Wigner-Kirkwood Expansion Research Articles

Equilibrium theory of two-dimensional simple liquids

Using the Wigner-Kirkwood expansion and bare Lennard-Jones (LJ) (12-6) potential, an effectiveLJ potential is derived, which includes the quantum effects through the expressions of the effective diameter∂(T, λ) and well-depth \(\bar \varepsilon \left( {T,\lambda } \right)\)(T, λ). We use theWCA perturbation theory to calculate the free energy and pressure for theLJ and effectiveLJ potentials. Simple analytic expressions are given for the reference system and the first order correction calculated. The results are quite good at high density. The quantum effects on the free energy and pressure are also discussed.

Extended Thomas-Fermi theory at finite temperature

Abstract We present a generalization of the extended Thomas-Fermi (ETF) theory to finite temperatures T . Starting from the Wigner-Kirkwood expansion of the Bloch density in powers of , we derive the gradient expansion of the free energy and entropy density functionals F [ρ] and σ[ρ] up to fourth order with their correct temperature-dependent coefficients. (Effective mass and spin-orbit contributions are taken into account up to second order.) For a harmonic-oscillator potential we show that both the h-expansion of the free energy and the entropy and the gradient expansion of the functionals [ρ] and σ[ρ] converge very fast and yield the exact quantum-mechanical results for kT ≳ 3 MeV, where the shell effects are washed out. Finally we discuss the Euler variational equation obtained with the new functionals and use its numerical solutions for semi-infinite symmetric nuclear matter to test the quality of parametrized trial densities. As an application, we present liquid-drop model parameters, calculated with a realistic Skyrme interaction, as functions of the temperature.

Complement to the Wigner-Kirkwood expansion.

We extend the region of applicability of phase-space techniques, for the study of quantum systems, by developing an expansion for the Wigner distribution function and correlation functions, in powers of an interaction potential. These results are not restricted to the near-classical regime (expansion in powers of \ensuremath{\Elzxh}), in contrast to the Wigner-Kirkwood expansion. The relation to other perturbation methods (Lindstedt-Poincar\'e, Green's function) is explored.

Cumulant approximations and renormalized Wigner-Kirkwood expansion for quantum Boltzmann densities.

The logarithm of the quantum Boltzmann density \ensuremath{\Vert}X〉, where ${H}_{V}$=${p}^{2}$/2m+V, is expressed as a cumulant expansion in powers of v=V-W, where W(x)=V(X)+V'(X)(x-X)+(1/2)V''(X )(x-X${)}^{2}$ is the local quadratic approximant to V(x) at the point X. Where V''(X)>0, this expansion behaves ``nonsecularly'' as \ensuremath{\beta}\ensuremath{\rightarrow}\ensuremath{\infty} (all its terms \ensuremath{\sim}\ensuremath{\beta}), and thus remains a useful approximation scheme even as the temperature ${\ensuremath{\beta}}^{\mathrm{\ensuremath{-}}1}$\ensuremath{\rightarrow}0 (in that limit, it yields Rayleigh-Schr\"odinger perturbation expansions of the ground state of ${H}_{V}$). By Taylor expanding v(x) about X in the cumulant expansion, we obtain an expansion which is a resummation over powers of V''(X) of the Wigner-Kirkwood (WK) expansion of ln${\ensuremath{\rho}}_{V}$; this ``renormalized'' WK expansion, whose coefficients are simple functions of V''(X), is as simple to use as the ordinary WK expansion, yet more accurate where V''(X)\ensuremath{\ne}0, and usable down to zero temperature where V''(X)>0 (yielding, in that limit, WK-type expansions for the ground state of ${H}_{V}$). In lowest order, it yields an approximation initially proposed by Miller.

Generalisations of the pitzer—gwinn and feynman—hibbs approximations for quantum partition functions

By using an expression of Feynman and Hibbs for the partition function Z = Tr[exp[- β( p 2/2 m + V)]) in terms of integrals over paths with fixed means, and substituting for the potential V a local harmonic approximant (following Miller), we obtain an improved Pitzer—Gwinn approximation, as simple to use as the original one, but applicable to a much broader class of potentials. Systematic corrections, in the form of cumulant and Wigner—Kirkwood expansions are then obtained.

