Entropy Of Gas Research Articles

Is the Voronoi Entropy a True Entropy? Comments on "Entropy, Shannon's Measure of Information and Boltzmann's H-Theorem", Entropy 2017, 19, 48.

The goal of this comment note is to express our considerations about the recent paper by A. Ben Naim (Entropy 2017, 19, 48). We strongly support the distinguishing between the Shannon measure of information and the thermodynamic entropy, suggested in the paper. We demonstrate that the Voronoi entropy should also be clearly distinguished from the entropy of a two-dimensional gas. Actually, the Voronoi entropy being an intensive value is the averaged Shannon measure of ordering for a given pattern.

Another physical expression of the entropy for an ideal photon gas

An expression of the entropy for massless photon gas in a unit volume is provided. A quantity, (Δx)T(Δpx)T, is introduced, which indicates the magnitude of an average uncertainty at certain temperature T. The entropy of the photon gas can be written in terms of thermal uncertainty (Δx)T(Δpx)T. The analytical derivation is carried out on the basis of fluctuation theory, for a canonical system in thermal equilibrium at T. The emphasis is given to the intimate connection between the uncertainty in the quantum mechanics and the entropy in macroscopic world. This new entropy formula of a zero mass m=0 ideal photon gas is discussed in comparison with that of a mass particle (m≠0) ideal gas.

Universal thermodynamic properties of the intracluster medium over two decades in radius in the X-COP sample

Context.The hot plasma in a galaxy cluster is expected to be heated to high temperatures through shocks and adiabatic compression. The thermodynamical properties of the gas encode information on the processes leading to the thermalization of the gas in the cluster’s potential well and on non-gravitational processes such as gas cooling, AGN feedback, shocks, turbulence, bulk motions, cosmic rays and magnetic field.Aims.In this work we present the radial profiles of the thermodynamic properties of the intracluster medium (ICM) out to the virial radius for a sample of 12 galaxy clusters selected from thePlanckall-sky survey. We determine the universal profiles of gas density, temperature, pressure, and entropy over more than two decades in radius, from 0.01R500to 2R500.Methods.We exploited X-ray information fromXMM-Newtonand Sunyaev-Zel’dovich constraints fromPlanckto recover thermodynamic properties out to 2R500. We provide average functional forms for the radial dependence of the main quantities and quantify the slope and intrinsic scatter of the population as a function of radius.Results.We find that gas density and pressure profiles steepen steadily with radius, in excellent agreement with previous observational results. Entropy profiles beyondR500closely follow the predictions for the gravitational collapse of structures. The scatter in all thermodynamical quantities reaches a minimum in the range [0.2 − 0.8]R500and increases outward. Somewhat surprisingly, we find that pressure is substantially more scattered than temperature and density.Conclusions.Our results indicate that once accreting substructures are properly excised, the properties of the ICM beyond the cooling region (R > 0.3R500) follow remarkably well the predictions of simple gravitational collapse and require few non-gravitational corrections.

Planck's experiment on gas mixing, Gibbs' paradox and the definition of entropy in statistical mechanics

The interpretation of entropy provided by statistical thermodynamics is not adequate to represent the thermodynamic entropy of the gas of noninteracting particles considered in this theory. Planck's thought experiment on reversible mixing and Gibbs' paradox provide perhaps the best-known evidence of this. The assumption that the internal energy of an ideal gas depends only on its temperature is introduced both in the kinetic theory of gases and in the classical thermodynamics. Such an assumption is no doubt adequate to deal with real gases at appropriately low pressures and high temperatures. However, the present paper shows that the same assumption implies that the entropy of an ideal gas, like its internal energy, must also depend only on temperature. The paper calculates the expression of the entropy function that is consistent with the internal energy function of the gas. From this expression, the thermodynamic entropy of the ideal gas – as distinct from its statistical entropy – is finally expressed in terms of statistical mechanics variables.

A simple link of information entropy of quantum systems with Newtonian [formula omitted] dependence of Verlinde’s entropic force

It is shown that the entropic force formula Fe=−λ∂S∕∂A leads to a Newtonian r−2 dependence. Here we employ the universal property of the information entropy S=a+blnN (N is the number of particles of a quantum system and A is the area containing the system). This property was previously obtained for fermionic systems (atoms, atomic clusters, nuclei and infinite Fermi systems i.e. electron gas, liquid 3He and nuclear matter) and bosonic ones (correlated boson-atoms in a trap). A similar dependence of the entropic force has been derived very recently by Plastino et al with a Bose or Fermi gas entropy, inspired by Verlinde’s conjecture (Verlide, 2011) that gravity is an emergent entropic force.

