Inverse Power Law Potential Research Articles

Acceptable inverse power potential quintessence with n=18/7

We present a particle physics quintessence model which agrees well with existing cosmological data, including the position of the acoustic peaks. This model has an inverse power law potential (IPL) with n=18/7∼2.57 and it gives weff=−0.75, an acoustic scale lA=307 and a density contrast σ8=0.95.Models with n>1 have been said to be disfavored by the analysis of the acoustic peaks. However, the results are not correct. The main reason is that the tracker approximation has been used in deriving the IPL constrains and for n<5 the scalar field has not reached its tracker value by present day.The model can be derived from particle physics, using Affleck–Dine–Seiberg “ADS” superpotential, for a non-Abelian gauge group with Nc=8, Nf=1. The advantage of having Nf=1 is that there is only one degree of freedom below the condensation scale given by the condensate (quintessence) φ2=〈QQ〉 field. The condensation scale is at 1 GeV a very interesting scale since it connects the quintessence “Q” with the standard model “SM” scale. The similarity in energy scales between Q and SM scale gives an “explanation” to the coincidence problem. The fact that only recently the universe is accelerating is a natural consequence of the Q scale and the evolution of φ.

Variable sphere molecular model for inverse power law and Lennard-Jones potentials in Monte Carlo simulations

The variable sphere (VS) molecular model for the Monte Carlo simulation of rarefied gas flow is introduced to provide consistency for diffusion and viscosity cross-sections with those of any realistic intermolecular potential. It is then applied to the inverse power law (IPL) and Lennard-Jones (LJ) potentials. The VS model has a much simpler scattering law than either the variable hard sphere (VHS) or variable soft sphere (VSS) models; also, it has almost the same computational efficiency as the VHS and VSS models. A simulation of velocity relaxation in a homogeneous space and two comparative simulations of molecular diffusion in a homogeneous heat-bath gas and normal shock wave structure in a monatomic gas are made to examine VS model validity. The relaxation to a Maxwellian distribution function and equipartition between all degrees of freedom are well established; good agreement is shown in the molecular diffusion and shock wave structure between the VS model and the IPL and LJ potentials. The VS model is combined with the statistical inelastic cross-section (SICS) model and applied to simulation of translational and rotational energy relaxation in a homogeneous space. The VS model shows the relaxation of Maxwellian and Boltzmann distribution functions and equipartition between all degrees of freedom. Comparative calculation between the VS model with the SICS (VS-SICS) model and the VSS model with the SICS (VSS-SICS) model is made for rotational relaxation in a nitrogen normal shock wave. Good agreement is shown in the shock wave structure and rotational energy distribution function between the VS-SICS model and the VSS-SICS model. This study demonstrates that diffusion and viscosity cross-sections, rather than the scattering law of each molecular collision, affect macroscopic transport phenomena.

Future Destiny of Quintessential Universe and Constraint on Model from Deceleration Parameter**The project supported by National Natural Science Foundation of China under Grant Nos 10047004 and NKBRSF G.19990754

The evolution of the quintessence in various stages of the universe, i.e., the radiation-, matter-, and quintessence-dominated stages, is closely related with the tracking behavior and the deceleration parameter of the universe. We gave the explicit relation between the equation-of-state of the quintessence in the epoch of the matter-quintessence equality and the inverse power index of the quintessence potential, obtained the constraint on this potential parameter coming from the present deceleration parameter, i.e., a low inverse power index. We point out that the low inverse power-law potential with a single term cannot work for the tracking solution. In order to have both of the tracker and the suitable deceleration parameter it is necessary to introduce at least two terms in the quintessence potential. We give the future evolution of the quintessential universe.

