The virial expansion of ideal quantum gases reveals some interesting and amusing properties when considered as a function of dimensionality $d$. In particular, the convergence radius $ρ_c(d)$ of the expansion is particulary large at {\em exactly\/} $d=3$ dimensions, $ρ_c(3) = 7.1068\ldots \times \lim_{d\to3} ρ_c(d)$. The same phenomenon occurs in a few other special (non-integer) dimensions. We explain the origin of these facts, and discuss more generally the structure of singularities governing the asymptotic behavior of the ideal gas virial expansion.
It is shown that the interaction potentials for argon, krypton, xenon, carbon dioxide, ammonia, water, n-pentane, n-octane, 1-propanol, suggested in [1] cannot describe experimental second density virial coefficient data within their uncertainty over a wide temperature range.
We derive path-integral expressions for the second and third virial coefficients of monatomic quantum gases. Unlike previous work that considered only Boltzmann statistics, we include exchange effects (Bose-Einstein or Fermi-Dirac statistics). We use state-of-the-art pair and three-body potentials to calculate the third virial coefficient of 3He and 4He in the temperature range 2.6-24.5561 K. We obtain uncertainties smaller than those of the limited experimental data. Inclusion of exchange effects is necessary to obtain accurate results below about 7 K.
Optically-trapped two-component Fermi gases near a broad Feshbach resonance exhibit universal thermodynamics, where the properties of the gas are independent of the details of the two-body scattering interactions. We present a global proof that such a universal gas obeys the virial theorem for {\it any} trapping potential $U$ and any spin mixture, without assuming either the local density approximation or harmonic confinement. The total energy of the gas is given in scale invariant form by $E=<ε\partial U/\partialε>$, where $ε$ is an {\it arbitrary} energy scale in terms of which all length an…
We present the first framework for fully quantum calculation of the third dielectric virial coefficient $C_\varepsilon(T)$ of noble gases, including exchange effects. The quantum effects are taken into account with the path-integral Monte Carlo method. Calculations employing state-of-the-art pair and three-body potentials and pair polarizabilities yield results generally consistent with the few scattered experimental data available for helium, neon, and argon, but rigorous calculations with well-described uncertainties will require the development of surfaces for the three-body nonadditive pol…
In this paper, we generally expressed the virial expansion of ideal quantum gases by the heat kernel coefficients for the corresponding Laplace type operator. As examples, we give the virial coefficients for quantum gases in $d$-dimensional confined space and spheres, respectively. Our results show that, the relative correction from the boundary to the second virial coefficient is independent of the dimension and it always enhances the quantum exchange interaction. In $d$-dimensional spheres, however, the influence of the curvature enhances the quantum exchange interaction in two dimensions, b…
We report a formula for the dry adiabatic lapse rate that depends on the compressibility factor and the adiabatic curves. Then, to take into account the non-ideal behavior of the gases, we consider molecules that can move, rotate, and vibrate and the information of molecular interactions through the virial coefficients. We deduce the compressibility factor in its virial expansion form and the adiabatic curves within the virial expansion up to any order. With this information and to illustrate the mentioned formula, we write the lapse rate for the ideal gas, and the virial expansion up to the s…
The virial expansion of a gas is a correction to the ideal gas law that is usually discussed in advanced courses in statistical mechanics. In this note we outline this derivation in a manner suitable for advanced undergraduate and introductory graduate classroom presentations. We introduce a physically meaningful interpretation of the virial expansion that has heretofore escaped attention, by showing that the virial series is actually an expansion in a parameter that is the ratio of the effective volume of a molecule to its mean volume. Using this interpretation we show why under normal condit…
Strongly correlated Fermi system plays a fundamental role in very different areas of physics, from neutron stars, quark-gluon plasmas, to high temperature superconductors. Despite the broad applicability, it is notoriously difficult to be understood theoretically because of the absence of a small interaction parameter. Recent achievements of ultracold trapped Fermi atoms near a Feshbach resonance have ushered in enormous changes. The unprecedented control of interaction, geometry and purity in these novel systems has led to many exciting experimental results, which are to be urgently understoo…
