Power-law Potentials Research Articles

Energy Fluctuation of Ideal Fermi Gas Trapped under Generic Power Law Potential in d Dimensions

Energy fluctuation of ideal Fermi gas trapped under generic power law potential has been calculated in arbitrary dimensions. Energy fluctuation is scrutinized further in the degenerate limit μ ≫ KBT with the help of Sommerfeld expansion. The dependence of energy fluctuation on dimensionality and power law potential is studied in detail. Most importantly our general result can not only exactly reproduce the recently published result regarding free and harmonically trapped ideal Fermi gas in d = 3 but also can describe the outcome for any power law potential in arbitrary dimension.

Monte Carlo simulations of vector pseudospins for strains: Microstructures and martensitic conversion times

We present systematic temperature-quench Monte Carlo simulations on discrete-strain pseudospin model Hamiltonians to study microstructural evolutions in ferroelastic transitions with two-component vector order parameters (NOP=2) in 2-spatial dimensions. The zero value pseudospin is the single high-temperature phase while the low-temperature phase has Nv variants. Thus the number of nonzero values of pseudospin are triangle-to-centered rectangle (Nv=3), square-to-oblique (Nv=4) and triangle-to-oblique (Nv=6). The model Hamiltonians contain a transition-specific Landau energy term, a domain wall cost or Ginzburg term, and power-law anisotropic interaction potential, induced from a strain compatibility condition. On quenching below a transition temperature, we find behaviour similar to the previously studied square-to-rectangle transition (NOP=1,Nv=2), showing that the rich behaviour found, is generic. Thus we find for two-component order parameters that the same Hamiltonian can describe both athermal and isothermal martensite regimes for different material parameters. The athermal/isothermal/austenite parameter regimes and temperature–time–transformation diagrams are understood, as previously, through parametrization of effective-droplet energies. In the athermal regime, we find rapid conversions below a spinodal-like temperature and austenite-martensite conversion delays above it, as in the experiment. The delays show early incubation behaviour, and at the transition to austenite the delay times have Vogel–Fulcher divergences and are insensitive to Hamiltonian energy scales, suggesting that entropy barriers are dominant. Systematic temperature quench experiments can look for martensite formation and growth during conversion-incubations, divergences, and distributions close to the transition.

Crystalline structures in a one-dimensional two-component lattice gas with 1/r α interactions

We investigate the ground state of a one-dimensional lattice system that hosts two different kinds of excitations (species) which interact with a power-law potential. Interactions are only present between excitations of the same kind and the interaction strength can be species-dependent. For the case in which only one excitation is permitted per site we derive a prescription for determining the ground state configuration as a function of the filling fractions of the two species. We show that depending on the filling fractions compatible or incompatible phases emerge. Furthermore, we discuss in detail the case in which one species is strongly and the other one weakly interacting. In this case the configuration of the strongly interacting (strong) species can be considered frozen and forms an effective inhomogeneous lattice for the other (weak) species. In this limit we work out in detail the microscopic ground state configuration and show that by varying the density of the weak species a series of compatible–incompatible transitions occurs. Finally, we determine the stability regions of the weak species in the compatible phase and compare it with numerical simulations.

Cosmological perturbations in coherent oscillating scalar field models

The fact that fast oscillating homogeneous scalar fields behave as perfect fluids in average and their intrinsic isotropy have made these models very fruitful in cosmology. In this work we will analyse the perturbations dynamics in these theories assuming general power law potentials $V(\phi)=\lambda \vert\phi\vert^{n}/n$. At leading order in the wavenumber expansion, a simple expression for the effective sound speed of perturbations is obtained $c_{\text{eff}}^2 = \omega=(n-2)/(n+2)$ with $\omega$ the effective equation of state. We also obtain the first order correction in $k^2/\omega_{\text{eff}}^2$, when the wavenumber $k$ of the perturbations is much smaller than the background oscillation frequency, $\omega_{\text{eff}}$. For the standard massive case we have also analysed general anharmonic contributions to the effective sound speed. These results are reached through a perturbed version of the generalized virial theorem and also studying the exact system both in the super-Hubble limit, deriving the natural ansatz for $\delta\phi$; and for sub-Hubble modes, exploiting Floquet's theorem.