Fitting inverse power-law quintessence models using the SNAP satellite

We investigate the possibility of using the proposed SNAP satellite in combination with low-$z$ supernova searches to distinguish between different inverse power-law quintessence models. If the true model is that of a cosmological constant, we determine the prospects of ruling out the inverse power-law potential. We show that SNAP combined with e.g., the SNfactory and an independent measurement of the mass energy density to 17% accuracy can distinguish between an inverse power-law potential and a cosmological constant and put severe constraints on the power-law exponent.

Current constraints on the dark energy equation of state

We combine complementary datasets from cosmic microwave background (CMB) anisotropy measurements, high redshift supernovae (SNIa) observations and data from local cluster abundances and galaxy clustering (LSS) to constrain the dark energy equation of state parametrized by a constant pressure-to-density ratio ${w}_{Q}.$ Under the assumption of flatness, we find ${w}_{Q}&lt;\ensuremath{-}0.85$ at $68%$ C.L., providing no significant evidence for quintessential behavior different from that of a cosmological constant. We then generalize our result to show that the constraints placed on a constant ${w}_{Q}$ can be safely extended to dynamical theories. We consider a variety of quintessential dynamical models based on inverse power law, exponential and oscillatory scaling potentials. We find that SNIa observations are ``numbed'' to dynamical shifts in the equation of state, making the prospect of reconstructing $w(z)$ a challenging one indeed.

Future supernova probes of quintessence

We investigate the potential of a future supernovae data set, as might be obtained by the proposed SNAP satellite, to discriminate between two possible explanations for the observed dimming of the high redshift type IA supernovae: namely, either (i) a cosmological evolution for which the expansion of the universe has been accelerating for a substantial range of redshifts $z\ensuremath{\sim}1;$ or (ii) an unexpected supernova luminosity evolution over such a redshift range. By evaluating Bayes factors we show that within the context of spatially flat model universes with a dark energy the future SNAP data set should be able to discriminate these two possibilities. Our calculations assume particular cosmological models with a quintessence field in the form of a dynamical pseudo Nambu-Goldstone boson (PNGB), and a simple empirical model of the evolution of peak luminosities of the supernovae sources which has been recently discussed in the literature. We also show that the fiducial SNAP data set, simulated with the assumption of no source evolution, is able to discriminate the PNGB model from a number of other spatially flat quintessence models which have been widely studied in the literature, namely those with inverse power-law, simple exponential and double-exponential potentials.

Inflation, braneworlds and quintessence

Inflationary cosmology is developed in the second Randall-Sundrum braneworld scenario, where the accelerated expansion arises through potentials that are too steep to drive inflation in conventional cosmology. A relationship between the scalar and tensor perturbation spectra is derived that is independent of both the inflaton potential and the brane tension. It is found that a single field with an inverse power law potential can act as both the inflaton and the quintessence field for suitable values of the brane tension.

Crystal structures and freezing of dipolar fluids.

We investigate the crystal structure of classical systems of spherical particles with an embedded point dipole at T=0. The ferroelectric ground state energy is calculated using generalizations of the Ewald summation technique. Due to the reduced symmetry compared to the nonpolar case the crystals are never strictly cubic. For the Stockmayer (i.e., Lennard-Jones plus dipolar) interaction three phases are found upon increasing the dipole moment: hexagonal, body-centered orthorhombic, and body-centered tetragonal. An even richer phase diagram arises for dipolar soft spheres with a purely repulsive inverse power law potential approximately r(-n). A crossover between qualitatively different sequences of phases occurs near the exponent n=12. The results are applicable to electro- and magnetorheological fluids. In addition to the exact ground state analysis we study freezing of the Stockmayer fluid by density-functional theory.

K-1715 分子間衝突と輸送係数に関する考察(J08-4 熱流体現象の原子・分子論的アプローチ(4))(J08 熱流体現象の原子・分子論的アプローチ)

The effect of molecular scattering angle in the inelastic collision on the transport coefficient is examined by the test particle Monte Carlo method and null-collision direct simulation Monte Carlo method with the inverse-power-law (IPL) potential and Borgnakke-Larsen (B-L) model. The diffusion and viscosity coefficients of B-L model with isotropic scattering for the inelastic collision are less than those of monatomic gas with IPL potential. On the other hand the transport coefficients of B-L model with IPL scattering for the inelastic collision are in reasonable agreement with those of monatomic gas. It is confirmed that the scattering law or angle are very important for the transport phenomena and transport coefficients.

