Natural Initial Conditions Research Articles

Unsaturated deformable porous media flow with thermal phase transition

In this paper, a continuum model is introduced for fluid flow in a deformable porous medium, where the fluid may undergo phase transitions. Typically, such problems arise in modeling liquid–solid phase transformations in groundwater flows. The system of equations is derived here from the conservation principles for mass, momentum, and energy and from the Clausius–Duhem inequality for entropy. It couples the evolution of the displacement in the matrix material, of the capillary pressure, of the absolute temperature, and of the phase fraction. Mathematical results are proved under the additional hypothesis that inertia effects and shear stresses can be neglected. For the resulting highly nonlinear system of two PDEs, one ODE and one ordinary differential inclusion with natural initial and boundary conditions, existence of global in time solutions are proved by means of cut-off techniques and suitable Möser-type estimates.

Hyperuniformity of initial conditions and critical decay of a diffusive epidemic process belonging to the Manna class.

For a fixed-energy Manna sandpile model belonging to a Manna class in one dimension (d=1), we recently showed that the critical decay is different for random and regular initial conditions (ICs). Compared with previous results of natural IC for several models, we suggested for the Manna class that the critical decay depends on the characteristics of the three ICs. But the dependence on the random and regular ICs was shown only for a single model. In this work, we study the critical decay for the random and regular ICs for another model of the Manna class in d=1, a diffusive epidemic process. It is shown that the critical decay exponent agrees with the previous result for each IC, which verifies that IC dependence is a common feature of the Manna class. In addition, for the random and regular ICs, we measure the variance σ^{2}(r) of total particle density in a region of size r by increasing r up to system size and investigate its temporal evolution toward the value σ_{q}^{2}(r) of the quasisteady state at criticality. In d=1,σ^{2}(r) scales as σ^{2}(r)∼r^{-ψ} with ψ=1 for random distributions and 1<ψ≤2 for hyperuniform ones. The temporal evolution shows that σ^{2}(r) of the two ICs differently relax toward σ_{q}^{2}(r) and the regular IC becomes a hyperuniform distribution of ψ=2 in the beginning of the evolution. We estimate ψ=1.45(3) for both the quasisteady state and absorbing states, so the quasisteady state is also as hyperuniform as absorbing states. The hyperuniformity of the quasisteady state shows that the natural IC also should be hyperuniform as much as the quasisteady state, because the natural IC is obtained from particle configurations close to the quasisteady state. Consequently, the different ψ of the three ICs suggest that σ^{2}(r) can classify the characteristics of the three ICs in a unified way and the different degree of hyperuniformity of the ICs provides another explanation for the observed IC-dependent critical decay in a point of view of initial fluctuations and correlations.

Analytical and numerical study of travelling waves using the Maxwell-Cattaneo relaxation model extended to reaction-advection-diffusion systems.

Within the framework of the Maxwell-Cattaneo relaxation model extended to reaction-diffusion systems with nonlinear advection, travelling wave (TW) solutions are analytically investigated by studying a normalized reaction-telegraph equation in the case of the reaction and advection terms described by quadratic functions. The problem involves two governing parameters: (i) a ratio φ^{2} of the relaxation time in the Maxwell-Cattaneo model to the characteristic time scale of the reaction term, and (ii) the normalized magnitude N of the advection term. By linearizing the equation at the leading edge of the TW, (i) necessary conditions for the existence of TW solutions that are smooth in the entire interval of -∞<ζ<∞ are obtained, (ii) the smooth TW speed is shown to be less than the maximal speed φ^{-1} of the propagation of a substance, (iii) the lowest TW speed as a function of φ and N is determined. If the necessary condition of N>φ-φ^{-1} does not hold, e.g., if the magnitude N of the nonlinear advection is insufficiently high in the case of φ^{2}>1, then, the studied equation admits piecewise smooth TW solutions with sharp leading fronts that propagate at the maximal speed φ^{-1}, with the substance concentration or its spatial derivative jumping at the front. An increase in N can make the solution smooth in the entire spatial domain. Moreover, an explicit TW solution to the considered equation is found provided that N>φ. Subsequently, by invoking a principle of the maximal decay rate of TW solution at its leading edge, relevant TW solutions are selected in a domain of (φ,N) that admits the smooth TWs. Application of this principle to the studied problem yields transition from pulled (propagation speed is controlled by the TW leading edge) to pushed (propagation speed is controlled by the entire TW structure) TW solutions at N=N_{cr}=sqrt[1+φ^{2}], with the pulled (pushed) TW being relevant at smaller (larger) N. An increase in the normalized relaxation time φ^{2} results in increasing N_{cr}, thus promoting the pulled TW solutions. The domains of (φ,N) that admit either the smooth or piecewise smooth TWs are not overlapped and, therefore, the selection problem does not arise for these two types of solutions. All the aforementioned results and, in particular, the maximal-decay-rate principle or appearance of the piecewise smooth TW solutions, are validated by numerically solving the initial boundary value problem for the reaction-telegraph equation with natural initial conditions localized to a bounded spatial region.

