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Static Structures in Monatomic Fluids: Comparison
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Please note this is a comparison between Version 2 by Perry Fu and Version 1 by Luis M. Sesé.

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.

  • thermodynamic equilibrium
  • monatomic fluids
  • classical behavior
  • quantum spinless behaviors
  • pair correlation functions
  • pair structure factors
  • linear response
  • functional calculus
  • path integrals
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][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 ℏ𝒌; in inelastic scattering, momentum and also energy (ℏ𝜔) transfers occur (k = wave vector, 𝜔= circular frequency) [2,4][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 𝑔𝑛 and structure factors 𝑆(𝑛)), 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][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[2][5][7][9][12][13][14][16][23][27],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 𝑔𝑛 and 𝑆(𝑛) developments in the real and the reciprocal Fourier spaces, respectively (n refers to the elemental number of particles involved) [2,5][2][5]. As for the experimental side, note that beyond the pair level (𝑛=2), structural information remains unobtainable today [4,20][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][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][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][23][24][25][26][27][28][29][30][31]. Moreover, the set of interrelations among higher-level structure schemes was firmly established [5,27][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][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][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][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][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][7][12][19][34][35][36][37]. Furthermore, PIMC and PIMD experienced important advances early in the present century [13[13][16][17][18][38][39][40][41][42][43],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][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][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][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][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][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][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][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][5][7][9][12][16].

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