Tracking extended quintessence

We investigate the cosmological role of a tracking field $\ensuremath{\varphi}$ in extended quintessence scenarios, where the dynamical vacuum energy driving the acceleration of the universe today possesses an explicit coupling with the Ricci scalar R of the form $F(\ensuremath{\varphi})R/2,$ where $F(\ensuremath{\varphi})$ mimics general relativity today, $F({\ensuremath{\varphi}}_{0})=1/8\ensuremath{\pi}G.$ We analyze explicit nonminimally coupled (NMC) models where $F(\ensuremath{\varphi})=1/8\ensuremath{\pi}G+\ensuremath{\xi}({\ensuremath{\varphi}}^{2}\ensuremath{-}{\ensuremath{\varphi}}_{0}^{2}),$ with $\ensuremath{\xi}$ is the coupling constant and ${\ensuremath{\varphi}}_{0}$ is the Q value today. Tracker solutions for these NMC models, with inverse power-law potentials, possess an initial enhancement of the scalar field dynamics, named the R-boost, caused by the effective potential generated by the Ricci scalar in the Klein-Gordon equation. During this phase the field performs a ``gravitational'' slow rolling until the true potential becomes important. We give accurate analytic formulas describing the R-boost, showing that the quintessence energy in this phase scales with the redshift z as ${(1+z)}^{2}.$ When the R-boost ends, the field trajectory matches the tracker solution in minimally coupled theories. We compute perturbations in these tracking extended quintessence models, by integrating the full set of equations for the evolution of linear fluctuations in scalar-tensor theories of gravity, and assuming Gaussian scale-invariant initial perturbations. The integrated Sachs-Wolfe (ISW) effect on the cosmic microwave background (CMB) angular spectrum causes a change $\ensuremath{\delta}{C}_{l}{/C}_{l}\ensuremath{\simeq}6[1\ensuremath{-}8\ensuremath{\pi}GF({\ensuremath{\varphi}}_{\mathrm{dec}})]$ at $l\ensuremath{\lesssim}10,$ where ``dec'' stands for decoupling. Similarly, the CMB acoustic peak multipoles shift compared to ordinary tracking quintessence models by roughly an amount $\ensuremath{\delta}l/l\ensuremath{\simeq}[8\ensuremath{\pi}GF({\ensuremath{\varphi}}_{\mathrm{dec}})\ensuremath{-}1]/8.$ The turnover wave number ${k}_{\mathrm{turn}}$ in the matter power spectrum shifts by an amount $\ensuremath{\delta}{k}_{\mathrm{turn}}{/k}_{\mathrm{turn}}\ensuremath{\simeq}[1\ensuremath{-}8\ensuremath{\pi}GF({\ensuremath{\varphi}}_{\mathrm{eq}})]/2,$ where ``eq'' stands for matter-radiation equivalence. All these corrections may assume positive as well as negative values, depending on the sign of the NMC parameter $\ensuremath{\xi}.$ We show that the above effects can be as large as 10--30 % with respect to equivalent cosmological constant and ordinary tracking quintessence models, respecting all the existing experimental constraints on scalar-tensor theories of gravity. These results demonstrate that the playground where the data of the next decade will have their impact includes the nature of the dark energy in the Universe, as well as the structure of the theory of gravity.

