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Wigner-Kirkwood expansion for semi-infinite quantum fluids

Ladislav Šamaj, B. Jancovici

Published Jan 1, 2007• 3 citations
OpenAlex

Abstract

For infinite (bulk) quantum fluids of particles interacting via pairwise sufficiently smooth interactions, the Wigner-Kirkwood formalism provides a semiclassical expansion of the Boltzmann density in configuration space in even powers of the thermal de Broglie wavelength $\lambda$. This result permits one to generate an analogous $\lambda$-expansion for the bulk free energy and many-body densities. The present paper brings a technically nontrivial generalization of the Wigner-Kirkwood technique to semi-infinite quantum fluids, constrained by a plane hard wall impenetrable to particles. In contrast to the bulk case, the resulting Boltzmann density involves also position-dependent terms of type $\exp(-2x^2/\lambda^2)$ ($x$ denotes the distance from the wall boundary) which are non-analytic in $\lambda$. Under some condition, the analyticity in $\lambda$ is restored by integrating the Boltzmann density over configuration space; however, in contrast to the bulk free energy, the semiclassical expansion of the surface part of the free energy (surface tension) contains odd powers of $\lambda$, too. Explicit expressions for the leading quantum corrections in the presence of the boundary are given for the one-body and two-body densities. As model systems for explicit calculations, we use Coulomb fluids, in particular the one-component plasma defined in the $\nu$-dimensional (integer $\nu\ge 2$) space.

