Ideal Quantum Gases Research Articles

Bose e Einstein: do nascimento da estatística quântica à condensação sem interação I

Em 1924, motivado por um trabalho de S. Bose sobre a radiação do corpo negro, A. Einstein escreveu uma série de três artigos acerca das propriedades termodinâmicas de um gás ideal quântico. No segundo destes artigos, publicado em janeiro de 1925, ele previu o fenômeno da "condensação" dos átomos não interagentes no estado fundamental a partir de uma certa densidade crítica do gás, fenômeno este que veio a ser posteriormente conhecido como Condensação de Bose-Einstein (CBE). Neste primeiro artigo discuto detalhadamente o trabalho de Bose e as implicacções do método por ele desenvolvido sobre o trabalho de Einstein. Em um artigo posterior discutir-se-á o trabalho de Einstein.

A scaling approach to ideal quantum gases

The thermodynamic properties of ideal quantum gases are derived solely from dimensional arguments, the Pauli principle, and thermodynamic relations, without resorting to statistical mechanics.

Influence of quantum degeneracy on the performance of a Stirling refrigeration cycle

Based on the equation of state of the ideal quantum gas, regenerative characteristics of the quantum Stirling refrigeration cycle is analyzed. The general expression for the coefficient of performance is derived. Under the condition of strong and weak gas degeneracy, the special expressions for the coefficient of performance are discussed. These results will be significant for the research of a Stirling refrigerator in the low temperature range.

The positive-entropy constraint for the classical ideal gas

The problem of determining the state parameters' sub-domain where the behaviour of the classical ideal gas approximates that of the Bose and Fermi ideal gases is tutorially discussed. The entropy of any quantum system being always positive, the classical approximation can only be satisfactory within the parameters' sub-domain where the classical entropy turns out to be positive. We show that the sub-domain determined by this condition is close to that where de Broglie's thermal wavelength is smaller than the mean interparticle distance. The exact determination of the state parameters' region, where the particle number density, the grand potential and the entropy of quantum ideal gases differ from those of the classical gas less than a specified quantity, is also illustrated.

The Lerch function and the thermodynamical functions of the ideal quantum gases

The unified description of the main thermodynamical functions of the Bose and Fermi ideal gases, obtained by Lee [J. Math. Phys. 36, 1217 (1995)] in terms of the polylogarithmic functions, can also be obtained by analytic continuation in the chemical potential owing to the analytic properties of the Lerch function that is simply related to the polylogarithmic ones. By this procedure we also show that the Fourier coefficients of the thermal Green function of the ideal Bose gas convert into those of the Fermi one.

Geometry effects in confined space.

In this paper we calculate some exact solutions of the grand partition functions for quantum gases in confined space, such as ideal gases in two- and three-dimensional boxes, in tubes, in annular containers, on the lateral surface of cylinders, and photon gases in three-dimensional boxes. Based on these exact solutions, which, of course, contain the complete information about the system, we discuss the geometry effect which is neglected in the calculation with the thermodynamic limit V--> infinity, and analyze the validity of the quantum statistical method which can be used to calculate the geometry effect on ideal quantum gases confined in two-dimensional irregular containers. We also calculate the grand partition function for phonon gases in confined space. Finally, we discuss the geometry effects in realistic systems.

Time correlations in a confined magnetized free-electron gas

The time-dependent pair correlation functions for a degenerate ideal quantum gas of charged particles in a uniform magnetic field are studied on the basis of equilibrium statistics. In particular, the influence of a flat hard wall on the correlations is investigated, both for a perpendicular and a parallel orientation of the wall with respect to the field. The coherent and incoherent parts of the time-dependent structure function in position space are determined from an expansion in terms of the eigenfunctions of the one-particle Hamiltonian. For the bulk of the system, the intermediate scattering function and the dynamical structure factor are derived by taking successive Fourier transforms. In the vicinity of the wall the time-dependent coherent structure function is found to decay faster than in the bulk. For coinciding positions near the wall the form of the structure function turns out to be independent of the orientation of the wall. Numerical results are shown to corroborate these findings.

Discrete Bose‐Einstein systems in a box with low adiabatic invariant

AbstractThe Bose‐Einstein energy spectrum of an ideal quantum gas confined in a box is discrete and strongly dependent on the box geometry and temperature, for low product of the atomic mass number, Aat and the adiabatic invariant, TV2/3, i.e. on γ = Aat TV2/3. Even within the approximation of non‐interacting particles in the gas, the calculation of the thermodynamic properties of Bose‐Einstein systems turns out to be a difficult mathematical problem. The present study compares the total number of particles, the total energy and the specific heat obtained by sums over the discrete states to the results of the approximate integrals (for defined values of γ) and shows the noticeable errors of the last ones at low adiabatic invariant (around condensation). Then, more rigorous and precise analytical approximations of sums are found in the case of finite number of atoms (correlated with the existence of a zero energy level) and the finite volume of the gas. The corrected thermodynamic functions depend on γ (for fixed fugacity). The condensation temperature is corrected also in order to describe more accurately the discrete Bose‐Einstein systems. Under these circumstances, the analysis of the specific heat leads to the conclusion that it shows no discontinuity, for the entire range of γ values.