Quintessence and gravitational waves

We investigate some aspects of quintessence models with a nonminimally coupled scalar field and in particular we show that it can behave as a component of matter with $\ensuremath{-}3\ensuremath{\lesssim}P/\ensuremath{\rho}\ensuremath{\lesssim}0.$ We study the properties of gravitational waves in this class of models and discuss their energy spectrum and the cosmic microwave background anisotropies they induce. We also show that gravitational waves are damped by the anisotropic stress of the radiation and that their energy spectrum may help to distinguish between the inverse power law potential and supergravity motivated potential. We finish with a discussion on the constraints arising from their density parameter ${\ensuremath{\Omega}}_{\mathrm{GW}}.$

New cosmological tracker solution for quintessence

In this paper we propose a quintessence model with the potential $V(\Phi )=V_{o}[ \sinh {(\alpha \sqrt{\kappa_{o}}\Delta \Phi})] ^{\beta}$, which asymptotic behavior corresponds to an inverse power-law potential at early times and to an exponential one at late times. We demonstrate that this is a tracker solution and that it could have driven the Universe into its current inflationary stage. The exact solutions and the description for a complete evolution of the Universe are also given. We compare such model with the current cosmological observations.

Tracker fields from nonminimally coupled theory

We extend the concept of quintessence to a flat nonminimally coupled scalar - tensor theories of gravity. By means of Noether's symmetries for the cosmological pointlike Lagrangian L, it is possible to exhibit exact solutions for a class of models depending on a free parameter s. This parameter comes out in the relationship existing between the coupling F(\phi) and the potential V(\phi) because of such a symmetry for L. When inverse power law potentials are taken in account, a whole family of exact solutions parametrized by such an s is proposed as a class of tracker fields, and some considerations are made about them.

Cosmological scaling solutions of nonminimally coupled scalar fields

We study the existence and stability of cosmological scaling solutions of a nonminimally coupled scalar field evolving in either an exponential or inverse power law potential. We show that, for inverse power law potentials, there exist scaling solutions the stability of which does not depend on the coupling constant $\ensuremath{\xi}.$ We then study the more involved case of exponential potentials and show that the scalar field will asymptotically behave as a baryotropic fluid when $\ensuremath{\xi}\ensuremath{\ll}1.$ The general case $\ensuremath{\xi}\ensuremath{\ll}/1$ is then discussed and we illustrate these results by some numerical examples.

Finite Size Scaling in Quantum Mechanics

The finite size scaling ansatz is combined with the variational method to extract information about critical behavior of quantum Hamiltonians. This approach is based on taking the number of elements in a complete basis set as the size of the system. As in statistical mechanics, the finite size scaling can then be used directly in the Schrodinger equation. This approach is general and gives very accurate results for the critical parameters, for which the bound-state energy becomes absorbed or degenerate with a continuum. To illustrate the applications in quantum calculations, we present detailed calculations for both short- and long-range potentials. I. Introduction In statistical mechanics, the singularities in thermodynamic functions associated with a critical point occur only in the thermodynamic limit, when all the dimensions of the system under consideration tend to infinity. Strictly speaking, there are no phase transitions in a finite system at nonzero temper- ature, and yet, experiments as well as numerical calculations all use finite systems. 1 To address this problem, the finite size scaling method was formulated by Fisher 2 and others 3 to extrapolate information obtained from a finite system to the thermodynamic limit. In quantum mechanics, when using variation methods, one encounters the same finite size problem in studying the critical behavior of a quantum Hamiltonian H(I1,...,Ik) as a function of its set of parameters {Ii}. In this context, critical means the values of {Ii} for which a bound-state energy is nonanalytic. In many cases, as in this study, this critical point is the point where a bound-state energy becomes absorbed or degenerate with a continuum. In this case, the finite size corresponds not to the spatial dimension but to the number of elements in a complete basis set used to expand the exact wave function of a given Hamiltonian. Recently, we used the finite size scaling and phenomenologi- cal renormalization equations for calculations of the critical charges for two- 4,5 and three-electron systems. 6 This approach is based on taking the lowest eigenvalues of a quantum Hamiltonian as the leading eigenvalues of a transfer matrix of a classical pseudosystem. In this paper we will assume that there exists a scaling function for the truncated mean value of a given operator, and with the help of the Hellmann-Feynman theorem we can obtain a direct finite size scaling approach to the Schrodinger equation. 7 This approach is general and can be used to study critical behavior of a quantum Hamiltonian as a function of its parameters. To illustrate this approach, we include detailed calculations for the critical parameters for two cases with qualitatively different behavior: one with short-range interaction, the Yukawa potential, and one with a long-range interaction, the inverse power law potential.

