Classical Ideal Gas Research Articles

Phase Space Cell in Nonextensive Classical Systems

We calculate the phase space volume $\Omega$ occupied by a nonextensive system of $N$ classical particles described by an equilibrium (or steady-state, or long-term stationary state of a nonequilibrium system) distribution function, which slightly deviates from Maxwell-Boltzmann (MB) distribution in the high energy tail. We explicitly require that the number of accessible microstates does not change respect to the extensive MB case. We also derive, within a classical scheme, an analytical expression of the elementary cell that can be seen as a macrocell, different from the third power of Planck constant. Thermodynamic quantities like entropy, chemical potential and free energy of a classical ideal gas, depending on elementary cell, are evaluated. Considering the fractional deviation from MB distribution we can deduce a physical meaning of the nonextensive parameter $q$ of the Tsallis nonextensive thermostatistics in terms of particle correlation functions (valid at least in the case, discussed in this work, of small deviations from MB standard case). pacs: 05.20.-y, 05.70.-a keywords: Classical Statistical Mechanics, Thermodynamics

On the Statistics of an Ideal Gas with Mass Fluctuations and the Boltzman Ergodic Theorem: The Combined Boltzman–Tsallis Statistics

In this letter, we evaluate the effective velocities probability distribution associated to a classical ideal gas in thermodynamical equilibrium, but in the presence of fluctuations of the (inertial) mass parameter of the molecules. We obtain in the context of linear response theory of the mass fluctuations and the usual Boltzman framework the combined statistics of Boltzman and Tsallis as an outcome of the study for our effective Fractal gas distribution probability. This result explains from first principles the Tsallis-inverse power law distribution previously only speculated on in the literature. Finally, we give a new mathematical proof of the Boltzman ergodic theorem in classical statistics mechanics.

THERMODYNAMICS OF A CLASSICAL IDEAL GAS AT ARBITRARY TEMPERATURES

We propose a fundamental relation for a classical ideal gas that is valid at all temperatures with remarkable accuracy. All thermodynamical properties of classical ideal gases can be deduced from this relation at arbitrary temperature.

A non-linear determination of the distribution function of degenerate gases with an application to semiconductors

In this paper we consider an extended thermodynamic model for degenerate gases in which the first twenty moments of the gas distribution function are used as state variables. Exploiting the maximum entropy principle, we determine an analytic expression for the non-equilibrium distribution function of Bose and Fermi gases by means of a non-linear expansion with respect to the local thermodynamical equilibrium. Once the distribution function is given, we find the constitutive functions which appear in the fluxes of the balance equations. Explicit results can be obtained for classical ideal gases and strongly degenerate Bose and Fermi gases. We also consider an application to silicon semiconductors for which we test the accuracy of the closure relations by a comparison with Monte Carlo results.

Joule–Thomson coefficients of quantum ideal-gases

The temperature drop of a gas divided by its pressure drop under constant enthalpy conditions is called the Joule–Thomson coefficient (JTC) of the gas. The JTC of an ideal gas is equal to zero since its enthalpy depends on only temperature. On the other hand, this is only true for classical ideal gas which obeys the classical ideal gas equation of state, pV=mRT. Under sufficiently low-temperature or high-pressure conditions, the quantum nature of gas particles becomes important and an ideal gas behaves like a quantum ideal gas instead of a classical one. In such a case, enthalpy becomes dependent on both temperature and pressure. Therefore, JTC of a quantum ideal gas is not equal to zero. In this work, the contribution of purely quantum nature of gas particles on JTC is examined. JTCs of monatomic Bose and Fermi type quantum ideal gases are derived. Their variations with temperature are examined for different pressure values. It is shown that JTC of a Bose gas is always greater than zero. Minimum value of temperature is limited by the Bose–Einstein condensation phenomena under the constant enthalpy condition. On the other hand, it is seen that JTC of a Fermi gas is always lower than zero and there is not any limitation on its temperature. For high temperature values, JTCs of Bose and Fermi gases go to zero since the quantum nature of gas particles becomes negligible. Moreover, variation of temperature versus pressure under the constant enthalpy condition is also examined. Consequently, it is understood that the quantum nature of a Bose-type gas contributes to the positive values of JTC while the quantum nature of a Fermi type gas contributes to the negative values of JTC. Therefore, a Bose-type gas is more suitable for cryogenic refrigeration systems.