Harmonically trapped ideal quantum gases

We solve the problem of a Bose or Fermi gas in $d$-dimensions trapped by $% \delta \leq d$ mutually perpendicular harmonic oscillator potentials. From the grand potential we derive their thermodynamic functions (internal energy, specific heat, etc.) as well as a generalized density of states. The Bose gas exhibits Bose-Einstein condensation at a nonzero critical temperature $T_{c}$ if and only if $d+\delta >2$, and a jump in the specific heat at $T_{c}$ if and only if $d+\delta >4$. Specific heats for both gas types precisely coincide as functions of temperature when $d+\delta =2$. The trapped system behaves like an ideal free quantum gas in $d+\delta $ dimensions. For $\delta =0$ we recover all known thermodynamic properties of ideal quantum gases in $d$ dimensions, while in 3D for $\delta =$ 1, 2 and 3 one simulates behavior reminiscent of quantum {\it wells, wires}and{\it dots}, respectively.

Quantum gases and polylogs

Ideal quantum gases have been historically studied separately according to whether they obey Bose or Fermi statistics. It has been found that their statistical thermodynamics can be unified by means of polylogs, certain transcendental functions. The unified formalism is used to show that in two dimensions (2d), Bose and Fermi gases are equivalent. In null dimension (0d) there is no classical behavior.

Partition functions and symmetric polynomials

We find a close correspondence between the partition functions of ideal quantum gases and certain symmetric polynomials. From this correspondence, it can be shown that a number of thermodynamic identities that have recently been considered in the literature are essentially of combinatorial origin and have been known for a long time as theorems on symmetric polynomials. For example, a recurrence relation for partition functions in the textbook by P. Landsberg is Newton’s identity in disguised form. Conversely, a theorem on symmetric polynomials translates into a new and unexpected relation between fermion and boson partition functions, which can be used to express the former by means of the latter and vice versa.

Quantum degeneracy effect on the work output from a Stirling cycle

The effect of quantum degeneracy on the work output from a Stirling cycle working at quantum degeneracy conditions (QDCs) is analyzed. Expressions for net work outputs of Stirling power cycles working with monatomic ideal Bose and Fermi gases are derived by using the quantum ideal gas equation of state. Ratios of net work outputs of Stirling cycles working with Bose and Fermi gases to the net work output of a classical Stirling cycle (RWB and RWF, respectively) are obtained. Variations of RWB and RWF with TH are examined for a given temperature ratio (τ=TL/TH) and a specific volume ratio (rν=νH/νL). At QDC, it is seen that RWB has a maximum value, which is greater than unity. On the other hand, there is no maximum or minimum point for RWF and RWF⩽1 for any values of TH. Consequently, the use of Bose gas as a working fluid in a Stirling cycle provides an advantage since it causes the net work output per cycle to increase by consuming more heat energy. This fact is seen to be in the opposite direction for a Stirling cycle working with Fermi gas.

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.

Ideal quantum gases in D-dimensional space and power-law potentials

We investigate ideal quantum gases in D-dimensional space and confined in a generic external potential by using the semiclassical approximation. In particular, we derive density of states, density profiles and critical temperatures for Fermions and Bosons trapped in isotropic power-law potentials. From such results, one can easily obtain those of quantum gases in a rigid box and in a harmonic trap. Finally, we show that the Bose–Einstein condensation can set up in a confining power-law potential if and only if D/2+D/n>1, where D is the space dimension and n is the power-law exponent.

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.

Chemical fluctuations in high-energy nuclear collisions

Fluctuations of the chemical composition of the hadronic system produced in nuclear collisions are discussed using the Φ-measure which has been earlier applied to study the transverse momentum fluctuations. The measure is expressed through the moments of the multiplicity distribution and then the properties of Φ are discussed within a few models of multiparticle production. A special attention is paid to the fluctuations in the equilibrium ideal quantum gas. The system of kaons and pions, which is particularly interesting from the experimental point of view, is discussed in detail.

Thermodynamic fermion–boson symmetry in harmonic oscillator potentials

A remarkable thermodynamic fermion–boson symmetry is found for the canonical ensemble of ideal quantum gases in harmonic oscillator potentials of odd dimensions. The bosonic partition function is related to the fermionic one extended to negative temperatures, and vice versa.

On the power cycles working with ideal quantum gases: I. The Ericsson cycle

The Ericsson power cycles working with ideal Bose and Fermi monoatomic gases are examined. They are conveniently called the Bose and Fermi cycles. Efficiencies of Bose and Fermi cycles are derived ( and respectively). Variations of them with the temperature ratio and pressure ratio of the cycle are examined. A comparison of the efficiencies with each other and that of the classical Ericsson cycle is made. In the degenerate gas state it is seen that , although in the classical gas state. In a Bose cycle, it is shown that there is an optimum value for the lowest temperature at which the efficiency reaches its maximum value for a given pressure ratio. Furthermore, Bose-Einstein condensation restricts the value of of a Bose cycle for a given value of . In a Fermi cycle, there is no an optimum value for . However, goes to a finite value of less than unity when goes to zero.