A method is proposed to approximate the unlimited subcritical set of Mayer`s reducible cluster integrals (i.e., the power coefficients of the known virial expansions for pressure and density in powers of activity) on the basis of information about several irreducible integrals (virial coefficients) and the saturation activity. For the Lennard-Jones fluid, the first four irreducible integrals are actually sufficient to reproduce gas isotherms (including the flat phase-transition region) with high accuracy at temperatures between the triple and critical points. At low temperatures, even the usag…
The equation of state for a two-dimensional hard-sphere gas is difficult to calculate by usual methods. In this paper we develop an approach for calculating the equation of state of hard-sphere gases, both for two- and three-dimensional cases. By regarding a hard-sphere gas as an ideal gas confined in a container with a multi-core (excluded sphere) boundary, we treat the hard-sphere interaction in an interacting gas as the boundary effect on an ideal quantum gas; this enables us to treat an interacting gas as an ideal one. We calculate the equation of state for a three-dimensional hard-sphere …
The van-der-Waals version of the second virial coefficient is not far from being exact if the model parameters are appropriately chosen. It is shown how the van-der-Waals resemblance originates from the interplay of thermal averaging and superposition of scattering phase shift contributions. The derivation of the two parameters from the quantum virial coefficient reveals a fermion-boson symmetry in non-ideal quantum gases. Numerical details are worked out for the Helium 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.
In this paper, the effects of boundary and connectivity on ideal gases in two-dimensional confined space and three-dimensional tubes are discussed in detail based on the analytical result. The implication of such effects on the mesoscopic system is also revealed.
The particle and current densities are shown to display damping and undergo decoherence in ideal quantum gases. The damping is read off from the equations of motion reminiscent of the Navier-Stokes equations and shows some formal similarity with Landau damping. The decoherence leads to consistent density and current histories with characteristic length and time scales given by the ideal gas.
In terms of the s-wave phase shift of the two-body scattering at thermal wavelength, a systematic perturbative expansion of the virial coefficients is developed for a two-dimensional dilute system of bosons in its gaseous phase at low temperature. The thermodynamic functions are calculated to the second order of the expansion parameter. The observability of the universal low energy limit of the two-dimensional phase shift with a quasi-two dimensional atomic gas in an anisotropic trap is discussed.
We show that when the thermal wavelength is comparable to the spatial size of a system, thermodynamic observables like Pressure and Volume have quantum fluctuations that cannot be ignored. They are now represented by operators; conventional (classical) thermodynamics is no longer applicable. We continue the work in earlier papers where quantization rules for thermodynamics were developed by analogy with optics and mechanics, by working out explicitly the quantum theory of van der Waals gases. We find a wave equation satisfied by the thermodynamic wave function as well as solutions ({\em cohere…
In this paper we show that for a non-ideal Bose gas there exists an upper limit on the transition temperature above which Bose-Einstein condensation cannot occur regardless of the pressure applied. Such upper limits for some realistic Bose gases are estimated. This result implies that there may also exist an upper limit on the transition temperature of superconductors.
We investigate the fluctuations of the condensate in the ideal and weakly interacting Bose gases confined in a box of volume V within canonical ensemble. Canonical ensemble is developed to describe the behavior of the fluctuations when different methods of approximation to the weakly interacting Bose gases are used. Research shows that the fluctuations of the condensate exhibit anomalous behavior for the interacting Bose gas confined in a box.
The departure from ideal gas behavior is described by several known equations of state (EoS), developed from a combination of theoretical considerations and experimental correlations. In this work a different approach is proposed, in which from a pairwise potential of the Lennard-Jones type, interaction between molecules is accounted for by means of a self-consistent force field. An EoS is thus derived and compared to the Virial and to the van der Waals EoS.
We investigate theoretically the properties of an ideal trapped gas in a time-dependent harmonic potential. Using a scaling formalism, we are able to present simple analytical results for two important classes of experiments: free expansion of the gas upon release of the trap; and the response of the gas to a harmonic modulation of the trapping potential is investigated. We present specific results relevant to current experiments on trapped Fermions.
