Exact Self-consistent Solutions Research Articles

Stellar equilibrium on a physical vacuum soil

We show that the repulsive effects associated to the zero-point energies of quantum fields are capable of supporting ultracompact stars that overcome the compactness limits present in general relativity for any object in hydrostatic equilibrium. These objects are exact self-consistent solutions in semiclassical gravity that incorporate the backreaction of the renormalized stress-energy tensor (RSET) of quantum fields in vacuum. We arrive at stars of striking qualitative agreement through two independent modelings of the RSET, evidencing the generality and robustness of this result. The main physical properties of these novel black hole mimickers are reviewed.

Coherent oscillations in balanced neural networks driven by endogenous fluctuations.

We present a detailed analysis of the dynamical regimes observed in a balanced network of identical quadratic integrate-and-fire neurons with sparse connectivity for homogeneous and heterogeneous in-degree distributions. Depending on the parameter values, either an asynchronous regime or periodic oscillations spontaneously emerge. Numerical simulations are compared with a mean-field model based on a self-consistent Fokker-Planck equation (FPE). The FPE reproduces quite well the asynchronous dynamics in the homogeneous case by either assuming a Poissonian or renewal distribution for the incoming spike trains. An exact self-consistent solution for the mean firing rate obtained in the limit of infinite in-degree allows identifying balanced regimes that can be either mean- or fluctuation-driven. A low-dimensional reduction of the FPE in terms of circular cumulants is also considered. Two cumulants suffice to reproduce the transition scenario observed in the network. The emergence of periodic collective oscillations is well captured both in the homogeneous and heterogeneous setups by the mean-field models upon tuning either the connectivity or the input DC current. In the heterogeneous situation, we analyze also the role of structural heterogeneity.

Casimir force for the [formula omitted] model

In this work, we derive exact self-consistent solutions to the gap equations of the CPN−1 model on a finite interval with Dirichlet boundary conditions in the large-N approximation. The solution reproduces the confining phase in the infinite system by taking the appropriate limit. We compute the vacuum energy and the Casimir force and observe that the sign of the force is always attractive.

Novel Method for Evaluating Shale-Gas and Shale-Tight-Oil Reservoirs Using Advanced Well-Log Data

Summary We introduce a novel well-logging method for determining more-accurate total porosities, fluid volumes, and kerogen volumes in shale-gas and shale-tight-oil wells. Improved accuracy is achieved by self-consistently accounting for the effects of light hydrocarbons and kerogen on the log responses. The logging measurements needed to practice this method are bulk densities, nuclear-magnetic-resonance (NMR) total porosities, and total-organic-carbon (TOC) weight fractions. The TOC weight fractions and the matrix densities, which are used to interpret the bulk density measurements, are both derived from geochemical-tool measurements. Most unconventional shale-gas and shale-tight-oil reservoirs contain some nonproducible immobile hydrocarbons. When immobile hydrocarbons are present, our method requires prior knowledge of in-situ total water volumes. The water volumes can be estimated from dielectric-tool measurements. In special cases (e.g., in some mature shale-gas reservoirs) where no immobile hydrocarbons are present, a dielectric tool is not needed. In such cases total water volumes are outputs of the method. We discuss the response functions in shale reservoirs for measurements of bulk densities, NMR porosities, and TOC weight fractions and derive exact self-consistent solutions to the response equations. The algebraic solutions are used to compute shale total porosities, fluid volumes, and kerogen volumes. The predicted shale total porosities and fluid volumes are corrected for light-hydrocarbon effects on the measured bulk densities and NMR porosities and for kerogen effects on the bulk densities. It is shown that significant errors can be made in log-derived shale total porosities if NMR porosities or density-log porosities are assumed to represent true-shale porosities without applying proper corrections. We discuss the application of the method to the analysis of logging data acquired in a mature shale-gas well drilled in the Marcellus Shale in the northeastern United States and to data acquired in a shale-tight-oil well drilled in the Permian Basin in west Texas. A multifrequency dielectric tool is used to determine in-situ total water volumes in the tight oil well. The mature shale-gas reservoir does not contain immobile hydrocarbons, and, therefore, dielectric-logging measurements were not needed in this well. The results in both wells are shown to compare favorably with core data.

