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Recent advances in the asymptotic analysis of energy levels of potentials produce relative errors in eigenvalue sums of order 10−34, but few non‐trivial potentials have been solved numerically to such accuracy. We solve the general quartic potential (arbitrary linear combination of x2 and x4) beyond this level of accuracy using a basis of several hundred oscillator states. We list the lowest 20 eigenvalues for 9 such potentials. We confirm the known asymptotic expansion for the levels of the pure quartic oscillator, and extract the next two terms in the asymptotic expansion. We give analytic formulas for expansion in up to three even basis states. We confirm the virial theorem for the various energy components to similar accuracy. The sextic oscillator levels are also given. These benchmark results should be useful for extreme tests of approximations in several areas of chemical physics and beyond. [ABSTRACT FROM AUTHOR]

In this paper, we calculate kinetic and potential energy contributions to the electronic ground-state total energy of several isolated atoms (He, Be, Ne, Mg, Ar, and Ca) by using the local density approximation (LDA) in density functional theory, the Hartree-Fock approximation (HFA), and the self-consistent GW approximation (GWA). To this end, we have implemented self-consistent HFA and GWA routines in our all-electron mixed basis code, TOMBO. We confirm that virial theorem is fairly well satisfied in all of these approximations, although the resulting eigenvalue of the highest occupied molecular orbital level, i.e., the negative of the ionization potential, is in excellent agreement only in the case of the GWA. We find that the wave function of the lowest unoccupied molecular orbital level of noble gas atoms is a resonating virtual bound state, and that of the GWA spreads wider than that of the LDA and thinner than that of the HFA. [ABSTRACT FROM AUTHOR]

The electronic virial theorem is extended to molecular systems within the framework of the Polarizable Continuum Model (PCM) to describe solvation effects. The theorem is given in the form of a relation involving the components of the energy (kinetic and potential) of a molecular solute and its electrostatic properties (potential and field) at the boundary of the cavity in the continuum medium. The virial theorem is also derived in the presence of the Pauli repulsion component of the solutesolvent interaction. Furthermore, it is shown that these forms of the PCM virial theorem may be related to the virial theorem of more simple systems as a molecule in the presence of fixed point charges, and as an atom in a spherical box with confining potential. [ABSTRACT FROM AUTHOR]

Molecular dynamics simulations of a nanoscale liquid droplet on a solid surface are carried out in order to examine the pressure tensor field around the multiphase interfaces, and to explore the validity of Young's equation. By applying the virial theorem to a hemicylindrical droplet consisting of argon molecules on a solid surface, two-dimensional distribution of the pressure tensor is obtained. Tensile principal pressure tangential to the interface is observed around the liquid-vapor transition layer, while both tensile and compressive principal pressure tangential to the interface exists around the solid-liquid transition layer due to the inhomogeneous density distribution. The two features intermix inside the overlap region between the transition layers at the contact line. The contact angle is evaluated by using a contour line of the maximum principal pressure difference. The interfacial tensions are calculated by using Bakker's equation and Young-Laplace equation to the pressure tensor distribution. The relation between measured contact angle and calculated interfacial tensions turns out to be consistent with Young's equation, which is known as the description of the force balance at the three-phase interface. [ABSTRACT FROM AUTHOR]

The electron spin angular momentum features are revealed to be identical to those of a uniformly charged ring. The electric field for this configuration can capture the electron being accelerated toward a proton, thus explaining the stability of the hydrogen atom. In addition, the proposed atom structure describes the bonding of two hydrogen atoms, which illuminates the covalent bond formation, the electronic state, bond length, and binding energy. Additionally, important wave properties ascribed to photons are assigned to particles with mass such as wave energy, angular frequency, phase velocity, and quantum spin. [ABSTRACT FROM AUTHOR]

The discovery of 19 dwarf galaxies without dark matter (DM) provides, counterintuitively, strong support for the ΛCDM standard model of cosmology. Their presence is well accommodated in a scenario where the DM is in the form of cold dark particles. However, it is interesting to explore quantitatively what is needed from modified gravity models to accommodate the presence of these galaxies and what extra degree of freedom is needed in these models. To this end, we derive the dynamics at galaxy scales (Virial theorem) for a general class of modified gravity models. We distinguish between theories that satisfy the Jebsen–Birkhoff theorem, and those that do not. Our aim is to develop tests that can distinguish whether DM is part of the theory of gravity or a particle. The 19 dwarf galaxies discovered provide us with a stringent test for models of modified gravity. Our main finding is that there will always be an extra contribution to the Virial theorem coming from the modification of gravity, even if a certain galaxy shows very small, if not negligible, trace of DM, as has been reported recently. Thus, if these and more galaxies are confirmed as devoid (or negligible) of DM, while other similar galaxies have abundant DM, it seems interesting to find modifications of gravity to describe DM. Our result can be used by future astronomical surveys to put constraints on the parameters of modified gravity models at astrophysical scales where DM is described as such. [ABSTRACT FROM AUTHOR]

