Static Structures in Monatomic Fluids

Definition

The basic structural concepts in the study of monatomic fluids at equilibrium are presented in this entry. The scope encompasses both the classical and the quantum domains, the latter concentrating on the diffraction and the zero-spin boson regimes. The main mathematical objects for describing the fluid structures are the following n-body functions: the correlation functions in real space and their associated structure factors in Fourier space. In these studies, the theory of linear response to external weak fields, involving functional calculus, and Feynman’s path integral formalism are the key conceptual ingredients. Emphasis is placed on the physical implications when going from the classical domain (limit of high temperatures) to the abovementioned quantum regimes (low temperatures). In the classical domain, there is only one class of n-body structures, which at every n level consists of one correlation function plus one structure factor. However, the quantum effects bring about the splitting of the foregoing class into three path integral classes, namely instantaneous, total thermalized-continuous linear response, and centroids; each of them is associated with the action of a distinct external weak field and keeps the above n-level structures. Special attention is given to the structural pair level $n=2$, and future directions towards the complete study of the quantum triplet level $n=3$ are suggested.

1. Introduction

Nowadays, the fact that matter is composed of tiny particles, as are molecules, atoms, electrons, etc. (e.g., typical atom size $1 Å={10}^{-8}$ cm), is common knowledge [1]. So is the fact that the external forms of solids are reflections of their inner atomic structures [1]. However, the existence of inner structures arising from the atom statistical arrangements in fluid systems (i.e., gaseous or liquid phases) may sound intriguing to the non-specialist. After all, normal experience indicates that fluids are shapeless, since they adopt the form of their containers and, contrary to solids, do not show any external feature suggesting that they possess internal regularities. Despite this first impression, experiments involving elastic and inelastic scattering of radiation (X-ray and neutron diffraction, respectively) [1,2,3,4] lead to the identification of inner structures, not only in solids but also in fluids! As an aside, note that elastic scattering is related only to radiation-system momentum transfers $\hslash \mathit{k}$; in inelastic scattering, momentum and also energy $(\hslash \omega )$ transfers occur (k = wave vector, $\omega =$ circular frequency) [2,4].

Focusing on the thermodynamic equilibrium of the macroscopic many-body systems that fluids are, their structures, also known as static structures (i.e., correlation functions $g_n$ and structure factors $S^{\left(n\right)})$, are central to a large variety of purposes. The latter range from pure theory to experimental techniques to practical applications [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31] (all are strongly interrelated). The mentioned structures admit a consistent mathematical formulation within the field of statistical mechanics [2,5,7,9,12,13,14,16,23,27], thus having the “spatial” arrangements of the atoms codified in quantitative forms. As regards pure theory, one may mention, for example, the $g_n$ and $S^{\left(n\right)}$ developments in the real and the reciprocal Fourier spaces, respectively (n refers to the elemental number of particles involved) [2,5]. As for the experimental side, note that beyond the pair level $\left(n=2\right),$ structural information remains unobtainable today [4,20]. In connection with the practical applications, examples are the computation of thermodynamic (mechanical and thermal) properties [2,5,10,11,12,24,27] and the assessment of change-of-phase situations together with stability questions and other density-related behaviors [17,19,27].

In the classical domain (i.e., the limit of high temperatures), the structure concepts in monatomic fluids were thoroughly developed in the past 20th century, covering both the theoretical and the computational aspects. Although most of the applications were focused on the pair level, the extension to the triplet level was also successfully achieved [23,24,25,26,27,28,29,30,31]. Moreover, the set of interrelations among higher-level structure schemes was firmly established [5,27]. Going into more detail, among the theoretical motivations for this central research task, one may mention the following: (a) the need to deal with the hierarchical nature of structures [2,5,6,25]; (b) the development of computational methods (Monte Carlo (MC), molecular dynamics (MD), closures) that could complement/substitute the experimental techniques [10,11,25]; and (c) the connection with the many-body interactions that may be crucial when the system density increases [6,10,20,31]. Despite the experimental limitations mentioned above, one can state that an insightful understanding of this topic is attained.

The quantum fluid structural domain presents one with the same general questions found in the classical monatomic fluid but also adds new ones related to the radical change in description and extra features involved. Accordingly, the quantum fluid side is not currently at the same stage of development. It is worth pointing out that, in the last quarter of the past century, the application of Feynman’s path integral framework (PI) [32], coupled with computer simulation methods, was instrumental in the great advances achieved in condensed matter studies at low temperatures. This PI framework superseded the semiclassical methods [5,6,7,10,33] utilized to tackle the related quantum problems, thus giving a new impetus to the study of quantum fluids; the main focus was on the diffraction effects and zero-spin boson regimes. A good deal of equilibrium questions related to the pair level were successfully addressed in various PI ways (path integral Monte Carlo (PIMC), path integral molecular dynamics (PIMD), and PI effective quantum potentials) [7,12,19,34,35,36,37]. Furthermore, PIMC and PIMD experienced important advances early in the present century [13,16,17,18,38,39,40,41,42,43], and the PIMC tackling of triplet structural issues has also come into play [8,9,44,45]. Interestingly, the related PI computational methods may be regarded as appropriate “translations” of those existing in the classical domain, and the latter has further served as a source of complementary methods for quantum applications [7,8,9,17,43]. Nevertheless, the case of fermion fluids always remained as a pending numerical issue since, because of the “sign problem,” it was out of the reach of PI applications. Fortunately, for the study of fermion systems, an advanced version of PIMC, based on Wigner’s functions (WPIMC), has been reported lately [14,15]. At this point, for non-charged-particle systems, it seems worthwhile to stress the contrast between the studies of the quantum diffraction regime and of the quantum exchange statistics (Bose–Einstein and Fermi–Dirac): the generality of the former, which can be applied to the properties of every system, versus the singularity of the latter, which must deal with the impressive properties shown by a reduced number of systems (e.g., liquid helium-4 and liquid helium-3 at very low temperatures) [12,14,46,47,48].

This entry is devoted to the structures of 3D “monatomic” fluids at equilibrium, which implies that these systems are homogeneous and isotropic [2,6]. The present use of “monatomic” is a broad one, as it covers the proper monatomic fluids (with no isotopic mixture), e.g., Ar, Ne, ⁴He, or ³He, and in a less rigorous form, those which, not being strictly monatomic, can be described under certain conditions as composed of one-site particles, e.g., N₂, CH₄, para-H₂, or hard-sphere [7,17,18,19,21,22,49,50,51,52]. For the latter group, it is obvious that some approximations in the descriptions of structures are to be made (e.g., the use of the particle centers of gravity to determine the positions and further assumptions to extract the actual atom–atom structures) [7,40]. By reason of its nature, this entry is an informative overview dealing mainly with well-established structural issues. Thus, the emphasis is placed mainly on the theoretical aspects at the pair level of the statistical mechanics treatments, addressing both the classical and the spinless quantum behaviors, the latter including the zero-spin boson fluid. Neither the discussion nor the references can be exhaustive. Hopefully, the interested readers will find the presentation of facts useful for their own organization of the many concepts involved and the list of selected works as an introductory source to start to grasp in depth this fundamental research area. For comprehensive accounts, see References [2,5,7,9,12,16].

2. Conceptual Background

2.1. The Equilibrium Concept

(i)

The concept of thermodynamic equilibrium in many-body systems is a powerful idealization that allows one to devise theoretical methods (statistical ensembles, simulation methods, interparticle potentials, etc.) using statistical mechanics [5,10] and quantum mechanics/chemistry [31,49,50,51] for dealing with its related problems. Equilibrium retains its conventional meaning as a time-independent condition, the state(s), characterized by a number of state variables (e.g., temperature, pressure, volume, composition, etc.), which can be combined into the corresponding macroscopic equation of state. Therefore, at equilibrium, the (mean) values of the characteristic properties of the system, thermodynamic and structural, remain constant in time ($\tau$). In fact, the actual property values fluctuate over time, but their spontaneous fluctuations stay controlled, and the stability of the macroscopic state as perceived by an external observer is guaranteed. (Note that stationary states are also time-independent, but they are not equilibrium states because of the existence of fluxes and their intrinsic irreversibility [53]).

(ii)

It is worth realizing that the equilibrium concept is generally linked to especial time scales that correspond to (a) the experimental time ${\tau }_{exp}$ for conducting observations/measurements and (b) the relaxation times ${\left\{{\tau }_r\right\}}_{r=\mathrm{1,2},\dots }$ associated with the durations of the phenomena occurring in the system. In a particular application, whenever ${\tau }_{exp}\ll {\tau }_r$ or ${\tau }_{exp}\gg {\tau }_r$, one can apply safely the equilibrium concept and its corresponding methods, whilst if ${\tau }_{exp}~{\tau }_r$, one cannot, as time-dependent (nonequilibrium) features dominate the system behavior under study. (For a thorough discussion of this many-faceted subject see Reference [53].) Consequently, the study of many-body systems can be undertaken at different levels of complexity, depending on the interest in the phenomenon analyzed. This also implies the reduction of the problem to the significant variables that define the joint consideration of the system and phenomenon.