WKB and exact wave functions for inverse power-law potentials

We compare WKB and exact wave functions for inverse power-law potentials and derive a universal algebraic formula reproducing reflection phases and scattering phase shifts with uniform high accuracy for any wave length. Appropriately modified quantization gives competitive accuracy for the spiked harmonic oscillator. The asymptotic accuracy of WKB wave functions is discussed.

Integration of the Heisenberg equations for inverse power-law potentials.

The direct method of integration of the operator Heisenberg equations of motion is extended to the solution of quantum tunneling when the central potential is a sum of inverse powers of the radial distance. By obtaining the equation of motion for the Weyl-ordered basis set {l_brace}{ital S}{sub {ital m},{ital n}}({ital t}){r_brace} formed from {ital r}({ital t}) and {ital p}{sub {ital r}}({ital t}), one can express the time evolution of any member of the set as an infinite sum involving the operators {l_brace}{ital S}{sub {ital m},{ital n}}(0){r_brace}. The direct integration enables one to find the expectation values of the radial position operator and its conjugate momentum and higher moments of these operators as functions of time. {copyright} {ital 1996 The American Physical Society.}

Burnett's equations from a (13+9N)-field theory

The Burnett equations are determined from a (13+9N)-field theory and the successive approximations up to the fifth-order of the Burnett's coefficients are given for gases whose particles interact according to a Lennard-Jones 6–12 potential and to an inverse power law potential.

Neutron displacement damage cross sections for SiC

Calculations of neutron displacement damage cross sections for SiC are presented. We use Biersack and Haggmark's empirical formula in constructing the electronic stopping power, which combines Lindhard's model at low PKA energies and Bethe-Bloch's model at high PKA energies. The electronic stopping power for polyatomic materials is computed on the basis of Bragg's Additivity Rule. A continuous form of the inverse power law potential is used for nuclear scattering. Coupled intergro-differential equations for the number of displaced atoms j, caused by PKA i, are then derived. The procedure outlined above gives partial displacement cross sections, displacement cross sections for each specie of the lattice, and for each PKA type. The corresponding damage rates for several fusion and fission neutron spectra are calculated. The stoichiometry of the irradiated material is investigated by finding the ratio of displacements among various atomic species. The role of each specie in displacing atoms is also investigated by calculating the fraction of displacements caused by each PKA type. The study shows that neutron displacement damage rates of SiC in typical magnetic fusion reactor first walls will be ~ 10–15 dpa MW −1 m 2; in typical lead-protected inertial confinement fusion reactor first walls they will be ~ 15–20 dpa MW −1 m 2. For fission spectra, we find that the neutron displacement damage rate of SiC is ~ 74 dpa per 10 27 n/m 2 in FFTF, ~ 39 dpa per 10 27 n/m 2 in HFIR, and 25 dpa per 10 27 n/m 2 in NRU. Approximately 80% of displacement atoms are shown to be of the carbon-type.

On idealised collision cascades in power law potentials

Idealised elastic collision cascades of energetic projectiles governed by repulsive inverse power law potentials in amorphous targets are considered. Spatial bounds of the statistical ensemble of projectile cascades are characterised and utilised to stipulate conditions for the onset of sputtering. Analytic expressions for the first and second moments of projectile range distribution are derived by exploiting the fractal self-similarity and cylindrical symmetry of the ensemble. The results, incorporating a new prescription for the average energy transfer per elastic collision in a power law potential, are presented.