Fermi-Dirac Gas Research Articles

Ideal quantum gases: A geometrothermodynamic approach

We derive the fundamental thermodynamic equation for Fermi-Dirac and Bose-Einstein quantum gases, which contains the first order contribution due to the quantum nature of the gas particles. Then, we analyze the fundamental equation in the context of geometrothermodynamics. Although the corresponding Hamiltonian does not contain a potential, indicating the lack of classical thermodynamic interaction, we show that the curvature of the equilibrium space is non-zero and can be interpreted as a measure of the effective quantum interaction between the gas particles. In the limiting case of a classical Boltzmann gas, we show that the equilibrium space becomes flat, as expected from the physical viewpoint. In addition, we derive a thermodynamic fundamental equation for the Bose-Einstein condensation and, using the Ehrenfest scheme, we show that it can be considered as a first order phase transition which in the equilibrium space corresponds to a curvature singularity. This result indicates that the curvature of the equilibrium space can be used to measure in an invariant way the thermodynamic interaction in classical and quantum ideal gases.

Self-gravitating clusters of Fermi-Dirac gas with planar, cylindrical, or spherical symmetry: Evolution of density profiles with temperature.

We calculate density profiles for self-gravitating clusters of an ideal Fermi-Dirac gas with nonrelativistic energy-momentum relation and macroscopic mass at thermal equilibrium. Our study includes clusters with planar symmetry in dimensions D=1,2,3, clusters with cylindrical symmetry in D=2,3, and clusters with spherical symmetry in D=3. Wall confinement is imposed where needed for stability against escape. The length scale and energy scale in use render all results independent of total mass and prove adequate at all temperatures. We present exact analytic expressions for (fully degenerate) T=0 density profiles in four of the six combinations of symmetry and dimensionality. Our numerical results for T>0 describe the emergence, upon quasistatic cooling, of a core with incipient degeneracy surrounded by a more dilute halo. The equilibrium macrostates are found to depend more strongly on the cluster symmetry than on the space dimensionality. We demonstrate the mechanical and thermal stability of spherical clusters with coexisting phases.

Eigenstate thermalization and ensemble equivalence in few-body fermionic systems.

We investigate eigenstate thermalization from the point of view of vanishing particle and heat currents between a few-body fermionic Hamiltonian prepared in one of its eigenstates and an external, weakly coupled Fermi-Dirac gas. The latter acts as a thermometric probe, with its temperature and chemical potential set so that there is neither particle nor heat current between the two subsystems. We argue that the probe temperature can be attributed to the few-fermion eigenstate in the sense that (i) it varies smoothly with energy from eigenstate to eigenstate, (ii) it is equal to the temperature obtained from a thermodynamic relation in a wide energy range, (iii) it is independent of details of the coupling between the two systems in a finite parameter range, (iv) it satisfies the transitivity condition underlying the zeroth law of thermodynamics, and (v) it is consistent with Carnot's theorem. For the spinless fermion model considered here, these conditions are essentially independent of the interaction strength. When the latter is weak, however, orbital occupancies in the few-fermion system differ from the Fermi-Dirac distribution so that partial currents from or to the probe will eventually change its state. We find that (vi) above a certain critical interaction strength, orbital occupancies become close to the Fermi-Dirac distribution, leading to a true equilibrium between the few-fermion system and the probe. In that case, the coupling between the Fermi-Dirac gas and few-fermion system does not modify the state of the latter, which justifies our approach a posteriori. From these results, we conjecture that for few-body systems with sufficiently strong interaction, the eigenstate thermalization hypothesis is complemented by ensemble equivalence: individual many-body eigenstates define a microcanonical ensemble that is equivalent to a canonical ensemble with grand canonical orbital occupancies.

Thermodynamic Calculation of the Characteristics of Metal Thermionic Emission

The thermodynamic equilibria of the “Fermi–Dirac gas–Maxwell gas–Boltzmann gas” and “metal–saturated vapor” systems are considered (with allowance for charged particles). It is shown that the work function of a metal at 0 K is equal to the Mulliken electronegativity of its atoms. For the latter case, it is proposed to take into account the relationship between the work-function thermal coefficient and the thermodynamic functions of positive and negative metal ions.

