Virial theorem

About: Virial theorem is a research topic. Over the lifetime, 4027 publications have been published within this topic receiving 129672 citations.

Equation of State for Nonattracting Rigid Spheres

Abstract: A new equation of state for rigid spheres has been developed from an analysis of the reduced virial series. Comparisons with existing equations show that the new formula possesses superior ability to describe rigid‐sphere behavior.

Role of Repulsive Forces in Determining the Equilibrium Structure of Simple Liquids

Abstract: The different roles the attractive and repulsive forces play in forming the equilibrium structure of a Lennard‐Jones liquid are discussed. It is found that the effects of these forces are most easily separated by considering the structure factor (or equivalently, the Fourier transform of the pair‐correlation function) rather than the pair‐correlation function itself. At intermediate and large wave vectors, the repulsive forces dominate the quantitative behavior of the liquid structure factor. The attractions are manifested primarily in the small wave vector part of the structure factor; but this effect decreases as the density increases and is almost negligible at reduced densities higher than 0.65. These conclusions are established by considering the structure factor of a hypothetical reference system in which the intermolecular forces are entirely repulsive and identical to the repulsive forces in a Lennard‐Jones fluid. This reference system structure factor is calculated with the aid of a simple but accurate approximation described herein. The conclusions lead to a very simple prescription for calculating the radial distribution function of dense liquids which is more accurate than that obtained by any previously reported theory. The thermodynamic ramifications of the conclusions are presented in the form of calculations of the free energy, the internal energy (from the energy equation), and the pressure (from the virial equation). The implications of our conclusions to perturbation theories for liquids and to the interpretation of x‐ray scattering experiments are discussed.

Statistical Properties of X-ray Clusters: Analytic and Numerical Comparisons

Abstract: We compare the results of Eulerian hydrodynamic simulations of cluster formation against virial scaling relations between four bulk quantities: the cluster mass, the dark matter velocity dispersion, the gas temperature and the cluster luminosity. The comparison is made for a large number of clusters at a range of redshifts in three different cosmological models (CHDM, CDM and OCDM). We find that the analytic formulae provide a good description of the relations between three of the four numerical quantities. The fourth (luminosity) also agrees once we introduce a procedure to correct for the fixed numerical resolution. We also compute the normalizations for the virial relations and compare extensively to the existing literature, finding remarkably good agreement. The Press-Schechter prescription is calibrated with the simulations, again finding results consistent with other authors. We also examine related issues such as the size of the scatter in the virial relations, the effect of metallicity with a fixed pass-band, and the structure of the halos. All of this is done in order to establish a firm groundwork for the use of clusters as cosmological probes. Implications for the models are briefly discussed.

A Bond Path: A Universal Indicator of Bonded Interactions

Abstract: The quantum mechanics of proper open systems yields the physics that governs the local behavior of the electron density, ρ(r). The Ehrenfest force F(r) acting on an element of ρ(r) and the virial field ν(r) that determine its potential energy are obtained from equations of motion for the electronic momentum and virial operators, respectively. Each is represented by a “dressed” density, a distribution in real space that results from replacing the property in question for a single electron with a corresponding density that describes its average interaction with all of the remaining particles in the system. All bond paths, lines of maximum density linking neighboring nuclei in a system in stable electrostatic equilibrium, have a common physical origin in terms of F(r) and ν(r), regardless of the nature of the interaction. Each is homeomorphically mirrored by a virial path, a line of maximally negative potential energy density linking the same nuclei. The presence of a bond path and its associated virial path...

Statistical Mechanics of Rigid Spheres

Abstract: An equilibrium theory of rigid sphere fluids is developed based on the properties of a new distribution function G(r) which measures the density of rigid sphere molecules in contact with a rigid sphere solute of arbitrary size. A number of exact relations which describe rather fully the functional form of G(r) are derived. These are based on both geometrical considerations and the virial theorem. A knowledge of G(a) where a is the diameter of a rigid sphere enables one to arrive at the equation of state. The resulting analytical expression which is exact up to the third virial coefficient gives the fourth virial coefficient within 3% and the fifth, insofar as it is known, within 5%. Furthermore over the entire range of fluid density, the equation of state derived from theory agrees with that computed using machine methods. Theory also gives an expression for the surface tension of a hard sphere fluid in contact with a perfectly repelling wall. The dependence of surface tension on curvature is also given. ...