2.2. Monatomic Fluids and Linear Response

(iii)

Hereafter, the discussion will center on monatomic fluids. The relevant thermodynamic variables, such as temperature, density, etc., will always be assumed to accompany the single term “fluid” when used. In statistical mechanics, the related thermodynamic and structural studies are performed at the level of the atoms (j), their position variables ${\mathit{r}}_j$ being taken as those of their nuclei (or of their centers of gravity for one-site particles). This is consistent with the wavelength of the radiation employed in scattering experiments: typical values are of the order of the atomic spacings in condensed matter $~1 Å$ [2,4]. Therefore, the Hamiltonian for modelling the fluid composed of N identical atoms reads as $H_0^{\left(N\right)}={\sum }_{j=1}^N{\mathit{p}}_j^2/2m+V^{\left(N\right)}\left({\mathit{r}}_1,{\mathit{r}}_2,\dots ,{\mathit{r}}_N\right)$ [5], where ${\mathit{p}}_j$ stands for the momentum (classical function or quantum operator) of atom j, m is the atom mass, and, since atoms interact with one another, the whole potential energy is defined as an effective internuclear function $V^{\left(N\right)}$, where the information related to the electronic degrees of freedom is quantum-mechanically averaged within the framework of the Born–Oppenheimer approximation [54]. The simplest interaction situation is that a pair of atoms, 1 and 2, interact through a pair potential $v_2\left(r_{12}\right),$ where $r_{12}=\left|{\mathit{r}}_1-{\mathit{r}}_2\right|$ is their distance and r the position vector $\left(x,y,z\right)$ of the indicated particle. For larger sets composed of N atoms, $V^{\left(N\right)}({\mathit{r}}_1,{\mathit{r}}_2,\dots ,{\mathit{r}}_N)$ is a highly involved function, which can be approximated via a many-body truncated expansion [5,6,10] given by the sum of all the atom interactions in the form of pairs $v_2\left(r_{jl}\right)$, plus triplets $v_3\left(r_{jl},r_{j{l}^{\prime }},r_{{ll}^{\prime }}\right)$, plus quadruplets $v_4\left(r_{jl},r_{{jl}^{\prime }},r_{{jl}^{″}},\dots \right)$, and so on. Note that combinations label/count the n-particle subsets. The basic interaction units $v_n$ are hence quantum in origin, and simple illustrative forms for $v_2$ and $v_3$ are the 6–12 Lennard–Jones [10,22] and the Axilrod–Teller triple-dipole [31], respectively. More general expressions for $v_n$ can be determined with quantum computational methods [49,50,51]. In addition, one can also utilize geometrical models for building the units $v_n$ (e.g., the hard-sphere potential for $v_2$ [10,17,21,36,55]). All these constructions must lead to a potential energy $V^{\left(N\right)}$ that must be “non-collapsing” and “tempered” [6]. It is interesting to realize that the pairwise approach $V^{\left(N\right)}\approx {\sum }_{j<l}{v}_2\left(r_{jl}\right),$ involving effective two-body potentials, has been widely utilized in a successful way. Furthermore, the expressions for $V^{\left(N\right)}$ can be supplemented with experimental information to further fit the resulting parameters defining the final formulas [10]. Statistical mechanics relies heavily on the quality of these units for producing accurate results, as compared to experimental data, when studying real systems [10,11,12].

(iv)

A fluid undergoing the action of an external field $\Psi$ responds to the perturbation exerted. If such perturbation is weak, the response can be described in terms of the equilibrium structures of the fluid in the absence of the field, i.e., $\Psi =0,$ or in this context, the isolated-from-$\Psi$ fluid condition [2,6]. This is a central result of the so-called linear response theory, which plays a fundamental role in explaining, for example, the responses from the fluid to a time-independent weak field $\Psi \ne \Psi \left(\tau \right).$ Note that linear response is a useful aspect of the very general fluctuation–dissipation framework [2,6], its essence in plain language being summarized in sentences [6] like this: the fluid “does not know” if what is occurring is due to an external force or is the result of its own spontaneous (random) fluctuations. Accordingly, the new equilibrium state reached by the fluid under a weak $\Psi \ne \Psi \left(\tau \right)$ has internal properties that arise from modifications of their counterparts in the absence of $\Psi .$ The action of the field within the current N-atom/-nucleus context is expressed through the potential energy contribution ${\sum }_{j=1}^N\Psi \left({\mathit{r}}_j\right),$ which is to be added to the isolated-from-$\Psi$ Hamiltonian $H_0^{\left(N\right)}.$

(v)

By focusing on the equilibrium structural issues, linear response theory serves at least a double purpose: (a) fixing the $\delta -$variations in certain significant density-related fluid properties that are induced by the $\delta \Psi$ variations, and (b) obtaining an independent access to distinctive features of the elastic approximations to the scattering of radiation (i.e., significant factors that appear in the intensities of the scattered radiation in different directions, or that are part of the differential cross sections) [2]. Therefore, the fluid $\Psi$-independent structures at the pair level can be fixed directly using standard experiments involving weak fields $\Psi ,$ a fact that opens the way to comparison with theoretical calculations.

2.3. The Classical Domain

(vi)

It is instructive to consider a basic description of a closed macroscopic monatomic fluid at constant volume in a thermal bath. At sufficiently high temperatures, such a description can be achieved via the classical canonical ensemble $\left(N,\mathcal{V},T\right),$ in which the number of particles N, the volume $\mathcal{V},$ and the temperature T are held fixed [2,5,6]. By “classical”, one means the conceptual framework of classical mechanics: particles have definite positions and momenta at any instant of time. Conditions for the validity of this approach can be ascertained by using the number density, ${\rho }_N={\displaystyle \frac{N}{\mathcal{V}}},$ and the de Broglie wavelength of an atom,${\lambda }_B={\displaystyle \frac{h}{\sqrt{2\pi m{k}_{B}T}}},$ where $h$ is Planck’s constant, $k_B$ is Boltzmann’s constant, and m is the mass particle. The classical approach can be applied if ${\lambda }_B\ll {\rho }_N^{-1/3},$ the latter quantity giving the typical interatomic distance in the fluid. In this context, the central quantity that gives access to the properties of the fluid is the canonical partition function $Z_C$, which reads as:

$$Z_C\left(N,\mathcal{V},T\right)={\displaystyle \frac{1}{N!}}{\left({\displaystyle \frac{m}{2\pi \beta {\hslash }^2}}\right)}^{3N/2}\int d{\mathit{r}}^N exp\left[-\beta V^{\left(N\right)}\left({\mathit{r}}^N\right)\right],$$

(1)

where $\beta ={\displaystyle \frac{1}{k_{B}T}}, \hslash =h/2\pi$ is Planck’s constant in Dirac’s form, and $d{\mathit{r}}^N=d{\mathit{r}}_{1}d{\mathit{r}}_2\dots d{\mathit{r}}_N$ is the 3N-dimensional volume element of the configuration space ${\mathit{r}}^N=\left\{{\mathit{r}}_1,{\mathit{r}}_2,\dots ,{\mathit{r}}_N\right\}$ associated with the N-particle fluid. The probability density of this ensemble can be cast as:

$$f_{N,\mathcal{V},T}\left({\mathit{r}}_1,{\mathit{r}}_2,\dots ,{\mathit{r}}_N\right)={\displaystyle \frac{exp\left[-\beta V^{\left(N\right)}\left({\mathit{r}}^N\right)\right]}{\int d{\mathit{r}}^{N}exp\left[-\beta V^{\left(N\right)}\left({\mathit{r}}^N\right)\right]}},$$

(2)

which means that $f_{N,\mathcal{V},T}\left({\mathit{r}}^N\right)d{\mathit{r}}^N$ is the probability that atom 1 is in ${(\mathit{r}}_1,{\mathit{r}}_1+d{\mathit{r}}_1)$, atom 2 in ${(\mathit{r}}_2,{\mathit{r}}_2+d{\mathit{r}}_2),$ …, and so on. Equations (1) and (2) contain the simplified form of Maxwell–Boltzmann (MB) statistics for classical fluids composed of identical and “distinguishable” atoms (for the present purposes, the momenta have been integrated out when writing Equations (1) and (2)). The reader, however, should note the presence of $N!$ and $\hslash$ in Equation (1), factors that come from the unavoidable quantum nature of atoms [5,6]: $N!$ comes from the permutational symmetry among the N atoms, and $\hslash$ from Heisenberg’s uncertainty principle for conjugate position–momentum variables. Without these two elements, which correct a purely classical partition function, the latter would be meaningless (i.e., dimensionality problems and uselessness at fixing thermal properties such as entropies and free energies) [6].

