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We are used to measuring temperature with a thermometer, and we know from everyday life that different types of thermometers measure the same temperature. This experience can be based on equilibrium thermodynamics, which explains the equivalence of different possibilities to define temperature. In contrast, for systems out of equilibrium such as active matter, measurements performed with different thermometers can generally lead to different temperature values. In the present work, we systematically compare different possibilities to define temperature for active systems. Based on simulations and theory for inertial active Brownian particles, we find that different temperatures generally lead to different temperature values, as expected. Remarkably, however, we find that different temperatures not only lead to the same values near equilibrium (low Péclet number or high particle mass) but also even far from equilibrium, several different temperatures approximately coincide. In particular, we find that the kinetic temperature, the configurational temperature, and temperatures based on higher moments of the velocity distribution constitute a class of temperatures that all assume very similar values over a wide parameter range. Notably, the effective temperature and temperatures exploiting the virial theorem, the Stokes–Einstein relation, or a harmonic confinement form a second class of temperatures whose values approximately coincide with each other but which strongly differ from those of the first class. Finally, we identify advantages and disadvantages of the different possibilities to define temperature and discuss their relevance for measuring the temperature of active systems. [ABSTRACT FROM AUTHOR]

The generalization of the virial theorem, introduced by Clausius in statistical mechanics, has recently been carried out in the framework of geometric approaches to Hamiltonian and Lagrangian theories and it has been formulated in an intrinsic way. It is here revisited not only in its more general situation of a generic vector field but mainly from the perspective of contact and cosymplectic geometry. The previous generalizations allowing virial like relations from one-parameter groups of non-strictly canonical transformations and the rôle of Killing and conformal Killing vector fields for Lagrangians of a mechanical type are here completed with the theory for contact Hamiltonian, as well as gradient and evolution, vector fields. The corresponding theories in the framework of cosymplectic geometry and the particular case of time-dependent vector fields are also developed. [ABSTRACT FROM AUTHOR]

The article considers the issue of changes in the total energy of baryonic objects and their systems in the presence of dark energy, whose carrier, by definition, interacts with all baryonic objects. It is emphasized that dark energy was discovered due to such interaction that accelerates the universe's expansion. Since dark energy (dark energy carrier) is uniformly distributed, the physical consequences of interaction between the dark energy carrier and baryonic matter are considered at all hierarchical levels of baryonic structures. Drawing on the second law of thermodynamics, the author concludes that under the effect of dark energy, the energy of all baryonic systems gradually increases and their stability declines while the virial value becomes positive and continues to grow for any structures considered to be stable. A similar analysis performed for atomic nuclei reveals that over time, nuclear binding energy decreases, nuclear stability declines, and the nuclear mass increases. This effect gives a new perspective on Ambartsumian's hypothesis about the existence of superdense matter clumps in the nuclei of stars and galaxies and enables its new interpretation. [ABSTRACT FROM AUTHOR]

The solutions of the radial Schrödinger equation for a multiparameter potential of the exponential type are obtained using two traditional methods. The energy equation obtained was used to study both virial theorem and thermodynamic properties. The energy eigenvalues obtained for the potential and its subset potentials arecompletely bounded. The computed virial theorem depends on the parameters A, B, and V. The result also shows that the three parameters A, B, and V exhibit different features on the thermodynamic properties with V having the same variation for all the thermodynamic functions. [ABSTRACT FROM AUTHOR]

I propose a possible way to introduce the effect of temperature (defined through the virial theorem) into Einstein's theory of general relativity. This requires the computation of a path integral on a ten-dimensional flat space in a four-dimensional spacetime lattice. Standard path integral Monte Carlo methods can be used to compute this. [ABSTRACT FROM AUTHOR]

I propose a possible way to introduce the effect of temperature (defined through the virial theorem) into Einstein’s theory of general relativity. This requires the computation of a path integral on a ten-dimensional flat space in a four-dimensional spacetime lattice. Standard path integral Monte Carlo methods can be used to compute this.

The accurate prediction of bandgap energy Eg is crucial for the future development of semiconductors. Ab initio simulation studies have been undertaken to unravel the intricacies of compound semiconductors. However, traditional density functional theory estimates Eg to be mostly 30–50% smaller than experimental values. To reconcile these disparities, fitting parameters such as U have been employed, albeit at the expense of violating the virial theorem's essential conditions. In our pursuit of a more accurate approach, utilizing a computational method that adheres to virial's theorem without resorting to fitting parameters is proposed. Employing the self‐consistent Green‐function vertex (GW) approximation calculation, standard phenomenological results are built upon as an initial condition. This novel methodology successfully resolves the long‐standing issue surrounding Eg of InN, a nitride semiconductor InGaAlN known for blue light‐emitting diodes. The numerical results from our calculations demonstrate a remarkable alignment with experimental values across the entire InxGa1−xN range, with an impressive accuracy of 0.1 eV. This innovative method holds promise for application to various semiconductors, serving as a potent tool for predicting new semiconductors with small Eg. This calculation method is also applied to Eg of In1−xAlxN and Ga1−xAlxN. [ABSTRACT FROM AUTHOR]

Gravity and electromagnetic interactions are the only fundamental physical interactions (outside the nuclear domain). In this work, we shall concentrate on Hamiltonians containing gravitational interaction, which according to general relativity must be retarded. In recent years, retarded gravity has explained many of the mysteries surrounding the "missing mass" related to galactic rotation curves, the Tully–Fisher relations, and gravitational lensing phenomena. Indeed, a recent paper analyzing 143 galaxies has demonstrated that retarded gravity will suffice to explain galaxies' rotation curves without the need to postulate dark matter for multiple types of galaxies. Moreover, it also demystified the "missing mass" related to galactic clusters and elliptic galaxies in which excess matter was derived through the virial theorem. Here, we give a mathematical criterion that specifies the cases in which retardation is important for gravity (and when it is not). The criterion takes the form of an inequality. [ABSTRACT FROM AUTHOR]