The relativistic electro-vortical field—revisiting magneto-genesis and allied problems

Following the idea of MagnetoFluid unification [S. M. Mahajan, Phys. Rev. Lett. 90, 035001 (2003)], a very general Electro-Vortical (EV) field is constructed to describe the dynamics of a perfect relativistic fluid. Structurally similar to the electromagnetic field Fμν, the Electro-Vortical field Mνμ unifies the macroscopic forces into a single grand force that is the weighted sum of the electromagnetic and the inertial/thermal forces. The new effective force may be viewed either as a vortico-thermal generalization of the electromagnetic force or as the electromagnetic generalization of the vortico-thermal forces that a fluid element experiences in course of its evolution. Two fundamental consequences follow from this grand unification: (1) emergences of a new helicity that is conserved for arbitrary thermodynamics and (2) the entire dynamics is formally expressible as an MHD (magnetohydrodynamics) like ideal Ohm's law in which the “electric” and “magnetic” components of the EV field replace the standard electric and magnetic fields. In the light of these more and more encompassing conserved helicities, the “scope and significance” of the classical problem of magneto-genesis (need for a seed field to get a dynamo started) is reexamined. It is shown that in models more advanced than MHD, looking for exotic seed-generation mechanisms (like the baroclinic thermodynamics) should not constitute a fundamental pursuit; the totally ideal dynamics is perfectly capable of generating and sustaining magnetic fields entirely within its own devices. For a specified thermodynamics, a variety of exact and semi exact self-consistent analytical solutions for equilibrium magnetic and flow fields are derived for a single species charged fluid. The scale lengths of the fields are determined by two natural scale lengths: the skin depth and the gradient length of the thermodynamic quantities. Generally, the skin depth, being the shorter (even much shorter) than the gradient length, will characterize the kinetic-magnetic reservoir of short scale energy that will drive the dynamo as well as reverse dynamo action—the creation of large scale magnetic and flow fields.

Effective coefficient of thermal expansion of n-layered composite sphere model: Exact solution and its finite element validation

Using the method of homogeneous solutions, we derive exact self-consistent solution for effective thermal expansion coefficient of n-layered composite sphere model. Assuming representative thermoelastic phase properties and using unstructured morphology-adaptive tetrahedron meshes, we conduct numerical finite element predictions of effective thermal expansion coefficient of periodic random computer microstructures with 27 non-overlapping spherical core–shell alumina, silica and hybrid latex-silica particles dispersed in aluminum and polymer matrices. We compare finite element estimates with corresponding predictions of n-layered composite sphere model and demonstrated that the derived solution is indeed very accurate.

Self-Consistent Multiple Complex-Kink Solutions in Bogoliubov–de Gennes and Chiral Gross-Neveu Systems

We exhaust all exact self-consistent solutions of complex-valued fermionic condensates in the (1+1)-dimensional Bogoliubov-deGennes and chiral Gross-Neveu systems under uniform boundary conditions. We obtain n complex (twisted) kinks, or gray solitons, with 2n parameters corresponding to their positions and phase shifts. Each soliton can be placed at an arbitrary position while the self-consistency requires its phase shift to be quantized by π/N for N flavors.