From a practical point of view, the canonical ensemble, as defined by Equation (1), is a most convenient tool, although other options better adapted to deal with different constraints are available (e.g., the isothermal–isobaric ensemble $(N,p,T)$, or the grand canonical ensemble $(\mu ,\mathcal{V},T)$, where p = pressure and µ = chemical potential are also held fixed) [5,10,11]. Computational methods for obtaining the values of thermodynamic and static structural properties in the various ensembles have been developed throughout the years [10,11]. These methods (computer simulations) fall into two main categories: Monte Carlo and molecular dynamics, the latter being also applicable to time-dependent properties. Quite interestingly, these computer simulations effectively break off the hierarchical structure of the correlation functions (i.e., to fix $g_n$, one would need first to know $g_{n+1})$ [2,5,6]. In both cases, MC and MD, one uses a reduced sample size: for example, in the canonical ensemble, one selects conditions $(N_S,{\mathcal{V}}_S,T),$ where $N_S$ and ${\mathcal{V}}_S$ (usually a cubic box) are extremely small as compared to the macroscopic fluid values but in such a way that the bulk number density is preserved ${\rho }_N={\displaystyle \frac{N}{V}}={\displaystyle \frac{N_S}{{\mathcal{V}}_S}}.$ The number of particles $N_S$ for the simulation can be optimized so as to obtain significant results for the macroscopic fluid intensive properties (i.e., “constancy” within a certain accuracy with increasing $N_S)$. With the inclusion of adequate approximations (i.e., periodic boundary conditions) and corrections (e.g., continuum, grand-ensemble, etc.) [10,11,28], this can be typically achieved with $N_S~{10}^2-{10}^3$. (Nowadays, ${10}^2$ is rather small a number for most purposes; clearly, the larger the $N_S$ is, the better the results are.) In the end, the simulation results are to be consistent with the so-called thermodynamic limit, which is defined as the theoretical limit: $\underset{N , \mathcal{V}\to \mathcal{\infty }}{\underbrace{\mathrm{T}-lim}}$ ${\displaystyle \frac{N}{\mathcal{V}}}={\rho }_N=$ finite-nonzero value. Also, there is equivalence between the different ensembles in the $\mathrm{T}-lim$ [5,6]. The latter may help to tackle general theoretical questions (e.g., number fluctuations using the canonical ensemble and singling out a large subsystem [6]), although for many reasons a proper use of each ensemble under its specific constraints should always be preferable [9,10,27].

2.4. The Quantum Domain

(vii)

In the foregoing discussion, T was assumed to be sufficiently high as to make the ever-present quantum effects negligible. This is a reason for the success of the classical approach in statistical mechanics. Nevertheless, as T is lowered, quantum effects become appreciable and play a decisive role in any system behavior; these effects get stronger if the density ${\rho }_N$ increases [12,17,18,19]. Thus, one first finds diffraction (or dispersion) effects, since atoms delocalize and interfere with each other, which cannot be neglected whenever ${\lambda }_B~{\rho }_N^{-1/3}.$ This kind of quantum effect can be described, in practice, by a sort of statistics resembling the classical MB but conceptually radically different from it [12,32,34]. Moreover, one may deal with especial monatomic fluids, which, upon lowering T further and further, enter the quantum statistics regimes of Bose–Einstein (BE) or Fermi–Dirac (FD). Both involve spin arguments (i.e., exchange between indistinguishable atoms/particles) and belong to the realm of the very-low temperatures [12,46]. Examples are liquid ⁴He that is composed of zero-spin atoms (BE statistics, in which the system is a superfluid that flows with a negligibly small viscosity for $T\lesssim 2 \mathrm{K})$ [12,32,46] and liquid ³He that is composed of one-half spin atoms (FD statistics for $2.7m\mathrm{K}\lesssim T<1 \mathrm{K}$, with changes to BE superfluidity for $T<2.7m\mathrm{K}$!) [46]. In all the quantum cases, the language of operators [6,7,12,13,32,42,48] replaces that of the classical dynamical functions [6].

(viii)

The equilibrium quantum ensemble concepts remain essentially the same as those stated above for the classical case, although large variations arise in the formulations. For example, the general form of the quantum canonical partition function for a monatomic fluid, with all the atoms in the same spin state, can be cast in the coordinate representation as [12,32,34]:

$$Z_Q(N,\mathcal{V},T)={\displaystyle \frac{1}{N!}}{\sum }_{\mathcal{P}}{\pi }_{\mathcal{P}}\int d{\mathit{r}}^N⟨{\mathit{r}}^N|exp\left[-\beta H_0^{\left(N\right)}\right]|{\mathcal{P}\mathit{r}}^N⟩,$$

(3)

where ${\mathit{r}}^N$ and ${d\mathit{r}}^N$ retain their meanings given in Equation (1), albeit they apply here to quantum atoms/particles, $\mathcal{P}$ runs over the whole group of $N!$ permutations, $|{\mathit{r}}^N⟩$ denotes the quantum state (ket) $|{\mathit{r}}_1,{\mathit{r}}_2,\dots ,{\mathit{r}}_N⟩$, a particular permutation is defined to act as $|{\mathcal{P}\mathit{r}}^N⟩=|\mathcal{P}{\mathit{r}}_1,\mathcal{P}{\mathit{r}}_2,\dots ,\mathcal{P}{\mathit{r}}_N⟩$, where a given $\mathcal{P}{\mathit{r}}_j$ is the corresponding position state onto which the permutation carries the atom label j, ${\pi }_{\mathcal{P}}$ is the parity factor of permutation $\mathcal{P}$, that is, if BE: $+1$ for every $\mathcal{P}$, or if FD: $\pm 1$ depending on the parity ($+1$ for even $\mathcal{P}$, $-1$ for odd $\mathcal{P}),$ and $H_0^{\left(N\right)}=T^{\left(N\right)}+V^{\left(N\right)}$ is the Hamiltonian operator for modelling the isolated-from-$\Psi$ fluid, which is built as the summation of the kinetic energy operator $T^{\left(N\right)}=-{(\hslash }^2/2m)\sum _j{\nabla }_j^2$ and the potential energy operator $V^{\left(N\right)}$, the latter adopting the same general mathematical form as that mentioned earlier. Also, note that the multidimensional integral in Equation (3) involves the density matrix $exp\left[-\beta H_0^{\left(N\right)}\right]$ elements in the coordinate representation [32]. A most powerful framework for giving operational forms to Equation (3) is Feynman’s path integrals (PI) [32]: no inconsistencies are present, the accuracy of its methods can be increased arbitrarily, and the classical limit given by Equation (1) is retrieved in an elegant and direct way.

(ix)

The case of diffraction effects, which corresponds to neglecting any type of quantum exchange, is obtained from Equation (3) by keeping only the identity permutation. One finds the canonical partition function as follows [12,32,34]:

$$Z_{Q,D}\left(N,\mathcal{V},T\right)={\displaystyle \frac{1}{N!}}\int d{\mathit{r}}^N⟨{\mathit{r}}^N|exp\left[-\beta H_0^{\left(N\right)}\right]|{\mathit{r}}^N⟩.$$

(4)

By applying well-established PI mathematical transformations [12], one arrives at a description of the actual N-atom fluid via a model composed of $N\times P$ “particles” (beads) subjected to a set of “interconnections” among them. Note first that every quantum atom j is represented by a closed elastic necklace composed of P beads, which are labelled as $t=\mathrm{1,2},\dots ,P;$ this label t marks “instants” in imaginary time $\beta \hslash .$ The appealing fact is that $Z_{Q,D}$ can be approximated by a (semi-)classical-like partition function $Z_{PI},$ which reads as:

$$Z_{Q,D}\approx Z_{PI}\left(N,\mathcal{V},T;P\right)={\displaystyle \frac{1}{N!}}{\left({\displaystyle \frac{mP}{2\pi \beta {\hslash }^2}}\right)}^{{\displaystyle \frac{3NP}{2}}}\int {\prod }_{j=1}^N{\prod }_{t=1}^{P}d{\mathit{r}}_j^t \times \mathrm{e}\mathrm{x}\mathrm{p}\left[-\beta W_{NP}\right],$$