Recently, two-component spherical galaxy models have been presented, where the stellar profile is described by a Jaffe law, and the total density by another Jaffe law, or by an r −3 law at large radii. We extend these two families to their ellipsoidal axisymmetric counterparts: the JJe and J3e models. The total and stellar density distributions can have different flattenings and scale lengths, and the dark matter halo is defined by difference. First, the analytical conditions required to have a nowhere negative dark matter halo density are derived. The Jeans equations for the stellar component are then solved analytically, in the limit of small flattenings, also in the presence of a central BH. The azimuthal velocity dispersion anisotropy is described by the Satoh k -decomposition. Finally, we present the analytical formulae for velocity fields near the centre and at large radii, together with the various terms entering the virial theorem. The JJe and J3e models can be useful in a number of theoretical applications, e.g. to explore the role of the various parameters (flattening, relative scale lengths, mass ratios, rotational support) in determining the behaviour of the stellar kinematical fields before performing more time-expensive integrations with specific galaxy models, to test codes of stellar dynamics and in numerical simulations of gas flows in galaxies. [ABSTRACT FROM AUTHOR]

Generally the virial theorem provides a relation between various components of energy integrated over a system. This helps us to understand the underlying equilibrium. Based on the virial theorem we can estimate, for example, the maximum allowed magnetic field in a star. Recent studies have proposed the existence of highly magnetized white dwarfs (B-WDs), with masses significantly higher than the Chandrasekhar limit. Surface magnetic fields of such white dwarfs could be more than |$10^{9}$|  G with the central magnitude several orders higher. These white dwarfs could be significantly smaller in size than their ordinary counterparts (with surface fields restricted to about |$10^9$|  G). In this paper, we reformulate the virial theorem for non-rotating B-WDs in which, unlike in previous formulations, the contribution of the magnetic pressure to the magnetohydrostatic balance cannot be neglected. Along with the new equation of magnetohydrostatic equilibrium, we approach the problem by invoking magnetic flux conservation and by varying the internal magnetic field with the matter density as a power law. Either of these choices is supported by previous independent work and neither violates any important physics. They are useful while there is no prior knowledge of field profile within a white dwarf. We then compute the modified gravitational, thermal, and magnetic energies and examine how the magnetic pressure influences the properties of such white dwarfs. Based on our results we predict important properties of these B-WDs, which turn out to be independent of our chosen field profiles. [ABSTRACT FROM AUTHOR]

Photon-induced electronic excitations are ubiquitously observed in organic chromophore. In this context, we present a simple, alternative time-independent DFT procedure, for the computation of single-particle excitation energies, in particular, the lower bound excited singlet states, which are of primary interest in photochemistry. This takes inspiration from recently developed Becke's exciton model, where a key step constitutes the accurate evaluation of correlated singlet–triplet splitting energy. It introduces a non-empirical model, from both "adiabatic connection theorem" and "virial theorem" to analyze the role of 2e - integral in such calculations. The latter quantity is efficiently mapped onto a real grid and computed accurately using a purely numerical strategy. Illustrative calculations are performed on 10 π -electron organic chromophores within a Cartesian grid implementation of pseudopotential Kohn–Sham (KS) DFT, developed in our laboratory, taking SBKJC-type basis functions within B3LYP approximation. The triplet and singlet excitation energies corresponding to first singly excited configuration are found to be in excellent agreement with TD-B3LYP calculations. Further, we perform the same for a set of larger molecular systems using the asymptotically corrected LC-BLYP, in addition to B3LYP. A systematic comparison with theoretical best estimates demonstrates the viability and suitability of current approach in determining optical gaps, combining predictive accuracy with moderate computational cost. [ABSTRACT FROM AUTHOR]

We study the modelling of the halo occupation distribution (HOD) for the eBOSS DR16 emission line galaxies (ELGs). Motivated by previous theoretical and observational studies, we consider different physical effects that can change how ELGs populate haloes. We explore the shape of the average HOD, the fraction of satellite galaxies, their probability distribution function (PDF), and their density and velocity profiles. Our baseline HOD shape was fitted to a semi-analytical model of galaxy formation and evolution, with a decaying occupation of central ELGs at high halo masses. We consider Poisson and sub/super-Poissonian PDFs for satellite assignment. We model both Navarro–Frenk–White and particle profiles for satellite positions, also allowing for decreased concentrations. We model velocities with the virial theorem and particle velocity distributions. Additionally, we introduce a velocity bias and a net infall velocity. We study how these choices impact the clustering statistics while keeping the number density and bias fixed to that from eBOSS ELGs. The projected correlation function, w p, captures most of the effects from the PDF and satellites profile. The quadrupole, ξ2, captures most of the effects coming from the velocity profile. We find that the impact of the mean HOD shape is subdominant relative to the rest of choices. We fit the clustering of the eBOSS DR16 ELG data under different combinations of the above assumptions. The catalogues presented here have been analysed in companion papers, showing that eBOSS RSD+BAO measurements are insensitive to the details of galaxy physics considered here. These catalogues are made publicly available. [ABSTRACT FROM AUTHOR]

