSTI2019

Machine learning for analysis of atomic spectral data

Physics based forward models are the basis on which many experimental diagnostics are interpreted. For some diagnostics, models can be computationally expensive which precludes their use in real time analysis. Reduced models have the potential to capture sufficient physics thereby enabling the desired real time analysis. Using statistical inference and machine learning techniques the application of reduced models for inversion of atomic spectral data used to diagnose magnetic fields in a plasma will be examined. Two approaches are considered, (a) a reduction of the forward model where traditional inversion can be performed on the proxy model, and (b) a reduction of the direct inverse where parameters are a function of measured signal. The resulting inversion is sufficiently fast to be utilized in an online context for digital twinning, and ultimately real-time prediction, design, and control of plasma systems, such as tokamaks. Furthermore, these methods will be demonstrated on both simulated and experimentally measured data.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2018

Universality for 1d Random Band Matrices: Sigma-Model Approximation

The paper continues the development of the rigorous supersymmetric transfer matrix approach to the random band matrices started in (J Stat Phys 164:1233–1260, 2016; Commun Math Phys 351:1009–1044, 2017). We consider random Hermitian block band matrices consisting of W×W random Gaussian blocks (parametrized by j,k∈Λ=[1,n]{sup d}∩ℤ{sup d}) with a fixed entry’s variance J{sub jk}=δ{sub j,k}W{sup −1}+βΔ{sub j,k}W{sup −2}, β>0 in each block. Taking the limit W→∞ with fixed n and β, we derive the sigma-model approximation of the second correlation function similar to Efetov’s one. Then, considering the limit β,n→∞, we prove that in the dimension d=1 the behaviour of the sigma-model approximation in the bulk of the spectrum, as β≫n, is determined by the classical Wigner–Dyson statistics.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

Extensions to models of the galaxy–halo connection

ABSTRACT We explore two widely used empirical models for the galaxy–halo connection, subhalo abundance matching (SHAM) and the halo occupation distribution (HOD), and compare them with the hydrodynamical simulation IllustrisTNG (TNG) for multiple statistics quantifying the galaxy distribution at $n_{\rm gal}\approx 1.3\times 10^{-3}\, ({\rm Mpc}\,h^{-1})^{-3}$. We observe that in their most straightforward implementations, both models fail to reproduce the two-point clustering measured in TNG. We find that SHAM models that use the relaxation velocity, Vrelax, and the peak velocity, Vpeak, perform best, and match the clustering reasonably well, although neither captures adequately the one-halo clustering. Splitting the total sample into sub-populations, we discover that SHAM overpredicts the clustering of high-mass, blue, star-forming, and late-forming galaxies and underpredicts that of low-mass, red, quiescent, and early-forming galaxies. We also study various baryonic effects, finding that subhaloes in the dark-matter-only simulation have consistently higher values of their SHAM-proxy properties than their full-physics counterparts. We then consider a 2D implementation of the HOD model augmented with a secondary parameter (environment, velocity anisotropy, σ2Rhalf-mass, and total potential) tuned so as to match the two-point clustering of the IllustrisTNG galaxies on large scales. We analyse these galaxy populations adopting alternative statistical tools such as galaxy–galaxy lensing, void–galaxy cross-correlations, and cumulants of the density field, finding that the hydrodynamical galaxy distribution disfavours σ2Rhalf-mass and the total potential as secondary parameters, while the environment and velocity anisotropy samples are consistent with full physics across all statistical probes examined. Our results demonstrate the power of examining multiple statistics for determining the secondary parameters that are vital for understanding the galaxy–halo connection.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2026

Improving statistical precision in Monte Carlo samples with negative weights via reweighting and uncertainty quantification

High statistical precision is critical for Monte Carlo (MC) samples in high energy physics and is degraded by negatively weighted events. This paper investigates a procedure to learn the relationship between the negative and positive weight distributions of any sample, allowing the reduction of statistical uncertainty by reweighting kinematically equivalent events with the same sign. A robust uncertainty quantification method is required for the practical application of such method. Two methods for the estimation of the reweighting uncertainty are developed: one at the event and another one at the final observable level. The latter method is strongly favored. The gains in statistical precision are then quantified. The method is demonstrated on Sherpa vector boson plus jets samples when using all generated events and when restricted to the signal region of a mock analysis. It is demonstrated to significantly reduce stochastic behavior in sparse MC samples while decreasing the overall uncertainty with a sufficiently well-known reweighting function.

Monte Carlo methods↗
DOE OSTI2018

Towards New Physics Through Precision Measurements in B$^0_S$ and $\tau$ Decays

Despite the correct and consistent description of the elementary particle interactions, there is evidence that the Standard Model (SM) is an incomplete theory. Solving the issues faced by the SM requires the introduction of new interactions and, if such interactions exist, they can be observed as deviations in the SM predictions or forbidden processes. Particularly of interest for this work are the B$^0_s$ mesons and $\tau$ leptons produced in the DØ and Belle II experiments, respectively, where the luminosity provides the statistics for high precision SM measurements and searches for new physics. In this work, the lifetime measurement of the B$^0_s$ meson in the decay channel $B^0_s \to J/\Psi f_0(980)$ is reported, which is a CP-odd final state. Using 10.4 fb$^{-1}$ of data collected with the DØ detector in Run II of the Tevatron, the lifetime of the CP-odd component of the B$^0_s$ meson is measured to be $\tau(B^0_s)$ = 1.70$\pm$0.14 (stat)$\pm$0.05 (syst) ps. This measurement is closely related to the $B^0_s - \bar{B^0_s}$ mixing and searches for new CP violation mechanisms. Additionally, the $\tau$ lepton physics program of Belle II is reviewed and two sensitivity studies are shown: (i) $\tau^- \to \eta \pi^-\pi^0 \nu_{\tau}$, which in the isospin symmetry limit SU(2) is a cross-check of consistency with the cross section $\sigma(e^+e^- \to \eta \pi^+ \pi^-$) at low energies; (ii) $\tau^- \to \eta \pi^- \nu_{\tau}$, which has never been observed and, given the suppression by G-parity, it allows the searches of scalar and tensor interactions. Estimations from 1 to 20 ab$^{-1}$ are shown, corresponding to the integrated luminosity expected during the first years of operation of Belle II.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2018

Higher order corrected thermodynamics and statistics of Kerr–Newman–Gödel black hole

In this paper, we consider the rotating charged Gödel black hole and study the effect of the higher order corrections of the entropy on the thermodynamics and statistics quantities of the Kerr–Newman–Gödel black hole. The leading order correction is logarithmic while higher-order terms are proportional to the inverse of the area of the black hole. We obtain modified thermodynamics and statistics and find that correction terms are important in the stability of the black hole.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