(5)

where $W_{NP}$ is a sort of “effective” potential that depends on the coordinates ${\mathit{r}}_j^t$ of every bead in the model and, also, explicitly on parameters such as the atom mass, the temperature, and Planck’s constant i.e., $W_{NP}\left({\mathit{r}}_1^1,{\mathit{r}}_1^2,\dots , {\mathit{r}}_1^P, \dots ,{\mathit{r}}_N^1,{\mathit{r}}_N^2,\dots , {\mathit{r}}_N^P; m, \beta ,\hslash \right). { W}_{NP}$ contains all the types of bead interconnections in a necklace and between different necklaces, e.g., terms related to the kinetic and potential energies [34], special terms connected to the kinetic energy that accelerate convergence [12,36,55], further potential energy interaction features [12,13,17,36,38,39], etc. (Figure 1a displays elemental features related to Equation (5)). Thus, one finds that (a) in each necklace j, any bead is harmonically coupled to its two adjoining neighbors, (b) between necklaces $j\ne l$, only equal t-label beads do interact, and (c) other features may play a role involving forces acting on the beads or correlation terms between beads, etc. The former 3N-dimensional configurational space of the N actual atoms is expanded into the 3NP-dimensional configurational space for the $N\times P$ model. In this regard, P is a positive integer number that needs to be optimized for obtaining an accurate description of the actual fluid (P is termed Trotter’s discretization). Note that $P=1$ gives the classical limit, whereas $P⟶\infty$ retrieves exactly the actual quantum fluid $Z_{Q,D}$ function [12,32,35]. One can find several forms for $W_{NP},$ which are built with the so-called propagators. The simplest propagator is the primitive one [12,32,34], for which all the beads in the sample are equivalent, and there is translational invariance in imaginary time. This propagator is the least efficient numerically (error-order $O\left(P^{-3}\right))$, and more efficient and accurate options are available [12,13,36,38,39]. However, the primitive propagator is perfectly suited for theoretical discussions [12], which is a great advantage in the basic understanding of PI. From the formal closeness between Equations (1) and (5) (i.e., the classical isomorphism [34]), computational methods to study diffraction effects can be expected to be obtained by adapting the well-known classical ones. In relation to this, it is easy to identify in Equation (5) the probability density for the bead positions. The main computational methods are PIMC and PIMD; both use reduced sample sizes ${(\mathrm{i}.\mathrm{e}.,N}_S)$ and seek for $\mathrm{T}-lim$ consistency. PIMD is an equilibrium method based on a fictitious dynamics, which bears no connection to the actual quantum dynamics. PIMC and PIMD admit elaborate variants and can be applied not only to fluids but also to the study of quantum condensed matter in general [7,12,13,17,18,19,37].

(x)

In the canonical ensemble, the BE partition function for a monatomic fluid composed of zero-spin atoms can be derived from Equation (3) and reads as [12,32]:

$$Z_{Q,BE}\approx Z_{PI}^{BE}\left(N,\mathcal{V},T;P\right)={\displaystyle \frac{1}{N!}}\int {\prod }_{j=1}^N{\prod }_{t=1}^P{d\mathit{r}}_j^t \times {\sum }_{\mathcal{P}}C\left(\mathcal{P}\right)exp\left[-\beta W_{NP}\left(\mathcal{P}\right)\right],$$

(6)

where $C\left(\mathcal{P}\right)$ are positive numbers and $W_{NP}\left(\mathcal{P}\right)$ the effective potentials, which depend on all the bead coordinates in the forms associated with each permutation (there are also dependences on $m, \beta ,\hslash )$. In building $W_{NP}\left(\mathcal{P}\right),$ the abovementioned propagators appropriately dealt with can be utilized. Therefore, in Equation (6), the $\mathcal{P}-$sum $\left({=\Omega }_{N,\mathcal{P}}^{BE}\right)$ is nonnegative everywhere and, once divided by $N!Z_{PI}^{BE},$ plays the role of a probability density in the bead configurational space. Also, in this exchange context, the basic facts regarding P and label t remain akin to those given above, but now the partition function consists of the “individual-permutation terms,” in which not only closed but also open necklaces that may interlink with one another may be present in a variety of ways (Figure 1b). Clearly, the case of the zero-spin boson fluid is amenable to direct simulation work (e.g., PIMC in different ensembles), albeit the computational methods are especial, and accuracy issues do arise [12,41,42]. Nevertheless, if one tries to tackle via PI a fermionic FD fluid (i.e., composed of half-odd-integer spin atoms) [32], the form of the partition function presents formidable numerical problems because of the permutational “alternating” sum in Equation (3) (i.e., the sign problem, for which, see also Reference [48]). Hence, even more involved methods are required to treat the FD case via path integrals [14,15].

3. Monatomic Fluid n−Body Structures at Equilibrium

By selecting the canonical ensemble $\left(N,\mathcal{V},T\right)$, in a homogeneous and isotropic monatomic fluid at equilibrium, the one-body elemental structure is ${\rho }_N^{\left(1\right)}\left({\mathit{r}}_1;\Psi =0\right)={\rho }_N{g}_1\left(\mathit{r}\right)={\rho }_N.$ The corresponding structural function is then $g_1\left(\mathit{r}\right)=1$ [5,6]. Note that the Fourier transform of ${\rho }_N$ is directly related to a Dirac-$\delta .$ The conceptually revealing structural functions appear when analyzing higher correlation levels $\left(n\ge 2\right):$ ${\rho }_N^{\left(n\right)}\left({\mathit{r}}_1,{\mathit{r}}_2,\dots ,{\mathit{r}}_n;\Psi =0\right)=$ ${\rho }_N^n{g}_n({\mathit{r}}_1,{\mathit{r}}_2{,\dots ,\mathit{r}}_n$). Therefore, one finds (a) the correlation functions in real space (for brevity and regardless of its dimensionality, r-space) $g_n({\mathit{r}}_1,{\mathit{r}}_2{,\dots ,\mathit{r}}_n$), which depend on the interatomic distances, i.e., $g_2\left(r_{12}\right),$ $g_3\left(r_{12},r_{13},r_{23}\right)$, etc., and (b) the structure factors $S^{\left(n\right)}\left(\mathit{k},{\mathit{k}}^{\prime },\dots ,{\mathit{k}}^{{\left(n-1\right)}^{\prime }}\right),$ which depend on the wave vectors (and/or the wavenumbers $k=\left|\mathit{k}\right|$, etc.) defining the associated reciprocal Fourier space (for brevity and regardless of its dimensionality, k-space), i.e., $S^{\left(2\right)}\left(k\right),$ $S^{\left(3\right)}\left(k,k^{\prime },\mathit{cos}\left[\mathit{k},{\mathit{k}}^{\prime }\right]\right),$ etc. The structures $g_n$ and $S^{\left(n\right)}$ are real-valued, intensive, and dimensionless properties of the fluid, which are connected through (nontrivial) Fourier transforms. Apart from their dependence on the natural variables in their formulations, they also depend on the fluid density ${\rho }_N$ and the temperature $T$, although these dependences are not explicitly written unless necessary. Also, note the fact that the wave vectors k serve to label the momentum transfers field-fluid $\hslash \mathit{k}$ in radiation scattering experiments [2,6].