The problem on the spectrum structure of motion equations in a mechanical system linearized near equilibrium in a nonpotential force field is considered. Particular attention is paid to the case when the force field is circulatory and there are also viscous friction forces acting on the system. The stability problem solution is based on the search of invariant subspaces that are uniquely projected onto the configuration space of the system. [ABSTRACT FROM AUTHOR]

Most introductory books on quantum mechanics discuss the particle-in-a-box problem through solutions of the Schrödinger equation, at least, in the one-dimensional case. When introducing the virial theorem, however, its discussion in the context of this simple model is not considered and students ponder the question of the validity of the virial theorem for a system with, apparently, no forces. In this work, we address this issue by solving the particle in a finite box and show that the virial theorem is fulfilled when the appropriate Cauchy boundary conditions are taken into account. We also illustrate how, in the limit of the infinite potential box, the virial theorem holds as well. As a consequence, it is possible to determine the averaged force exerted by the walls on the particle. Finally, a discussion of these results in the classical limit is provided. [ABSTRACT FROM AUTHOR]

Based on the virial theorem, the influence of the following factors to characteristics of mechanical support systems of toroidal magnetic energy storage is considered: the configuration of the coils, the support system elements location under compressive forces, the mechanical properties of the material, the distribution of the cross-sectional areas of the support rings and the corresponding distribution of radial forces. It is established that the volume of structure requirement depends only on the radii from which the coils transmit forces to the support rings. The found general relation, in contrast to the existing ones, establishes a relationship between all the parameters that determine the volume of the support structures. The characteristics of support systems are analyzed both for identical mechanical stresses and for identical mechanical properties of all support elements. The characteristics for specific examples of toroidal systems with circular coils are found. It is shown that systems with support elements in compression located on the side of the vertical axis of the torus have a significantly smaller volume of structure compared to other locations. The influence of the relative radial size of the torus cross-section and the distribution of the cross-sectional areas of the support rings on the structure requirement is analyzed. The relationship between dimensionless characteristics that determine the volume of elements subjected to mechanical stresses of compression and tension is noted. References 37, figures 8, table 1.

Osmosis is one of the most important physical phenomena in living and soft matter systems. While the thermodynamics of osmosis is well understood, the underlying microscopic dynamical mechanisms remain the subject of discussion. Unravelling these mechanisms is a prerequisite for understanding osmosis in non-equilibrium systems. Here, we investigate the microscopic basis of osmosis, in a system at equilibrium, using molecular dynamics simulations of a minimal model in which repulsive solute and solvent particles differ only in their interactions with an external potential. For this system, we can derive a simple virial-like relation for the osmotic pressure. Our simulations support an intuitive picture in which the solvent concentration gradient, at osmotic equilibrium, arises from the balance between an outward force, caused by the increased total density in the solution, and an inward diffusive flux caused by the decreased solvent density in the solution. While more complex effects may occur in other osmotic systems, our results suggest that they are not required for a minimal picture of the dynamic mechanisms underlying osmosis. [ABSTRACT FROM AUTHOR]

We present a detailed non-relativistic study of the atoms H, He, C and K and the molecule CH 4 in the centre of a spherical soft confinement potential of the form V N (r) = (r / r 0) N with stiffness parameter N and confinement radius r 0 . The soft confinement potential approaches the hard-wall limit as N → ∞ , giving a more detailed picture of spherical confinement. The confined hydrogen atom is considered as a base model: it is treated numerically to obtain ground- and excited-state energies and nodal positions of the eigenstates to study the convergence towards the hard-wall limit. We also derive some important analytical relations. The use of Gaussian basis sets is analysed. We find that, for increasing stiffness parameter N, the convergence towards the basis-set limit becomes problematic. As an application, we report dipole polarisabilities for different values of N and r 0 of hydrogen. For helium, we determine electron correlation effects with varying N and r 0 , and discuss the virial theorem for both soft and hard confinements in the limit r 0 → 0. For carbon, a change in the orbital population from 2s 2 2p 2 to 2s 0 2 p 4 is observed with decreasing r 0 , while, for potassium, we observe a change from the 2 S to 2 D ground state at small r 0 values. For CH 4 , we show that the one-particle density becomes more spherical with increasing confinement. A possible application of soft confinement to atoms and molecules under high pressure is discussed Prof. Jürgen Gauss observing Schrödinger's cat under quantum confinement. [ABSTRACT FROM AUTHOR]