It is widely known that porous structure design is an important way to reduce the weight of matrix materials. However, there is still a lack of systematic understanding of how factors such as the shape, size, and concentration of pores affect the dynamic response of materials. This study investigated the elastic–plastic behavior and failure characteristics of nanoporous Al from a molecular dynamics perspective, taking into account columnar voids with a diameter of 2–18 nm and two types of arrangement configurations. The results show that all samples undergo elastic deformation for a strain range of ∼−4% to 5%. In this range, the amplitude of temperature and stress changes with strain decreases sequentially as the sample density decreases. The corresponding yield stress of the void sample under compression and tension is calculated according to the virial theorem. During the compression process, local plastic deformation and collapse mechanisms of voids can occur in low porosity samples, while strain localization and slip thickening mechanisms can occur in the transverse ligaments between large voids. During the stretching process, local plastic deformation and lateral expansion mechanisms of voids can occur in low porosity samples, while strain localization and necking fracture mechanisms can occur in the transverse ligaments between large voids. Finally, the transformation law of deformation mechanism with porosity was given based on the amount of plastic deformation. [ABSTRACT FROM AUTHOR]

Context: We demonstrate that the minimum of the reaction force curve of a diatomic or polyatomic molecule undergoing bond dissociation is significant in several respects. As has been pointed out in the past, it is the point at which the force opposing dissociation is strongest. It marks the boundary between the primarily structural stage of a bond dissociation (stretching) and the transition region between the stretched bond and independent atoms. We now show that the reaction force minimum is also where the kinetic and potential energy curves tend to change direction abruptly. At this point, the total energy E(R) has increased by about 27% of the dissociation energy, for both diatomic and polyatomic molecules. Methods: Dissociation curves are analyzed at the UHF/daug-cc-pV5Z level of theory using Gaussian 16. [ABSTRACT FROM AUTHOR]

The incomplete understanding of electron correlation is still profound due to the lack of exact solutions of the Schrödinger equation of many electron systems. In this work, we present the correlation-induced changes in the calculated many-electron systems beyond the standard residual. To locate the minimum of the Rayleigh quotient, each iteration is to seek the lowest eigenpairs in a subspace spanned by the current wave function and its gradient of the Rayleigh-quotient as well as the upcoming higher-order residual. Consequently, as the upcoming errors can be introduced and circumvented with the search in the higher-order residual, a concomitant improved performance in terms of number of iterations, convergence rate, and total elapsed time is very significant. The correlation energy components obtained with the original residual are corrected with the higher-order residual application, satisfying the correlation virial theorem with much improved accuracy. The comparison with the original residual, the higher-order residual significantly improves the electron binding, favoring the localization of electrons' distribution, revealed with the increasing peak of the distribution and correlation function and the reduced interelectron distance and its angle. [ABSTRACT FROM AUTHOR]

At sufficiently large radii dark energy modifies the behavior of (a) bound orbits around a galaxy and (b) virialized gas in a cluster of galaxies. Dark energy also provides a natural cutoff to a cluster's dark matter halo. In (a) there exists a maximum circular orbit beyond which periodic motion is no longer possible, and orbital evolution near critical binding is analytically calculable using an adiabatic invariant integral. The finding implicates the study of wide galaxy pairs. In (b), dark energy necessitates the use of a generalized Virial Theorem to describe gas at the outskirts of a cluster. When coupled to the baryonic escape condition, aided by dark energy, the results is a radius beyond which the continued establishment of a hydrostatic halo of thermalized baryons is untenable. This leads to a theoretically motivated virial radius. We use this theory to probe the structure of a cluster's baryonic halo and apply it to X-ray and weak-lensing data collected on cluster Abell 1835. We find that gas in its outskirts deviates significantly from hydrostatic equilibrium beginning at ∼ 1.3 Mpc , the 'inner' virial radius. We also define a model dependent dark matter halo cutoff radius to A1835. The dark matter cutoff gives an upper limit to the cluster's total mass of ∼ 7 × 10 15 M ⊙ . Moreover, it is possible to derive an 'outer' hydrostatic equilibrium cutoff radius given a dark matter cutoff radius. A region of cluster gas transport and turbulence occurs between the inner and outer cutoff radii. [ABSTRACT FROM AUTHOR]

The virial theorem is established in the framework of resolution-scale relativity for stochastic dynamics characterized by a diffusion constant . It only relies on a simple time average just like the classical virial theorem, while the quantum mechanical virial theorem involves the expectation values of the observables. Nevertheless, by the emergence of a quantum-like potential term, the resolution-scale relativity virial theorem also encompasses quantum mechanical dynamics under the identification ℏ ↔ 2 m . This provides an illustration of the resolution-scale relativistic approach to the foundation of standard quantum mechanics. Furthermore, it is pointed out that if the resolution-scale relativity principle is implemented in macroscopic systems that are complex and/or chaotic, then the application of the classical virial theorem in the analysis of the dynamics of astrophysical systems neglects the contribution from a resolution-scale relativistic quantum-like potential. It is shown that this quantum-like potential could account for some fraction of the dark matter hypothesis. [ABSTRACT FROM AUTHOR]

We study several different aspects of the energy equipartition principle for water waves. We prove a virial identity that implies that the potential energy is equal, on average, to a modified version of the kinetic energy. This is an exact identity for the complete nonlinear water-wave problem, which is valid for arbitrary solutions. As an application, we obtain nonperturbative results about the free-surface Rayleigh--Taylor instability, for any nonzero initial data. We also derive exact virial identities involving higher order energies. We illustrate these results by an explicit computation for standing waves. As an aside, we prove trace inequalities for harmonic functions in Lipschitz domains which are optimal with respect to the dependence in the Lipschitz norm of the graph. [ABSTRACT FROM AUTHOR]

The boundaries of galaxy groups and clusters are defined by the interplay between the Newtonian attractive force and the decoupling from the local expansion of the Universe. This work extends the definition of a zero radial acceleration surface (ZRAS) and the turnaround surface (TS) for a general distribution of the masses in an expanding background, governed by the cosmological constant. We apply these definitions to different galaxy groups in the local Universe, mapping these groups up to ten megaparsec distances. We discuss the dipole and the quadrupole rate for the Local Group of Galaxies and the implementations on the Hubble diagram correction and galaxy groups virialization. With these definitions, we present the surfaces showing the interplay between the local expansion vs the local Newtonian attraction for galaxy groups in the local Universe. Further, we estimate the masses of different galaxy groups and show that the inclusion of the Cosmological Constant in the analysis predicts these masses to be higher by 5-10%. For instance the Local Group of galaxies is estimated to be (2.47±0.08)⋅1012M⊙. For the groups with enough tracers close to the TS, the contribution of the Cosmological Constant makes the masses to be even higher. The results show the importance of including the local cosmic expansion in analyzing the Cosmic Flow of the local Universe.