Re-Optimisation of Otto Power Cycles Working with Ideal Quantum Gases

To analyse the effect of quantum degeneracy of a working gas on the work optimisation of an ideal gas cycle, Otto power cycles working with ideal Bose and Fermi type quantum gases are considered. They are called Bose and Fermi Otto cycles respectively. Expressions for their net work outputs per cycle are derived by using the corrected ideal gas equation of state, which considers the quantum degeneracy of the working gas. For Bose and Fermi Otto cycles, the maximum net work output condition (MWOC) is re-derived. It is seen that this condition is different from that of the classical Otto cycle, which works with classical ideal gases. Therefore, expressions for maximum work output and efficiency at MWOC become different from their classical expressions. These new expressions are more general expressions which are valid under both classical and quantum ideal gas conditions. It is shown that maximum work output and efficiency at MWOC of a Bose Otto cycle can be higher than that of a classical Otto cycle. On the other hand, both quantities of the Fermi Otto cycle are always lower than their classical values.

Brayton refrigeration cycles working under quantum degeneracy conditions

At sufficiently low temperatures, quantum degeneracy of gas particles becomes important and an ideal gas deviates from the classical ideal-gas behaviour. In such a case, an ideal gas is called a quantum ideal gas. For quantum ideal gases, a corrected equation of state, which considers the quantum behaviour of gas particles, is used instead of the classical one. It is valid for both quantum and classical ideal-gases and it is reduced to a classical ideal-gas equation-of-state, under the classical gas conditions. There are two types of quantum ideal-gases. One of them is the Bose type and the other is the Fermi type. Here, Brayton refrigeration cycles working with Bose and Fermi type ideal quantum gases are considered and they are called Bose and Fermi Brayton cycles respectively. Coefficients of performance and refrigeration loads of these cycles are derived by using the corrected equation of state. It is seen that refrigeration loads are different from those of the classical Brayton cycle, which works with the classical ideal gas. On the other hand, coefficients of performance of these cycles are not effected by the quantum degeneracy of the refrigerant and they are the same as that of the classical cycle. Variations of the refrigeration load with low temperature ( T L) and low pressure ( P L) are examined. Under the quantum degeneracy conditions, it is shown that the refrigeration load of the Bose Brayton cycle is always greater than that of the classical Brayton cycle. On the contrary, the refrigeration load of the Fermi Brayton cycle is always lower than that of the classical one. Moreover, the minimum value of T L for the Bose Brayton cycle is restricted by the Bose–Einstein condensation temperature for a given value of P L.

The improvement effect of quantum degeneracy on the work from a Carnot cycle

The efficiency of a Carnot ( η C) cycle is independent of the physical properties of the working gas. Therefore, η C does not change due to quantum degeneracy of the gas. On the other hand, cycle work depends on the physical properties of the working gas since it is determined by the equation of state of the gas. Therefore, cycle work can be influenced by the quantum degeneracy of working gas. Here, Carnot power cycles working with ideal Bose and Fermi gases are examined under quantum degeneracy conditions. They are called Bose and Fermi Carnot cycles respectively. Cycle works of Bose and Fermi Carnot cycles ( W B and W F) are derived. By dividing these works into the work of the classical Carnot power cycle ( W C), which works with classical ideal gas, work ratios are defined as R W B= W B W C and R W F= W F W C . Variations of R W B and R W F with high temperature of the cycle ( T H) are examined for a given temperature ratio τ= T L T H and specific volume ratio r v = v H v L . It is shown that R W B >1 for some values of T H while R W F <1 for all values of T H. At high T H values, R W B and R W F go to unity. For an optimum value of T H, it is seen that R W B has a maximum, which is greater than unity and cannot be predicted by the classical ideal gas approximation. At the quantum degeneracy conditions, consequently, it is possible to produce more network output per cycle when a Bose gas is used as the working gas in a Carnot power cycle. For refrigeration and heat pump Carnot cycles, the use of a Bose gas persists to provide an advantage since it causes the heat pumped per cycle to increase. However, the use of a Fermi gas is always disadvantageous since it causes the network output in a power cycle and the heat pumped in the refrigeration and heat pump cycles to decrease.