A proof of the Biswas–Mitra–Bhattacharyya conjecture for the ideal quantum gas trapped under the generic power law potential in d dimensions

.The well known relation for the ideal classical gas , which does not remain valid for quantum systems, is revisited. A new connection is established between the energy fluctuation and specific heat for quantum gases, which is valid in both the classical limit and the degenerate quantum regime. Most importantly, the proposed Biswas–Mitra–Bhattacharyya conjecture (Biswas et al 2015 J. Stat. Mech. P03013) relating the hump in energy fluctuation and the discontinuity of specific heat is essentially proved and made precise in this manuscript.

Local Existence of Weak Solutions to Kinetic Models of Granular Media

We prove in any dimension \({d \geqq 1}\) a local in time existence of weak solutions to the Cauchy problem for the kinetic equation of granular media, $$\partial_t f+v\cdot \nabla_x f = {div}_v[f(\nabla W *_v f)]$$ when the initial data are nonnegative, integrable and bounded functions with compact support in velocity, and the interaction potential \({W}\) is a \({C^2({\mathbb{R}}^d)}\) radially symmetric convex function. Our proof is constructive and relies on a splitting argument in position and velocity, where the spatially homogeneous equation is interpreted as the gradient flow of a convex interaction energy with respect to the quadratic Wasserstein distance. Our result generalizes the local existence result obtained by Benedetto et al. (RAIRO Model Math Anal Numer 31(5):615–641, 1997) on the one-dimensional model of this equation for a cubic power-law interaction potential.

Classical and quantum solutions in Brans-Dicke cosmology with a perfect fluid

We consider the application of group invariant transformations in order to constrain a flat isotropic and homogeneous cosmological model, containing of a Brans-Dicke scalar field and a perfect fluid with a constant equation of state parameter $w$, where the latter is not interacting with the scalar field in the gravitational action integral. The requirement that the Wheeler-DeWitt equation be invariant under one-parameter point transformations provides us with two families of power-law potentials for the Brans-Dicke field, in which the powers are functions of the Brans-Dicke parameter $\omega_{BD}$ and the parameter $w$. The existence of the Lie symmetry in the Wheeler-DeWitt equation is equivalent to the existence of a conserved quantity in field equations and with oscillatory terms in the wavefunction of the universe. This enables us to solve the field equations. For a specific value of the conserved quantity, we find a closed-form solution for the Hubble factor, which is equivalent to a cosmological model in general relativity containing two perfect fluids. This provides us with different models for specific values of the parameters $\omega_{BD},$ and $w$. Finally, the results hold for the specific case where the Brans-Dicke parameter $\omega_{BD}$ is zero, that is, for the O'Hanlon massive dilaton theory, and consequently for $f\left( R\right) $ gravity in the metric formalism.

Intermediate accelerated solutions as generic late-time attractors in a modified Jordan-Brans-Dicke theory