The thermodynamic potential for a homogeneous and isotropic equilibrium nonrelativistic system of electrons and nuclei interacting with each other according to the Coulomb law is considered. Using the virial theorem and exact relations for the Green functions, it is shown that the thermodynamic functions of the Coulomb system are uniquely determined by the single-particle Green functions of electrons and nuclei outside the framework of perturbation theory. On this basis, self-consistency conditions are formulated for the thermodynamic functions that determine the applicability of the approximations used for single-particle Green functions. [ABSTRACT FROM AUTHOR]

The ground state solution of hydrogen molecule and ions are numerically obtained as an application of our scheme to solve many-electron multi-center potential Schrödinger equation by using one-dimensional hydrogen wavefunctions as basis functions. The all-electron sparse Hamiltonian matrix for the given system is generated with the standard order finite-difference method, then the electronic trial wavefunction to describe the ground state is constructed based on the molecular orbital treatment, and finally an effective and accurate iteration process is implemented to systematically improve the result. Many problems associated with the evaluation of the matrix elements of the Hamiltonian in more general basis and potential are circumvented. Compared with the standard results, the variationally obtained energy of H2+ is within 0.1 mhartree accuracy, while that of H2 and H3+ include the electron correlation effect. The equilibrium bond length is highly consistent with the accurate results and the virial theorem is satisfied to an accuracy of −V/T = 2.0. [ABSTRACT FROM AUTHOR]

Molecular clouds are a fundamental ingredient of galaxies: they are the channels that transform the diffuse gas into stars. The detailed process of how they do it is not completely understood. We review the current knowledge of molecular clouds and their substructure from scales ∼ 1 kpc down to the filament and core scale. We first review the mechanisms of cloud formation from the warm diffuse interstellar medium down to the cold and dense molecular clouds, the process of molecule formation and the role of the thermal and gravitational instabilities. We also discuss the main physical mechanisms through which clouds gather their mass, and note that all of them may have a role at various stages of the process. In order to understand the dynamics of clouds we then give a critical review of the widely used virial theorem, and its relation to the measurable properties of molecular clouds. Since these properties are the tools we have for understanding the dynamical state of clouds, we critically analyse them. We finally discuss the ubiquitous filamentary structure of molecular clouds and its connection to prestellar cores and star formation. [ABSTRACT FROM AUTHOR]

The kinetic energy of electrons and the potential energy of a nitrogen atom with open electron shell are calculated in this work in the basis of the Slater functions using the Hartree–Fock–Roothaan (HFR) method. Analytical expressions are derived for the kinetic and potential energies of the nitrogen atom with open electron shell. Twenty independent determinant wave functions corresponding to the 1s22s22p3 electronic configuration of the nitrogen atom have been found by summing the moment projections. The results obtained have been verified using the virial theorem and the expression V ¯ / T ¯ = − 2 . [ABSTRACT FROM AUTHOR]

Paper presents the results of the in-silico experiments concerning simulation of the tension and failure dynamics of two-dimensional (2D) titanium carbide Ti2C under different types of tensile loading. The behavior of 2D nanosheet was studied within classical molecular dynamics (MD) methods. Two different loading methods, namely axial displacement and uniform tensile strain were considered in experiments. The first loading method consists in a consecutive shift of atoms in the right edge of the sample along the x-axis while atoms in the left edge of the sample are held fixed. The uniform tensile strain was performed by shifting left and right parts of the sample in opposite directions. During the simulations, atomistic configurations of the 2D Ti2C nanosheet at different strain values were built for both loading methods. As it follows from the obtained data, different loading procedures lead to different fracture dynamics and crack formation in the studied sample. As calculated atomistic configurations show, in the case of axial displacement the fracture begins from the formation of cracks at the lateral edges of the sample. Cracks appear along the layers of constrained atoms at both fixed and shifted edges of the nanosheet, while at uniform tensile strain Ti2C sample undergoes uniform stretching up to the critical strain where the crack starts to form. The strain-stress curves for both axial displacement and homogenous strain were calculated through the virial theorem. Strain-stress dependencies obtained for different loading procedures for Ti2C sample overlap in the area of elastic deformation. Calculated data also show that plastic deformation and following destruction of the Ti2C sample occur at strain ε > 0.04 for both methods of loading. [ABSTRACT FROM AUTHOR]