Entropy of hard square lattice gas with k distinct species of particles: coloring problems and vertex models

Coloring the faces of a two-dimensional square lattice with k distinct colors such that no two adjacent faces have the same color is considered by establishing a connection between the k-coloring problem and a generalized vertex model. Associating the colors with k distinct species of particles with an infinite repulsive force between nearest neighbors of the same type and zero chemical potential μ associated with each species, the number of ways [W(k)]N for large N is related to the entropy of the hard square lattice gas at close packing of the lattice, where N is the number of lattice sites. We discuss the evaluation of using the transfer matrix method with non-periodic boundary conditions imposed along at least one direction, and show the characteristic Toeplitz block structure of the transfer matrix. Using this result, we present some analytical calculations for non-periodic models that remain finite in one dimension. The case k = 3 is found to approach the exact result obtained by Lieb for the residual entropy of ice with periodic boundary conditions. Finally, we show, by explicit calculation of the contribution of subgraphs and the series expansion of W(k), that the generalized Pauling type estimate (which is based on a mean-field approximation) dominates at large values of k. We thus also provide an alternative series expansion for the chromatic polynomial of a regular square graph.

High entropy and evidence for a merger in the low surface brightness cluster Abell 2399

Abstract We present results of the X-ray analyses of the nearby cluster of galaxies Abell 2399 ($z$ = 0.058) using the XMM–Newton and Suzaku satellites. This cluster is classified as a low surface brightness (LSB) cluster. It has a bimodal structure in X-rays, and the X-ray-emission peaks are significantly offset from the peaks in gas temperature. By de-projecting the annular spectra, the temperature and electron density in the central r < 0.1 r200 region are measured to be 3.6 keV and 1.4 × 10−3 cm−3, respectively. This yields a very high gas entropy, ∼300 keV cm2, in the central region, comparable to the values observed in other LSB clusters. The scaled entropy of Abell 2399 is the highest among the REXCESS cluster sample. The spatial distribution of the member galaxies exhibits multiple substructures, the locations of which are significantly different from those in the X-ray image. We discovered a clear discontinuity in the X-ray brightness and temperature distributions in the western gas clump, which provides evidence of the presence of a cold front. Therefore, our results strongly suggest that A2399 has experienced a merger and that the high central entropy originates from the merger activity.

AGN feedback and multiphase gas in giant elliptical galaxies

Recent observations have found extended multi-phase gas in a significant fraction of massive elliptical galaxies. We perform high-resolution three-dimensional hydrodynamical simulations of two idealized elliptical galaxies -- one representing a typical galaxy characterized by initial conditions conducive to the development of thermal instability and the other one less likely to develop thermal instability -- in order to study the development of thermal instability and the formation of multi-phase structures. We analyze the interplay between radiative cooling, momentum-driven AGN feedback, star formation, and stellar feedback from both young and old stars. We find that in one class of elliptical galaxies, the entropy of the hot halo gas rises sharply as a function of radius, and the hot halo is thermally stable and run-away cooling can only happen in the very center of the galaxy. In other class of ellipticals, the hot halo gas has a cooling to free-fall time ratio close to 10, and the non-linear perturbation driven by AGN feedback can cause the hot gas to frequently precipitate into extended multi-phase filaments. Both multi- and single-phase elliptical galaxies experience cooling-driven AGN feedback cycles. Interestingly, AGN feedback maintains the multi- or single-phase nature of the halo but does not turn multi-phase galaxies into single-phase ones or vice versa. Some of the extended cold gas in the multi-phase galaxy also forms young stars. The level of star formation and its spatial distribution are in excellent agreement with {\it Hubble} observations of nearby elliptical galaxies.

Spin-disorder resistivity of random fcc-NiFe alloys

The spin-disorder resistivity (SDR) of a disordered fcc-(${\mathrm{Ni}}_{1\ensuremath{-}x},{\mathrm{Fe}}_{x}$) alloy is determined from first principles. We identify the SDR at and above the critical temperature with the residual resistivity of the corresponding paramagnetic state evaluated in the framework of the disordered local moment (DLM) model. The underlying electronic structure is determined by means of the tight-binding linear muffin-tin orbital method, which employs the coherent potential approximation (CPA) to describe both the DLM state and the chemical disorder in alloys. An extension of the DLM fixed-spin moment method for two independent magnetic moments is used and combined with the paramagnetic lattice gas entropy to determine local moments by minimizing the corresponding free energy. The effect of phonon scattering is included through the mapping of static atomic displacements into a multicomponent random alloy which is then treated in the CPA. Finally, the Kubo-Greenwood-CPA approach is employed to estimate the SDR. We also address the problem of the validity of the Matthiessen rule at the Curie point. Good agreement of calculated and measured SDR is obtained over the whole studied concentration range; the results point to the importance of nonzero Ni magnetic moments in the limit of pure nickel.