There are three main lines of thought that lead to the foregoing equilibrium structures [2,5,6,9,27,45] and give the complete physical meaning of such static n-body functions. The first two stem from the partition function framework via (1) the ensemble probabilistic arguments related to the spatial distribution of any n atoms, irrespective of the configuration of the rest $N-n,$ in the correspondingly reduced configuration space of the fluid $\left\{{\mathit{r}}_1,{\mathit{r}}_2{,\dots ,\mathit{r}}_n\right\}$, and (2) the taking of functional derivatives with respect to the action of an external weak field, acting as $\Psi ={\sum }_j\Psi \left({\mathit{r}}_j\right).$ The $\Psi$-effect, after having been properly added to $H_0^{\left(N\right)}$, defines a new partition function $Z\left(\Psi \right)$ for the perturbed fluid situation. The formal standard results within the canonical ensemble begin with ${\Gamma }_1\left({\mathit{r}}_1;\Psi \right)={\rho }_N^{\left(1\right)}\left({\mathit{r}}_1;\Psi \right)=-k_{B}T{\displaystyle \frac{\delta lnZ\left(\Psi \right)}{\delta \Psi \left({\mathit{r}}_1\right)}},$ and by iterating the differentiations, one finds the n-th order quantities under the conditions $\Psi \ne 0$ and $\Psi =0$, e.g., ${\Gamma }_n\left({\mathit{r}}_1,{\mathit{r}}_2{,\dots ,\mathit{r}}_n;\Psi =0\right)={(-k_{B}T)}^n{\left.{\displaystyle \frac{{\delta }^{n}lnZ\left(\Psi \right)}{\delta \Psi \left({\mathit{r}}_1\right)\delta \Psi \left({\mathit{r}}_2\right)\dots \delta \Psi \left({\mathit{r}}_n\right)}}\right|}_{\Psi =0}.$ Application of linear response theory utilizes ${\Gamma }_n(\Psi =0)$ quantities for approximating behaviors under $\Psi \ne 0$. Albeit this functional derivative formalism enjoys a perfectly definite mathematical status in the classical domain [2,6,27], there are some subtle difficulties in the quantum case (e.g., the “collapse” of the particle thermal quantum packets under localization experiments, or dealing with inverse functional derivatives) [7,9,45]. However, at the pair level $n=2$, the general concepts in the r- and k-spaces arising from the foregoing two lines of thought may be related, exactly or to a great accuracy, with those of a third line: (3) the theoretical analysis of experimental facts (e.g., elastic and inelastic scattering of radiation experiments [2,4,6,20,44]), which form the necessary link with observable reality. It is worthwhile to stress that the latter experiments form a very special field of research full of intricacies [2,4,20]. Recall that beyond the pair level, no structure can be determined experimentally. Also, note that the raw data obtained through both types of experiments need careful processing for fixing the k-space results. Thus, in X-ray scattering the interaction mechanism is the photon–electron collision, and corrections for Compton scattering effects, the atomic form factor, etc., are made. In neutron scattering, the mechanism is the neutron–nucleus collision, and corrections for inelastic and recoil effects, analyses with sum rules, etc., are applied.

A given structural function $S^{\left(n\right)}$ is the Fourier transform of ${\Gamma }_n\left(\Psi =0\right),$ which turns out to be an operation involving the $\left\{g_{n^{\prime }}\right\}$ correlations up to level n $(n^{\prime }\le n)$. Put in other words, from the analysis in k-space, $S^{\left(n\right)}$ appears as a linear response function from the fluid that is defined in terms of the correlation functions in the isolated-from-$\Psi$ fluid. Thus, for $n\ge 2,$ $S^{\left(n\right)}$ arises as a proportionality factor between the variations in the global structural properties $\delta {\Gamma }_{\mathit{k}}^{\left(n-1\right)}\left(\Psi \right)$ and the variations in the external field $\delta {\Psi }_{\mathit{k}}$. The treatment of the pair level has become a standard task [7,10,11,12,13,16,17,20]. Beyond the pair level $(n\ge 3)$, the determination of the structural functions is a highly involved theoretical/numerical task based on computer simulations and/or qualitative approaches (i.e., the so-called closures, which try to describe approximately structural properties at level n in terms of the information available at lower levels $n^{\prime }<n).$ For works on the $n=3$ triplet structures, the reader is referred to References [23,24,25,26,27,29,30] for the classical case and to References [8,9,33,45] for the quantum diffraction case. To grasp the essence of the foregoing ideas, it may be sufficient to consider the structural functions at the pair level $n=2$ in the canonical ensemble. The discussion starts with the classical fluid, and then the quantum cases of diffraction effects and of zero-spin boson exchange follow.

3.1. The Classical Monatomic Fluid at Level $n=2$

In the classical domain, there is only one class of structures ${\left\{g_n,S^{\left(n\right)}\right\}}_C$ (Cn), which, at the pair level, consists of $g_2$ and $S^{\left(2\right)}$ [2,5,6]. The pair radial correlation function $g_2\left(r\right)$ of the isolated-from-$\Psi$ fluid in the canonical ensemble is given by the following definition:

$${\rho }_N^{\left(2\right)}\left({\mathit{r}}_1,{\mathit{r}}_2;\Psi =0\right)= {\rho }_N^2{g}_2(r)={\displaystyle \frac{N!}{\left(N-2\right)!}}{\displaystyle \frac{\int d{\mathit{r}}_{3}d{\mathit{r}}_4\dots {d\mathit{r}}_N exp\left[-\beta V^{\left(N\right)}\left({\mathit{r}}^N\right)\right]}{\int d{\mathit{r}}^{N}exp\left[-\beta V^{\left(N\right)}\left({\mathit{r}}^N\right)\right]}}$$

(7)

where $r=\left|{\mathit{r}}_1-{\mathit{r}}_2\right|.$ The normalization condition is customarily cast as the following spherical average:

$$4\pi {\rho }_N{\int }_0^{\infty }{g}_2\left(r\right)r^{2}dr=N-1.$$

(8)

In this connection, the following facts are worth remarking. (a) The combinatorial factors in front of the quotient of integrals in Equation (7) that indicate how pairs of atoms are selected. (b) $g_2\left(r\right)$ is related to the probability of finding an atom at a distance r from any other atom in the fluid; as written, such probability is not normalized to unity. (c) The limits of integration in the normalization could have been taken according to the volume $\mathcal{V}$ of the container, but note that in Equation (8), the upper limit in the spherical average is set to infinity, while the lower limit that is set to zero can be increased to a value $\sigma >0.$ (d) The foregoing upper limit is a mathematically convenient assumption (rooted in the $\mathrm{T}-lim$) [5,6], since the macroscopic fluid is vastly larger than a single atom, and a lower limit $\sigma$ arises from the strong repulsions between atoms at short distances, which make $g_2\left(r<\sigma \right)=0$. (e) Any atom in the fluid “sees” $N-1$ atoms around it. (f) ${\rho }_N{g}_2\left(r\right)$ yields the average local density about a given atom as a function of the distance. (g) At very long distances, two atoms are uncorrelated, and one expects the theoretically correct behavior $g_2\left(r\to \infty \right)\to 1$ (see below) [5]. (h) Last but not least, no surface effects are considered.

Furthermore, it is useful to give a powerful formulation of Equation (7), which can be cast as the following ensemble average $〈\dots 〉$ involving Dirac-$\delta$ functions [6]:

$${\rho }_N^2{g}_2\left(r\right)=\langle \sum _{j\ne l}\delta ({\mathit{q}}_1-{\mathit{r}}_j)\delta ({\mathit{q}}_2-{\mathit{r}}_l)\rangle =$$

(9)

$${\displaystyle \frac{\int d{\mathit{r}}_{1}d{\mathit{r}}_{2}d{\mathit{r}}_3\dots {d\mathit{r}}_N \left(\sum _{j\ne l}\delta ({\mathit{q}}_1-{\mathit{r}}_j)\delta ({\mathit{q}}_2-{\mathit{r}}_l)\right) exp\left[-\beta V^{\left(N\right)}\left({\mathit{r}}^N\right)\right]}{\int d{\mathit{r}}^{N}exp\left[-\beta V^{\left(N\right)}\left({\mathit{r}}^N\right)\right]}}$$

where the indices j and l run over $\mathrm{1,2},\dots ,N,$ avoiding equal values (this summation convention will be followed hereafter). Note that for notational convenience use is made of arbitrary vectors ${\mathit{q}}_1$ and ${\mathit{q}}_2,$ which are constrained by $r=\left|{\mathit{q}}_1-{\mathit{q}}_2\right|$. At higher levels, the formulas for $g_n$ can be easily generalized [6], and the use of the q-auxiliary vectors turns out to be a great help in setting and manipulating the related equations [6]. (In the structure topic, $\delta$-symbols may appear even within the same formula with two different meanings [2,6,9,27]: Dirac-$\delta$’s and functional $\delta -$variations; they must not be confused with each other.)