Single fermion Green's function in the quantum ordered Fermi-system: Analytic solution

An exact self-consistent solution for a finite temperature quantum-ordered state of correlated electron system found previously (Mukhin, 2009, 2011) is used to derive the fermionic single-particle Green's function. The quantum order parameter (QOP) found in the form of a periodic (elliptic Jacoby) function of the Matsubara's imaginary time (Mukhin, 2009), plays the role of effective scattering potential seen by electrons. The analytic solution for the Green's function demonstrates the following new features: (1) the pseudo-gap behavior of the single-electron density of states (DOS) near the (shifted) Fermi-level;(2) the side-bands of decreasing intensity away from the Fermi-level; (3) scaling of the quasi-particle energies with the QOP amplitude; (4) fermionic quasi-particles in the QOP state are combined from two confined “odd” and “even” fermions that separately would be unstable. The false-color plot of single-fermion DOS in the limit of a periodic kink-like Matsubara time-dependence of QOP is presented and could be used as prediction for the ARPES experiments. The plot of the DOS transfer between different energies at the “fermi-surface” momentum for a given kink-like QOP is also presented. Some possibly observable consequences of the found finger-prints are discussed.

Two-fluid tokamak equilibria with reversed magnetic shear and sheared flow

AbstractThe aim of the present work is to investigate tokamak equilibria with reversed magnetic shear and sheared flow, which may play a role in the formation of internal transport barriers (ITBs), within the framework of the two-fluid model in cylindrical geometry. The study is based on exact self-consistent solutions by means of which the impact of the magnetic shear, s, and the ‘toroidal’ (axial) and ‘poloidal’ (azimuthal) ion velocity components, viz and viθ, on the radial electric field, Er, its shear, |dEr/dr|, and the shear of the E×B velocity, ωE×B≡|d/dr(E× B/B2)|, is examined. For a wide parametric regime of experimental concern it turns out that the contributions of the viz, viθ and pressure gradient (∇ Pi) terms to Er, |Er′| and ωE×B are of the same order of magnitude. The contribution of the ∇ Pi term is missing in the framework of magnetohydrodynamics (MHD) (G. Poulipoulis et al. 2004 Plasma Phys. Control. Fusion46, 639). The impact of s on ωE×B through the ∇ Pi term is stronger than that through the velocity terms; in particular for constant Bz the ion pressure gradient contribution to ωE×B at the point where dEr/dr=0 scales as (1−s)(2−s), whereas the ion flow contributions to ωE×B scale as (1−s). The results indicate that, like MHD, the magnetic shear and the sheared toroidal and poloidal velocities act synergetically in producing electric fields and therefore ωE×B profiles compatible with the ones observed in discharges with ITBs; owing to the ∇ Pi term, however, the impact of s on Er, |Er′| and ωE×B is stronger than that in MHD.

A nonstationary probe that is self-consistently negatively charged and embedded in plasma

The dynamics of ions in a plasma layer adjacent to the conducting surface of an electrically insulated negatively charged probe is studied. Exact self-consistent analytical solutions to the nonlinear system of equations of ion hydrodynamics are obtained. The solutions satisfy nonstationary boundary conditions on the probe’s surface. The intensity of the electric field and the velocity, concentration, and density of the ion current are found as functions of the initial position of a particle and the distance to the probe’s surface. The rate of change in the probe’s charge is determined and it is shown that, with time, the parameters of the system considered approach their asymptotical values. For a spherical probe, the time dependence of the ion-current density, i.e., the probe’s transient response (which is an analogue of the current—voltage characteristic of the Langmuir probe with a constant negative bias), is obtained.

Nonlinear spinor field in Bianchi type-I cosmology: Inflation, isotropization, and late time acceleration

A self-consistent system of interacting nonlinear spinor and scalar fields within the scope of a Bianchi type-I cosmological model filled with perfect fluid is considered. Exact self-consistent solutions to the corresponding field equations are obtained. A role of the spinor field in the evolution of the Universe is studied. It is shown that the spinor field gives rise to an accelerated mode of expansion of the Universe. Early in the evolution, the spinor field nonlinearity generates an accelerated mode of expansion. The rapid growth of the Universe in this case results in the earlier isotropization. At the later evolution the acceleration of the Universe is provided with a spinor mass.