In this paper we explore the effects that the presence of a fraction of binary stars has in the determination of a star cluster mass via the virial theorem. To reach this aim in an accurate and consistent way, we run a set of simulations using the direct summation, high precision, code NBODY7. By means of this suite of simulations we are able to quantify the overestimate of open-star-cluster-like models' dynamical masses when making a straight application of the virial theorem using available position and radial velocity measurements. The mass inflation caused by the binary "heating" contribution to the measured velocity dispersion depends, of course, on the initial binary fraction, fb0 and its following dynamical evolution. For an fb (evolved up to 1.5 Gyr) in the range 8%–42% the overestimate of the mass done using experimentally sounding estimates for the velocity dispersion can be up to a factor of 45. We provide a useful fitting formula to correct the dynamical mass determination for the presence of binaries, and underline how neglecting the role of binaries in stellar systems might lead to erroneous conclusions about their total mass budget. If this trend remains valid for larger systems like dwarf spheroidal galaxies, which are still far out of reach for high-precision dynamical simulations taking their binaries into account, it would imply an incorrect overestimation of their dark matter content, as inferred by means of available velocity dispersion measurements. [ABSTRACT FROM AUTHOR]

Early-type galaxies – slow and fast rotating ellipticals (E-SRs and E-FRs) and S0s/lenticulars – define a Fundamental Plane (FP) in the space of half-light radius Re , enclosed surface brightness Ie , and velocity dispersion σ e . Since Ie and σ e are distance-independent measurements, the thickness of the FP is often expressed in terms of the accuracy with which Ie and σ e can be used to estimate sizes Re. We show that: (1) The thickness of the FP depends strongly on morphology. If the sample only includes E-SRs, then the observed scatter in Re is |$\sim 16{{\ \rm per\ cent}}$|⁠ , of which only |$\sim 9{{\ \rm per\ cent}}$| is intrinsic. Removing galaxies with M * < 1011 M⊙ further reduces the observed scatter to |$\sim 13{{\ \rm per\ cent}}$| (⁠|$\sim 4{{\ \rm per\ cent}}$| intrinsic). The observed scatter increases to |$\sim 25{{\ \rm per\ cent}}$| usually quoted in the literature if E-FRs and S0s are added. If the FP is defined using the eigenvectors of the covariance matrix of the observables, then the E-SRs again define an exceptionally thin FP, with intrinsic scatter of only 5 per cent orthogonal to the plane. (2) The structure within the FP is most easily understood as arising from the fact that Ie and σ e are nearly independent, whereas the Re − Ie and Re −σ e correlations are nearly equal and opposite. (3) If the coefficients of the FP differ from those associated with the virial theorem the plane is said to be 'tilted'. If we multiply Ie by the global stellar mass-to-light ratio M */ L and we account for non-homology across the population by using Sérsic photometry, then the resulting stellar mass FP is less tilted. Accounting self-consistently for M */ L gradients will change the tilt. The tilt we currently see suggests that the efficiency of turning baryons into stars increases and/or the dark matter fraction decreases as stellar surface brightness increases. [ABSTRACT FROM AUTHOR]

We study a sample of 148 early-type galaxies in the Coma cluster using SDSS photometry and spectra, and calibrate our results using detailed dynamical models for a subset of these galaxies, to create a precise benchmark for dynamical scaling relations in high-density environments. For these galaxies, we successfully measured global galaxy properties, modelled stellar populations, and created dynamical models, and support the results using detailed dynamical models of 16 galaxies, including the two most massive cluster galaxies, using data taken with the SAURON IFU. By design, the study provides minimal scatter in derived scaling relations due to the small uncertainty in the relative distances of galaxies compared to the cluster distance. Our results demonstrate low (≤55 per cent for 90th percentile) dark matter fractions in the inner 1 R e of galaxies. Owing to the study design, we produce the tightest, to our knowledge, IMF–σe relation of galaxies, with a slope consistent with that seen in local galaxies. Leveraging our dynamical models, we transform the classical Fundamental Plane of the galaxies to the Mass Plane. We find that the coefficients of the Mass Plane are close to predictions from the virial theorem, and have significantly lower scatter compared to the Fundamental Plane. We show that Coma galaxies occupy similar locations in the (M *– R e) and (M *−σe) relations as local field galaxies but are older. This, and the fact we find only three slow rotators in the cluster, is consistent with the scenario of hierarchical galaxy formation and expectations of the kinematic morphology–density relation. [ABSTRACT FROM AUTHOR]

In order to simulate rigidly rotating polytropes, we have simulated systems of N point particles, with N up to 1800. Two particles at a distance r interact by an attractive potential -1=r and a repulsive potential 1=r2. The repulsion simulates the pressure in a polytropic gas of polytropic index 3=2. We take the total angular momentum L to be conserved, but not the total energy E. The particles are stationary in the rotating coordinate system. The rotational energy is L2=ð2IÞ where I is the moment of inertia. Configurations, where the energy E has a local minimum, are stable. In the continuum limit N !1, the particles become more and more tightly packed in a finite volume, with the interparticle distances decreasing as N-1=3. We argue that N-1=3 is a good parameter for describing the continuum limit. We argue further that the continuum limit is the polytropic gas of index 3=2. For example, the density profile of the nonrotating gas approaches that computed fromthe Lane-Emden equation describing the nonrotating polytropic gas. In the case of maximum rotation, the instability occurs by the loss of particles from the equator, which becomes a sharp edge, as predicted by Jeans in his study of rotating polytropes. We describe the minimum energy nonrotating configurations for a number of small values of N. [ABSTRACT FROM AUTHOR]

