Number Of Possible Microstates Research Articles

Quantum Entropy and the Mechanism of Gravity

Quantum entropy measures the number of possible microstates of position and momentum due to quantum uncertainty. Black hole entropy is an example of quantum entropy. Classical entropy, like the Gibbs entropy of a gas, does not depend on quantum uncertainty. This paper is the first to define the quantum entropy of ordinary matter, termed “atomic entropy”. The atomic entropy at the surface of a spherical object depends on its mass and is inversely proportional to its surface area. The entropy scale factor (ESF) is a theory that all changes in the scale of spacetime are due to differences in entropy. Defining atomic entropy allows the ESF to make predictions for systems including ordinary matter. In the ESF, gravitational objects are attracted to one another because gravitational entropy, a form of quantum entropy, increases as the distance between these objects decreases. The increase in gravitational entropy increases the total entropy of the system, in accordance with the second law of thermodynamics. This paper is also the first to use the gradients in the scale of space and time predicted by the ESF to derive Newtonian gravity. Explaining the mechanism of gravity in this way provides a link between quantum physics and gravity.

New Numerical Approach to Calculate Microstates of Equivalent and Non-Equivalent Electrons

A term symbol is used to describe atomic microstate states, which give the multiplicity and total angular momentum of the atomic state. Russel Sauder coupling scheme is used to generate terms of equivalent and non-equivalent electronic configurations. For equivalent electrons, the terms are calculated using Pauli’s principle, and the number of terms is limited and is calculated by the combination rule. The total possible electrons and total available electrons are used in the combination formula. In case of non-equivalent electrons, the number of terms are found by the permutation rule. The number of terms for equivalent electrons is less than the terms for non-equivalent electrons. The number of possible microstates for p2 and d5 configurations are 15 and 252 respectively. While the number of final microstates for 1p2p and 3d4d configurations are 36 and 100. In the proposed study, a Python programme was developed that generates the microstate according to filled and half-filled subshell electronic configurations for equivalent, non-equivalent, and combinations of both. Examples of microstates for non-equivalent electrons of configuration 1s2s, sp, sd, ss, 2p3p, pd, pf, 3d4d, df, 4f5f and for equivalent electrons of configuration su, pv, dx, and f y are presented.

Phase diagram of brittle fracture in the semi-grand-canonical ensemble.

We present a simulation method to assess the quasistatic fracture resistance of materials. Set within a semi-grand-canonical Monte Carlo (SGCMC) simulation environment, an auxiliary field-the bond rupture potential-is introduced to generate a sufficiently large number of possible microstates in the semi-grand-canonical ensemble, and associated energy and bond fluctuations. The SGCMC approach permits identifying the full phase diagram of brittle fracture for harmonic and nonharmonic bond potentials, analogous to the gas-liquid phase diagram, with the equivalent of a liquidus line ending in a critical point. The phase diagram delineates a solid phase, a fractured phase, and a gas phase, and provides clear evidence of a first-order phase transition intrinsic to fracture. Moreover, energy and bond fluctuations generated with the SGCMC approach permit determination of the maximum energy dissipation associated with bond rupture, and hence of the fracture resistance of a widespread range of materials that can be described by bond potentials.

Some thermodynamics modifications by the least length assumption via the microcanonical scheme

Here, we consider a quantum gravity effect that leads to the correction of the number of possible microstates for some microcanonical ensembles. Via the quantum gravity theories, it is existed a physical least length, that as a presupposition, modifies the momentums of the constituent elements of a statistical ensemble. This generalized momentum modifies the ordinary volume element of the phase space and affects the density of the present microstates. Consequently, the macroscopic properties of the ensemble are corrected which especially could be seen for high energy systems. Here, the induced modifications to the density of states are obtained for the microcanonical ensembles of the ideal gas, harmonic oscillator, and ultrarelativistic ideal gas. In each case, the changes induced to the volume of the momentum hyperspace are approximately calculated and their outgoings thermodynamics are also discussed. The results are compared with the corresponding canonical formalism. Besides, as an application, we investigate the modification of the spectrum of blackbody radiation, by using the Bose–Einstein statistics.