Partial resummation of the Wigner—Kirkwood expansion for quantum Boltzmann densities

The logarithmic Wigner—Kirkwood (WK) expansion for the one-dimensional quantum Boltzmann density < X|exp[−β( p 2/2 m + V)]| X> is resummed over powers of V′'( X) = d 2 V/d X 2. The resummed expansion is as simple to use as the original WK expansion, but stays usable down to zero temperature where V′'( X) > 0. In lowest order, it yields an approximation initially introduced by Miller.

Equation for the WKB approximation and Wigner-Kirkwood expansions for systems with magnetic field

Equation for the WKB approximation and Wigner-Kirkwood expansions for systems with magnetic field

Semiclassical expansion of the microscopic nucleon-nucleus optical potential

A semiclassical expansion is given for the microscopic optical potential in ladder-type approximations. The problem has been approached by starting with the formulation of the optical potential within the Green's function scheme. By a reordering of the WignerKirkwood expansion it has been possible to obtain a general expansion, which agrees in zeroth order with the standard nuclear matter approach. The next order, which we discuss in more detail, contains the curvature corrections.

Molecular-dynamics calculation of the quantum corrections to the pair-distribution function for a Lennard-Jones fluid

We report the first molecular-dynamic calculation of the quantum corrections of the pair-distribution function for a Lennard-Jones fluid. The method is based on the Wigner-Kirkwood expansion and is useful for "almost classical" many-body systems.

SEMICLASSICAL CALCULATION OF THE WIGNER TRANSFORM OF THE SINGLE PARTICLE DENSITY MATRIX AND SEMICLASSICAL HARTREE-FOCK

It is shown how the smooth part of the Wigner transform of ρ corresponding to a Woods-Saxon potential can be calculated semiclassically. We find that its surface thickness in phase space is appreciable but always much smaller than the Fermi energy thus justifying the Wigner- Kirkwood _ expansion. Using this fact we propose and perform a new semiclassical Hartree-Fock method directly based on the Wigner-Kirkwood expansion. No functional τ [ρ] is therefore needed.

Wigner-Kirkwood expansion ofN-body Green's function: The case with magnetic field

A systematic method is given to compute the semiclassical Wigner-Kirkwood expansion of the off-diagonal thermal kernel of an N-body quantum system. © 1983 The American Physical Society.

Semiclassical statistical mechanics of a two-dimensional fluid

Expansions are obtained for the radial distribution function and free energy for a two-dimensional hard-core fluid in the semiclassical limit, using the ‘‘modified’’ Wigner–Kirkwood expansion method. These results are used to obtain expressions for the density-independent part of the radial distribution function and the first-order density correction to it. Quantum corrections to the second and third virial coefficients are discussed in detail. Explicit results are given for the Sutherland, Yukawa-tail, Wood–Saxan, square-well, and Lennard-Jones (12-6) pair potential models.

Equilibrium statistical mechanics of a nearly classical one-component plasma in a magnetic field, in three or two dimensions1–3)

Abstract The behaviour of a nearly classical one-component plasma in a magnetic field is studied. The Wigner distribution function obeys a Bloch equation which can be cast into an explicitly gauge-invariant form. If the de Broglie wavelength λ is small compared to all other characteristics length, Planck's constant ħ is a small parameter, and a straight Wigner-Kirkwood expansion in powers of ħ 2 can be used. In a strong field, the Landau radius l becomes smaller than λ, and a modified expansion must be devised. Exchange effects give only exponentially small corrections. The leading term of the induced magnetization is independent of the interactions between the particles; the next term is structure-independent for the three-dimensional case. A strong magnetic field quenches the position quantum fluctuations transverse to it. In two dimensions, in the strong field limit, the plasma is a system of point-like Landau orbits, which behave like classical particles; as a consequence, a classical Wigner crystallization can be induced by the magnetic field. The model can be used for describing electron layers at the surface of liquid helium or in a metal-oxide-semiconductor sandwich (MOS).