The dissemination of models across disciplinary lines has become a phenomenon of interest to philosophers of science. To account for this phenomenon, philosophers have invented two units of analysis. The first identifies to the thing that transfers, model templates. The second identifies the thing to which transferable templates apply, landing zones. There exists a dynamic between the thing that is transferred and the thing to which transferrable templates apply. The use of a transferable template in a new domain requires reconception of domain-specific phenomena. This paper examines two cases of model transfer, the use of the ideal gas law in biology by R.A. Fisher and the use of the virial theorem in chemistry by Richard Bader. These two stories of model transfer in biology and chemistry indicate a dimension to conceptual progress related to this dynamic. Using discourse on model transfer affords philosophers a novel approach for depicting the invention of, for instance, chemical concepts and resulting disputes. [ABSTRACT FROM AUTHOR]

In the framework of the Gross-Pitaevskii equation, we consider the problem of the expansion of quantum gases into a vacuum. For them, the chemical potential μ has a power-law dependence on the density n with the exponent ν = 2/D, whereD is the space dimension. For gas condensates of Bose atoms as the temperature T → 0, s scattering gives the main contribution to the interaction of atoms in the leading order in the gas parameter. Therefore, the exponent ν = 1 for an arbitrary D. In the three-dimensional case, ν = 2/3 is realized for condensates of Fermi atoms in the so-called unitary limit. For ν = 2/D, the Gross-Pitaevskii equation has an additional symmetry under Talanov transformations of the conformal type, which were first found for the stationary self-focusing of light. A consequence of this symmetry is the virial theorem relating the average size R of an expanding gas cloud to its Hamiltonian. The quantity R asymptotically increases linearly with time as t →∞. In the semiclassical limit, the equations of motion coincide with those of the hydrodynamics of an ideal gas with the adiabatic exponent γ = 1+2/D. In this approximations, self-similar solutions describe angular deformations of the gas cloud against the background of the expanding gas in the framework of equations of the Ermakov-Ray-Reid type. [ABSTRACT FROM AUTHOR]

In this article, we study the origin of precursor flare activity and investigate its role towards triggering the eruption of a flux rope which resulted into a dual-peak M-class flare (SOL2015-06-21T02:36) in the active region NOAA 12371. The flare evolved in two distinct phases with peak flux levels of M2.1 and M2.6 at an interval of ≈ 54 min. The active region exhibited striking moving magnetic features (MMFs) along with sunspot rotation. Nonlinear force-free field (NLFFF) modelling of the active region corona reveals a magnetic flux rope along the polarity inversion line in the trailing sunspot group which is observationally manifested by the co-spatial structures of an active region filament and a hot channel identified in the 304 and 94 Å images, respectively, from the Atmospheric Imaging Assembly (AIA). The active region underwent a prolonged phase of flux enhancement followed by a relatively shorter period of flux cancellation prior to the onset of the flare which led to the build up and activation of the flux rope. Extreme ultra-violet (EUV) images reveal localised and structured pre-flare emission, from the region of MMFs, adjacent to the location of the main flare. Our analysis reveals strong, localised regions of photospheric currents of opposite polarities at the precursor location, thereby making the region susceptible to small-scale magnetic reconnection. Precursor reconnection activity from this location most likely induced a slipping reconnection towards the northern leg of the hot channel which led to the destabilisation of the flux rope. The application of magnetic virial theorem suggests that there was an overall growth of magnetic free energy in the active region during the prolonged pre-flare phase which decayed rapidly after the hot channel eruption and its successful transformation into a halo coronal mass ejection (CME). [ABSTRACT FROM AUTHOR]

Hamilton's principle is applied to obtain the equations of motion for fully relativistic collision-free plasma. The variational treatment is presented in both the Eulerian and Lagrangian frameworks. A Clebsch representation of the plasma fluid equations shows the connection between the Lagrangian and Eulerian formulations, clarifying the meaning of the multiplier in Lin's constraint. The existence of a fully relativistic hydromagnetic Cauchy invariant is demonstrated. The Lagrangian approach allows a straightforward determination of the Hamiltonian density and energy integral. The definitions of momentum, stress, and energy densities allow one to write the conservation equations in a compact and covariant form. The conservation equations are also written in an integral form with an emphasis on a generalized virial theorem. The treatment of boundary conditions produces a general expression for energy density distribution in plasma fluid. [ABSTRACT FROM AUTHOR]

Explicit exact expressions for the thermodynamic functions of the electron liquid via the single-particle Green's function are obtained and self-consistency conditions for these functions are formulated using the virial theorem and the equations of motion for the single-particle Green's function. [ABSTRACT FROM AUTHOR]