How correlation suppresses density fluctuations in the uniform electron gas of one, two, or three dimensions

The particle number N fluctuates in a spherical volume fragment Ω of a uniform electron gas. In an ideal classical-gas or “Hartree” model, the fluctuation is strong, with (ΔNΩ)2=NΩ. We show in detail how this fluctuation is reduced by exchange in the ideal Fermi gas and further reduced by Coulomb correlation in the interacting Fermi gas. Besides the mean particle number NΩ and mean square fluctuation (ΔNΩ)2=(N2)Ω−(NΩ)2, we also examine the full probability distribution PΩ(N). The latter is approximately Gaussian, and exactly Gaussian for \documentclass{article}\pagestyle{empty}\begin{document}$N_{\Omega}\gg1$\end{document}. More precisely, for any NΩ it is a Poisson distribution for the ideal classical gas and a modified Poisson distribution for the ideal or interacting Fermi gases. While most of our results are for nonzero densities and three dimensions, we also consider fluctuations in the low-density or strictly correlated limit and in the electron gas of one or two dimensions. In one dimension, the electrons may be strictly correlated at all finite densities. Fulde's fluctuation-based index of correlation strength applies to the uniform gas in any number of dimensions. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 819–830, 2000

N-body classical systems with rα interparticle potential and two-particle correlation function using Tsallis statistics

We investigate the relation between the two-particle correlation function of N-body classical systems with r α interparticle potential and the generalized mean value introduced by Tsallis. In previous work, Plastino et al. [J. Phys. A 27 (1994) 5707] have shown that the generalized partition function appears in the generalized mean value of N-body classical ideal gas and have derived the generalized version of equipartition law. We show the N-body generalized partition function appears also from the interaction part of the generalized mean value. We also review the generalized version of the virial theorem.

Detailed balance, quantum distribution functions, and equilibrium of mixtures

We consider systems of nearly free particles (or quasiparticles) interacting by scattering, emission and absorption of radiation, or by physical or chemical transformation. The condition of detailed balance yields the appropriate distribution function for each species, the equality of their temperatures, and a relation for their chemical potentials associated with particle transformations. For example, antiparticles coexisting in equilibrium have opposite chemical potentials, and excitations above the Bose–Einstein condensate have zero chemical potential. For mixtures of classical ideal gases, the law of mass action is obtained.

Pauli paramagnetic gas in the framework of Riemannian geometry.

We investigate the thermodynamic curvature resulting from a Riemannian geometry approach to thermodynamics for the Pauli paramagnetic gas which is a system of identical fermions each with spin 1 / 2, and also for classical ideal paramagnetic gas. We find that both the curvature of classical ideal paramagnetic gas and the curvature of the Pauli gas in the classical limit reduce to that of a two-component ideal gas. On the other hand, it is seen straightforwardly that the curvature of classical gas satisfies the geometrical equation exactly. Also a simple relationship between the curvature of Pauli gas and the correlation volume is obtained. We see that it is only in the classical and semiclassical regime that the absolute value of the thermodynamic curvature can be interpreted as a measure of the stability of the system.

Riemann scalar curvature of ideal quantum gases obeying Gentile's statistics

The scalar curvature (R) of ideal quantum gases obeying Gentile's statistics is investigated by the method of information geometrical theory. The R value is specified by the fugacity and the maximum number, p, of particles in a state. The lowest case p = 1, corresponds to Fermi-Dirac statistics and the unbounded case, p, to Bose-Einstein statistics. In contrast to R = 0 for ideal classical gases obeying Boltzmann statistics, we find R = (2)1/2/32 for p2 and R = -(2)1/2/32 for p = 1, in 0 which is the classical limit. This means that a quantum statistical character is left in R, in the classical limit. Also, a correlation between the sign of R and a quantum mechanical exchange effect is recognized for 0 and >>1. Furthermore, we obtain results that support the instability interpretation of R proposed by Janyszek and Mrugala.

Towards a Physical Interpretation for the Stephani Universes

A physically reasonable interpretation is provided for the perfect fluid, spherically symmetric, conformally flat "Stephani Universes". The free parameters of this class of exact solutions are determined so that the ideal gas relation p = nk B T is identically fulfilled, while the full equation of state of a classical monatomic ideal gas and a matter-radiation mixture holds up to a good approximation in a near dust, matter dominated regime. Only the models having spacelike slices with positive curvature admit a regular evolution domain that avoids an unphysical singularity. In the matter dominated regime these models are dynamically and observationally indistinguishable from "standard" FLRW cosmology with a dust source.