A Universal Entropy Profile for the Hot Atmospheres of Galaxies and Clusters within R2500

Abstract We present atmospheric gas entropy profiles for 40 early-type galaxies and 110 clusters spanning several decades of halo mass, atmospheric gas mass, radio jet power, and galaxy type. We show that within ∼0.1R 2500 the entropy profiles of low-mass systems, including ellipticals, brightest cluster galaxies, and spiral galaxies, scale approximately as K ∝ R 2/3. Beyond ∼0.1R 2500 entropy profiles are slightly shallower than the K ∝ R 1.1 profile expected from gravitational collapse alone, indicating that heating by active galactic nuclei (AGN) feedback extends well beyond the central galaxy. We show that the K ∝ R 2/3 entropy profile shape indicates that thermally unstable cooling is balanced by heating where the inner cooling and free-fall timescales approach a constant ratio. Hot atmospheres of elliptical galaxies have a higher rate of heating per gas particle compared to those of central cluster galaxies. This excess heating may explain why some central cluster galaxies are forming stars while most early-type galaxies have experienced no significant star formation for billions of years. We show that the entropy profiles of six lenticular and spiral galaxies follow the R 2/3 form. The continuity between central galaxies in clusters, giant ellipticals, and spirals suggests perhaps that processes heating the atmospheres of elliptical and brightest cluster galaxies are also active in spiral galaxies.

Suzaku observations of low surface brightness cluster Abell 1631

Abstract We present analysis results for a nearby galaxy cluster Abell 1631 at z = 0.046 using the X-ray observatory Suzaku. This cluster is categorized as a low X-ray surface brightness cluster. To study the dynamical state of the cluster, we conduct four-pointed Suzaku observations and investigate physical properties of the Mpc-scale hot gas associated with the A 1631 cluster for the first time. Unlike relaxed clusters, the X-ray image shows no strong peak at the center and an irregular morphology. We perform spectral analysis and investigate the radial profiles of the gas temperature, density, and entropy out to approximately 1.5 Mpc in the east, north, west, and south directions by combining with the XMM-Newton data archive. The measured gas density in the central region is relatively low (a few ×10−4 cm−3) at the given temperature (∼2.9 keV) compared with X-ray-selected clusters. The entropy profile and value within the central region (r < 0.1 r200) are found to be flatter and higher (≳400 keV cm2). The observed bolometric luminosity is approximately three times lower than that expected from the luminosity–temperature relation in previous studies of relaxed clusters. These features are also observed in another low surface brightness cluster, Abell 76. The spatial distributions of galaxies and the hot gas appear to be different. The X-ray luminosity is relatively lower than that expected from the velocity dispersion. A post-merger scenario may explain the observed results.

Thermal and magnetic properties of electron gas in toroidal quantum dot

One-electron states in a toroidal quantum dot in the presence of an external magnetic field have been considered. The magnetic field operator and the Schrodinger equation have been written in toroidal coordinates. The dependence of one-electron energy spectrum and wave function on the geometrical parameters of a toroidal quantum dot and magnetic field strength have been studied. The energy levels are employed to calculate the canonical partition function, which in its turn is used to obtain mean energy, heat capacity, entropy, magnetization, and susceptibility of noninteracting electron gas. The possibility to control the thermodynamic and magnetic properties of the noninteracting electron gas via changing the geometric parameters of the QD, magnetic field, and temperature, was demonstrated.

Information Entropy of Liquid Metals.

Correlations reduce the configurational entropies of liquids below their ideal gas limits. By means of first-principles molecular dynamics simulations, we obtain accurate pair correlation functions of liquid metals, then subtract the mutual information content of these correlations from the ideal gas entropies to predict the absolute entropies over a broad range of temperatures. We apply this method to liquid aluminum and copper and demonstrate good agreement with experimental measurements; then, we apply it to predict the entropy of a liquid aluminum-copper alloy. Corrections due to electronic entropy and many-body correlations are discussed.

An alternative expression to the Sackur-Tetrode entropy formula for an ideal gas

An expression for the entropy of a monoatomic classical ideal gas is known as the Sackur-Tetrode equation. This pioneering investigation about 100 years ago incorporates quantum considerations. The purpose of this paper is to provide an alternative expression for the entropy in terms of the Heisenberg uncertainty relation. The analysis is made on the basis of fluctuation theory, for a canonical system in thermal equilibrium at temperature T. This new formula indicates manifestly that the entropy of macroscopic world is recognized as a measure of uncertainty in microscopic quantum world. The entropy in the Sackur-Tetrode equation can be re-interpreted from a different perspective viewpoint. The emphasis is on the connection between the entropy and the uncertainty relation in quantum consideration.