Born-Kothari Condensation for Fermions

In the spirit of Bose–Einstein condensation, we present a detailed account of the statistical description of the condensation phenomena for a Fermi–Dirac gas following the works of Born and Kothari. For bosons, while the condensed phase below a certain critical temperature, permits macroscopic occupation at the lowest energy single particle state, for fermions, due to Pauli exclusion principle, the condensed phase occurs only in the form of a single occupancy dense modes at the highest energy state. In spite of these rudimentary differences, our recent findings [Ghosh and Ray, 2017] identify the foregoing phenomenon as condensation-like coherence among fermions in an analogous way to Bose–Einstein condensate which is collectively described by a coherent matter wave. To reach the above conclusion, we employ the close relationship between the statistical methods of bosonic and fermionic fields pioneered by Cahill and Glauber. In addition to our previous results, we described in this mini-review that the highest momentum (energy) for individual fermions, prerequisite for the condensation process, can be specified in terms of the natural length and energy scales of the problem. The existence of such condensed phases, which are of obvious significance in the context of elementary particles, have also been scrutinized.

Exact solution of the (0+1)-dimensional Boltzmann equation for massive Bose–Einstein and Fermi–Dirac gases

We present the exact solution of the (0+1)-dimensional Boltzmann equation for massive Bose–Einstein and Fermi–Dirac gases. For the initial conditions used typically in ultra-relativistic heavy-ion collisions, we find that the effects of quantum statistics are very small for thermodynamics-like functions such as the effective temperature, energy density or transverse and longitudinal pressures. Similarly, the inclusion of quantum statistics affects very little the shear viscosity. On the other hand, the quantum statistics becomes important for description of the phenomena connected with the bulk viscosity.

Special treatment of the optical potential of a spherical attractive fermi gas as a Saxon-Woods potential

Within a special scheme providing significant new results, we show that the (attractive) optical potential due to a spherical (non-relativistic) dilute and degenerate Fermi-Dirac gas becomes a Saxon-Woods potential which, in turn, is approximated by a truncated Dirac delta function near the center of the above spherical gas. This approximation is suitable to investigate phenomena in which fermions are restricted to move in relatively small spatial domains. In relation to this, we determine the corresponding fermion stationary wavefunctions which are found to be proportional to the aforementioned delta function if twice the fermion rest-mass multiplied by the total electron energy is much larger than the square of the reduced Planck constant. If this relationship is not fulfilled, the above fermion system is found to be roughly equivalent to fermions in an infinite one-dimensional potential well. In addition, the force field derived from the optical potential in question is determined as well as the chemical potential of the gas. Finally, application of our formulation to study electron transport in nanostructures is outlined.

The Hartree Equation for Infinitely Many Particles I. Well-Posedness Theory

We show local and global well-posedness results for the Hartree equation $$i\partial_t\gamma=[-\Delta+w*\rho_\gamma,\gamma],$$ where $\gamma$ is a bounded self-adjoint operator on $L^2(\R^d)$, $\rho_\gamma(x)=\gamma(x,x)$ and $w$ is a smooth short-range interaction potential. The initial datum $\gamma(0)$ is assumed to be a perturbation of a translation-invariant state $\gamma_f=f(-\Delta)$ which describes a quantum system with an infinite number of particles, such as the Fermi sea at zero temperature, or the Fermi-Dirac and Bose-Einstein gases at positive temperature. Global well-posedness follows from the conservation of the relative (free) energy of the state $\gamma(t)$, counted relatively to the stationary state $\gamma_f$. We indeed use a general notion of relative entropy, which allows to treat a wide class of stationary states $f(-\Delta)$. Our results are based on a Lieb-Thirring inequality at positive density and on a recent Strichartz inequality for orthonormal functions, which are both due to Frank, Lieb, Seiringer and the first author of this article.

On the construction of point processes in statistical mechanics

We present a new approach to the construction of point processes of classical statistical mechanics as well as processes related to the Ginibre Bose gas of Brownian loops and to the dissolution in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^d$\end{document}Rd of Ginibre's Fermi-Dirac gas of such loops. This approach is based on the cluster expansion method. We obtain the existence of Gibbs perturbations of a large class of point processes. Moreover, it is shown that certain “limiting Gibbs processes” are Gibbs in the sense of Dobrushin, Lanford, and Ruelle if the underlying potential is positive. Finally, Gibbs modifications of infinitely divisible point processes are shown to solve a new integration by parts formula if the underlying potential is positive.