Correlation induced by Tsallis’ nonextensivity

In Tsallis’ generalized statistical mechanics, correlation is induced by nonextensivity even if the microscopic degrees of freedom are dynamically independent. Here, using the classical ideal gas model, the generalized variance, covariance and correlation coefficient regarding the particle energies are calculated and their properties discussed. It is shown that the correlation is suppressed for a large number of particles. This demonstrates the validity of the independent particle picture for a dense gas rather than for a dilute gas. It is also found that, in the thermodynamic limit, the correlation again vanishes and the generalized variance exhibits a power-law behavior with respect to the particle number density. Relevance of these results to the zeroth law of thermodynamics in nonextensive statistical mechanics is pointed out.

Open three‐dimensional unhindered molecular chains: Partition function

AbstractAn open molecular chain, formed by N classical particles moving in three dimensions and having N − 1 bonds of constant lengths between successive neighbours, is considered. Hamiltonian methods with manifest rotational invariance allow to characterize all constraints. The classical partition function, Z0, for a large open chain in thermal equilibrium is analyzed in general. Simpler integral representations for Z0 are obtained, with the following advantages: (i) explicit rotational invariance in the integrands, (ii) only tridiagonal matrices appear and, moreover, approximations for their determinants can be obtained. For a two‐dimensional open chain, an approximate factorized formula for Z0 is presented, and its essentials are generalized to the three‐dimensional case. In both cases, the features of the partition functions bear certain similarities to that for a classical ideal gas. The approximate partition functions lead to approximate analytical computations of the correlations between pairs of different bond vectors, of the squared end‐to‐end distance, of the probability distribution for the end‐to‐end vector and of the structure factor, which display some novel features. Some comparisons with the corresponding results for the Gaussian chain are made.

Microcanonical ensemble study of a gas column under gravity

The study of a classical ideal gas column of finite height H in a uniform gravitational field g is made by the microcanonical ensemble at energy E. The primary functions of this ensemble, the phase volume and the density of states, are derived. Related statistical quantities, such as the entropy, the temperature and the heat capacity, are also reported. The equivalence in the thermodynamic limit between the calculated microcanonical expressions and those obtained from the canonical ensemble is shown numerically. The expression for the temperature is used to analyze the temperature change when the gas is permitted to expand into an evacuated region increasing the height of the column from H1 to H2. The microcanonical single-particle momentum and height distributions are also reported.

Thermodynamics of a Charged Fermion Layer at High rs Values.

The chemical potential $\ensuremath{\mu}$ of a hole layer is mapped as a function of temperature and density $p$. We present the first investigation of the crossover from the strongly interacting degenerate regime to the classical ideal gas limit. The low temperature inverse compressibility, ${\ensuremath{\kappa}}^{\ensuremath{-}1}\ensuremath{\equiv}{p}^{2}\ensuremath{\partial}\ensuremath{\mu}/\ensuremath{\partial}p$, is negative and temperature independent. At $T\ensuremath{\approx}17\mathrm{K}$ temperature dependence commences abruptly: ${\ensuremath{\kappa}}^{\ensuremath{-}1}$ turns less negative with rising $T$ and eventually reverses sign. An ideal classical gas is attained at temperatures comparable to the interaction energy, more than an order of magnitude larger than the Fermi energy.

The thermodynamics of a charged fermion layer at high rs values

We report the measurement of the chemical potential of a two-dimensional hole layer in wide temperature and density ranges. Due to their heavy mass, the holes are in a regime where the ratio of Coulomb to Fermi energies r s is very high, thus facilitating the first detailed study of the full temperature-dependent crossover from the strongly interacting degenerate regime to the classical ideal gas limit. The measured low-temperature chemical potential and inverse compressibility are negative and uninfluenced by temperature. At T ≈ 17K they simultaneously and abruptly become temperature-dependent for all carrier concentrations. As the temperature is further raised, the inverse compressibility becomes less negative and reverses sign. An ideal classical gas behavior is attained only at temperatures of the order of the interaction energy, more than an order of magnitude larger than the Fermi energy.

Generalized statistical mechanics: Extension of the Hilhorst formula and application to the classical ideal gas

Within the framework of the Tsallis generalized statistical mechanics, by means of an integral representation of the gamma function in the complex plane, it is possible to find, following along the lines of the Hilhorst formula, a relationship between the q < 1 and q = 1 partition functions. We illustrate its usefulness through the calculation of the generalized specific heat of the classical ideal gas.