We calculate the systematic average speed of the adiabatic piston and a thermal Brownian motor, introduced in [Van den Broeck, Kawai and Meurs, \emph{Microscopic analysis of a thermal Brownian motor}, to appear in Phys. Rev. Lett.], by an expansion of the Boltzmann equation and compare with the exact numerical solution.
An operator that governs the discrete time evolution of the velocity distribution of an out-of-equilibrium ideal gas will be presented. This nonlinear map, which conserves the momentum and the energy of the ideal gas, has the Maxwellian Velocity Distribution (MVD) as an asymptotic equilibrium. Moreover, the system displays the increasing of the entropy during the decay to the MVD.
Using a high temperature virial expansion, we present a controllable study of the thermodynamics of strongly correlated Fermi gases near the BEC-BCS crossover region. We propose a practical way to determine the expansion coefficients for both harmonically trapped and homogeneous cases, and calculate the third order coefficient $b_{3}(T)$ at finite temperatures $T$. At resonance, a $T$-independent coefficient $b_{3,\infty}^{\hom}\approx-0.29095295$ is determined in free space. These results are compared with a recent thermodynamic measurement of $^{6}$Li atoms, at temperatures below the degener…
The fluctuation-dissipation theorem together with the exact density response spectrum for ideal quantum gases has been utilized to yield a new expression for the static structure factor, which we use to derive exact analytical expressions for the temperature{dependent pair distribution function g(r) of the ideal gases. The plots of bosonic and fermionic g(r) display "Bose pile" and "Fermi hole" typically akin to bunching and antibunching as observed experimentally for ultracold atomic gases. The behavior of spin-scaled pair correlation for fermions is almost featureless but bosons show a rich …
Recursion formulae of the N-particle partition function, the occupation numbers and its fluctuations are given using the single-particle partition function. Exact results are presented for fermions and bosons in a common one-dimensional harmonic oscillator potential, for the three-dimensional harmonic oscillator approximations are tested. Applications to excited nuclei and Bose-Einstein condensation are discussed.
Using a field-theoretic approach, we systematically generalize the usual semiclassical approximation for a harmonically trapped ideal Bose gas in such a way that its range of applicability is essentially extended. With this we can analytically calculate thermodynamic properties even for small particle numbers. In particular, it now becomes possible to determine the critical temperature as well as the temperature dependence of both heat capacity and condensate fraction in low-dimensional traps, where the standard semiclassical approximation is not even applicable.
We theoretically examine equilibrium properties of the harmonically trapped ideal Bose and Fermi gases in the quantum degeneracy regime. We analyze thermodynamic characteristics of gases with a finite number of atoms by means of the known semiclassical approach and perform comparison with exact numerical results. For a Fermi gas, we demonstrate deviations in the Fermi energy values originating from a discrete level structure and show that these are observable only for a small number of particles. For a Bose gas, we observe characteristic softening of phase transition features, which contrasts …
We investigate perturbative thermodynamic geometry of nonextensive ideal Classical, Bose and Fermi gases.We show that the intrinsic statistical interaction of nonextensive Bose (Fermi) gas is attractive (repulsive) similar to the extensive case but the value of thermodynamic curvature is changed by nonextensive parameter. In contrary to the extensive ideal classical gas, the nonextensive one may be divided to two different regimes. According to deviation parameter of the system to the nonextensive case, one can find a special value of fugacity, $z^{*}$, where the sign of thermodynamic curvatur…
In this work, a new model in kinetic gas theory for deriving the Maxwellian Velocity Distribution (MVD) is proposed. We construct an operator that governs the discrete time evolution of the velocity distribution. This operator, which conserves the momentum and the energy of the ideal gas, has the MVD as a fixed point. Moreover, for any initial out-of-equilibrium velocity distribution, it is shown that the gas decays to the equilibrium distribution, that is, to the MVD.