Magnetohydrodynamic self-consistent exact helical solutions

We consider the idealized case of a one-component plasma with aligned fluid velocity and current density. Constant density and pressure as well as zero external magnetic field are also assumed. We show that suitably determined axially symmetric helical current densities within a straight infinite cylinder are exact self-consistent solutions of magnetohydrodynamics. Self-consistent here means that the magnetic field is the field produced by the current density itself. The equation of motion gives a nonlinear differential equation that relates the axial vz(ρ) and the azimuthal v(ρ) velocities as functions of radial distance ρ. Prescribing one of these gives a specific solution for the other. The solutions can be understood as a set of helix-shaped charged particle trajectories that spiral self-consistently through the magnetic field that they themselves give rise to. Four different specific exact solutions are given: (i) a single particle outside a rectilinear line current, (ii) current on a thin cylinder, (iii) current density with constant angular velocity and (iv) current density with constant axial velocity, both within a cylinder of finite radius.

Nonlinear spinor field in cosmology

Within the scope of Bianchi type VI (BVI) model the self-consistent system of nonlinear spinor and gravitational fields is considered. Exact self-consistent solutions to the spinor and gravitational field equations are obtained for some special choice of spatial inhomogeneity and nonlinear spinor term. The role of inhomogeneity in the evolution of spinor and gravitational field is studied. Oscillatory mode of expansion of the BVI universe is obtained for some special choice of spinor field nonlinearity.

New self-consistent approach for inelastic materials based on translated fields

In real heterogeneous materials, the simultaneous presence of instantaneous mechanisms (elasticity) and time dependent ones (non linear visco-plasticity) leads to a complex space-time coupling between the mechanical fields, difficult to represent in a simple and efficient way. In this work, a new micro-macro model is proposed, starting from the integral equation and translated strain rate fields. The chosen translated field is the (compatible) viscoplastic strain rate of the (fictitious) viscoplastic heterogeneous medium submitted to a uniform (unknown) boundary condition. The self-consistency condition allows to define these boundary conditions so that a relative simple and compact strain rate concentration equation is obtained. The model is first applied to the case of a two-phase composite with isotropic, linear and incompressible visco-elastic properties. In that case, an exact self-consistent solution using the Laplace-Carson transform is available. The agreement between both approaches appears quite good. Results for elastic-viscoplastic FCC polycrystals are also presented and compared with those obtained from other models.

A simple quantum model for the MOS structure in accumulation mode

A simple model for the accumulation mode of a MOS structure is proposed. The model regards the ground quantum level and is based on the approximated empirical expression for the band profile in semiconductor. The results for the dependency of a surface potential on the electric field in oxide were screened with the exact self-consistent solution and shown to be in a satisfactory agreement with it.

A new class of micro–macro models for elastic–viscoplastic heterogeneous materials

Abstract The determination of the effective behavior of heterogeneous materials from the properties of the components and the microstructure constitutes a major task in the design of new materials and the modeling of their mechanical behavior. In real heterogeneous materials, the simultaneous presence of instantaneous mechanisms (elasticity) and time dependent ones (non-linear viscoplasticity) leads to a complex space–time coupling between the mechanical fields, difficult to represent in a simple and efficient way. In this work, a new self-consistent model is proposed, starting from the integral equation for a translated strain rate field. The chosen translated field is the (compatible) viscoplastic strain rate of the (fictitious) viscoplastic heterogeneous medium submitted to a uniform (unknown) boundary condition. The self-consistency condition allows to define these boundary conditions so that a relative simple and compact strain rate concentration equation is obtained. This equation is explained in terms of interactions between an inclusion and a matrix, which lead to interesting conclusions. The model is first applied to the case of two-phase composites with isotropic, linear and incompressible viscoelastic properties. In that case, an exact self-consistent solution using the Laplace–Carson transform is available. The agreement between both approaches appears quite good. Results for elastic–viscoplastic BCC polycrystals are also presented and compared with results obtained from Kroner–Weng's and Paquin et al. (Arch. Appl. Mech. 69 (1999) 14)'s model.