THERMODYNAMICS AND EXTRA DIMENSIONS

We consider the effects of extra dimensions on the thermodynamics of classical ideal gases, Bose–Einstein gases and Fermi–Dirac gas. Assuming a q-dimensional torus for the extra dimensions, we compute the thermodynamic functions such as the equation of state, the average energy and the specific heat at constant volume for the three systems. We show that the corrections due to the extra dimensions are small, proportional to [Formula: see text].

A gas-kinetic BGK scheme for semiclassical Boltzmann hydrodynamic transport

A class of gas-kinetic BGK schemes for solving quantum hydrodynamic transport based on the semiclassical Boltzmann equation with the relaxation time approximation is presented. The derivation is a generalization to the development of Xu [K. Xu, A gas-kinetic BGK scheme for the Navier–Stokes equations and its connection with artificial dissipation and Godunov method, from gas-kinetic theory, J. Comput. Phys. 171 (2001) 289–335] for the classical gas. Both Bose–Einstein and Fermi–Dirac gases are considered. Some new features due to the quantum equilibrium distributions are delineated. The first-order Chapman–Enskog expansion of the quantum BGK-Boltzmann equation is derived. The coefficients of shear viscosity and thermal conductivity of a quantum gas are given. The van Leer’s limiter is used to interpolate and construct the distribution on interface to achieve second-order accuracy. The present quantum gas-kinetic BGK scheme recovers the Xu’s scheme when the classical limit is taken. Several one-dimensional quantum gas flows in a shock tube are computed to illustrate the present method.

A numerical study of oblique shock wave reflections over wedges in an ideal quantum gas

The various oblique shock wave reflection patterns generated by a moving incident shock on a planar wedge using an ideal quantum gas model are numerically studied using a novel high resolution quantum kinetic flux splitting scheme. With different incident shock Mach numbers and wedge angles as flow parameters, four different types of reflection patterns, namely, the regular reflection, simple Mach reflection, complex Mach reflection and the double Mach reflection as in the classical gas can be classified and observed. Both Bose–Einstein and Fermi–Dirac gases are considered.

High-order kinetic flux vector splitting schemes in general coordinates for ideal quantum gas dynamics

A class of high-order kinetic flux vector splitting schemes are presented for solving ideal quantum gas dynamics based on quantum statistical mechanics. The collisionless quantum Boltzmann equation approach is adopted and both Bose–Einstein and Fermi–Dirac gases are considered. The formulas for the split flux vectors are derived based on the general three-dimensional distribution function in velocity space and formulas for lower dimensions can be directly deduced. General curvilinear coordinates are introduced to treat practical problems with general geometry. High-order accurate schemes using weighted essentially non-oscillatory methods are implemented. The resulting high resolution kinetic flux splitting schemes are tested for 1D shock tube flows and shock wave diffraction by a 2D wedge and by a circular cylinder in ideal quantum gases. Excellent results have been obtained for all examples computed.

Kinetic Flux Vector Splitting Schemes for Ideal Quantum Gas Dynamics

Novel quantum kinetic flux vector splitting schemes are presented for the computation of ideal quantum gas dynamical flows. The quantum Boltzmann equation approach is adopted and both Bose-Einstein and Fermi-Dirac gases are considered. Formulas for one spatial dimension are first derived, and the resulting kinetic flux vector splitting scheme is tested for shock tube flows. A modified flux vector splitting based on the BGK model is also presented to reduce the numerical diffusion. Implementation of the efficient weighted essentially nonoscillatory method to yield a class of high resolution methods is also devised. Both the classical limit and the nearly degenerate limit are computed. The flow structures can all be accurately captured by the present kinetic flux vector splitting schemes. Results for one-dimensional boson gas under external potential are also included.