The static structure factor $S^{\left(2\right)}\left(k\right)$ is defined essentially by the Fourier transform of $g_2\left(r\right)$, but in full detail, it must be written as [2]:

$$S^{\left(2\right)}\left(k\right)=1+{\rho }_N\int d\mathit{r}\mathit{exp}\left(i\mathit{k}·\mathit{r}\right)\left(g_2\left(r\right)-1\right),$$

(10)

an expression that can be derived from the elastic approach to scattering of radiation experiments (X-ray, neutron diffraction) [2,4,6]. Here, it is pertinent to comment on two related facts. First, from the study of elastic X-ray scattering [2], one finds that the intensity of the scattered radiation, at an angle $\theta$ to the incoming beam direction, is directly proportional to $S^{\left(2\right)}\left(k\right)$ (warning: there is an additional subtraction of forward scattering). Second, Equation (10) can also be derived by applying the elastic sum rule to the dynamic structure factor $S_{dyn.}^{\left(2\right)}(\mathit{k},\omega )$ obtained through inelastic neutron scattering [2]. Furthermore, application of Yvon’s linear response theory [2] to the second functional derivative ${\Gamma }_2\left({\mathit{r}}_1,{\mathit{r}}_2;\Psi \right)={(-k_{B}T)}^2{\displaystyle \frac{{\delta }^{2}lnZ\left(\Psi \right)}{\delta \Psi \left({\mathit{r}}_1\right)\delta \Psi \left({\mathit{r}}_2\right)}}$ leads, through its Fourier transform, directly to the k-space relationship $\delta {\rho }_N^{\left(1\right)}\left(\mathit{k};\Psi \right)\approx -\beta {\rho }_N{S}^{\left(2\right)}\left(k\right)\delta \Psi \left(\mathit{k}\right),$ which quantifies the loss of homogeneity in the fluid due to the field. It goes without saying that the latter relationship is proof of the power of this sort of general reasoning. Alternatively, $S^{\left(2\right)}\left(k\right)$ can be expressed via the following canonical ensemble average [2,6]:

$$S^{\left(2\right)}\left(k\right)=N^{-1}〈{\sum }_{j=1}^N{\sum }_{l=1}^{N}exp\left(i\mathit{k}·\left[{\mathit{r}}_j-{\mathit{r}}_l\right]\right)〉-{\left(2\pi \right)}^3{\rho }_N\delta \left(\mathit{k}\right),$$

(11)

where the second term on the right-hand side is related to the forward scattering alluded to above. If higher correlation levels are to be investigated, the use of functional calculus always provides the final correct formulations regarding the proper inclusion of Dirac-$\delta$’s [27].

It must be remarked that the canonical correlation functions $g_n$ suffer from an incorrect asymptotic behavior: for very long interparticle distances in the fluid, terms of the order $O({\displaystyle \frac{1}{N}})$ play an undesired role [5], for example, $g_2\left(r\to \infty \right)\to 1+O\left(N^{-1}\right).$ Moreover, the accurate numerical evaluation of the Fourier transform in Equation (10) depends critically on the precise knowledge of the long-ranged oscillatory behavior of $g_2\left(r\right)$ about unity [10]. All these drawbacks in augmented form are also present at higher levels, their formal solutions being attained by working in the grand canonical ensemble [5,26,27]. In the latter ensemble $g_2\left(r\to \infty \right)\to 1$ [5], and one can introduce auxiliary structural functions in a fully consistent way, i.e., the n-body direct correlation functions $c_n\left({\mathit{r}}_1{,\mathit{r}}_2,\dots ,{\mathit{r}}_n\right),$ via inverse functional derivative procedures [2,6,26,27]. These $c_n$ functions, through their associated Ornstein–Zernike schemes, may help to circumvent the numerical problems with the Fourier transforms and, also, the MC/MD simulation costs (e.g., Equation (11)) [10,27,28,29,30]. In relation to this, the example at the pair level is the clearest: Equation (10), which reads formally the same in the grand canonical ensemble (in the open system, ${\rho }_N$ is not constant but is the mean density $〈N〉/\mathcal{V})$ can be recast as $S^{\left(2\right)}\left(k\right)={\left(1-{\rho }_N{c}^{\left(2\right)}\left(k\right)\right)}^{-1}$, where high accuracy in the Fourier transform $c_2\left(k\right)$ of the short-ranged $c_2\left(r\right)$ can be achieved [10]. In this connection, there is the crucial thermodynamic relationship $S^{\left(2\right)}\left(k=0\right)={\rho }_N{k}_{B}T{\chi }_T$, where ${\chi }_T$ is the isothermal compressibility of the fluid (i.e., a way to the equation of state!) [2,6,10]. Also, the asymptotic behavior at large k wave vectors is $S^{\left(2\right)}\left(k\to \infty \right)\to 1,$ which is reached in an oscillatory way about unity. As mentioned above, the computational process involved in fixing $S^{\left(2\right)}\left(k\right)$ via Equation (11) is costly, its major drawback being the determination of the $k=0$ component, which requires extrapolation procedures and hence additional computational effort [10]. A further remark: within the usual statistical mechanics context, for a truly monatomic fluid, there is no difference between the pair response functions $S^{\left(2\right)}\left(k\right)$ fixed with X-ray or with neutron diffraction; clearly, there may be slight differences between their respective results owing to the different interaction mechanisms involved and the corrections utilized [4,20].

3.2. The Quantum Monatomic Fluid up to Level $n=2$ Under the Quantum Diffraction and the Zero-Spin Boson Exchange Regimes

The structural study of monatomic fluids at equilibrium and with spinless quantum behavior can be performed in ways somewhat close to that of the classical fluid. The related regimes are quantum diffraction effects (i.e., no consideration of spin at all) and bosonic exchange for indistinguishable zero-spin atoms. According to the theorem of spin-statistics, a fluid composed of zero-spin atoms (e.g., helium-4) belongs to the boson type and must be described by wavefunctions symmetric under the permutations of the atoms (the density matrix is built with such wavefunctions, and permutational symmetry must hold) [32]. Nonzero-spin atom fluids, either the boson type (composed of integer spin atoms) or the fermion type (composed of half-odd-integer spin atoms), do necessitate treatments that fall out of the subject of this entry; the reader is referred, for instance, to the recent PIMC literature on fermion systems [14,15].

Within the framework defined for this entry, the mathematical tools arise from Feynman path integrals PI [32] and are embodied in Equations (5) and (6) [12,34]. The form of $W_{NP}$ will be the theoretically convenient and simplest choice: that based on the primitive propagator [12,34]. The experimental techniques that reveal the quantum structures (if possible) are the same as those mentioned in the classical case: X-ray and neutron diffraction.

3.2.1. Quantum Diffraction Regime

The thermal quantum diffraction conditions bring about the delocalization of the particles in any system: the lower the temperature T and the particle mass m are, the larger the delocalization ${(\lambda }_B)$ becomes. Therefore, by “switching on” diffraction effects, the single class ${\left\{g_n,S^{\left(n\right)}\right\}}_C$ in the classical monatomic fluid must undergo significant changes. Within the PI framework, the study of the structures defined by the beads-in-necklace picture reveals that ${\left\{g_n,S^{\left(n\right)}\right\}}_C$ splits into three physically significant classes, namely instantaneous ${\left\{g_n,S^{\left(n\right)}\right\}}_{ETn}$ (ETn), total thermalized-continuous linear response ${\left\{G_n,S^{\left(n\right)}\right\}}_{TLRn}$ (TLRn), and centroids ${\left\{g_n,S^{\left(n\right)}\right\}}_{CMn}$ (CMn), where a centroid is the “center-of-mass” of a PI necklace [7,9,13,43,44,45]. Each of these classes is related to the linear response from the fluid to the action of a type of external field: ETn is related to localizing fields (e.g., as in X-ray or neutron diffraction), TLRn to general continuous fields acting as $\Psi ={\sum }_j\Psi \left({\mathit{r}}_j\right),$ and CMn specifically to continuous fields of constant strength $\mathit{f}$, such that $\Psi ={\Psi }_F={\sum }_j\mathit{f}·{\mathit{r}}_j.$ Up to the pair level, the first two responses (ET2 and TLR2) can be determined directly through experiments [4], whereas the third (CM2) cannot. Interestingly, the foregoing three classes of structures can also be identified in the zero-spin boson exchange regime, although their distinct features differ obviously from those in the diffraction regime.

The form of the PI canonical quantum partition function of a monatomic fluid in the diffraction regime is given in Equation (5). In what follows, Trotter’s discretization P is assumed to be the optimal from the statistical standpoint. $W_{NP}$ is given by the simple primitive formula [12,34]:

$$W_{NP}={\displaystyle \frac{mP}{2{\beta }^2{\hslash }^2}}{\sum }_{j=1}^N{\sum }_{t=1}^{P*}{\left({\mathit{r}}_j^t{-\mathit{r}}_j^{t+1}\right)}^2+P^{-1}{\sum }_{j<l}{\sum }_{t=1}^P{v}_2\left(r_{jl}^t\right),$$

(12)

where $P^{*}$ indicates that the summation is cyclic (i.e., $t+1=P+1 \equiv 1)$. The harmonic couplings between adjacent beads mentioned earlier are contained in the first term; they are related to the kinetic contribution to the internal energy of the fluid. The pairwise approximation for the potential energy $V^{\left(N\right)}$ is used in the second term, where $r_{jl}^t=\left|{\mathit{r}}_j^t-{\mathit{r}}_l^t\right|$ (Figure 1a). As seen, all the beads in the primitive-propagator sample are treated formally on an equal footing (depending on the propagator, this may not be so [7,13]).