Uniqueness of the null solution to a nonlinear partial differential equation satisfied by the explosion probability of a branching diffusion

AbstractThe explosion probability before time t of a branching diffusion satisfies a nonlinear parabolic partial differential equation. This equation, along with the natural boundary and initial conditions, has only the trivial solution, i.e. explosion in finite time does not occur, provided the creation rate does not grow faster than the square power at ∞.

Inflation with the Starobinsky potential in loop quantum cosmology

A self-consistent pre-inflationary extension of the inflationary scenario with the Starobinsky potential, favored by Planck data, is studied using techniques from loop quantum cosmology (LQC). The results are compared with the quadratic potential previously studied. Planck scale completion of the inflationary paradigm and observable signatures of LQC are found to be robust under the change of the inflaton potential. The entire evolution, from the quantum bounce all the way to the end of inflation, is compatible with observations. Occurrence of desired slow-roll phase is almost inevitable and natural initial conditions exist for both the background and perturbations for which the resulting power spectrum agrees with recent observations. There exist initial data for which the quantum gravitational corrections to the power spectrum are potentially observable.

Critical behavior of a fixed-energy Manna sandpile model for regular initial conditions in one dimension.

For a fixed-energy (FE) Manna sandpile model in one dimension, we investigate the critical behavior for regular initial conditions in which activities are distributed at regular intervals on average. The FE Manna model conserves the density ρ of total particles and undergoes an absorbing phase transition at a critical ρ(c). For the regular initial conditions, we show via extensive simulations that the dynamical scaling behaviors differ from those of the random and the natural initial conditions. Off-critical scaling exponents β and ν(⊥) are also measured and shown to agree well with the values of the directed percolation (DP) class as reported by Basu et al. [Phys. Rev. Lett. 109, 015702 (2012)]. Our results suggest that the dynamical scaling behaviors depend on the characteristics of initial conditions, but the off-critical scaling behaviors in the steady state are independent of initial conditions and belong to the DP class.

Cosmological attractors and initial conditions for inflation

Inflationary $\alpha$-attractor models in supergravity, which provide excellent fits to the latest observational data, are based on the Poincare disk hyperbolic geometry. We refine these models by constructing Kahler potentials with built-in inflaton shift symmetry and by making a canonical choice of the goldstino Kahler potential. The refined models are stable with respect to all scalar fields at all $\alpha$, no additional stabilization terms are required. The scalar potential V has a nearly Minkowski minimum at small values of the inflaton field $\varphi$, and an infinitely long dS valley of constant depth and width at large $\varphi$. Because of the infinite length of this shift-symmetric valley, the initial value of the inflaton field at the Planck density is expected to be extremely large. We show that the inflaton field $\varphi$ does not change much until all fields lose their energy and fall to the bottom of the dS valley at large $\varphi$. This provides natural initial conditions for inflation driven by the inflaton field slowly rolling along the dS valley towards the minimum of the potential at small $\varphi$. A detailed description of this process is given for $\alpha$-attractors in supergravity, but we believe that our general conclusions concerning naturalness of initial conditions for inflation are valid for a broad class of inflationary models with sufficiently flat potentials.