Neutrino crown of a protoneutron star and analysis of its convective instability

A numerical solution to the problem of the structure of the neutrino crown of a protoneutron star that is formed upon an iron-star-core collapse, which is peculiar to all massive stars at the end of their thermonuclear evolution, is given. The structure of a neutrino crown, which is semitransparent to neutrino radiation from a spherical layer between the neutrinosphere and the front of the accretion shock wave, is determined by a set of nonlinear ordinary differential equations of spherically symmetric neutrino hydrodynamics with allowance for a complete set of beta processes in a Boltzmann free-nucleon gas and an ultrarelativistic Fermi-Dirac electron-positron gas that form neutrino-crown matter. The problem of consistently taking into account nonequilibrium neutrino-absorption and neutrino-emission processes and the problem of formulating boundary conditions for a neutrino crown were the main problems in constructing the numerical solution in question, which was obtained by means of a dedicated algorithm. The problem at hand features a number of parameters: the protoneutron-star mass, M0; the rate of accretion of the outer layers of the collapsing star being considered, ⊙M0; the effective temperature of the neutrinosphere and the effective neutrino chemical potential there, Tveff and ψveff, respectively; and, finally, the total neutrino emissivity of the neutrinosphere, \(L_{v\tilde v} \). Two of these parameters, M0 and \(L_{v\tilde v} \), are varied within broad intervals in accordance with the hydrodynamic theory of a collapse. On one hand, the numerical solutions constructed in the present study give an idea of the physical conditions in the immediate vicinity of a protoneutron star in the course of its continuing gravitational collapse; on the other hand, they make it possible to obtain exhaustive information about its convective instability, which is the most important property of a so-called soundless collapse—that is, a collapse not accompanied by an explosion of a supernova scale. The increment of the development of a convective instability is obtained at a linear stage, this giving sufficient grounds to introduce the hypothesis that the instability in question plays a key role in the origin of observed gamma-ray bursts. More precisely, these bursts may result from the development of the instability at the subsequent nonlinear stage, which has yet to be studied theoretically—in particular, on the basis of non-one-dimensional numerical models of neutrino hydrodynamics.

A kinetic beam scheme for ideal quantum gas dynamics

A novel kinetic beam scheme for the ideal quantum gas is presented for the computation of quantum gas dynamical flows. The quantum Boltzmann equation approach is adopted and the local thermodynamic equilibrium quantum distribution is assumed. Both Bose–Einstein and Fermi–Dirac gases are considered. Formulae for one spatial dimension is first derived and the resulting beam scheme is tested for shock tube flows. Implementation of high-order methods is also outlined. We only consider the system in the normal phase consisting of particles in excited states and both the classical limit and the nearly degenerate limit are computed. The flow structures can all be accurately captured by the present beam scheme. Formulations for multiple spatial dimensions are also included.

Optical detection of fractional particle number in an atomic Fermi-Dirac gas.

We study theoretically a Fermi-Dirac atomic gas in a one-dimensional optical lattice coupled to a coherent electromagnetic field with a topologically nontrivial soliton phase profile. We argue that the resulting fractional eigenvalues of the particle number operator can be detected via light scattering. This could be a truly quantum mechanical measurement of particle number fractionalization in a dilute atomic gas.

Particle number fractionalization of an atomic Fermi-Dirac gas in an optical lattice.

We show that a dilute two-species gas of Fermi-Dirac alkali-metal atoms in a periodic optical lattice may exhibit fractionalization of the particle number when the two components are coupled via a coherent electromagnetic field with a topologically nontrivial phase profile. This results in fractional eigenvalues of the spin operator with vanishing fluctuations. The fractional part can be accurately controlled by modifying the effective detuning of the electromagnetic field.

Scattering of light and atoms in a Fermi-Dirac gas with Bardeen-Cooper-Schrieffer pairing

We theoretically study the optical properties of a Fermi-Dirac gas in the presence of a superfluid state. We calculate the leading quantum-statistical corrections to the standard column density result of the electric susceptibility. We also consider the Bragg diffraction of atoms by means of light-stimulated transitions of photons between two intersecting laser beams. Bardeen-Cooper-Schrieffer pairing between atoms in different internal levels magnifies incoherent scattering processes. The absorption linewidth of a Fermi-Dirac gas is broadened and shifted. Bardeen-Cooper-Schrieffer pairing introduces a collisional local-field shift that may dramatically dominate the Lorentz-Lorenz shift. For the case of the Bragg spectroscopy the static structure function may be significantly increased due to superfluidity in the near-forward scattering.

Optical response of a superfluid state in dilute atomic Fermi-Dirac gases

We theoretically study the propagation of light in a Fermi-Dirac gas in the presence of a superfluid state. BCS pairing between atoms in different hyperfine levels may significantly increase the optical linewidth and line shift of a quantum degenerate Fermi-Dirac gas and introduce a local-field correction that, under certain conditions, dramatically dominates over the Lorentz-Lorenz shift. These optical properties could possibly unambiguously sign the presence of the superfluid state and determine the value of the BCS order parameter.