The Hartree–Fock (HF) approximation has been an important tool for quantum-chemical calculations since its earliest appearance in the late 1920s and remains the starting point of most single-reference methods in use today. Intuition suggests that the HF kinetic energy should not exceed the exact kinetic energy; but no proof of this conjecture exists, despite a near century of development. Beginning from a generalized virial theorem derived from scaling considerations, we derive a general expression for the kinetic energy difference that applies to all systems. For any atom or ion, this trivially reduces to the well-known result that the total energy is the negative of the kinetic energy and, since correlation energies are never positive, proves the conjecture in this case. Similar considerations apply to molecules at their equilibrium bond lengths. We use highly precise calculations on Hooke's atom (two electrons in a parabolic well) to test the conjecture in a nontrivial case and to parameterize the difference between density functional and HF quantities, but find no violations of the conjecture. [ABSTRACT FROM AUTHOR]

This work presented a new model, Polarizable Fragment Density Model (PFDM), for the fast energy estimation of peptides, proteins, or other large molecular systems. By introducing an analogous relation to the virial theorem, the kinetic energy in Kohn–Sham Density Functional Theory (DFT) is approximated to the corresponding potential energy multiplied by a scale factor. Furthermore, the error due to this approximation together with the exchange–correlation energy is approximated as a second order Taylor's expansion about density. The PFDM energy is expressed as a functional of electronic density with system-dependent model parameters, such as a scaling factor c and a series of atomic pairwise KAB. The electron density in PFDM consists of a frozen part retaining chemical bonding information and a polarizable part to describe polarization effects, both of which are expanded as a linear expansion of Gaussian basis functions. The frozen density can be pre-calculated by fitting the DFT calculated density of fragments, as well as the polarizable density is optimized to solve PFDM energy. The PFDM energy is a quadratic function of the expansion coefficients of polarizable density and can be solved without expensive iteration process and numerical integrals. PFDM is especially suitable for the energy calculation of large molecular system with identical subunits, such as proteins, nucleic acids, and molecular clusters. Applying the PFDM method to the proteins, the results show that the accuracy is comparable to the PM6 semi-empirical method, and the efficiency is one order of magnitude faster than PM6. [ABSTRACT FROM AUTHOR]

Disk galaxies have a typical dimension of a few tens of kiloparsecs. It follows from the theory of general relativity that any signal originating from the galactic center will be noticed at the outskirts of the galaxy only tens of thousands of years later. This retardation effect, however, is absent in modelling used to calculate rotation curves throughout entire galaxies and their external gas. The considerable differences between Newtonian predictions and observed velocities are currently removed either by assuming dark matter or by suggesting various modifications to the laws of gravity, MOND being a long standing alternative to Newtonian gravity. In previous papers we have shown that by applying general relativity in a rigorous fashion, without neglecting retardation, one can explain the rotational velocities of galactic matter without modifying gravity or adding dark matter. Moreover, it was shown that dark matter effects, as they appear in gravitational lensing, the Tully-Fisher relation, and mass estimations based on the virial theorem could also be explained as retarded-gravity effects. It must be noted, however, that the proposed theory relies on the existence of a mass flow (of a changing rate) into the galaxy; a requirement that was not directly observed. In the original paper on the subject only one galaxy (M33) was analysed in detail. This was later amended with a published study of eleven galaxies. Here we give a more comprehensive retardation analysis of 143 galaxies of different types from the SPARC Galaxy collection. We show that in most cases we obtain very accurate fits to the data. [ABSTRACT FROM AUTHOR]

Our understanding of the formation and interaction processes of astronomical structures in the Universe and in galaxies has significantly expanded due to recent progress in improving the quality of observations, brought about by the emergence of new telescopes. Such processes are associated with the release of a significant amount of energy and may influence our vital activities. One example of energy flow release is magnetic storms, which are associated with processes occurring on the Sun. Such storms lead to disruptions of communication facilities, impact people, and can cause satellite and aircraft failures. There are objects in both near and deep space that are much more powerful than our star. In such objects, enormous energy flows are generated and released in the form of space jets. These energy flows occur in quasars, black holes, protostars, and during supernova explosions of types Ia, Ib, Ic, and II. During a supernova explosion, an immense amount of energy is released in the form of energy and high-energy particles. Such processes can have a significant impact on the safety of space flights. In this paper, the processes of formation of stellar systems within a turbulent interstellar medium are considered. This process is the result of complex processes that occur in the interstellar medium and denser gas formations – molecular clouds. At the same time, there are nonlinear interactions between turbulence and gravity, collisions between molecular clouds, interactions with shock waves, and so on. Supersonic turbulence is one of the main reasons for the formation of dynamic prestellar structures. The evolution of superdense substances formation begins when they gather in turbulent flows or are formed by supersonic collisions between molecular clouds. This process continues until these areas reach prestellar density. Depending on various factors, these superdense formations can either collapse and form new star systems or disintegrate, returning the substance to the interstellar medium. Until recently, the prevailing opinion was that a significant portions of molecular clouds were not gravitationally bound. Conclusions about this were made based on the virial theorem, which expresses the relationship between gravitational and kinetic energy. However, recent studies have refuted this. In this paper, the results of a simulation carried out to study the collision of large ellipsoidal and spherical molecular clouds in a three-dimensional setting. The simulation was conducted on high-resolution grids and utilized the adaptive mesh refinement (AMR) method. • Morphological analysis of supersonic turbulence arising from the collision of spherical and prolate molecular clouds. • Modeling processes leading to the possible formation of new stellar systems. • Study of complex shock wave loading leading to the emergence of supersonic turbulence, Kelvin-Helmholtz instability and the formation of clumps. • Comparison of Chandrasekhar's analytical and numerical solution for an prolate molecular cloud. [ABSTRACT FROM AUTHOR]