In this paper we investigate the evolution of a Jordan-Brans-Dicke scalar field, Φ, with a power-law potential in the presence of a second scalar field, ϕ, with an exponential potential, in both the Jordan and the Einstein frames. We present the relation of our model with the induced gravity model with power-law potential and the integrability of this kind of models is discussed when the quintessence field ϕ is massless, and has a small velocity. The fact that for some fine-tuned values of the parameters we may get some integrable cosmological models, makes our choice of potentials very interesting. We prove that in Jordan-Brans-Dicke theory, the de Sitter solution is not a natural attractor. Instead, we show that the attractor in the Jordan frame corresponds to an ``intermediate accelerated'' solution of the form a(t) ≃ eα1 tp1, as t → ∞ where α1 > 0 and 0 < p1 < 1, for a wide range of parameters. Furthermore, when we work in the Einstein frame we get that the attractor is also an ``intermediate accelerated'' solution of the form \U0001d51e(\U0001d531) ≃ eα2 \U0001d531p2 as \U0001d531 → ∞ where α2 > 0 and 0<p2<1, for the same conditions on the parameter space as in the Jordan frame. In the special case of a quadratic potential in the Jordan frame, or for a constant potential in the Einstein's frame, the above intermediate solutions are of saddle type. These results were proved using the center manifold theorem, which is not based on linear approximation. Finally, we present a specific elaboration of our extension of the induced gravity model in the Jordan frame, which corresponds to a particular choice of a linear potential of Φ. The dynamical system is then reduced to a two dimensional one, and the late-time attractor is linked with the exact solution found for the induced gravity model. In this example the ``intermediate accelerated'' solution does not exist, and the attractor solution has an asymptotic de Sitter-like evolution law for the scale factor. Apart from some fine-tuned examples such as the linear, and quadratic potential U(Φ) in the Jordan frame, it is true that ``intermediate accelerated'' solutions are generic late-time attractors in a modified Jordan-Brans-Dicke theory.

Investigation of Bose Condensation in Ideal Bose Gas Trapped under Generic Power Law Potential in d Dimension

The changes in characteristics of Bose condensation of ideal Bose gas due to an external generic power law potential are studied carefully. Detailed calculation of Kim et al. (J. Phys. Condens. Matter 11 (1999) 10269) yielded the hierarchy of condensation transitions with changing fractional dimensionality. In this manuscript, some theorems regarding specific heat at constant volume CV are presented. Careful examination of these theorems reveal the existence of hidden hierarchy of the condensation transition in trapped systems as well.

Derivation of the Linear Landau Equation and Linear Boltzmann Equation from the Lorentz Model with Magnetic Field

We consider a test particle moving in a random distribution of obstacles in the plane, under the action of a uniform magnetic field, orthogonal to the plane. We show that, in a weak coupling limit, the particle distribution behaves according to the linear Landau equation with a magnetic transport term. Moreover, we show that, in a low density regime, when each obstacle generates an inverse power law potential, the particle distribution behaves according to the linear Boltzmann equation with a magnetic transport term. We provide an explicit control of the error in the kinetic limit by estimating the contributions of the configurations which prevent the Markovianity. We compare these results with those ones obtained for a system of hard disks in \cite{BMHH}, which show instead that the memory effects are not negligible in the Boltzmann-Grad limit.

Prediction of binary nanoparticle superlattices from soft potentials.

Driven by the hypothesis that a sufficiently continuous short-ranged potential is able to account for shell flexibility and phonon modes and therefore provides a more realistic description of nanoparticle interactions than a hard sphere model, we compute the solid phase diagram of particles of different radii interacting with an inverse power law potential. From a pool of 24 candidate lattices, the free energy is optimized with respect to additional internal parameters and the p-exponent, determining the short-range properties of the potential, is varied between p = 12 and p = 6. The phase diagrams contain the phases found in ongoing self-assembly experiments, including DNA programmable self-assembly and nanoparticles with capping ligands assembled by evaporation from an organic solvent. The resulting phase diagrams can be mapped quantitatively to existing experiments as a function of only two parameters: Nanoparticle radius ratio (γ) and softness asymmetry.

Observational constraints on varying neutrino-mass cosmology

We consider generic models of quintessence and we investigate the influence of massive neutrino matter with field-dependent masses on the matter power spectrum. In case of minimally coupled neutrino matter, we examine the effect in tracker models with inverse power-law and double exponential potentials. We present detailed investigations for the scaling field with a steep exponential potential, non-minimally coupled to massive neutrino matter, and we derive constraints on field-dependent neutrino masses from the observational data.