Here we suggest an alternative understanding of the surface gravity wave propagation mechanism based on the baroclinic torque, which operates to translate the interfacial vorticity anomalies at the air–water interface. We demonstrate how the non‐Boussinesq term of the baroclinic torque acts against the Boussinesq one to hinder wave propagation. By standard vorticity inversion and mirror imaging, we then show how the existence of the bottom boundary affects the two types of torque. Since the opposing non‐Boussinesq torque results solely from the mirror image, it vanishes in the deep‐water limit and its magnitude is half of the Boussinesq torque in the shallow‐water limit. This reveals that the Boussinesq approximation is valid in the deep‐water limit, even though the density contrast between air and water is large. The mechanistic roles played by the Boussinesq and non‐Boussinesq parts of the baroclinic torque remain obscured in the standard derivation where the time‐dependent Bernoulli equation is implemented instead of the interfacial vorticity equation. Finally, we note on passing that the Virial theorem for surface gravity waves can be obtained solely from considerations of the dynamics at the air–water interface. [ABSTRACT FROM AUTHOR]

The dynamical properties of spherically symmetric galaxy models, where a Jaffe stellar density profile is embedded in a total mass density decreasing as r −3 at large radii, are presented. The orbital structure of the stellar component is described by the Osipkov–Merritt anisotropy; the dark matter halo is isotropic, and a black hole is added at the centre of the galaxy. First, the conditions for a nowhere negative and monotonically decreasing dark matter halo density profile are derived; this profile can be made asymptotically coincident with an NFW profile at the centre and large radii. Then, the minimum value of the anisotropy radius for phase-space consistency is derived as a function of the galaxy parameters. The Jeans equations for the stellar component are solved analytically; the projected velocity dispersion at the centre and large radii is also obtained, for generic values of the anisotropy radius. Finally, analytical expressions for the terms entering the Virial Theorem are derived, and the fiducial anisotropy limit required to prevent the onset of Radial Orbit Instability is determined as a function of the galaxy parameters. The presented models, built following an approach already adopted in our previous works, can be a useful starting point for a more advanced modelling of the dynamics of elliptical galaxies, and can be easily implemented in numerical simulations requiring a realistic dynamical model of a galaxy. [ABSTRACT FROM AUTHOR]

The force virial is a construct combining information about the current configuration of a mechanical system in motion with information about the acting forces; its long-time average turns out to be proportional to the long-time average of the system's kinetic energy, that is, the virial theorem holds true; a version of this theorem is obtained when a connection between kinetic energy and temperature is established, as it happens in the kinetic theory of gases or in classical equilibrium statistical mechanics. In fact, a virial theorem holds in whatever mechanics. This paper is an exposition of its various formulations, from the simplest deterministic formulation for a single massy particle in Newtonian motion to the fairly more complicated statistical formulation, which makes the theorem a corollary of the equipartition theorem. [ABSTRACT FROM AUTHOR]

Self-diffusion of colloidal particles confined to a cylindrical microchannel is considered theoretically and numerically. Virial expansion of the self-diffusion coefficient is performed. Two-body and three-body hydrodynamic interactions are evaluated with high precision using the multipole method. The multipole expansion algorithm is also used to perform numerical simulations of the self-diffusion coefficient, valid for all possible particle packing fractions. Comparison with earlier results shows that the widely used method of reflections is insufficient for calculations of hydrodynamic interactions even for small packing fractions and small particles radii, contrary to the prevalent opinion. [ABSTRACT FROM AUTHOR]

Ensemble non-interacting kinetic energy functional is constructed for spherically symmetric systems. The differential virial theorem is derived for the ensemble. A first-order differential equation for the functional derivative of the ensemble non-interacting kinetic energy functional and the ensemble Pauli potential is presented. This equation can be solved and a special case of the solution provides the original non-interacting kinetic energy of the density functional theory. [ABSTRACT FROM AUTHOR]

We have examined the role of the exchange in describing the electrostatic potential in the fragment molecular orbital method and showed that it should be included in the total Fock matrix to obtain an accurate one-electron spectrum; however, adding it to the Fock matrices of individual fragments and pairs leads to very large errors. For the error analysis we have used the virial theorem; numerical tests have been performed for solvated phenol at the Hartree–Fock level with the 6-31G* and 6-311G** basis sets. [ABSTRACT FROM AUTHOR]

