Incomplete nonextensive statistics and the zeroth law of thermodynamics

On the basis of the entropy of incomplete statistics （IS） and the joint probability factorization condition, two controversial problems existing in IS are investigated： one is what expression of the internal energy is reasonable for a composite system and the other is whether the traditional zeroth law of thermodynamics is suitable for IS. Some new equivalent expressions of the internal energy of a composite system are derived through accurate mathematical calculation. Moreover, a self-consistent calculation is used to expound that the zeroth law of thermodynamics is also suitable for IS, but it cannot be proven theoretically. Finally, it is pointed out that the generalized zeroth law of thermodynamics for incomplete nonextensive statistics is unnecessary and the nonextensive assumptions for the composite internal energy will lead to mathematical contradiction.

Outlier Detection of Air Quality for Two Indian Urban Cities Using Functional Data Analysis

Human living would be impossible without air quality. Consistent advancements in practically every aspect of contemporary human life have harmed air quality. Everyday industrial, transportation, and home activities turn up dangerous contaminants in our surroundings. This study investigated two years’ worth of air quality and outlier detection data from two Indian cities. Studies on air pollution have used numerous types of methodologies, with various gases being seen as a vector whose components include gas concentration values for each observation per-formed. We use curves to represent the monthly average of daily gas emissions in our technique. The approach, which is based on functional depth, was used to find outliers in the city of Delhi and Kolkata’s gas emissions, and the outcomes were compared to those from the traditional method. In the evaluation and comparison of these models’ performances, the functional approach model studied well.

New Probability Distributions in Astrophysics: IV. The Relativistic Maxwell-Boltzmann Distribution

Two relativistic distributions which generalize the Maxwell Boltzman (MB) distribution are analyzed: the relativistic MB and the Maxwell-Jüttner (MJ) distribution. For the two distributions, we derived in terms of special functions the constant of normalization, the average value, the second moment about the origin, the variance, the mode, the asymptotic behavior, approximate expressions for the average value as function of the temperature and the connected inverted expressions for the temperature as function of the average value. Two astrophysical applications to the synchrotron emission in presence of the magnetic field and the relativistic electrons are presented.

New Probability Distributions in Astrophysics: III. The Truncated Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann (MB) distribution for velocities in ideal gases is usually defined between zero and infinity. A double truncated MB distribution is here introduced and the probability density function, the distribution function, the average value, the rth moment about the origin, the root-mean-square speed and the variance are evaluated. Two applications are presented: 1) a numerical relationship between root-mean-square speed and temperature, and 2) a modification of the formula for the Jeans escape flux of molecules from an atmosphere.

Wave Mechanics or Wave Statistical Mechanics

By comparison between equations of motion of geometrical optics and that of classical statistical mechanics, this paper finds that there should be an analogy between geometrical optics and classical statistical mechanics instead of geometrical mechanics and classical mechanics. Furthermore, by comparison between the classical limit of quantum mechanics and classical statistical mechanics, it finds that classical limit of quantum mechanics is classical statistical mechanics not classical mechanics, hence it demonstrates that quantum mechanics is a natural generalization of classical statistical mechanics instead of classical mechanics. Thence quantum mechanics in its true appearance is a wave statistical mechanics instead of a wave mechanics.

Population I Cepheids and understanding star formation history of the Small Magellanic Cloud

In this paper, we study the age and spatial distributions of Cepheids in the Small Magellanic Cloud （SMC） as a function of their ages using data from the OGLE III photometric catalogue. A period - age relation derived for Classical Cepheids in the Large Magellanic Cloud （LMC） has been used to find the ages of Cepheids. The age distribution of the SMC Classical Cepheids is found to have a peak at log（Age） ： 8.40 ± 0.10 which suggests that a major star formation event might have occurred in the SMC about 250 4- 50 Myr ago. It is believed that this star forming burst had been triggered by close interactions of the SMC with the LMC and/or the Milky Way. A comparison of the observed spatial distributions of the Cepheids and open star clusters has also been carried out to study the star formation scenario in the SMC.

Spatial-temporal variability of the fluctuation of soil temperature in the Babao River Basin, Northwest China

The Babao River Basin is the "water tower" of the Heihe River Basin.The combination of vulnerable ecosystems and inhospitable natural environments substantially restricts the existence of humans and the sustainable development of society and environment in the Heihe River Basin.Soil temperature(ST) is a critical soil variable that could affect a series of physical,chemical and biological soil processes,which is the guarantee of water conservation and vegetation growth in this region.To measure the temporal variation and spatial pattern of ST fluctuation in the Babao River Basin,fluctuation of ST at various depths were analyzed with ST data at depths of 4,10 and 20 cm using classical statistical methods and permutation entropy.The study results show the following: 1) There are variations of ST at different depths,although ST followed an obvious seasonal law.ST at shallower depths is higher than at deeper depths in summer,and vice versa in winter.The difference of ST between different depths is close to zero when ST is near 5℃ in March or –5℃ in September.2) In spring,ST at the shallower depths becomes higher than at deeper depths as soon as ST is above –5℃;this is reversed in autumn when ST is below 5℃.ST at a soil depth of 4 cm is the first to change,followed by ST at 10 and 20 cm,and the time that ST reaches the same level is delayed for 10–15 days.In chilling and warming seasons,September and February are,respectively,the months when ST at various depths are similar.3) The average PE values of ST for 17 sites at 4 cm are 0.765 in spring > 0.764 in summer > 0.735 in autumn > 0.723 in winter,which implies the complicated degree of fluctuations of ST.4) For the variation of ST at different depths,it appears that Max,Ranges,Average and the Standard Deviation of ST decrease by depth increments in soil.Surface soil is more complicated because ST fluctuation at shallower depths is more pronounced and random.The average PE value of ST for 17sites are 0.863 at a depth of 4 cm > 0.818 at 10 cm > 0.744 at 20 cm.5) For the variation of ST at different elevations,it appears that Max,Ranges,Average,Standard Deviation and ST fluctuation decrease with increasing elevation at the same soil depth.And with the increase of elevation,the decrease rates of Max,Range,Average,Standard Deviation at 4 cm are –0.89℃/100 m,–0.94℃/100 m,–0.43℃/100 m,and –0.25℃/100 m,respectively.In addition,this correlation decreased with the increase of soil depth.6) Significant correlation between PE values of ST at depths of 4,10 and 20 cm can easily be found.This finding implies that temperature can easily be transmitted within soil at depths between 4 and 20 cm.7) For the variation of ST on shady slope and sunny slope sides,it appears that the PE values of ST at 4,10 and 20 cm for 8 sites located on shady slope side are 0.868,0.824 and 0.776,respectively,whereas they are 0.858,0.810 and 0.716 for 9 sites located on sunny slope side.