Jacobi Stability Analysis of Scalar Field Models with Minimal Coupling to Gravity in a Cosmological Background

We study the stability of the cosmological scalar field models by using the Jacobi stability analysis, or the Kosambi-Cartan-Chern (KCC) theory. In this approach, we describe the time evolution of the scalar field cosmologies in geometric terms, by performing a “second geometrization” and considering them as paths of a semispray. By introducing a nonlinear connection and a Berwald-type connection associated with the Friedmann and Klein-Gordon equations, five geometrical invariants can be constructed, with the second invariant giving the Jacobi stability of the cosmological model. We obtain all the relevant geometric quantities, and we formulate the condition for Jacobi stability in scalar field cosmologies. We consider the Jacobi stability properties of the scalar fields with exponential and Higgs type potential. The Universe dominated by a scalar field exponential potential is in Jacobi unstable state, while the cosmological evolution in the presence of Higgs fields has alternating stable and unstable phases. We also investigate the stability of the phantom quintessence and tachyonic scalar field models, by lifting the first-order system to the tangent bundle. It turns out that in the presence of a power law potential both of these models are Jacobi unstable during the entire cosmological evolution.

Quantum correlations between each two-level system in a pair of atoms and general coherent fields

The quantitative description of the quantum correlations between each two-level system in a two-atom system and the coherent fields initially defined in a coherent state in the framework of power-law potentials (PLPCSs) is considered. Specifically, we consider two atoms locally interacting with PLPCSs and take into account the different terms of interactions, the entanglement and quantum discord are studied including the time-dependent coupling and photon transition effects. Using the monogamic relation between the entanglement of formation and quantum discord in tripartite systems, we show that the control and preservation of the different kinds of quantum correlations greatly benefit from the combination of the choice of the physical quantities. Finally, we explore the link between the dynamical behavior of quantum correlations and nonclassicality of the fields with and without atomic motion effect.

Self-similar nonequilibrium dynamics of a many-body system with power-law interactions.

The influence of power-law interactions on the dynamics of many-body systems far from equilibrium is much less explored than their effect on static and thermodynamic properties. To gain insight into this problem we introduce and analyze here an out-of-equilibrium deposition process in which the deposition rate of a given particle depends as a power law on the distance to previously deposited particles. This model draws its relevance from recent experimental progress in the domain of cold atomic gases, which are studied in a setting where atoms that are excited to high-lying Rydberg states interact through power-law potentials that translate into power-law excitation rates. The out-of-equilibrium dynamics of this system turns out to be surprisingly rich. It features a self-similar evolution which leads to a characteristic power-law time dependence of observables such as the particle concentration, and results in a scale invariance of the structure factor. Our findings show that in dissipative Rydberg gases out of equilibrium the characteristic distance among excitations-often referred to as the blockade radius-is not a static but rather a dynamic quantity.

Thawing quintessence from the inflationary epoch to today

Using the latest observational data we obtain a lower bound on the initial value of the quintessence field in thawing quintessence models of dark energy. For potentials of the form V(\phi) \phi^{\pm2} we find that the initial value |\phi_i|>7x10^{18}\gev. We then relate \phi_i to the duration of inflation by assuming that the initial value of the quintessence field is determined by quantum fluctuations of the quintessence field during inflation. From the lower bound on $\phi_i$ we obtain a lower bound on the number of e-foldings of inflation, namely, N>2x10^{11}. We obtain similar bounds for other power law potentials for which too we obtain |\phi_{i}|>O(M_{P}.

Galaxy clustering, CMB and supernova data constraints on ϕCDM model with massive neutrinos