Reversible formation of weakly associated protein oligomers or clusters is a key early step in processes such as protein aggregation and colloidal phase separation. A previously developed cell-based, quasichemical model for lattice fluids [T. M. Young and C. J. Roberts, J. Chem. Phys. 127, 165101 (2007)] is extended here to treat continuous-space systems. It is illustrated using two simplified limiting cases for globular proteins at the isoelectric point: spherical square-well (SW) particles with an isotropic short-ranged attraction and screened dipolar particles with SW attractions and square-shoulder repulsions. Cluster free energies (ΔAi) and structures are analyzed as a function of the reduced second virial coefficient b2*. ΔAi values and the average structures of clusters up to pentamers have distinct differences due to the anisotropic nature of the dipolar interactions. However, ΔAi values can be mapped semiquantitatively between the two cases if compared at common values of b2*. Free energy landscapes of oligomerization are constructed, illustrating significant differences in landscape ruggedness for small clusters of dipolar versus SW fluids, and suggesting a possible molecular interpretation for empirical models of nucleation-dependent aggregation of proteins. [ABSTRACT FROM AUTHOR]

The extension of Onsager’s second-virial theory [L. Onsager, Ann. N.Y. Acad. Sci. 51, 627 (1949)] for the orientational ordering of hard rods to mixtures of nonspherical hard bodies with finite length-to-breadth ratios is examined using the decoupling approximations of Parsons [Phys. Rev. A 19, 1225 (1979)] and Lee [J. Chem. Phys. 86, 6567 (1987); 89, 7036 (1988)]. Invariably the extension of the Parsons–Lee (PL) theory to mixtures has in the past involved a van der Waals one-fluid treatment in which the properties of the mixture are approximated by those of a reference one-component hard-sphere fluid with an effective diameter which depends on the composition of the mixture and the molecular parameters of the various components; commonly this is achieved by equating the molecular volumes of the effective hard sphere and of the components in the mixture and is referred to as the PL theory of mixtures. It is well known that a one-fluid treatment is not the most appropriate for the description of the thermodynamic properties of isotropic fluids, and inadequacies are often rectified with a many-fluid (MF) theory. Here, we examine MF theories which are developed from the virial theorem and the virial expansion of the Helmholtz free energy of anisotropic fluid mixtures. The use of the decoupling approximation of the pair distribution function at the level of a multicomponent hard-sphere reference system leads to our MF Parsons (MFP) theory of anisotropic mixtures. Alternatively the mapping of the virial coefficients of the hard-body mixtures onto those of equivalent hard-sphere systems leads to our MF Lee (MFL) theory. The description of the isotropic-nematic phase behavior of binary mixtures of hard Gaussian overlap particles is used to assess the adequacy of the four different theories, namely, the original second-virial theory of Onsager, the usual PL one-fluid theory, and the MF theories based on the Lee (MFL) and Parsons (MFP) approaches. A comparison with the simulation data for the mixtures studied by Zhou et al. [J. Chem. Phys. 120, 1832 (2004)] suggests that the Parsons MF description (MFP) provides the most accurate representation of the properties of the isotropic-nematic ordering transition and density (pressure) dependence of the order parameters. [ABSTRACT FROM AUTHOR]

A recently derived method [R. D. Rohrmann and A. Santos, Phys. Rev. E 76, 051202 (2007)] to obtain the exact solution of the Percus–Yevick equation for a fluid of hard spheres in (odd) d dimensions is used to investigate the convergence properties of the resulting virial series. This is done both for the virial and compressibility routes, in which the virial coefficients Bj are expressed in terms of the solution of a set of (d-1)/2 coupled algebraic equations which become nonlinear for d≥5. Results have been derived up to d=13. A confirmation of the alternating character of the series for d≥5, due to the existence of a branch point on the negative real axis, is found and the radius of convergence is explicitly determined for each dimension. The resulting scaled density per dimension 2η1/d, where η is the packing fraction, is wholly consistent with the limiting value of 1 for d→∞. Finally, the values for Bj predicted by the virial and compressibility routes in the Percus–Yevick approximation are compared with the known exact values [N. Clisby and B. M. McCoy, J. Stat. Phys. 122, 15 (2006)]. [ABSTRACT FROM AUTHOR]

The virial expansion technique to determine the dielectric constant ε of dipolar hard-sphere fluid is developed. It is shown that the formalism allows to bring into agreement the results of Debye’s, Onsager’s, and Langevin’s to the problem. The third virial coefficient of ε is considered as a series over dipolar parameter λ=m2/d3kT. The terms up to O(λ11) are calculated analytically providing a correct description of the third virial coefficient for small and intermediate values of λ (0≤=λ≤=4). The results of the dielectric virial series are compared with the Monte Carlo data for ε found by Matyushov and Ladanyi [J. Chem. Phys. 110, 994 (1999)]. The theory is in agreement with simulations only at small values of λ≤=2. At higher polarities, the virial series diverges. Realization of the renormalization procedure permits to enlarge the range of applicability of the virial series. In this way, the new expression for the dielectric constant as a function of two dipolar parameters, λ and y=4πnm2/9kT, has been found explicitly. The expression gives a perfect upper bound of the dielectric constant and is more reliable for determination of ε than the previously known ones. [ABSTRACT FROM AUTHOR]