A Kratzer‐Fues potential is coupled with its self to form a double Kratzer‐Fues potential. The solutions of the radial Schrödinger equation with the combined potential are obtained via two different methodologies. Using the supersymmetric approach, the energy equation for a non‐approximated centrifugal term is obtained. The energy equation for the approximated centrifugal term is obtained using the parametric Nikiforov‐Uvarov method. Numerical results are computed for five molecules using MATLAB software program. The computed numerical values are compared with the experimental values and the ground state energy obtained from Herzberg's energy level equation. The study revealed that the results of the approximated centrifugal term are better than the results of the non‐approximated centrifugal term for four molecules. [ABSTRACT FROM AUTHOR]

We present dynamical scaling relations, combined with the stellar population properties, for a subsample of about 6000 nearby galaxies with the most reliable dynamical models extracted from the full Mapping Nearby Galaxies at Apache Point Observatory (MaNGA) sample of 10 000 galaxies. We show that the inclination-corrected mass plane for both early-type galaxies (ETGs) and late-type galaxies (LTGs), which links dynamical mass, projected half-light radius R e, and the second stellar velocity moment σe within R e, satisfies the virial theorem and is even tighter than the uncorrected one. We find a clear parabolic relation between |$\lg (M/L)_{\rm e}$|⁠ , the total mass-to-light ratio (M / L) within a sphere of radius R e, and |$\lg \sigma _{\rm e}$|⁠ , with the M / L increasing with σe and for older stellar populations. However, the relation for ETGs is linear and the one for the youngest galaxies is constant. We confirm and improve the relation between mass-weighted total density slopes |$\overline{\gamma _{_{\rm T}}}$| and σe: |$\overline{\gamma _{_{\rm T}}}$| become steeper with increasing σe until |$\lg (\sigma _{\rm e}/{\rm km\, s^{-1}})\approx 2.2$| and then remain constant around |$\overline{\gamma _{_{\rm T}}}\approx 2.2$|⁠. The |$\overline{\gamma _{_{\rm T}}}\!-\!\sigma _{\rm e}$| variation is larger for LTGs than ETGs. At fixed σe the total density profiles steepen with galaxy age and for ETGs. We find generally low dark matter fractions, median f DM(< R e) = 8 per cent, within a sphere of radius R e. However, we find that f DM(< R e) depends on σe better than stellar mass: dark matter increases to a median f DM(< R e) = 33 per cent for galaxies with |$\sigma _{\rm e}\lesssim 100$|  km s−1. The increased f DM(< R e) at low σe explains the parabolic |$\lg (M/L)_{\rm e}\!-\!\lg \sigma _{\rm e}$| relation. [ABSTRACT FROM AUTHOR]

The article shows that in a magnetic system with a thin-walled balanced winding close to a force-free one, a significant increase in the parameter θ = WMγ/MσM, is possible, which, according to the virial theorem, characterizes the ratio of the energy of the magnetic system WM to the weight of equipment with a material density γ, where under the action of electromagnetic forces there appears a mechanical stress σM. In a quasi-force- free magnetic system, the main part of the winding is in a state of local equilibrium, and only a relatively small part of the equipment is subject to stress. This part determines the weight of the entire system, and this weight can be minimized. The configurations of balanced thin-walled windings are developed, at the boundaries two boundary conditions are fulfilled simultaneously—the absence of the induction component normal to the boundary and the constancy of the product of induction and radius. The authors consider an example of a system consisting of a main part—a sequence of balanced "transverse" modules in the form of flat discs and end parts, consisting of a combination of "transverse" modules and "longitudinal" ones, having the form of rings elongated along the axis with balanced end parts. It is shown that in the system under consideration, the characteristic dimensionless parameter θ with an unlimited increase in the number of elements of the main part can reach a value of about 24, and when the number of these elements changes within 20–40, it changes from 6 to 9. [ABSTRACT FROM AUTHOR]

Recently, it has been proved that a set of spherically symmetric non-degenerate densities determines uniquely the Coulomb external potential. This theory is now extended to degenerate states. Euler equations and Kohn–Sham equations are derived and a novel form of the virial theorem is presented. It is emphasized that degenerate states of atoms can be rigorously treated as spherically symmetric when a subspace density is used with equal weighting factors. [ABSTRACT FROM AUTHOR]

Derivation and some applications of the scalar virial theorem to physical problems in classical mechanics, classical statistical mechanics, and non-relativistic quantum mechanics have been dealt with lucidly. Besides, brief comments about the history of its development and its versions in the special theory of relativity, in the systems enclosed in a container of finite volume, and in quantum statistical mechanics have been made. Also included are remarks bringing out its main limitations. [ABSTRACT FROM AUTHOR]

Galaxy Clusters are huge physical systems having a generic size of tens of millions of light years. Thus any modification at the center of the cluster will affect the outskirts only tens of millions of years afterwards. Those retardation considerations seem to be neglected in present day analysis used to estimate the total mass of the galaxy cluster, including those estimates which are based on the virial theorem. The significant differences between the velocities predicted by Newtonian action at a distance theory and observations are usually dealt with by either assuming an unobservable type of matter or by modifying gravity. Here we demonstrate that considering general relativistic effects one can explain the apparent excess matter appearing in galaxy clusters virial estimates. [ABSTRACT FROM AUTHOR]

Available inventories of baryonic mass in the universe are based largely on galactic data and empirical calculations made >20 years ago. Values falling below cosmological estimates underlie proposals that certain rarified gassy regions could have extremely high T, which motivated absorption measurements and hydrodynamic models. Yet, the shortfall remains. We inventory the total baryonic mass, focusing on gravitational interactions and updated measurements. A recent analytical inverse method for analyzing galactic rotation curves quantified how baryon mass and associated volumetric density (ρ) depend on distance (r) from galactic centers. The model is based on the dynamical consequences of the observed oblate shape of galaxies and the Virial Theorem. The parameter-free solution provides ρ(r) ∝ 1/r2 which describes star-rich galactic interiors, gas-rich outer discoids, circumgalactic media, and gradation into intergalactic media. Independent observational determinations of baryonic ρ validate that our 1/r2 result describes baryons alone. This solution shows that total baryonic mass associated with any galaxy is 2.4 to 40 times detectable luminosity, depending on galaxy size and spacing. Luminosity data within 50 Mpc show that Andromeda equivalents separated by ~1 Mpc represent the local universe. Combining the above yields (6 ± 2) × 10−25 kg m−3 for the present-day universe. Three other approaches support this high density: (1) evaluating trends and luminosity data near 1000 Mpc; (2) using a recent estimate for the number of galaxies in the universe; (3) calculating an energy balance. We discuss uncertainties in the critical density. Implications of large baryonic ρ are briefly discussed. [ABSTRACT FROM AUTHOR]