Classical and quantum initial conditions for Higgs inflation

We investigate whether Higgs inflation can occur in the Standard Model starting from natural initial conditions or not. The Higgs has a non-minimal coupling to the Ricci scalar. We confine our attention to the regime where quantum Einstein gravity effects are small in order to have results that are independent of the ultraviolet completion of gravity. At the classical level we find no tuning is required to have successful Higgs inflation, provided the initial homogeneity condition is satisfied. On the other hand, at the quantum level we obtain that the renormalization for large non-minimal coupling requires an additional degree of freedom, unless a tuning of the initial values of the running parameters is made. In order to see that this effect may change the predictions we finally include such degree of freedom in the field content and show that Starobinsky's R2 inflation dominates over Higgs inflation.

Approximate solutions to the initial value problem for some compressible flows

For the natural initial conditions L 1 in the density field (more generally a positive bounded Radon measure) and L ∞ in the velocity field, we obtain global approximate solutions to the Cauchy problem for the 3-D systems of isothermal and isentropic gases and the 2-D shallow water equations. We obtain a sequence of functions which are differentiable in time and continuous in space and tend to satisfy the equations in the sense of distributions in the space variables and in the strong sense in the time variable. The method of construction relies on the study of a specific family of two ODEs in a classical Banach space (one for the continuity equation and one for the Euler equation). Standard convergent numerical methods for the solution of these ODEs can be used to provide concrete approximate solutions. It has been checked in numerous cases in which the solutions of systems of fluid dynamics are known that our construction always gives back the known solutions. It is also proved that it gives the classical analytic solutions in the domain of application of the Cauchy–Kovalevskaya theorem.

On the transient response of viscoelastic beams and plates on viscoelastic medium

In the present study, free vibration of isotropic viscoelastic beams and plates on viscoelastic medium is investigated. The Boltzmann superposition integral model with Dynamic Mechanical Analysis (DMA) results is considered for viscoelastic material of the beam and plate to describe more accurately the behavior of material rather than simple complex constants model. The viscoelastic beam and plate on viscoelastic medium modeling are used in railroad vibration with the damping layer on the lateral surface of the rail and the polymeric flooring on the floor as a viscoelastic medium for the beam and the plate, respectively. Weighted residual method and QZ algorithm are applied to obtain beam and plate natural frequency. In addition, medium properties׳ effects on the natural frequencies of the beam and plate are discussed. For the first time, critical damping of viscoelastic beam is presented due to the greater effect of medium viscosity at all boundary conditions. The unknown coefficients of dynamic response function are calculated by Fourier transform. Closed form expressions are obtained for lateral displacement of the beam and plate with various boundary conditions in terms of complex natural frequency, mode numbers, geometry parameters, initial conditions, material and medium properties. Results show excellent agreement with relative studies.

A uniqueness theorem for Frobenius manifolds and Gromov–Witten theory for orbifold projective lines

Abstract We prove that the Frobenius structure constructed from the Gromov–Witten theory for an orbifold projective line with at most three orbifold points is uniquely determined by the Witten–Dijkgraaf–Verlinde–Verlinde equations with certain natural initial conditions.

Thick brane isotropization in a generalized 5D anisotropic standing wave braneworld model

We study a smooth cosmological solution within a generalized 5D standing wave braneworld modeled by gravity and a phantom scalar field. In this model the 3-brane is anisotropically warped along its spatial dimensions and contains a novel time-dependent scale factor that multiplies the anisotropic spatial interval of the 5D metric, a fact that allows us to study cosmological effects. By explicitly solving the bulk field equations we found a natural mechanism which isotropizes the braneworld for a wide class of natural initial conditions. We are able to give a physical interpretation of the anisotropic dissipation: as the anisotropic energy of the 3-brane rapidly leaks into the bulk through the nontrivial components of the nonlocal Weyl tensor projected to the brane, the bulk becomes less isotropic. At the same time, under the action of the 4D cosmological constant, the anisotropic braneworld super-exponentially isotropizes by itself, rendering a 3-brane with de Sitter symmetry embedded in a 5D de Sitter space-time, while the phantom scalar field exponentially vanishes.

On boundary conditions for the gravity wave equations

In this study, the reasons why mathematically well posed problems for gravity wave equations with quite natural initial and boundary conditions can produce physically meaningless solutions are examined. The mechanism of generating such solutions is analyzed and general conditions on initial and boundary functions are found under which the solutions have at least linear growth with respect to time. Different examples of smooth bounded input functions are given, which lead to unbounded growth of the respective solutions. The same problem can rise in numerical models for one- and two-dimensional gravity wave and shallow water equations, but origin of the problem is hard to be found without analysis of the primitive differential problems. Based on the performed analysis and numerical experiments, some recommendations for choosing the boundary conditions are given to avoid this unphysical behavior of differential and numerical solutions.

