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In the absence of work, the exchange of heat of a sample of matter corresponds to the change of its internal energy, given by the kinetic energy of random translational motion of all its constituent atoms or molecules relative to the center of mass of the sample, plus the excitation of quantum states, such as vibration and rotation, and the energy of electrons in excess to their ground state. If the sample of matter is equilibrated it is described by Boltzmann’s statistical thermodynamics and characterized by a temperature T. Monotonic motion such as that of the stars of an expanding universe is work against gravity and represents the exchange of kinetic and potential energy, as described by the virial theorem, but not an exchange of heat. Heat and work are two distinct properties of thermodynamic systems. Temperature is defined for the radiative cosmic background and for individual stars, but for the ensemble of moving stars neither temperature, nor pressure, nor heat capacities are properly defined, and the application of thermodynamics is, therefore, not advised. For equilibrated atomic nanoclusters, in contrast, one may talk about negative heat capacities when kinetic energy is transformed into potential energy of expanding bonds.

We presently extend the virial theorem for both discrete and continuous systems of material points with variable mass, relying on developments presented in Ganghoffer (Int J Solids Struct 47:1209–1220, 2010 ). The developed framework is applicable to describe physical systems at very different scales, from the evolution of a population of biological cells accounting for growth to mass ejection phenomena occurring within a collection of gravitating objects at the very large astrophysical scales. As a starting basis, the field equations in continuum mechanics are written to account for a mass source and a mass flux, leading to a formulation of the virial theorem accounting for non-constant mass within the considered system. The scalar and tensorial forms of the virial theorem are then written successively in both Lagrangian and Eulerian formats, incorporating the mass flux. As an illustration, the averaged stress tensor in accreting gravitating solid bodies is evaluated based on the generalized virial theorem.

A bstract We review and examine in detail recent developments regarding the question of the nucleon mass decomposition. We discuss in particular the virial theorem in quantum field theory and its implications for the nucleon mass decomposition and mechanical equilibrium. We reconsider the renormalization of the QCD energy-momentum tensor in minimal-subtraction-type schemes and the physical interpretation of its components, as well as the role played by the trace anomaly and Poincaré symmetry. We also study the concept of “quantum anomalous energy” proposed in some works as a new contribution to the nucleon mass. Examining the various arguments, we conclude that the quantum anomalous energy is not a genuine contribution to the mass sum rule, as a consequence of translation symmetry.

Polytropes, Virial theorem, evolutionary time scales, degrees of freedom, radiative transport, molecular weight, degeneracy, Jeans mass, and Eddington luminosity are basic ingredients to describe the physics of stars. In the present paper they will be presented in details as long with their role in the stellar evolution. PACS. PACS-key stellar evoluion - PACS-key basic physics

REBCO tapes are strong candidates for high-field magnets because they keep a higher critical current density in a high magnetic field range. In addition, the Hastelloy substrate of the REBCO tapes can be used as support structures for electromagnetic forces. However, the winding configuration of superconducting coils should be optimized to enhance the stress limit caused by electromagnetic forces. Force-balanced coils can balance the electromagnetic forces through a helically wound configuration and minimize the required mass of support structures for energy storage based on virial theorem. This work discusses the design considerations of the force-balanced coil windings for SMES using REBCO tapes. The force-balanced coils with a geodesic winding pitch will be a feasible solution for HTS-SMES using REBCO tapes. The geodesic pitch enables minimizing the in-plane curvature of helical windings, which applies the edgewise bending strain to REBCO tapes and reduces the perpendicular magnetic field to 5% of the maximum toroidal magnetic field. These features will prevent a decrease in the critical current of REBCO tapes in a high magnetic field range.

A bstract We derive simple bootstrap bounds on correlation functions of the BFSS matrix theory/D0-brane quantum mechanics. The result strengthens and extends Polchinski’s virial theorem bound to finite energies and gives the first non-trivial bound on ⟨Tr X 2 ⟩. Despite their simplicity, the bounds hint at some features of the dual black hole geometry. Our best lower bounds are already a factor of ∼ 2 from existing Monte Carlo results.

Topological optimization uses two main optimization conditions aimed at achieving the maximum stiffness at minimum weight of the loaded object, while not exceeding the allowable stress. This process naturally creates complex structures with varying degrees of density. There is a certain regularity between the density of the structure and stiffness, with the optimal density being related to the golden ratio. This study contributes to materials modeling and their characterization by introducing a mathematical theory related to the virial theorem as a predictive framework for understanding stiffness–density relationships in additively manufactured structures. The definition of virial stability and the methodology for deriving this stability from the kinetic and potential components of a random signal are introduced. The proposed virial-based model offers a generalizable tool for materials characterization, applicable not only to topological optimization but also to broader areas of materials science and advanced manufacturing.

We derive a class of mesoscopic virial equations governing energy partition between conjugate position and momentum variables of individual degrees of freedom. They are shown to apply to a wide range of nonequilibrium steady states with stochastic (Langevin) and deterministic (Nosé-Hoover) dynamics, and to extend to collective modes for models of heat-conducting lattices. A macroscopic virial theorem ensues upon summation over all degrees of freedom. It allows for the derivation of generalised (nonequilibrium) equations of state that involve average dissipative heat flows besides genuine state variables, as exemplified for inertial Brownian motion with solid friction and overdamped active Brownian particles subject to inhomogeneous pressure.

The stress tensor is described as a symmetric tensor in all classical continuum mechanics theories and in most existing statistical mechanics formulations. In this work, we examine the theoretical origins of the symmetry of the stress tensor and identify the assumptions and misinterpretations that lead to its symmetric property. We then make a direct measurement of the stress tensor in molecular dynamics simulations of four different material systems using the physical definition of stress as force per unit area acting on surface elements. Simulation results demonstrate that the stress tensor is asymmetric near dislocation cores, phase boundaries, holes and even in homogeneous material under a shear loading. In addition, the atomic virial stress and Hardy stress formulae are shown to significantly underestimate the stress tensor in regions of stress concentration.

In this study, we investigate the gravitational collapses of rotating stellar systems accounting for gamma-ray burst jet progenitors. Based on the virial theorem of hadron collisional relaxations and Newtonian slow-rotating approximation, we analyze the conversion of gravitational binding energy into kinetic energy of hadrons, whose collisions produce photons and electron-positron pairs forming fireballs. Our qualitative analysis implies that rotation effects collimated and spinning fireballs with nontrivial angular momenta along the propagating direction, thus making ultra-relativistic jets. Results reveal the possible trends that the fireball becomes more collimated and the jet angle decreases as the total angular momentum and mass ratio J / M of the slow-rotating collapsing core increases. Discussing the extrapolation of these trends to fast-rotating collapsing systems, we speculate that the ratio J / M should be a key quantity for differentiating long bursts (massive core collapses) from short bursts (binary coalescence). We derive the intrinsic correlations of collimated fireball quantities that should be imprinted on a large sample of observed GRB data as empirical correlations.