A hydrodynamic model for small acoustic oscillations in a cloud of stars is built, taking into account the self-consistent gravitational field in equilibrium with a non-zero second correlation moment. It is assumed that the momentum flux density tensor should include the analog of the anisotropic pressure tensor and the second correlation moment of both longitudinal and transverse gravitational field strength. The non-relativistic temporal equation for the second correlation moment of the gravitational field strength is derived from the Einstein equations using the first-order post-Newtonian approximation. One longitudinal and two transverse branches of acoustic oscillations are found in a homogeneous and isotropic star cloud. The requirement for the velocity of transverse oscillations to be zero provides the boundary condition for the stability of the cloud. The critical radius of the spherical cloud of stars is obtained, which is precisely consistent with the virial theorem. [ABSTRACT FROM AUTHOR]

Feedback from active galactic nuclei (AGNs) and supernovae can affect measurements of integrated Sunyaev–Zeldovich (SZ) flux of haloes (Y SZ) from cosmic microwave background (CMB) surveys, and cause its relation with the halo mass (Y SZ– M) to deviate from the self-similar power-law prediction of the virial theorem. We perform a comprehensive study of such deviations using CAMELS, a suite of hydrodynamic simulations with extensive variations in feedback prescriptions. We use a combination of two machine learning tools (random forest and symbolic regression) to search for analogues of the Y – M relation which are more robust to feedback processes for low masses (⁠|$M\lesssim 10^{14}\, \mathrm{ h}^{-1} \, \mathrm{ M}_\odot$|⁠); we find that simply replacing Y → Y (1 + M */ M gas) in the relation makes it remarkably self-similar. This could serve as a robust multiwavelength mass proxy for low-mass clusters and galaxy groups. Our methodology can also be generally useful to improve the domain of validity of other astrophysical scaling relations. We also forecast that measurements of the Y – M relation could provide per cent level constraints on certain combinations of feedback parameters and/or rule out a major part of the parameter space of supernova and AGN feedback models used in current state-of-the-art hydrodynamic simulations. Our results can be useful for using upcoming SZ surveys (e.g. SO, CMB-S4) and galaxy surveys (e.g. DESI and Rubin) to constrain the nature of baryonic feedback. Finally, we find that the alternative relation, Y – M *, provides complementary information on feedback than Y – M. [ABSTRACT FROM AUTHOR]

We formulate and study a generalized virial theorem for contact Hamiltonian systems. Such systems describe mechanical systems in the presence of simple dissipative forces such as Rayleigh friction, or the vertical motion of a particle falling through a fluid (quadratic drag) under the action of constant gravity. We find a generalized virial theorem for contact Hamiltonian systems which is distinct from that obtained earlier for the symplectic case. The 'contact' generalized virial theorem is shown to reduce to the earlier result on symplectic manifolds as a special case. Various examples of dissipative mechanical systems are discussed. We also formulate a generalized virial theorem in the contact Lagrangian framework. [ABSTRACT FROM AUTHOR]

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 X2⟩. 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. [ABSTRACT FROM AUTHOR]

We apply Newtonian fractional-dimension gravity (NFDG), an alternative gravitational model, to some notable cases of galaxies with little or no dark matter. In the case of the ultra-diffuse galaxy AGC 114905, we show that NFDG methods can effectively reproduce the observed rotation curve using a variable fractional dimension D R , as was performed for other galaxies in previous studies. For AGC 114905, we obtain a variable dimension in the range D ≈ 2.2–3.2, but our fixed D = 3 curve can still fit all the experimental data within their error bars. This confirms other studies indicating that the dynamics of this galaxy can be described almost entirely by the baryonic mass distribution alone. In the case of NGC 1052-DF2, we use an argument based on the NFDG extension of the virial theorem applied to the velocity dispersion of globular clusters showing that, in general, discrepancies between observed and predicted velocity dispersions can be attributed to an overall fractal dimension D < 3 of the astrophysical structure considered, and not to the presence of dark matter. For NGC 1052-DF2, we estimate D ≈ 2.9 , thus confirming that this galaxy almost follows standard Newtonian behavior. We also consider the case of the Bullet Cluster merger (1E0657-56), assumed to be one of the strongest proofs of dark matter existence. A simplified but effective NFDG model of the collision shows that the observed infall velocity of this merger can be explained by a fractional dimension of the system in the range D ≃ 2.4–2.5, again, without using any dark matter. [ABSTRACT FROM AUTHOR]

An alternative to the well-known complete form of the Mie–Grüneisen equation of state (EOS) for water is suggested. A closed analytical description of the self-consistent EOS for an arbitrary medium based only on the first law of thermodynamics and on a new form of virial theorem is obtained. This form of the virial theorem (allowing a variable power-law exponent of the particles interaction potential) is a result of the generalization of the known method of similarity [Feynman et al., "Equation of state of elements based on the generalized Fermi–Thomas theory," Phys. Rev. 75, 1561 (1949)]. In the new EOS, the description of the internal potential energy as a solution of a nonlinear Riemann–Hopf type equation is proposed. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In the recent study, the authors have proposed an integral equation for solving the inverse Kohn–Sham problem. In the present paper, the integral equation is numerically solved for one‐dimensional model of a He atom and an H2 molecule in the electronic ground states. For this purpose, we propose an iterative solution algorithm avoiding the inversion of the kernel of the integral equation. To quantify the numerical accuracy of the calculated exchange‐correlation potentials, we evaluate the exchange and correlation energies based on the virial theorem as well as the reproduction of the exact ground‐state electronic energy. The results demonstrate that the numerical solutions of our integral equation for the inverse Kohn–Sham problem are accurate enough in reproducing the Kohn–Sham potential and in satisfying the virial theorem. [ABSTRACT FROM AUTHOR]