Boltzmann Entropy for the Spatial Information of Raster Data

Abstract. Entropy is an important concept that originated in thermodynamics. It is the subject of the famous Second Law of Thermodynamics, which states that “the entropy of a closed system increases continuously and irrevocably toward a maximum” (Huettner 1976, 102) or “the disorder in the universe always increases” (Framer and Cook 2013, 21). Accordingly, it has been widely regarded as an ideal measure of disorder. Its computation can be theoretically performed according to the Boltzmann equation, which was proposed by the Austrian physicist Ludwig Boltzmann in 1872. In practice, however, the Boltzmann equation involves two problems that are difficult to solve, that is the definition of the macrostate of a system and the determination of the number of possible microstates in the microstate. As noted by the American sociologist Kenneth Bailey, “when the notion of entropy is extended beyond physics, researchers may not be certain how to specify and measure the macrostate/microstate relations” (Bailey 2009, 151). As a result, this entropy (also referred to as Boltzmann entropy and thermodynamic entropy) has remained largely at a conceptual level. In practice, the widely used entropy is actually proposed by the American mathematician, electrical engineer, and cryptographer Claude Elwood Shannon in 1948, hence the term Shannon entropy. Shannon entropy was proposed to quantify the statistical disorder of telegraph messages in the area of communications. The quantification result was interpreted as the information content of a telegraph message, hence also the term information entropy. This entropy has served as the cornerstone of information theory and was introduced to various fields including chemistry, biology, and geography. It has been widely utilized to quantify the information content of geographic data (or spatial data) in either a vector format (i.e., vector data) or a raster format (i.e., raster data). However, only the statistical information of spatial data can be quantified by using Shannon entropy. The spatial information is ignored by Shannon entropy; for example, a grey image and its corresponding error image share the same Shannon entropy. Therefore, considerable efforts have been made to improve the suitability of Shannon entropy for spatial data, and a number of improved Shannon entropies have been put forward. Rather than further improving Shannon entropy, this study introduces a novel strategy, namely shifting back from Shannon entropy to Boltzmann entropy. There are two advantages of employing Boltzmann entropy. First, as previously mentioned, Boltzmann entropy is the ideal, standard measure of disorder or information. It is theoretically capable of quantifying not only the statistical information but also the spatial information of a data set. Second, Boltzmann entropy can serve as the bridge between spatial patterns and thermodynamic interpretations. In this sense, the Boltzmann entropy of spatial data may have wider applications. In this study, Boltzmann entropy is employed to quantify the spatial information of raster data, such as images, raster maps, digital elevation models, landscape mosaics, and landscape gradients. To this end, the macrostate of raster data is defined, and the number of all possible microstates in the macrostate is determined. To demonstrate the usefulness of Boltzmann entropy, it is applied to satellite remote sensing image processing, and a comparison is made between its performance and that of Shannon entropy.

STATISTICAL MECHANICS OF COLLISIONLESS ORBITS. I. ORIGIN OF CENTRAL CUSPS IN DARK-MATTER HALOS

We present an equilibrium statistical mechanical theory of collisionless self-gravitational systems with isotropic velocity distributions. Compared to existing standard theories, we introduce two changes: (1) the number of possible microstates is computed in energy (orbit) space rather than phase space and (2) low occupation numbers are treated more appropriately than using Stirling's approximation. Combined, the two modifications predict that the relaxed parts of collisionless self-gravitating systems, such as dark-matter halos, have a differential energy distribution N(E) ~ [exp(phi_0 - E) - 1], dubbed "DARKexp". Such systems have central power-law density cusps rho(r) ~ r^-1, which suggests a statistical mechanical origin of cusps in simulated dark-matter halos.

A comprehensive thermodynamic theory of the cross-effect in isothermal diffusion (“Hertz effect”) in a multicomponent gas, liquid, or solid

The cross-effect between diffusion fluxes in a nonuniform, multicomponent mixture is the cause of diffusion of a component against its own concentration gradient. Gustav Hertz was the first to use this cross-effect for the separation of isotopes. The calculation of the magnitude of the cross-effect, which we call Hertz effect, is possible if all coefficients of the diffusion tensor of the mixture are known. For dilute gases, kinetic gas theory enables the calculation of the diffusion tensor, but calculation methods for dense gases, liquids, and solids require input of measured binary or tracer diffusion coefficients. This paper presents for the first time a calculation method for the magnitude of the Hertz effect without using input of any measured diffusion coefficient. Input is required from the equation-of-state of the mixture and from kinetic gas theory. The theory is based on the principle of a maximum number of possible microstates in the stationary, nonequilibrium state. The theory has been compared with all measured diffusion data of mixtures that can be represented by our phase-behavior package. There is fair to good agreement between measurement and theory, but the Hertz effect in these mixtures is small. A test against diffusion data of highly nonideal mixtures is therefore recommended.

A thermodynamic theory of the Soret effect in a multicomponent liquid

A phenomenological theory of the Soret effect in a multicomponent mixture is presented. The theory is applicable to mixtures in which the Soret effect is determined mainly by the energetic interaction between the molecules, such as liquids and dense gases, and it is based on the assumption that the steady state has a maximum number of possible microstates. The equations derived for the separation enable the thermal diffusion factor αT of each component to be calculated using only an equation of state for the mixture and the standard state enthalpy. With a sufficiently accurate equation of state, the calculated αT values correspond well with the data on measured αT values of liquids and dense gases and show a better correspondence than Haase’s theory. The method presented may be applicable to other coupled transport phenomena in the steady state.