Quantum back reaction of massive fields and self-consistent semiclassical extreme black holes and acceleration horizons

We consider the effect of backreaction of quantized massive fields on the metric of extreme black holes (EBH). We find the analytical approximate expression for the stress-energy tensor for a scalar (with an arbitrary coupling), spinor and vector fields near an event horizon. We show that, independent of a concrete type of EBH, the energy measured by a freely falling observer is finite on the horizon, so that quantum backreaction is consistent with the existence of EBH. For the Reissner-Nordstrom EBH with a total mass M_{tot} and charge Q we show that for all cases of physical interest M_{tot}< Q. We also discuss different types of quantum-corrected Bertotti-Robinson spacetimes, find for them exact self-consistent solutions and consider situations in which tiny quantum corrections lead to the qualitative change of the classical geometry and topology. In all cases one should start not from a classical background with further adding quantum corrections but from the quantum-corrected self-consistent geometries from the very beginning.

Midgap edge states and pairing symmetry of quasi-one-dimensional organic superconductors

The singlet s-, d- and triplet p-wave pairing symmetries in quasi-one-dimensional organic superconductors can be experimentally discriminated by probing the Andreev bound states at the sample edges. These states have the energy in the middle of the superconducting gap and manifest themselves as a zero-bias peak in tunneling conductance into the corresponding edge. Their existence is related to the sign change of the pairing potential around the Fermi surface. We present an exact self-consistent solution of the edge problem showing the presence of the midgap states for p_x-wave superconductivity. The spins of the edge state respond paramagnetically to a magnetic field parallel to the vector d that characterizes triplet pairing.

Exact self-consistent plane-symmetric solutions of the spinor-field equation with a nonlinear term that depends on theS 2−P 2 invariant

Exact plane-symmetric solutions of the spinor-field equation with zero mass parameter and nonlinear term that depends arbitrarily on the S2−P2 invariant are derived with consideration of an intrinsic gravitational field. The existence of regular solutions with localized energy density among the solutions obtained is investigated. Equations with powerlaw and polynomial nonlinearity types are examined in detail. For the power-law nonlinearity, when the nonlinear term entering into the Lagrangian has the form LN=γIn, where γ is the nonlinearity parameter and n=const, it is shown that the initial system of Einstein and spinor-field equations has regular solutions with localized energy density only under the conditions λ=−Λ2 1. In this case, the examined field configuration posesses a negative energy. In the case of polynomial nonlinearity, regular solutions with localized energy density T 0 0 (x), positive energy $$E_f = \int\limits_{ - \infty }^{ + \infty } {T_0^0 } \sqrt { - ^3 g} dx$$ (upon integration over y and z between finite limits), and an everywhere regular metric that transforms into a two-dimensional space-time metric at spatial infinity are obtained. It is shown that the initial nonlinear spinor-field equations in two-dimensional space-time have no solutions with localized energy density. Thus, it is established that the intrinsic gravitational field plays a regularizing role in the frmation of regular localized solutions to the examined nonlinear spinor-field equations.

Exact self-consistent plane-symmetric solutions of the equations of two interacting fields (spinor field and scalar field)

A spinor field interacting with a zero-mass neutral scalar field is considered for the case of the simplest type of direct interaction, where the interaction Lagrangian has the formLint=1/2 ϕαϕ,αF(S) whereF(S) is an arbitrary function of the spinor field invariantS=ψψ. Exact solutions of the corresponding systems of equations that take into account the natural gravitational field in a plane-symmetric metric are obtained. It is proved that the initial system of equations has regular localized soliton-type solutions only if the energy density of the zero-mass scalar field is negative as it “disengages” from interaction with the spinor field. In two-dimensional space-time the system of field equations we are studying describes the configuration of fields with constant energy densityT00, i.e., no soliton-like solutions exist in this case.