We analyse the bulge/spheroid size–(stellar mass), R e, Sph − M *, Sph, relation and spheroid structural parameters for 202 local (predominantly |$\lesssim 110~\rm Mpc$|⁠) galaxies spanning |$M_*\sim 3\times 10^{9}\!-\!10^{12}~\rm M_{\odot }$| and |$0.1 \lesssim R_\mathrm{e, Sph}\lesssim 32~\rm kpc$| from multicomponent decomposition. The correlations between the spheroid Sérsic index (n Sph), central surface brightness (μ0, Sph), effective half-light radius (R e, Sph), absolute magnitude (⁠|$\mathfrak {M}_\mathrm{Sph}$|⁠), and stellar mass (M *, Sph) are explored. We also investigate the consequences of using different scale radii, |$R_{z,\rm Sph}$|⁠ , encapsulating a different fraction (z , from 0 to 1) of the total spheroid luminosity. The correlation strengths for projected mass densities, Σ z and 〈Σ〉 z , vary significantly with the choice of z. Spheroid size (⁠|$R_\mathrm{z, \rm Sph}$|⁠) and mass (M *, Sph) are strongly correlated for all light fractions z. We find |$\log (R_\mathrm{e,Sph}/\rm kpc) = 0.88\log (M_\mathrm{*,Sph}/\rm M_{\odot })-9.15$| with a small scatter of |$\Delta _{rms} = 0.24~\rm dex$| in the log (R e, Sph) direction. This result is discussed relative to the curved size–mass relation for early-type galaxies due to their discs yielding larger galaxy radii at lower masses. Moreover, the slope of our spheroid size–mass relation is a factor of ∼3, steeper than reported bulge size–mass relations, and with bulge sizes at M *, sph ∼3 × 109 M⊙ which are 2–3 times smaller. Our spheroid size–mass relation present no significant flattening in slope in the low-mass end (⁠|$M_{\rm *,sph}\sim 10^9 - 10^{10}\rm ~M_{\odot }$|⁠). Instead of treating galaxies as single entities, future theoretical and evolutionary models should also attempt to recreate the strong scaling relations of specific galactic components. Additional scaling relations, such as log (n Sph) − log (M *, Sph), log (Σ0, Sph) − log (n Sph), and log (n Sph) − log (R e, Sph), are also presented. Finally, we show that the local spheroids align well with the size-mass distribution of quiescent galaxies at z ∼ 1.25–2.25. In essence, local spheroids and high- z quiescent galaxies appear structurally similar, likely dictated by the virial theorem. [ABSTRACT FROM AUTHOR]

While dry mergers can produce considerable scatter in the (black hole mass, M bh)-(spheroid stellar mass, M *, sph) and M bh-(spheroid half-light radius, R e, sph) diagrams, the virial theorem is used here to explain why the scatter about the M bh–(velocity dispersion, σ) relation remains low in the face of such mergers. Its small scatter has been claimed as evidence of feedback from active galactic nuclei (AGNs). However, it is shown that galaxy mergers also play a significant role. The collision of two lenticular (S0) galaxies is expected to yield three types of merger product (a core-Sérsic S0, an ellicular ES,e or an elliptical E galaxy), depending on the remnant's orbital angular momentum. It is shown that the major merger of two S0 galaxies with M *, sph ∼ 1011 M⊙ advances the system along a slope of ∼5 in the M bh-σ diagram, while a major E+E galaxy merger moves a system slightly along a trajectory with a slope of ∼9. Mergers of lower-mass S0 galaxies with M *, sph ∼ 1010 M⊙ move slightly along a trajectory with a slope of ∼3, thereby further contributing to the steeper distribution for the E (and Es,e) galaxies in the M bh-σ diagram, reported here to have a slope of 7.27 ± 0.91, compared to the S0 galaxies that have a slope of 5.68 ± 0.60. This result forms an important complement to the AGN feedback models like that from Silk & Rees, providing a more complete picture of galaxy/(black hole) co-evolution. It also has important implications for nanohertz gravitational-wave research. [ABSTRACT FROM AUTHOR]

Polarons and bipolarons in two-dimensional (2 D) crystals with a simple conduction band are considered in the work. A study of the potential of pair interaction for the singlet states of polarons was carried out in the region intermediate coupling. The effective functional of exchange-coupled pair of polarons was obtained by the Buimistrov-Pekar method. Variational calculations of the singlet states energies were carried out taking into account the direct dependence of the wave function of the system on the interelectron distance. The variational functions have been selected in the form of a multiparameter system of Gaussian functions with exponentially correlated multipliers. The region of existence of a two-dimensional bipolaron was determined as a function of the crystal parameters. It has been shown that a single-center configuration (a configuration similar to a two-dimensional helium atom) is stable. The minimum corresponding to the two-center configuration ("polaron molecule") is secondary extremum that disappears when the electronic correlations associated with the direct dependence of the wave function of the system on the distance between electrons are taken into account. The main energy minimum corresponds to single-center configuration. The study of the fulfillment of the virial theorem for polaron and bipolaron functionals was carried out. The results of variational calculations were compared with the calculations performed by other authors. Two-center configuration of a 2D bipolaron is unstable. The only minimum corresponds to a one-center 2D bipolaron. Exchange energy of two 2D polarons is antiferromagnetic. [ABSTRACT FROM AUTHOR]

Modelling the hydrogen combustion reaction confined in carbon nanotubes (CNTs) implies the effective control of the initial conditions of position and velocities of the gas and of the nanocontainer, so as to reproduce the pressure and temperature of the system. In this work, the initial conditions of the gas particles were randomly generated according to the Maxwell-Boltzmann distributions at a given temperature. As for the CNT structure, we follow a procedure which relates the internal energy of the CNT to its temperature, which allowed us to define the kinetic energy to distribute in order to obtain a CNT at the same temperature as the reactive mixture. Results indicate that we can locate the carbon atoms at their equilibrium geometries and use 3 N k B T for their average kinetic energy. [ABSTRACT FROM AUTHOR]