We use the non-equilibrium statistical field theory for classical particles recently developed by Mazenko and Das and Mazenko, together with the free generating functional for particles initially correlated in phase space derived in Bartelmann et al. to study the impact of initial correlations on the equation of state of real gases. We first show that we can reproduce the well known van der Waals equation of state for uncorrelated initial conditions using this approach. We then impose correlated initial conditions and study their qualitative and quantitative effect on the equation of state of …
We show that the recently discovered thermodynamic equivalence between noninteracting Bose and Fermi gases in two dimensions, and between one-dimensional Bose and Fermi systems with linear dispersion, both in the grand-canonical ensemble, are special cases of a larger class of equivalences of noninteracting systems having an energy-independent single-particle density of states. We also conjecture that the same equivalence will hold in the grand-canonical ensemble for any noninteracting quantum gas with a discrete ladder-type spectrum whenever $σΔ/ N k_{\rm B} T$ is small, where $N$ is the aver…
We theoretically study the collective excitations of an ideal gas confined in an isotropic harmonic trap. We give an exact solution to the Boltzmann-Vlasov equation; as expected for a single-component system, the associated mode frequencies are integer multiples of the trapping frequency. We show that the expressions found by the scaling ansatz method are a special case of our solution. Our findings, however, are most useful in case the trap contains more than one phase: we demonstrate how to obtain the oscillation frequencies in case an interface is present between the ideal gas and a differe…
Thermodynamic length is a metric distance between equilibrium thermodynamic states that asymptotically bounds the dissipation induced by a finite time transformation of a thermodynamic system. By means of thermodynamic length, we first evaluate the departures from ideal to real gases in geometric thermodynamics with and without Legendre invariance. In particular, we investigate ideal and real gases in the Ruppeiner and geometrothermodynamic formalisms. Afterwards, we formulate a strategy to relate thermodynamic lengths to efficiency of thermodynamic systems in both the aforementioned framework…
Using a collective coordinate numerical optimization procedure, we construct ground-state configurations of interacting particle systems in various space dimensions so that the scattering of radiation exactly matches a prescribed pattern for a set of wave vectors. We show that the constructed ground states are, counterintuitively, disordered (i.e., possess no long-range order) in the infinite-volume limit. We focus on three classes of configurations with unique radiation scattering characteristics: (i)``stealth'' materials, which are transparent to incident radiation at certain wavelengths; (i…
The ideal uniform two-dimensional (2D) Fermi and Bose gases are considered both in the thermodynamic limit and the finite case. We derive May's Theorem, viz. the correspondence between the internal energies of the Fermi and Bose gases in the thermodynamic limit. This results in both gases having the same heat capacity. However, as we shall show, the thermodynamic limit is never truly reached in two dimensions and so it is essential to consider finite-size effects. We show in an elementary manner that for the finite 2D Bose gas, a pseudo-Bose-Einstein condensate forms at low temperatures, incom…
We calculate numerically and analytically the fluctuations of the fermionic condensate and of the number of particles above the condensate for systems of constant density of states. We compare the canonical fluctuations, obtained from the equivalent Bose condensate fluctuation, with the grandcanonical fermionic calculation. The fluctuations of the condensate are almost the same in the two ensembles, with a small correction comming from the total particle number fluctuation in the grandcanonical ensemble. On the other hand the number of particles above the condensate and its fluctuation is inse…
We derive a formula for the dry adiabatic lapse rate of atmospheres composed of real gases. We restrict our study to those described by a family of two-parameter cubic equations of state and the recent Guevara non-cubic equation. Since our formula depends on the adiabatic curves, we compute them all at once, considering molecules that can move, rotate, and vibrate, for any equation of state. To illustrate our results, we estimate the lapse rate of the troposphere of Titan, obtaining a better approximation to the observed data in some instances, when compared to the estimation provided by the v…
I investigate the possibility of condensation in ideal Fermi systems of general single particle density of states. For this I calculate the probability $w_{N_0}$ of having exactly $N_0$ particles in the condensate and analyze its maxima. The existence of such maxima at macroscopic values of $N_0$ indicates a condensate. An interesting situation occurs for example in 1D systems, where $w_{N_0}$ may have two maxima. One is at $N_0=0$ and another one may exist at finite $N_0$ (for temperatures bellow a certain condensation temperature). This suggests the existence of a first order phase transitio…