Walks with up to two monomers per site on the Bethe lattice: Interpolation between models with immediate reversals allowed and immediate reversals forbidden

We generalize earlier calculations of the RA (immediate reversals allowed) and RF (immediate reversals forbidden) model with multiple monomers per site, including a parameter in the model, so that both cases studied before are particular cases of the model considered here. Also, we calculate the bulk free energy of the model using Gujrati's prescription, which leads to corrections in the localization of coexistence lines and qualitative changes in the phase diagram of the model. Thus, this calculation provides an example in which the use of the method of iterating recursion relations starting with natural initial conditions to locate coexistence loci is not trustworthy and may lead to qualitatively different results. A continuous collapse transition appears in all cases as a tricritical point.

Old dark energy

Dark energy dynamics in the recent universe is influenced by its evolution through the long, matter dominated expansion history. A particular dynamical property, the flow variable, remains constant in several classes of scalar field models as long as matter dominates; the dark energy is only free to diverge in behavior at recent times. This gives natural initial conditions for Monte Carlo studies of dark energy dynamics. We propose a parametrization for the later evolution that covers a wide range of possible behaviors, is tractable in making predictions, and can be constrained by observations. We compare the approach to directly parametrizing the potential, which does not take into account the maturity of the dark energy dynamics.

THE EFFECTIVE THEORY OF INFLATION IN THE STANDARD MODEL OF THE UNIVERSE AND THE CMB+LSS DATA ANALYSIS

Inflation is today a part of the Standard Model of the Universe supported by the cosmic microwave background (CMB) and large scale structure (LSS) datasets. Inflation solves the horizon and flatness problems and naturally generates density fluctuations that seed LSS and CMB anisotropies, and tensor perturbations (primordial gravitational waves). Inflation theory is based on a scalar field φ (the inflaton) whose potential is fairly flat, leading to a slow-roll evolution. This review focuses on the following new aspects of inflation. We present the effective theory of inflation à la Ginsburg and Landau, in which the inflaton potential is a polynomial in the field φ and has the universal form [Formula: see text], where [Formula: see text], M ≪ M Pl is the scale of inflation and N ~ 60 is the number of e-folds since the cosmologically relevant modes exit the horizon till inflation ends. The slow-roll expansion becomes a systematic 1/N expansion and the inflaton couplings become naturally small as powers of the ratio (M/M Pl )2. The spectral index and the ratio of tensor/scalar fluctuations are [Formula: see text], [Formula: see text], while the running index turns out to be [Formula: see text] and therefore can be neglected. The energy scale of inflation M ~ 0.7 × 1016 GeV is completely determined by the amplitude of the scalar adiabatic fluctuations. A complete analytic study plus the Monte Carlo Markov chain (MCMC) analysis of the available CMB+LSS data (including WMAP5) with fourth degree trinomial potentials showed: (a) the spontaneous breaking of the φ → - φ symmetry of the inflaton potential; (b) a lower bound for r in new inflation: r &gt; 0.023 (95% CL) and r &gt; 0.046 (68 CL); (c) the preferred inflation potential is a double-well, even function of the field with a moderate quartic coupling yielding as the most probable values ns ≃ 0.964, r ≃ 0.051. This value for r is within reach of forthcoming CMB observations. The present data in the effective theory of inflation clearly prefer new inflation. Study of higher degree inflaton potentials shows that terms of degree higher than 4 do not affect the fit in a significant way. In addition, a horizon exit happens for [Formula: see text], making higher order terms in the potential w negligible. We summarize the physical effects of generic initial conditions (different from Bunch–Davies) on the scalar and tensor perturbations during slow roll and introduce the transfer function D(k), which encodes the observable initial condition effects on the power spectra. These effects are more prominent in the low CMB multipoles: a change in the initial conditions during slow roll can account for the observed CMB quadrupole suppression. Slow-roll inflation is generically preceded by a short, fast-roll stage. Bunch–Davies initial conditions are the natural initial conditions for the fast-roll perturbations. During fast roll, the potential in the wave equations of curvature and tensor perturbations is purely attractive and leads to a suppression of the curvature and tensor CMB quadrupoles. An MCMC analysis of the WMAP+SDSS data including fast roll shows that the quadrupole mode exits the horizon about 0.2 e-fold before fast roll ends and its amplitude gets suppressed. In addition, fast roll fixes the initial inflation redshift to be z init = 0.9 × 1056 and the total number of e-folds of inflation to be N tot ≃ 64. Fast roll fits the TT, the TE and the EE modes well, reproducing the quadrupole suppression. A thorough study of the quantum loop corrections reveals that they are very small and are controlled by powers of (H/M Pl )2 ~ 10-9, a conclusion that validates the reliability of the effective theory of inflation. The present review shows how powerful the Ginsburg–Landau effective theory of inflation is in predicting observables that are being or will soon be contrasted with observations.