First singlet (S1) excitations are of primary importance in the photoluminescence spectra of organic chromophores. However, due to the multi-determinantal nature of the singlet excited states, standard Kohn-Sham density-functional theory (DFT) is not applicable. While linear-response time-dependent DFT is the method of choice for the computation of excitation energies, it fails severely for excitations with charge-transfer character. Becke's recent virial exciton model [A. D. Becke, J. Chem. Phys. 148, 044112 (2018)] offers a promising solution to employ standard DFT for calculation of the S1 excitation energy in molecular systems. Here, it is shown that the virial exciton model is free of charge-transfer error. It is equally reliable for S1 excitations with significant charge-transfer character as for other classes of transitions. [ABSTRACT FROM AUTHOR]

A specific modification of Newtonian dynamics known as MOND has been shown to reproduce the dynamics of most astrophysical systems at different scales without invoking non-baryonic dark matter (DM). There is, however, a long-standing unsolved problem when MOND is applied to rich clusters of galaxies in the form of a deficit (by a factor around two) of predicted dynamical mass derived from the virial theorem with respect to observations. In this article, we approach the virial theorem using the velocity dispersion of cluster members along the line of sight rather than using the cluster temperature from X-ray data and hydrostatic equilibrium. Analytical calculations of the virial theorem in clusters for Newtonian gravity + DM and MOND are developed, applying pressure (surface) corrections for non-closed systems. Recent calibrations of DM profiles, baryonic ratio, and baryonic (β model or others) profiles are used, while allowing free parameters to range within the observational constraints. It is shown that solutions exist for MOND in clusters that give similar results to Newton + DM – particularly in the case of an isothermal β model for β = 0.55–0.70 and core radii r c between 0.1 and 0.3 times r 500 (in agreement with the known data). The disagreements found in previous studies seem to be due to the lack of pressure corrections (based on inappropriate hydrostatic equilibrium assumptions) and/or inappropriate parameters for the baryonic matter profiles. [ABSTRACT FROM AUTHOR]

In the case of the one-electron Dirac equation with a point nucleus, the virial theorem (VT) states that the ratio of the kinetic energy to potential energy is exactly − 1 , a ratio that can be an independent test of the accuracy of a computed solution. This paper studies the virial theorem for subshells of equivalent electrons and their interactions in many-electron atoms. This shows that the linear scaling of the dilation is achieved through the balancing of the contributions to the potential of an electron from inner and outer regions that some Slater integrals impose conditions on a single subshell, but others impose conditions between subshells. The latter slows the rate of convergence of the self-consistent field process in which radial functions are updated one at a time. Several cases are considered. Results are also extended to the nonrelativistic case. [ABSTRACT FROM AUTHOR]

In this work, we apply virial analysis to the model of self-gravitating turbulent cloud ensembles introduced by Donkov & Stefanov in two previous papers, clarifying some aspects of turbulence and extending the model to account not only for supersonic flows but for trans- and subsonic ones as well. Making use of the Eulerian virial theorem at an arbitrary scale, far from the cloud core, we derive an equation for the density profile and solve it in approximate way. The result confirms the solution ϱ(ℓ) = ℓ−2 found in the previous papers. This solution corresponds to three possible configurations for the energy balance. For trans- or subsonic flows, we obtain a balance between the gravitational and thermal energy (Case 1) or between the gravitational, turbulent, and thermal energies (Case 2) while for supersonic flows, the possible balance is between the gravitational and turbulent energy (Case 3). In Cases 1 and 2, the energy of the fluid element can be negative or zero; thus the solution is dynamically stable and shall be long lived. In Case 3, the energy of the fluid element is positive or zero, i.e. the solution is unstable or at best marginally bound. At scales near the core, one cannot neglect the second derivative of the moment of inertia of the gas, which prevents derivation of an analytic equation for the density profile. However, we obtain that gas near the core is not virialized and its state is marginally bound since the energy of the fluid element vanishes. [ABSTRACT FROM AUTHOR]

We investigate changes in stellar population age and metallicity ([Z/H]) scaling relations for quiescent galaxies from intermediate redshift (0.60 ≤   |$z$| ≤ 0.76) using the LEGA-C Survey to low redshift (0.014 ≤   |$z$| ≤ 0.10) using the SAMI Galaxy Survey. Specifically, we study how the spatially integrated global age and metallicity of individual quiescent galaxies vary in the mass–size plane, using the stellar mass M * and a dynamical mass proxy derived from the virial theorem M D  ∝ σ2  R e. We find that, similarly to at low redshift, the metallicity of quiescent galaxies at 0.60 ≤   |$z$| ≤ 0.76 closely correlates with M / R e (a proxy for the gravitational potential or escape velocity), in that galaxies with deeper potential wells are more metal-rich. This supports the hypothesis that the relation arises due to the gravitational potential regulating the retention of metals by determining the escape velocity for metal-rich stellar and supernova ejecta to escape the system and avoid being recycled into later stellar generations. Conversely, we find no correlation between age and surface density (⁠|$M/R_\mathrm{e}^2$|⁠) at 0.60 ≤   |$z$| ≤ 0.76, despite this relation being strong at low redshift. We consider this change in the age– |$M/R_\mathrm{e}^2$| relation in the context of the redshift evolution of the star-forming and quiescent mass–size relations, and find our results are consistent with galaxies forming more compactly at higher redshifts and remaining compact throughout their evolution. Furthermore, galaxies appear to quench at a characteristic surface density that decreases with decreasing redshift. The |$z$| ∼  0 age– |$M/R_\mathrm{e}^2$| relation is therefore a result of building up the quiescent and star-forming populations with galaxies that formed at a range of redshifts and therefore a range of surface densities. [ABSTRACT FROM AUTHOR]

In this paper, we theoretically investigate the stability of spin-wave solitons in Bose-Einstein condensates of repulsive magnons, confined by an inhomogeneous external magnetic field described by a Gaussian well. For this purpose, we use the quasi-one-dimensional Gross-Pitaevskii equation to describe the behavior of the condensate. In order to solve the Gross-Pitaevskii equation, we used two different approaches: one analytical (variational method) and another numerical (split-step Crank-Nicolson method). The stability of the solutions and the validation of the numerical results were confirmed, respectively, through the anti-VK criterion and the virial theorem. Furthermore, the simulations described the behavior of physical quantities of interest such as chemical potential, energy per magnon and central density as a function of the nonlinearity of the model (magnon-magnon interactions). The theoretical results provide subsidies for a better understanding of the nonlinear phenomena related to the Bose-Einstein condensates of magnons in ferromagnetic films. [ABSTRACT FROM AUTHOR]