Use of the Padé approximants for calculating the second virial coefficient

The expansions in reduced temperature, occurring in relations for the second virial coefficient of the power-law potentials were expressed by the Pade approximant. For the Kihara potential it is proved that the functions F1(T*), F2(T*) and F3(T*) and consequently the whole virial coefficient can bee expressed for reduced temperatures T* 0.6 with a good accuracy by means of a simple Pade approximant. To describe quantum effects on the second virial coefficient of low-molecular gases at low temperatures a simple approximant was formulated on the basis of the Wigner-Kirkwood expansion allowing to correlate data on the second virial coefficient even in the temperature range where the use of the Wigner-Kirkwood expansion already fails.

Variational principle for the equilibrium wigner distribution function

A variational principle is derived for the equilibrium Wigner distribution function of quantum statistical mechanics. This provides an alternative to the usual Wigner-Kirkwood expansion method of calculating quantum corrections to the structural and thermodynamic properties of systems. As an illustration, the method is applied to the second virial coefficient of a Lennard-Jones gas, and it is found that the simplest trial function gives results superior to the Wigner-Kirkwood series.

Magnetic properties of a nearly classical one-component plasma in three or two dimensions

The magnetic properties of the one-component plasma, in three and two dimensions, are studied in the nearly classical case. Since magnetism is an essentially quantum effect, Planck's constant ℏ can be used as a small parameter. A generalized Wigner-Kirkwood expansion in powers of ℏ 2 is derived. This expansion is used for computing the induced magnetization. The displacement of the liquid-solid phase transition of the model, when a magnetic field is applied, is discussed. The model is applicable to electrons deposited at the surface of liquid helium.

Quantum corrections to the second virial coefficient of a gas with a triangular-well potential

The quantum corrections to the second virial coefficient of a gas with a triangular-well potential are calculated using the modified Wigner–Kirkwood expansion method, developed by Derderian and Steele. The result is reported in an analytic form.

A technique for increasing the utility of the Wigner-Kirkwood expansion for the second virial coefficient

A number of techniques for accelerating the convergence of the semiclassical Wigner-Kirkwood expansion for the second virial coefficient are described and illustrated with numerical examples. The most successful technique increases the range of utility of the third-order semi-classical expansion from the temperature range T > 50 K to T > 20 K for 4He.

Quantum corrections to the second virial coefficient of a gas using hard sphere basis functions

The quantum corrections to the second virial coefficient for a gas at high temperature are calculated using the modified Wigner–Kirkwood expansion method, developed by Derderian and Steele. We find that the higher corrections depend both on the shape of the well and the value of the potential at the core. The general expression for the second virial coefficient of a gas at high temperature is given as Bdir=Bdirc + (2πa3/3) e−βup(a) [(3/2√2) (λ/a) + (BII/π)(λ/a)2 + BIII/16√2π) (λ/a)3 + ...], where BII and BIII are constants depending upon the shape and nature of the perturbation potential and equal to unity in the limit of hard sphere [i.e., up(r) →0].

Phase Shifts of the Static Screened Coulomb Potential

Phase shifts and their weighted sums over angular momentum states have been obtained for the static screened Coulomb potential (SSCP). Results are given for both an attractive and a repulsive SSCP. A procedure is given for obtaining the phase shifts which uses direct numerical integration near the origin and the first- and second-order WKB approximation at larger distances. At high energy a simple asymptotic fit to the weighted phase-shift sums is obtained. The results are applied to the calculation of the second virial coefficient of plasmas that have no bound states and of those that do have bound states. Very good agreement with the Wigner-Kirkwood expansion is found for a Boltzmann gas interacting through a repulsive SSCP, except that at high temperature, in agreement with DeWitt, the Wigner-Kirkwood expansion is shown to diverge for the SSCP.