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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. Full article

The current developments in the theory of quantum static triplet correlations and their associated structures (real r-space and Fourier k-space) in monatomic fluids are reviewed. The main framework utilized is Feynman’s path integral formalism (PI), and the issues addressed cover quantum diffraction effects and zero-spin bosonic exchange. The structures are associated with the external weak fields that reveal their nature, and due attention is paid to the underlying pair-level structures. Without the pair, level one cannot fully grasp the triplet extensions in the hierarchical ladder of structures, as both the pair and the triplet structures are essential ingredients in the triplet response functions. Three general classes of PI structures do arise: centroid, total continuous linear response, and instantaneous. Use of functional differentiation techniques is widely made, and, as a bonus, this leads to the identification of an exact extension of the “classical isomorphism” when the centroid structures are considered. In this connection, the direct correlation functions, as borrowed from classical statistical mechanics, play a key role (either exact or approximate) in the corresponding quantum applications. Additionally, as an auxiliary framework, the traditional closure schemes for triplets are also discussed, owing to their potential usefulness for rationalizing PI triplet results. To illustrate some basic concepts, new numerical calculations (path integral Monte Carlo PIMC and closures) are reported. They are focused on the purely diffraction regime and deal with supercritical helium-3 and the quantum hard-sphere fluid. Full article

In this note, I derive the Chandrasekhar instability of a fluid sphere in (N + 1)-dimensional Schwarzschild–Tangherlini spacetime and take the homogeneous (uniform energy density) solution for illustration. Qualitatively, the effect of a positive (negative) cosmological constant tends to destabilize (stabilize) the sphere. In the absence of a cosmological constant, the privileged position of (3 + 1)-dimensional spacetime is manifest in its own right. As it is, the marginal dimensionality in which a monatomic ideal fluid sphere is stable but not too stable to trigger the onset of gravitational collapse. Furthermore, it is the unique dimensionality that can accommodate stable hydrostatic equilibrium with a positive cosmological constant. However, given the current cosmological constant observed, no stable configuration can be larger than ${10}^{21}{\mathrm{M}}_{\odot }$. On the other hand, in (2 + 1) dimensions, it is too stable either in the context of Newtonian Gravity (NG) or Einstein’s General Relativity (GR). In GR, the role of negative cosmological constant is crucial not only to guarantee fluid equilibrium (decreasing monotonicity of pressure) but also to have the Bañados–Teitelboim–Zanelli (BTZ) black hole solution. Owing to the negativeness of the cosmological constant, there is no unstable configuration for a homogeneous fluid disk with mass $0<\mathcal{M}\le 0.5$ to collapse into a naked singularity, which supports the Cosmic Censorship Conjecture. However, the relativistic instability can be triggered for a homogeneous disk with mass $0.5<\mathcal{M}\lesssim 0.518$ under causal limit, which implies that BTZ holes of mass ${\mathcal{M}}_{\mathrm{BTZ}}>0$ could emerge from collapsing fluid disks under proper conditions. The implicit assumptions and implications are also discussed. Full article