With improved measurement and modelling technology, variability has emerged as an essential feature in non-equilibrium processes. While traditionally, mean values and variance have been heavily used, they are not appropriate in describing extreme events where a significant deviation from mean values often occurs. Furthermore, stationary Probability Density Functions (PDFs) miss crucial information about the dynamics associated with variability. It is thus critical to go beyond a traditional approach and deal with time-dependent PDFs. Here, we consider atmospheric data from the Whole Atmosphere Community Climate Model (WACCM) and calculate time-dependent PDFs and the information length from these PDFs, which is the total number of statistically different states that a system passes through in time. Time-dependent PDFs are shown to be non-Gaussian in general, and the information length calculated from these PDFs shed us a new perspective of understanding variabilities, correlation among different variables and regions. Specifically, we calculate time-dependent PDFs and information length and show that the information length tends to increase with the altitude albeit in a complex form. This tendency is more robust for flows/shears than temperature. Also, much similarity among flows and shears in the information length is found in comparison with the temperature. This means a stronger correlation among flows/shears because of a strong coupling through gravity waves in this particular WACCM model. We also find the increase of the information length with the latitude and interesting hemispheric asymmetry for flows/shears/temperature, a stronger anti-correlation (correlation) between flows/shears and temperature at a higher (low) latitude. These results also suggest the importance of high latitude/altitude in the information budge in the Earth's atmosphere, the spatial gradient of the information as a useful proxy for the transport of physical quantities.

Many systems in nature and laboratories are far from equilibrium and exhibit significant fluc- tuations, invalidating the key assumptions of small fluctuations and short memory time in or near equilibrium. A full knowledge of Probability Distribution Functions (PDFs), especially time- dependent PDFs, becomes essential in understanding far-from-equilibrium processes. We consider a stochastic logistic model with multiplicative noise, which has gamma distributions as stationary PDFs. We numerically solve the transient relaxation problem, and show that as the strength of the stochastic noise increases the time-dependent PDFs increasingly deviate from gamma distributions. For sufficiently strong noise a transition occurs whereby the PDF never reaches a stationary state, but instead forms a peak that becomes ever more narrowly concentrated at the origin. The addition of an arbitrarily small amount of additive noise regularizes these solutions, and re-establishes the existence of stationary solutions. In addition to diagnostic quantities such as mean value, standard deviation, skewness and kurtosis, the transitions between different solutions are analyzed in terms of entropy and information length, the total number of statistically distinguishable states that a system passes through in time.

This study addresses the central question that how can a precise measurement of the gluon Parton Distribution Function (PDF) be attained, thereby enhancing our understanding of the intricate quark-gluon composition within nucleons? Through a dual approach involving Quantum Chromodynamics (QCD) and high-energy particle collisions, the research combines computational tools and experimental techniques, focusing on protons and neutrons. Scattering experiments and advanced detectors are utilized to probe nucleon structure. Theoretical frameworks like QCD equations and the Momentum Sum Rule investigate Parton Distribution Functions at specific momentum fractions and energy scales. The primary goal is to achieve an accurate measurement of the gluon PDF, crucial for unraveling strong force complexities and internal nucleon dynamics. The study's theoretical plots and equations illustrate hypothetical behaviors of gluon distribution, quark distribution, and splitting functions, contributing to a deeper understanding of fundamental particles.