We present the full thermodynamics of a fluid confined by an arbitrary external potential based on the virial expansion of the grand potential. The fluid may be classical or quantum and it is assumed that interatomic interactions are pairwise additive. We indicate how the appropriate "generalized" volume and pressure variables, that replace the usual volume and hydrostatic pressure, emerge for a given confining potential in the thermodynamic limit. A discussion of the physical meaning and of the measurement of these variables is presented. We emphasize that this treatment yields the correct eq…
The mean ground state occupation number and condensate fluctuations of interacting and non-interacting Bose gases confined in a harmonic trap are considered by using a canonical ensemble approach. To obtain the mean ground state occupation number and the condensate fluctuations, an analytical description for the probability distribution function of the condensate is provided directly starting from the analysis of the partition function of the system. For the ideal Bose gas, the probability distribution function is found to be a Gaussian one for the case of the harmonic trap. For the interactin…
We study the roles of the dynamical high order perturbation and statistically non-linear infrared fluctuation/correlation in the virial equation of state for the Fermi gas in the unitary limit. Incorporating the quantum level crossing rearrangement effects, the spontaneously generated entropy departing from the mean-field theory formalism leads to concise thermodynamical expressions. The dimensionless virial coefficients with complex non-local correlations are calculated up to the fourth order for the first time. The virial coefficients of unitary Fermi gas are found to be proportional to thos…
The necessary conditions to derive the quantum VdW EoS with hard-core repulsion from the quantum partition are discussed. On a plausible example it is shown that an alternative way to account correctly for the 3-rd virial coefficient of classical hard spheres leads to inconsistencies. The multicomponent formulation of the quantum VdW EoS with hard-core repulsion is derived within a self-consisting approximation. For practical applications it is simplified, extended to higher densities and generalized to the case of hard convex bodies of any dimension $D\ge 2$.
We study the thermodynamic properties of an ideal gas of fermions in a harmonic oscillator confining potential. The analogy between this problem and the de Haas-van Alphen effect is discussed and used to obtain analytical results for the chemical potential and specific heat in the case of both isotropic and anisotropic potentials. Step-like behaviour in the chemical potential, first noted in numerical studies, is obtained analytically and shown to result in an oscillatory behaviour of the specific heat when the particle number is varied. The origin of these oscillations is that part of the the…
The energy route to the equation of state of hard-sphere fluids is ill-defined since the internal energy is just that of an ideal gas and thus it is independent of density. It is shown that this ambiguity can be avoided by considering a square-shoulder interaction and taking the limit of vanishing shoulder width. The resulting hard-sphere equation of state coincides exactly with the one obtained through the virial route. Therefore, the energy and virial routes to the equation of state of hard-sphere fluids can be considered as equivalent.
We consider a heavy piston in an infinite cylinder surrounded by ideal gases on both sides. The piston moves under elastic collisions with gas atoms. We assume here that the gases always exert equal pressures on the piston, hence the piston remains at the so called mechanical equilibrium. However, the temperatures and densities of the gases may differ across the piston. In that case some earlier studies by Gruber, Piasecki and others reveal a very slow motion (drift) of the piston in the direction of the hotter gas. At the same time the hotter gas slowly transfers its energy (heat) across the …
The ideal gas laws are derived from the democritian concept of corpuscles moving in vacuum plus a principle of simplicity, namely that these laws are independent of the laws of motion aside from the law of energy conservation. A single corpuscle in contact with a heat bath and submitted to a $z$ and $t$-invariant force $-w$ is considered, in which case corpuscle distinguishability is irrelevant. The non-relativistic approximation is made only in examples. Some of the end results are known but the method appears to be novel. The mathematics being elementary the present paper should facilitate t…
We derive an expression for the reduced density matrices of ideal Bose and Fermi gases in the canonical ensemble, which corresponds to the Bloch--De Dominicis (or Wick's) theorem in the grand canonical ensemble for normal-ordered products of operators. Using this expression, we study one- and two-body correlations of homogeneous ideal gases with $N$ particles. The pair distribution function $g^{(2)}(r)$ of fermions clearly exhibits antibunching with $g^{(2)}(0)=0$ due to the Pauli exclusion principle at all temperatures, whereas that of normal bosons shows bunching with $g^{(2)}(0)\approx 2$, …