The instantaneous ET2 pair correlation function is defined via the equal-t-label ensemble average $(r=\left|{\mathit{q}}_1-{\mathit{q}}_2\right|)$ [12,34]:

$${\rho }_N^2{g}_{ET2}\left(r\right)={P^{-1}\langle {\sum }_{j\ne l}{\sum }_{t=1}^P\delta ({\mathit{q}}_1-{\mathit{r}}_j^t)\delta ({\mathit{q}}_2-{\mathit{r}}_l^t)\rangle }_{PI},$$

(13)

which can be understood as the P-average of the pair radial structures existing in the model fluid and defined for every equal-t set of N beads: each one of these beads belongs to a different necklace $\left(j=\mathrm{1,2},\dots ,N\right),$ and all of them share the same label $\left(t=\mathrm{1,2},\dots ,P\right).$ This structure of the fluid in the absence of $\Psi$ is associated with its instantaneous structure factor, which takes the classical-like form [2,4,12]:

$$S_{ET}^{\left(2\right)}\left(k\right)=1+{\rho }_N\int d\mathit{r} \mathit{exp}\left(i\mathit{k}·\mathit{r}\right)\left(g_{ET2}\left(r\right)-1\right).$$

(14)

and can be deduced from standard quantum mechanical reasoning [2]. It is worth pointing out that the ET2 class cannot be derived from functional arguments based on an external field and analogous with those applied to the classical fluid. This is due to the singularity of the atom–radiation interactions involved in the corresponding scattering experiments [2]. On the one hand, in X-ray elastic scattering, the electrons are the actual entities localized by the photons, defining in this way the atom (nucleus) r-positions; as a result, in addition to $S_{ET}^{\left(2\right)}\left(k\right)$, an independent one-body term (atomic form factor) is produced, its treatment being carried out congruously with the pair-level information contained in Equation (14) [2]. On the other hand, disregarding spin effects, in inelastic neutron scattering, the atom nuclei are the entities localized, since the neutron–nucleus interaction is essentially given by the Fermi pseudopotential (a Dirac-$\delta ),$ and there is no atomic form factor [2,4]. Therefore, one faces in both cases the localization of the atom and the “collapse” of its thermal quantum packet, a fact that is not consistent with the PI framework utilized. Also, for truly monatomic fluids, the $S_{ET}^{\left(2\right)}\left(k\right)$ obtained in X-ray and in neutron scattering (via a sum rule involving the dynamic structure factor $S_{dyn.}^{\left(2\right)}\left(\mathit{k},\omega \right)$ [4]) enjoy the same statistical mechanics status, as occurred in the classical $S^{\left(2\right)}\left(k\right).$ In this time-independent context, using the necessarily condensed language of theory for encompassing highly complex and distinct experimental realities, one may broadly say that “collision/localization/elastic arguments” are involved in the $S_{ET}^{\left(2\right)}\left(k\right)$ discussion [7,9].

In sharp contrast, the functional derivative developments [34] can be utilized for identifying the TLRn and CMn classes [7,8,9,43,44,45]. One must add the continuous field $\Psi =\sum _{j=1}^N\Psi \left({\mathit{r}}_j\right)$ to $H_0^{\left(N\right)}$ and proceed accordingly with the partition function. The new PI equilibrium (primitive) partition function in the presence of the field reads as [7]:

$$Z_{PI}\left(N,\mathcal{V},T;P;\Psi \right)={\displaystyle \frac{1}{N!}}{\left({\displaystyle \frac{mP}{2\pi \beta {\hslash }^2}}\right)}^{{\displaystyle \frac{3NP}{2}}}\int {\prod }_{j=1}^N{\prod }_{t=1}^{P}d{\mathit{r}}_j^t\times exp\left[-\beta W_{NP}-{\displaystyle \frac{\beta }{P}}{\sum }_{j,t}\Psi \left({\mathit{r}}_j^t\right)\right],$$

(15)

where, in the exp-factor, j runs over the values $\mathrm{1,2},\dots ,N$, t runs over $\mathrm{1,2},\dots ,P,$ and the field couples with every bead in the sample (note the P-average). By carrying out the first two functional derivatives of $ln{Z}_{PI}\left(\Psi \right)$ with respect to $\Psi \left({\mathit{r}}_j^t\right),$ and by applying linear response arguments, one arrives at the TLR2 functions of the isolated-from-$\Psi$ fluid $(r=\left|{\mathit{q}}_1-{\mathit{q}}_2\right|)$ [7,17]:

$$\begin{gathered} {\left(P{\rho }_N\right)}^2{G}_{TLR2}\left(r\right)={\langle {\sum }_{j=1}^N {\sum }_{t\left(j\right)=1}^P{}_{\underset{t\ne t^{\prime }}{⏟}}{\sum }_{t^{\prime }\left(j\right)=1}^P\delta \left({\mathit{q}}_1-{\mathit{r}}_j^t\right)\delta \left({\mathit{q}}_2-{\mathit{r}}_j^{t^{\prime }}\right) \rangle }_{PI}+ \\ {\langle {\sum }_{j\ne l}{\sum }_{t\left(j\right)=1}^P{\sum }_{t^{\prime }\left(l\right)=1}^P\delta \left({\mathit{q}}_1-{\mathit{r}}_j^t\right)\delta \left({\mathit{q}}_2-{\mathit{r}}_l^{t^{\prime }}\right)\rangle }_{PI}={\left(P{\rho }_N\right)}^2\left(s_{SC,1}\left(r\right)+g_{LR2}\left(r\right)\right). \end{gathered}$$

(16)

$$S_{TLR}^{\left(2\right)}\left(k\right)=P^{-1}+{\rho }_N\int d\mathit{r} exp(i\mathit{k}·\mathit{r}) \left(G_{TLR2}\left(r\right)-1\right),$$

(17)

where $G_{TLR2}\left(r\right)$ is the overall pair correlation function between beads in the $N\times P$ model (at density $P{\rho }_N)$, a function that can be decomposed into bead–bead correlations in the same necklace (self-correlations $s_{SC,1},$ with the restriction $t\ne t^{\prime }$) and bead–bead correlations between pairs of necklaces ($g_{LR2},$ with no restrictions on the t labels). From Equations (16) and (17) one can identify for TLR2 a sort of thermal quantum “form factor” associated with a delocalized nucleus (atom) [17]. Note that an experimental determination of $S_{TLR}^{\left(2\right)}\left(k\right)$ can be extracted from $S_{dyn.}^{\left(2\right)}\left(\mathit{k},\omega \right)$ through a further sum rule [4].

By selecting the especial field of constant strength ${\Psi }_F=\sum _{j=1}^N\mathit{f}·{\mathit{r}}_j$ as a particular case of the continuous field, use of the PI centroid variable ${\mathit{R}}_{j,CM}=P^{-1}{\sum }_{t=1}^P{\mathit{r}}_j^t$ allows one to rewrite Equation (15) by incorporating the centroid coordinates. Although the centroids as such cannot physically couple with the field, the functional derivatives with respect to ${\Psi }_F\left({\mathit{R}}_{j,CM}\right)$ can be formally obtained. Application of linear response theory gives the centroid pair structures $(R=\left|{\mathit{q}}_1-{\mathit{q}}_2\right|)$[7,17,43]:

$${\rho }_N^2{g}_{CM2}\left(R\right)={\langle {\sum }_{j\ne l}\delta ({\mathit{q}}_1-{\mathit{R}}_{j,CM})\delta ({\mathit{q}}_2-{\mathit{R}}_{l,CM})\rangle }_{PI},$$

(18)

$$S_{CM}^{\left(2\right)}\left(k\right)=1+{\rho }_N\int d\mathit{R} \mathit{exp}\left(i\mathit{k}·\mathit{R}\right)\left(g_{CM2}\left(R\right)-1\right),$$

(19)

both expressions being classical-like.

The foregoing ET2 and CM2 functions behave in ways analogous to the classical C2 functions (i.e., same general features at short- and long-range distances), whereas TLR2 shows distinct features that arise from its mixed character combining self and pair correlations [7,17]. Thus, for $G_{TLR2}\left(r\right)$, one observes that (a) $s_{SC,1}$ increases with P as $r\to 0$, whereas it always tends to zero as $r\to \infty$ [44]$,$ and (b) $g_{LR2}\left(r\right)$ may not tend to zero for $r\to 0$, albeit it tends to unity as r increases [17]. As regards $S_{TLR}^{\left(2\right)}\left(k\right)$, it tends to zero for large k wave vectors [17,44]. A joint consideration of the three structure factors, using the grand canonical ensemble, leads to the extended isothermal compressibility theorem [43]:

$$S_{ET}^{\left(2\right)}\left(k=0\right)=S_{TLR}^{\left(2\right)}\left(k=0\right)=S_{CM}^{\left(2\right)}\left(k=0\right)={\rho }_N{k}_{B}T{\chi }_T,$$

(20)

which results from the fact that the number fluctuations in the fluid [6], which give the isolated-from-$\Psi$ fluid isothermal compressibility, are independent of the pair-level structural function selected for their counting. Alternatively, Equation (20) can be viewed as the uniqueness of the value of the fluid response functions when no momentum transfer occurs $(\hslash \mathit{k}=0)$.