A recent virial-theorem-based model of the singlet-triplet splitting in singly excited configurations [A. D. Becke, J. Chem. Phys. 148, 044112 (2018)] offers an accurate and economical tool for the computation of optical gaps in large molecules. Two single-determinant density-functional-theory calculations, one on the (closed-shell) ground state and another on the (restricted open-shell) HOMO → LUMO triplet excited state, followed by a simple two-electron integral evaluation, are all we need. Here the method is used to compute the optical gaps of trans-polyenes containing up to 60 carbon atoms, approaching the polyacetylene limit. Comparisons with previous computations, and experiment, are made. We also explore changes of the exact-exchange mixing fraction in the underlying density functional. Its effect on the optical gap, and also the exciton size, is enormous. Thus we face the vexing, often asked, question: how much exact exchange should be used? [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. [ABSTRACT FROM AUTHOR]

Photoluminescence effects in organic chromophores are of significant importance and requires precise description of low lying excited states. In this communication, we put forward an alternative time‐independent DFT scheme for computing lowest single‐particle excitation energy, especially for singlet excited state. This adopts a recently developed "virial"‐theorem based model of singlet‐triplet splitting which requires a DFT calculation on closed shell ground state and a restricted open‐shell triplet excited state, followed by a simple 2e- integral evaluation. This produces vertical excitation energies in small molecules, linear and non‐linear polycyclic aromatic hydrocarbon and organic dyes in comparable accuracy to the TDDFT. We also explore the functional dependency of present method with three different functionals (B3LYP, wB97X and CAM‐B3LYP) for polyenes and linear acenes. A systematic comparison with literature value illustrates the validity and usefulness of the present scheme in determining optical gap with fair computational cost. [ABSTRACT FROM AUTHOR]

In the case of the one-electron Dirac equation with a point nucleus, the virial theorem (VT) states that the ratio of the kinetic energy to potential energy is exactly −1, a ratio that can be an independent test of the accuracy of a computed solution. This paper studies the virial theorem for subshells of equivalent electrons and their interactions in many-electron atoms. This shows that the linear scaling of the dilation is achieved through the balancing of the contributions to the potential of an electron from inner and outer regions that some Slater integrals impose conditions on a single subshell, but others impose conditions between subshells. The latter slows the rate of convergence of the self-consistent field process in which radial functions are updated one at a time. Several cases are considered. Results are also extended to the nonrelativistic case.

We present high spatial resolution (≈12 pc) Atacama Large Millimeter/submillimeter Array 12CO(J  = 3–2) observations of the nearby lenticular galaxy NGC 4429. We identify 217 giant molecular clouds within the 450 pc radius molecular gas disc. The clouds generally have smaller sizes and masses but higher surface densities and observed linewidths than those of Milky Way disc clouds. An unusually steep size–linewidth relation (⁠|$\sigma \propto R_{\rm c}^{0.8}$|⁠) and large cloud internal velocity gradients (0.05–0.91 km s−1 pc−1) and observed virial parameters (〈αobs,vir〉 ≈ 4.0) are found, which appear due to internal rotation driven by the background galactic gravitational potential. Removing this rotation, an internal virial equilibrium appears to be established between the self-gravitational (U sg) and turbulent kinetic (E turb) energies of each cloud, i.e. |$\langle \alpha _{\rm sg,vir}\equiv \frac{2E_{\rm turb}}{\vert U_{\rm sg}\vert }\rangle \approx 1.3$|⁠. However, to properly account for both self and external gravity (shear and tidal forces), we formulate a modified virial theorem and define an effective virial parameter |$\alpha _{\rm eff,vir}\equiv \alpha _{\rm sg,vir}+\frac{E_{\rm ext}}{\vert U_{\rm sg}\vert }$| (and associated effective velocity dispersion). The NGC 4429 clouds then appear to be in a critical state in which the self-gravitational energy and the contribution of external gravity to the cloud's energy budget (E ext) are approximately equal, i.e. |$\frac{E_{\rm ext}}{\vert U_{\rm sg}\vert }\approx 1$|⁠. As such, 〈αeff,vir〉 ≈ 2.2 and most clouds are not virialized but remain marginally gravitationally bound. We show this is consistent with the clouds having sizes similar to their tidal radii and being generally radially elongated. External gravity is thus as important as self-gravity to regulate the clouds of NGC 4429. [ABSTRACT FROM AUTHOR]

Using the formalism of stochastic embedding developed by Cresson and Darses [J. Math. Phys. 48, 072703 (2007)], we study how the dynamics of the classical Newton equation for a force deriving from a potential is deformed under the assumption that this equation can admit stochastic processes as solutions. We focus on two definitions of a stochastic Newton equation called differential and variational. We first prove a stochastic virial theorem that is a natural generalization of the classical case. The stochasticity modifies the virial relation by adding a potential term called the induced potential, which corresponds in quantum mechanics to the Bohm potential. Moreover, the differential stochastic Newton equation naturally provides an action functional that satisfies a stochastic Hamilton–Jacobi equation. The real part of this equation corresponds to the classical Hamilton–Jacobi equation with an extra potential term corresponding to the induced potential already observed in the stochastic virial theorem. The induced potential has an explicit form depending on the density of the stochastic process solutions of the stochastic Newton equation. It is proved that this density satisfies a nonlinear Schrödinger equation. Applying this formalism for the Kepler potential, one proves that the induced potential coincides with the ad hoc "dark potential" used to recover a flat rotation curve of spiral galaxies. We then discuss the application of the previous formalism in the context of spiral galaxies following the proposal and computations given by Da Rocha and Nottale [Chaos, Solitons Fractals, 16(4), 565–595 (2003)] where the emergence of the "dark potential" is seen as a consequence of the fractality of space in the context of the scale relativity theory. [ABSTRACT FROM AUTHOR]