Properties of the path-integral quantum hard-sphere fluid in k space

Abstract

The properties of quantum fluids in Fourier space, as the system response functions to weak external fields, are analyzed taking the quantum hard-sphere fluid as a probe. This serves to clarify the physical meaning of the different radial correlation functions that can be defined in a path-integral quantum fluid, since these functions are the r-space counterparts of the response functions. The basic feature of the external field relevant to this discussion is connected with its localizing/nonlocalizing effect on the quantum particles composing the fluid (i.e., a localizing field causes the collapse of the particle thermal packet). Fields that localize the quantum particles reveal the so-called instantaneous quantities (e.g., the conventional static structure factor), which are related with the diagonal elements of the density matrix. Fields that do not localize the quantum particles show the so-called linear response quantities, which are related to the diagonal and the off-diagonal density matrix elements. To perform this study the path-integral formalism is considered from the functional analysis approach. Given that the Gaussian Feynman–Hibbs effective potential picture is known to represent well many structural features of the quantum hard-sphere fluid, the parallel study of the response functions within this picture is also presented. In particular, the latter picture provides an accurate Ornstein–Zernike scheme that can be used for numerical calculations of response functions over a wide range of conditions, and also gives fine estimates for quantities difficult to compute with the path integral. Results for the quantum hard-sphere fluid obtained within the latter scheme are reported, tests of consistency are given, and the possibility of approximating the instantaneous response function by means of the coherent part of the linear response function is assessed.

On static triplet structures in fluids with quantum behavior

“…63 Although the instantaneous ET3 and centroid CM3 cases resemble formally the classical formulation, the total thermalized-continuous linear response TLR3 case presents added features that are brought about explicitly by the particle self-correlations (i.e., the thermal delocalization). Furthermore, for reasons stated elsewhere, 43,63 the CM3 and TLR3 cases can be dealt with via functional differentiation techniques, whilst for ET3 the developments rely on operator calculus.…”

“…63 Although the instantaneous ET3 and centroid CM3 cases resemble formally the classical formulation, the total thermalized-continuous linear response TLR3 case presents added features that are brought about explicitly by the particle self-correlations (i.e., the thermal delocalization). Furthermore, for reasons stated elsewhere, 43,63 the CM3 and TLR3 cases can be dealt with via functional differentiation techniques, whilst for ET3 the developments rely on operator calculus. 64 It is important to point out that the connections with the corresponding quantum structure factors in k-space go beyond the Fourier transform of the "simple" three-particle correlation function g 3 (r,s,u)" which is parallel to the well-known situation in classical statistical mechanics.…”

“…In the foregoing equations, the spatial quantities are as follows: (a) G TLR2 (r) is the well-known overall significant bead-bead correlation function in the PI sample, 43,44,63 which is the sum of the bead-bead correlation function g LR2 (beads in different necklaces) plus the self-correlation function s SC 1 (beads in the same necklace); (b) g LR3 is the triple-bead correlation functions with each significant t * bead in a different necklace (j k l j, as in g LR2 : no restrictions exist on the t * labels); (c) s SC 1,3 is the triple-bead self-correlation function with the three beads in the same necklace (j = k = l, t * t * t * t * ); and (d) Σ LR2, 3 is the (bead 1 -bead 2 )-bead 3 mixed correlation function, with the first two beads in the same necklace and the third bead in a different necklace [j = k l, t * (j) t * (j), and no restrictions on the t * (l) labels].…”

Cryogenic Noble Gas Separation without Distillation: The Effect of Carbon Surface Curvature on Adsorptive Separation

“…On the other hand, liquid 4 He is an important cooling agent used in nuclear plants, basic scientific research, medical imaging, and other cryogenic systems. − Because the electron shells of noble gases are completely filled, they are inert, that is, do not normally form chemical compounds . Moreover, 20 Ne and 4 He exhibit significant quantum effects throughout their liquid range. − As has been shown by Beenakker et al and others, − nanoscale confinement additionally impacts the space/momentum localization of quantum particles, which amplify the importance of quantum effects at finite temperatures. Thus, the accurate treatment of the phase behavior of these fluids under strong confinement must include quantum effects.…”

“…Therefore, the localization of quantum particle by hard cores of surrounding neighbors reduces the size of the Gaussian wave packet. In contrast, following the Feynman–Hibbs (FH) approximations, − the size of the Gaussian wave packet (i.e., the spreading of the probability function in the positional space) does not change with the intermolecular distance, r , which is incorrect as r → 0. Note that in the nanoscale confinement, the quantum particles are tightly packed and compressed by strong surface forces.…”