I discuss ideal and interacting quantum gases obeying general fractional exclusion statistics. For systems with constant density of single-particle states, described in the mean field approximation, the entropy depends neither on the microscopic exclusion statistics, nor on the interaction. Such systems are called {\em thermodynamically equivalent} and I show that the microscopic reason for this equivalence is a one-to-one correspondence between the excited states of these systems. This provides a method, different from the bosonisation technique, to transform between systems of different excl…
In an ideal Bose gas that is driven into a steady state far from thermal equilibrium, a generalized form of Bose condensation can occur. Namely, the single-particle states unambiguously separate into two groups: the group of Bose-selected states, whose occupations increase linearly with the total particle number, and the group of all other states whose occupations saturate [Phys. Rev. Lett. 111, 240405 (2013)]. However, so far very little is known about how the number of Bose-selected states depends on the properties of the system and its coupling to the environment. The answer to this questio…
We study the thermodynamics of ideal Bose gas as well as the transport properties of non interacting bosons and fermions in a one dimensional quasi-periodic potential, namely Aubry-André (AA) model at finite temperature. For bosons in finite size systems, the effect of quasi-periodic potential on the crossover phenomena corresponding to Bose-Einstein condensation (BEC), superfluidity and localization phenomena at finite temperatures are investigated. From the ground state number fluctuation we calculate the crossover temperature of BEC which exhibits a non monotonic behavior with the strength …
We aim at deriving an equation of motion for specific sums of momentum mode occupation numbers from models for electrons in periodic lattices experiencing elastic scattering, electron-phonon scattering or electron-electron scattering. These sums correspond to "grains" in momentum space. This equation of motion is supposed to involve only a moderate number of dynamical variables and/or exhibit a sufficiently simple structure such that neither its construction nor its analyzation/solution requires substantial numerical effort. To this we end compute, by means of a projection operator technique, …
The analytical probability distribution of the quasi-2D (and purely 2D) ideal and interacting Bose gas are investigated by using a canonical ensemble approach. Using the analytical probability distribution of the condensate, the statistical properties such as the mean occupation number and particle number fluctuations of the condensate are calculated. Researches show that there is a continuous crossover of the statistical properties from a quasi-2D to a purely 2D ideal or interacting gases. Different from the case of a 3D Bose gas, the interaction between atoms changes in a deep way the nature…
An ideal Maxwell-Boltzmann gas confined in various rectangular nano domains is considered under quantum size effects. Thermodynamic quantities are calculated from their relations with partition function which consists of triple infinite summations over momentum states in each direction. To get analytical expressions, summations are converted to integrals for macro systems by continuum approximation which fails at nanoscales. To avoid both from the numerical calculation of summations and the failure of their integral approximations at nanoscale, a method which gives an analytical expression for…
Explicit density expansions of non-equilibrium probability distribution functions for molecular Brownian particle in ideal gas are obtained in original form what visually implies (is exact solution to) the previously established dynamical virial relations. Role of these relations in unbiased analysis of molecular chaos properties in many-particle statistical mechanics, including the mobility 1/f noise, is newly investigated in clear terms of conditional probabilities and averages.
Motivated by recent experiments on rotating Bose-Einstein condensates, we investigate a rotating, polarized Fermi gas trapped in an anharmonic potential. We apply a semiclassical expansion of the density of states in order to determine how the thermodynamic properties depend on the rotation frequency. The accuracy of the semiclassical approximation is tested and shown to be sufficient for describing typical experiments. At zero temperature, rotating the gas above a given frequency $Ω_{\rm DO}$ leads to a `donut'-shaped cloud which is analogous to the hole found in two-dimensional Bose-Einstein…
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\to \infty $, and analyze the validity of the quantum statistical method which can be used to calculate the g…
The thermodynamics framework of an interacting quantum gas trapped by an arbitrary external potential is reviewed. We show that for each confining potential, in the thermodynamic limit, there emerge "generalized" volume and pressure variables ${\cal V}$ and ${\cal P}$, that replace the usual volume and hydrostatic pressure of a uniform system. This scheme is validated with the derivation of the virial expansion of the grand potential. We show that this approach yields experimentally amenable procedures to find the equation of state of the fluid, ${\cal P} = {\cal P}({\cal V}/N,T)$ with $N$ the…