Like the classical case, accuracy in the calculations of the foregoing pair structure factors entails either an extended r-behavior knowledge of the corresponding radial function or the costly k-space averages that are the adapted PI-translations [9] of the classical Equation (11). In any case, one must face increasing computational effort (recall the role of P in PIMC or PIMD) [7]. Under these circumstances, one can resort to direct correlation functions and their associated (classical-like) Ornstein–Zernike schemes, although this kind of mathematical tool is only formally exact in the centroid CM2 case [9,43,44]. Even though for ET2 and TLR2 such direct correlation function schemes are approximations, they have proven their worth, yielding excellent results under a wide range of conditions [7,17].

3.2.2. Zero-Spin Boson Regime

The zero-spin boson monatomic fluid retains the three PI structural classes just reviewed above: instantaneous, total thermalized-continuous linear response, and centroids. Although the general theoretical methods and experimental technique operations are formally the same as stated earlier, their precise nature changes dramatically. This is due to (a) the especial form of the canonical partition function Equation (6), which involves the $\mathcal{P}$ − sum over the $N!$ permutations among indistinguishable atoms, and (b) the more demanding very-low-temperature requirements.

The interesting fact from the theoretical standpoint is that the PI fluid model in the canonical ensemble still consists of $N\times P$ beads, which interact through the corresponding $v_n$ − potential units, if and only if they possess identical t-labels. Moreover, for a given bead configuration, the potential energy is the same regardless of the permutation considered, because of the internal symmetry in $V^{\left(N\right)}$. Furthermore, the beads interlink (harmonically) among them in each permutation in various forms. It is worth realizing that the previous PI-diffraction description forms the basic pattern for building the far-more-complicated bead interconnections of the boson fluid. Thus, one finds that (a) the identity permutation yields the fluid picture under diffraction effects, with N closed necklaces with P beads apiece; (b) there are permutations that produce n closed P-necklaces coexisting with other closed and larger necklaces sized $n^{\prime }P, n^{″}P,\dots ,$ $(n,n^{\prime },n^{″}\dots <N;$ total number of beads in every permutation = NP); and so on, up to reaching (c) the permutations that give rise to a whole NP-membered closed necklace. It is important to insist on the fact that the individual P-necklaces are not necessarily closed, but rather they are best viewed as open broken P-lines under certain permutations. If they must interlink, they interlink in a head–tail fashion: for example, a closed 2P-necklace arising from open necklace j and open necklace l is built from harmonically linking bead P(j) with bead 1(l) and bead 1(j) with bead P(l); the rest of the beads in j and in l keep their harmonic links with their corresponding adjacent neighbors. This qualitative $2P$ − description can be generalized in a very straightforward way (see Figure 1b for a further example), and the reader is referred to References [12,34,41] for specific details.

The definitions of the isolated-from-$\Psi$ boson–fluid pair structures must take into account that no atomic thermal quantum packet can be defined here and that the density probability obtainable from Equation (6) is to be employed [9,12,43]. For the instantaneous $g_{ET2}^{BE}\left(r\right)$ and $S_{ET}^{BE\left(2\right)}\left(k\right),$ the pairs of beads of the $N\times P$ sample sharing the same t-label are involved in the definitions, in ways similar to Equations (13) and (14). The physical meaning of these functions is the same as described in connection with the quantum diffraction regime (i.e., actual particle localization arguments for radiation scattering phenomena apply here). For the total thermalized-continuous linear response to an external weak field $\mathsf{\Psi }\left(\mathbf{r}\right)$, the functions $G_{TLR2}^{BE}\left(r\right)$ and $S_{TLR}^{BE\left(2\right)}\left(r\right)$ are formulated involving the pairs of beads in the $N\times P$ sample without any restriction on their t-labels, in ways similar to Equations (16) and (17) but without any separation into self and pair parts. Furthermore, the especial choice of a field of constant force ${\Psi }_F$ also leads to the pair centroid $g_{CM2}^{BE}\left(R\right)$ and $S_{CM}^{BE\left(2\right)}\left(R\right)$ structures. Here, it is necessary to keep from the start the general notion of N open P-necklaces (or P-membered broken lines) in order to define their associated centroid variables (i.e., the “centers of mass” of such broken P-lines) [56,57], with which the partition function that depends on ${\Psi }_F$ can be reorganized in a centroid-per-actual-atom suitable way [43]:

$$\begin{gathered} Z_{PI}^{BE}({\Psi }_F)={\displaystyle \frac{1}{N!}}\left\{\int {\prod }_{\begin{array}{c}j=1,\dots ,N\\ t=1,\dots ,P\end{array}}d{\mathit{r}}_j^t\times \int {\prod }_{j=1,\dots ,N}d{\mathit{R}}_j\delta ({\mathit{R}}_j-{\mathit{R}}_{j,CM})\times {\Omega }_{N,\mathcal{P}}^{BE}({\mathit{r}}_1^1,\dots ,{\mathit{r}}_N^P)\times \\ exp\left(-\beta {\sum }_{j=1,\dots ,N}{\Psi }_F\left({\mathit{R}}_j\right)\right)\right\}, \end{gathered}$$

(21)

where ${\Omega }_{N,\mathcal{P}}^{BE}\ge 0$ everywhere in the bead configurational space and contains all the BE permutational effects. (Note that the foregoing definition appears in Equation (A7b) in Reference [9] with the $\delta$ − centroid product integral misplaced outside the braces.) There are some theoretical and practical problems with this latter manipulation [56,57]. Yet, if one only wants to address questions related to number fluctuations (by utilizing the quantum grand canonical ensemble), the theoretical situation for such centroids becomes acceptable, although the computational side seems intricate and still remains unexplored. In this regard, one can extend the isothermal compressibility theorem as given in Equation (20) to the zero-spin boson fluids [9,43]. As for the general PI computational aspects, propagators more elaborate and efficient than the primitive choice [12] and advanced procedures, such as the worm algorithm [42], must be utilized for evaluating the boson fluid structures.

The fixing of the boson structure factors CM2, ET2, and TLR2, via the use of classical-like direct correlation functions, presents several drawbacks [9]. For CM2, the theoretical grounds of this procedure should be clarified to assess whether it is formally exact (as it is in the diffraction regime) or just only an approximation. For ET2 and TLR2, such procedures would be forceful approximations, which, for TLR2, could be expected to be based on far more drastic assumptions. In any case, on the condition that accurate pair radial functions are available, pilot numerical calculations might be carried out to check if one could expect some reliability from such alternate Ornstein–Zernike procedures.

4. Conclusions and Future Directions

In the classical domain, the topic of static structure functions in monatomic fluids was thoroughly developed in the past 20th century. Although further progress in any area of science can always be expected, one can state that an insightful understanding of this side of the wide fluid-structure studies has been achieved. Nonetheless, the advanced research on the quantum fluid structures utilizing path integrals, which began in the last quarter of the 20th century, is as yet unfinished, and due to recent path integral Monte Carlo calculations, it appears to be a promising research avenue.

In the coming quantum fluid structural study, which will generalize the prior pair-level studies described in this entry, the following facts may prompt an ample range of developments. (a) There are no experimental techniques today that yield fluid structural information beyond the pair level $n=2.$ (b) Computations are the only way to obtain access to the fluid structures for $n\ge 3.$ (c) In a detailed description of interatomic interactions, going beyond the standard pairwise approximation, the fluid thermodynamic properties depend on and can be calculated with the higher-level structures. (d) The grand canonical ensemble should be, in general, the preferred construction for tackling theoretical and computational studies of structures. (e) Higher-level structures may hold quantum fluid features signalling the onset of changes of phase (e.g., triplet features and quantum freezing). (f) Closure schemes at the level $n=3$ may work better than expected, and, for completeness, their meaning and usefulness are still to be clarified. (g) There is a further area of structural research associated with fermion fluids.

One also notes that the related triplet functions at equilibrium are 4D objects and that path integral methods are far more expensive than their classical counterparts. These facts imply that efficient calculations and significant strategies for visualizing the results will also be important issues in this challenge.