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For infinite (bulk) quantum fluids of particles interacting via pairwise sufficiently smooth interactions, the Wigner-Kirkwood formalism provides a semiclassical expansion of the Boltzmann density in configuration space in even powers of the thermal de Broglie wavelength λ\lambdaλ. This result permits one to generate an analogous λ\lambdaλ-expansion for the bulk free energy and many-body densities. The present paper brings a technically nontrivial generalization of the Wigner-Kirkwood technique to semi-infinite quantum fluids, constrained by a plane hard wall impenetrable to particles. In contrast to the bulk case, the resulting Boltzmann density involves also position-dependent terms of type exp⁡(−2x2/λ2)\exp(-2x^2/\lambda^2)exp(−2x2/λ2) (xxx denotes the distance from the wall boundary) which are non-analytic in λ\lambdaλ. Under some condition, the analyticity in λ\lambdaλ is restored by integrating the Boltzmann density over configuration space; however, in contrast to the bulk free energy, the semiclassical expansion of the surface part of the free energy (surface tension) contains odd powers of λ\lambdaλ, too. Explicit expressions for the leading quantum corrections in the presence of the boundary are given for the one-body and two-body densities. As model systems for explicit calculations, we use Coulomb fluids, in particular the one-component plasma defined in the ν\nuν-dimensional (integer ν≥2\nu\ge 2ν≥2) space.
This paper constructs a general method for obtaining series expansions of the logarithm of the configuration function FFF. For an NNN-body quantum system with Hamiltonian HHH and inverse temperature $\ensuremath{\beta}$, the configuration function is the ratio of the exact to the free coordinate-space heat kernels: $F=\frac{〈\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}|{e}^{\ensuremath{-}\ensuremath{\beta}H}|{\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}}^{\ensuremath{'}}〉}{〈\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}|{e}^{\ensuremath{-}\ensuremath{\beta}{H}_{0}}|{\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}}^{\ensuremath{'}}〉}$. Expansion of lnF\mathrm{ln}FlnF in Planck's constant hhh leads to the semiclassical expansion of Wigner and Kirkwood, whereas the series in the variable $\ensuremath{\beta}$ provides the high-temperature expansion. By the introduction of an appropriate linked-graph method, it is shown how to obtain explicit formulas for the coefficient functions that enter either of these two series. Further, it is established that the same results can be derived by using the Feynman-Kac path-integral description of the partition function.
For a system of point charges that interact through the three-dimensional electrostatic Coulomb potential (without any regularization) and obey the laws of nonrelativistic quantum mechanics with Bose or Fermi statistics, the static correlations are expressed in terms of Mayer-like diagrams. The exchange effects are taken into account systematically and the long-range Coulomb divergencies are exactly resummed in order to get finite diagrams. For this purpose, in the framework of the grand canonical ensemble, the matrix elements of the imaginary-time evolution operator are represented by the Feynman-Kac functional integral according to Ginibre's idea [J. Math. Phys. 6, 238 (1965); 6, 252 (1965), 6, 1432 (1965)] and we exhibit a correspondence between the correlations in the quantum system of point particles and the distribution functions in a classical fluid of "exchange" loops with Brownian random shapes. The size of a loop, which corresponds to the number of particles involved in the corresponding cyclic permutation, and the shape of a loop, which describes the quantum fluctuations, play the part of internal degrees of freedom that must be integrated over when calculating the distributions relative to the quantum point charges. The loops interact through a two-body potential that is different from the electrostatic interaction between two charged curves: each line element of a loop interacts only with a discrete number of line elements in every other loop. The linear response theory to an infinitesimal external charge distribution can be written in this formalism by a formula analogous to that of classical statistics, and the loop-fugacity and loop-density expansions of the loop-distribution functions are derived by the usual techniques of Mayer diagrams generalized to extended objects. At large distances rrr, the loop potential behaves as 1r\frac{1}{r}r1​ and, as in the classical case, every Mayer diagram diverges. However, this nonintegrable tail is independent of the shape of the loops and it is explicitly and exactly resummed by a generalization of the method developed by Meeron [J. Chem. Phys. 28, 630 (1958); Plasma Physics (McGraw-Hill, New York, 1961)], for classical fluids of point entities. Auxiliary 1r\frac{1}{r}r1​ bonds are introduced and subdiagrams involving chains of 1r\frac{1}{r}r1​ bonds are integrated over first in a systematic way. The new diagrams contain bonds between loops that decay either exponentially or algebraically, with a 1r3\frac{1}{{r}^{3}}r31​ leading term, and the new diagrams are at least conditionally integrable. The part of the quantum particle-particle correlation arising directly from exchange, which is derived from the density of loop, decays faster than any inverse power law, whereas, as shown in the following paper [Phys. Rev. E 53, 4595 (1996)], the whole quantum particle-particle correlation, which also involves the correlations between loops, decays only as 1r6\frac{1}{{r}^{6}}r61​.
Exploring the analogy between quantum mechanics and statistical mechanics, we formulate an integrated version of the Quantropy functional. With this prescription, we compute the propagator associated to Boltzmann–Gibbs statistics in the semiclassical approximation as K=F(T)exp(iScl/ℏ) . We determine also propagators associated to different nonadditive statistics; those are the entropies depending only on the probability S± and Tsallis entropy Sq . For S± , we obtain a power series solution for the probability vs. the energy, which can be analytically continued to the complex plane and employed to obtain the propagators. Our work is motivated by the work of Nobre et al. where a modified q-Schrödinger equation is obtained that provides the wave function for the free particle as a q-exponential. The modified q-propagator obtained with our method leads to the same q-wave function for that case. The procedure presented in this work allows to calculate q-wave functions in problems with interactions determining nonlinear quantum implications of nonadditive statistics. In a similar manner, the corresponding generalized wave functions associated to S± can also be constructed. The corrections to the original propagator are explicitly determined in the case of a free particle and the harmonic oscillator for which the semiclassical approximation is exact, and also the case of a particle with an infinite potential barrier is discussed.
We present analytical results for the nonlocal pair correlations in one-dimensional bosonic systems with repulsive contact interactions that are uniformly valid from the classical regime of high temperatures down to weak quantum degeneracy entering the regime of ultralow temperatures. By using the information contained in the short-time approximations of the full many-body propagator, we derive results that are nonperturbative in the interaction parameter while covering a wide range of temperatures and densities. For the case of three particles we give a simple formula for arbitrary couplings that is exact in the dilute limit while remaining valid up to the regime where the thermal de Broglie wavelength ${\ensuremath{\lambda}}_{T}$ is of the order of the characteristic length LLL of the system. We then show how to use this result to find analytical expressions for the nonlocal correlations for arbitrary but fixed particle numbers NNN including finite-size corrections. Neglecting the latter in the thermodynamic limit provides an expansion in the quantum degeneracy parameter $N{\ensuremath{\lambda}}_{T}/L$. We compare our analytical results with numerical Bethe ansatz calculations, finding excellent agreement.
Some model Hamiltonians are proposed for quantum-mechanical many-body systems with pair forces. In the case of an infinite system in thermal equilibrium, they lead to temperature-domain propagator expansions which are expressible by closed, formally exact equations. The expansions are identical with infinite subclasses of terms from the propagator expansion for the true many-body problem. The two principal models introduced correspond, respectively, to ring and ladder summations from the true propagator expansion, but augmented by infinite classes of self-energy corrections. The latter are expected to yield damping of single-particle excitations. The eigenvalues of the ring and ladder model Hamiltonians are real, and they are bounded from below if the pair potential obeys certain conditions. The models are formulated for fermions, bosons, and distinguishable particles. In addition to the ring and ladder models, two simpler types are discussed, one of which yields the Hartree-Fock approximation to the true problem. A novel feature of all the model Hamiltonians (except the Hartree-Fock) is that they contain an infinite number of parameters whose phases are fixed by random choices. Explicit closed expressions are obtained for the Helmholtz free energy of all the models in the classical limit.