We investigate a scalar field dark energy model (i.e., ϕCDM model) with massive neutrinos, where the scalar field possesses an inverse power-law potential, i.e., V(ϕ)∝ϕ−α (α>0). We find that the sum of neutrino masses Σmν has significant impacts on the CMB temperature power spectrum and on the matter power spectrum. In addition, the parameter α also has slight impacts on the spectra. A joint sample, including CMB data from Planck 2013 and WMAP9, galaxy clustering data from WiggleZ and BOSS DR11, and JLA compilation of Type Ia supernova observations, is adopted to confine the parameters. Within the context of the ϕCDM model under consideration, the joint sample determines the cosmological parameters to high precision: the angular size of the sound horizon at recombination, the Thomson scattering optical depth due to reionization, the physical densities of baryons and cold dark matter, and the scalar spectral index are estimated to be θ⁎=(1.0415−0.0011+0.0012)×10−2, τ=0.0914−0.0242+0.0266, Ωbh2=0.0222±0.0005, Ωch2=0.1177±0.0036, and ns=0.9644−0.0119+0.0118, respectively, at 95% confidence level (CL). It turns out that α<4.995 at 95% CL for the ϕCDM model. And yet, the ΛCDM scenario corresponding to α=0 is not ruled out at 95% CL. Moreover, we get Σmν<0.262 eV at 95% CL for the ϕCDM model, while the corresponding one for the ΛCDM model is Σmν<0.293 eV. The allowed scale of Σmν in the ϕCDM model is a bit smaller than that in the ΛCDM model. It is consistent with the qualitative analysis, which reveals that the increases of α and Σmν both can result in the suppression of the matter power spectrum. As a consequence, when α is larger, in order to avoid suppressing the matter power spectrum too much, the value of Σmν should be smaller.

The Vlasov–Maxwell–Boltzmann system with a uniform ionic background near Maxwellians

The two-species Vlasov–Maxwell–Boltzmann system is an important model for plasma physics describing the time evolution of dilute charged particles consisting of electrons and ions under the influence of the self-consistent internally generated Lorentz forces. In physical situations the ion mass is usually much larger than the electron mass so that the electrons move much faster than the ions. Thus, the ions are often described by a fixed ion background and only the electrons move. For such a simple case, the two-species Vlasov–Maxwell–Boltzmann system is reduced to the one-species Vlasov–Maxwell–Boltzmann system.Although the one-species Vlasov–Maxwell–Boltzmann system is a simplified model of the two-species Vlasov–Maxwell–Boltzmann system, its global well-posedness theory near a given global Maxwellian in the perturbative framework is more difficult than the two-species case, which is partially due to the slow-decay of the electromagnetic field and up to now, the problem on the construction of global in time solutions near a given global Maxwellian in the perturbative framework for the Cauchy problem of the one-species Vlasov–Maxwell–Boltzmann system with cutoff non-hard sphere intermolecular collisions remains unsolved. It is shown in this paper that the Cauchy problem of the one-species Vlasov–Maxwell–Boltzmann system with cutoff non-hard sphere intermolecular collisions including the cutoff inverse power law potentials is globally well-posed provided that the perturbative initial data satisfies certain regularity, smallness, and integrability conditions. Our analysis is based on a new time-velocity weighted energy method with two key technical parts: one is to introduce the exponentially weighted estimates into the cutoff Boltzmann operator and the other is to design a delicate temporal energy X(t)-norm to obtain its uniform bound. As a by-product of our analysis, we can also deduce certain temporal decay estimates on the global solutions constructed above for the same range of intermolecular collisions. Notice that even for the hard sphere model, although the global solutions have been successfully constructed in [8], its large time behaviors are unknown, cf. [8].

Scalar field deformations ofΛCDMcosmology

This paper treats nonrelativistic matter and a scalar field $\phi$ with a monotonically decreasing potential minimally coupled to gravity in flat Friedmann-Lema\^{i}tre-Robertson-Walker cosmology. The field equations are reformulated as a three-dimensional dynamical system on an extended compact state space, complemented with cosmographic diagrams. A dynamical systems analysis provides global dynamical results describing possible asymptotic behavior. It is shown that one should impose \emph{global and asymptotic} bounds on $\lambda=-V^{-1}\,dV/d\phi$ to obtain viable cosmological models that continuously deform $\Lambda$CDM cosmology. In particular we introduce a regularized inverse power-law potential as a simple specific example.

Inflation with non-minimally derivative coupling

We derive the second order correction to the scalar and tensor spectral tilts for the inflationary models with non-minimally derivative coupling. In the high friction limit, the quartic power law potential is consistent with the observational constraint at 95% CL because the amplitude of the primordial gravitational waves is smaller, and the inflaton excursion is sub-Planckian.