The pressure and entropy of a unitary Fermi gas with particle-hole fluctuation

We calculate the pressure and entropy of a unitary Fermi gas based on universal relations combined with our previous prediction of energy which was calculated within the framework of the non-self-consistent T-matrix approximation with particle-hole fluctuation. The resulting entropy and pressure are compared with the experimental data and the theoretical results without induced interaction. For entropy, we find good agreement between our results with particle-hole fluctuation and the experimental measurements reported by ENS group and MIT experiment. For pressure, our results suffer from a systematic upshift compared to MIT data.

Entropy of the Bose-Einstein-condensate ground state: Correlation versus ground-state entropy

Calculation of the entropy of an ideal Bose Einstein Condensate (BEC) in a three dimensional trap reveals unusual, previously unrecognized, features of the Canonical Ensemble. It is found that, for any temperature, the entropy of the Bose gas is equal to the entropy of the excited particles although the entropy of the particles in the ground state is nonzero. We explain this by considering the correlations between the ground state particles and particles in the excited states. These correlations lead to a correlation entropy which is exactly equal to the contribution from the ground state. The correlations themselves arise from the fact that we have a fixed number of particles obeying quantum statistics. We present results for correlation functions between the ground and excited states in Bose gas, so to clarify the role of fluctuations in the system. We also report the sub-Poissonian nature of the ground state fluctuations.

Canonical ensemble ground state and correlation entropy of Bose–Einstein condensate

Constraint of a fixed total number of particles yields a correlation between the fluctuation of particles in different states in the canonical ensemble. Here we show that, below the temperature of Bose–Einstein condensation (BEC), the correlation part of the entropy of an ideal Bose gas is cancelled by the ground-state contribution. Thus, in the BEC region, the thermodynamic properties of the gas in the canonical ensemble can be described accurately in a simplified model which excludes the ground state and assumes no correlation between excited levels.

Analysis of the Mean Field Free Energy Functional of Electrolyte Solution with Nonhomogenous Boundary Conditions and the Generalized PB/PNP Equations with Inhomogeneous Dielectric Permittivity

The energy functional, the governing partial differential equation(s) (PDE), and the boundary conditions need to be consistent with each other in a modeling system. In electrolyte solution study, people usually use a free energy form of an infinite domain system (with vanishing potential boundary condition) and the derived PDE(s) for analysis and computing. However, in many real systems and/or numerical computing, the objective domain is bounded, and people still use the similar energy form, PDE(s), but with different boundary conditions, which may cause inconsistency. In this work, (1) we present a mean field free energy functional for the electrolyte solution within a bounded domain with either physical or numerically required artificial boundary. Apart from the conventional energy components (electrostatic potential energy, ideal gas entropy term, and chemical potential term), new boundary interaction terms are added for both Neumann and Dirichlet boundary conditions. These new terms count for physical...

Two-phase thermodynamic model for computing entropies of liquids reanalyzed.

The two-phase thermodynamic (2PT) model [S.-T. Lin et al., J. Chem. Phys. 119, 11792-11805 (2003)] provides a promising paradigm to efficiently determine the ionic entropies of liquids from molecular dynamics. In this model, the vibrational density of states (VDoS) of a liquid is decomposed into a diffusive gas-like component and a vibrational solid-like component. By treating the diffusive component as hard sphere (HS) gas and the vibrational component as harmonic oscillators, the ionic entropy of the liquid is determined. Here we examine three issues crucial for practical implementations of the 2PT model: (i) the mismatch between the VDoS of the liquid system and that of the HS gas; (ii) the excess entropy of the HS gas; (iii) the partition of the gas-like and solid-like components. Some of these issues have not been addressed before, yet they profoundly change the entropy predicted from the model. Based on these findings, a revised 2PT formalism is proposed and successfully tested in systems with Lennard-Jones potentials as well as many-atom potentials of liquid metals. Aside from being capable of performing quick entropy estimations for a wide range of systems, the formalism also supports fine-tuning to accurately determine entropies at specific thermal states.

Entropy of a box of gas in an external gravitational field revisited

Earlier it was shown that the entropy of an ideal gas, contained in a box and moving in a gravitational field, develops an area dependence when it approaches the horizon of a static, spherically symmetric spacetime. Here we extend the above result in two directions; viz., to (a) the stationary axisymmteric spacetimes and (b) time dependent cosmological spacetimes evolving asymptotically to the de Sitter or the Schwarzschild de Sitter spacetimes. While our calculations are exact for the stationary axisymmetric spacetimes, for the cosmological case we present an analytical expression of the entropy when the spacetime is close to the de Sitter or the Schwarzschild de Sitter spacetime. Unlike the static spacetimes, there is no hypersurface orthogonal timelike Killing vector field in these cases. Nevertheless, the results hold and the entropy develops an area dependence in the appropriate limit.