An exact solution of Dirac's equation in the field of plane EM wave: Physical implications

Within the frameworks of Classical and Quantum Electrodynamics, the behavior of relativistic charged particle in the field of a plane electromagnetic wave is considered. It is shown that the imposition of natural initial conditions in relativistic motion equation of Lorentz and the choice of initial wave function in Dirac's equation describe the correspondence between the classical and quantum descriptions properly. The same initial conditions for free particle applied to the Lorentz-Dirac equation also result in physical solutions without any pre-acceleration.

Dynamical realization of democratic Yukawa matrices and alignment ofAterms

We study realization of the democratic form of Yukawa matrices by infrared fixed points. We investigate renormalization-group flows of Yukawa couplings in models with a single Yukawa matrix for three families, and up and down-sector Yukawa matrices. It is found that each model has its certain pattern of renormalization-group flows of Yukawa matrices. We apply them to the charged lepton sector and quark sector, and show that realistic results for the second and third families are obtained with natural initial conditions, e.g. $(V_{MNS})_{23} \sim 2/\sqrt{6}$, $m_s/m_b \sim V_{cb} \sim m_{\mu}/m_\tau$ and $m_c/m_t \sim (m_s/m_b)^{3/2}$. We also study corresponding A-terms. The A-terms approach toward the universal form with no physical CP-violating phase. Thus, constraints due to various neutral flavor changing processes except for $\mu \to e \gamma$ are found to be satisfied by this dynamics. In order to suppress the electric dipole moments as well as $\mu \to e \gamma$ sufficiently, more alignment of the A-terms with some reason is required.

Does inflation provide natural initial conditions for the universe

If our universe underwent inflation, its entropy during the inflationary phase was substantially lower than it is today. Because a low-entropy state is less likely to be chosen randomly than a high-entropy one, inflation is unlikely to arise through randomly-chosen initial conditions. To resolve this puzzle, we examine the notion of a natural state for the universe, and argue that it is a nearly-empty spacetime. If empty space has a small vacuum energy, however, inflation can begin spontaneously in this background. This scenario explains why a universe like ours is likely to have begun via a period of inflation, and also provides an origin for the cosmological arrow of time.

Realizations of hybrid inflation in supergravity with natural initial conditions

We present viable F-term realizations of the hybrid inflationary scenario in the context of supergravity addressing at the same time the well-known problems of the initial conditions and of the adequate suppression of the inflaton mass. An essential role in our construction is played by "decoupled" superheavy fields without superpotential acquiring large vevs due to D-terms associated with "anomalous" U(1) gauge symmetries. The naturalness of the initial conditions is achieved through a "chaotic" inflation starting at energies close to the Planck scale and driven by the "anomalous" D-terms. We discuss two distinct mechanisms leading to such an early "chaotic" D-term inflation which depend on the choice of the K\"ahler potential involving the superheavy fields. The one relies on a choice of the K\"ahler potential of the $SU(1,1)/U(1)$ K\"ahler manifold of the type encountered in no-scale supergravity whereas the other employs a more "conventional" choice for the K\"ahler potential of the $SU(1,1)/U(1)$ or $SU(2)/U(1)$ K\"ahler manifold but invokes rather specific valuesof the Fayet-Iliopoulos $\xi$ term.