Simple Ansatz Research Articles

Pairing in asymmetric many-fermion systems: Functional renormalisation group approach

Functional renormalisation group approach is applied to a imbalanced many-fermion system with a short-range attractive force. We introduce a composite boson field to describe pairing effects, and assume a simple ansatz for the effective action. A set of approximate flow equations for the effective coupling including boson and fermionic fluctuations is derived and solved. We identify the critical values of particle number density mismatch when the system undergoes a normal state. We determine the phase diagram both at unitarity and around. The obtained phase diagram is in a reasonable agreement with the experimental data.

A Study of Accelerated Expansion of the Universe and Time Varying Gravitational Constant in the Framework of Brans-Dicke Theory

A simple mathematical model has been proposed on the basis of generalized Brans-Dicke (BD) theory where, the dimensionless BD parameter is regarded as a function of the scalar field. The gravitational constant is reciprocal of this scalar field. The calculations in this model are based on a very simple ansatz , which has been chosen for the purpose of solving the differential equations most conveniently. This function has been chosen to account for an inter-conversion between the matter and dark energy. This inter-conversion is said to be connected to an interaction that causes the accelerated expansion of the universe. The requirement of a signature flip of the deceleration parameter , which is evident from other studies, sets the boundary conditions to be satisfied by the function , leading to the formulation of its time dependence. A simple empirical relation has been assumed to represent the time dependence of , and the constants in this expression have been determined from the boundary conditions. The parameter has been found to have a negative value. The dependence of upon has been shown graphically. The present model shows that a transition of the universe, from decelerated expansion to accelerated expansion, takes place due to a conversion of matter into dark energy. An increase of gravitational constant with time is found in the present study

Pair Densities in Density Functional Theory

The exact interaction energy of a many-electron system is determined by the electron pair density, which is not well-approximated in standard Kohn--Sham density functional models. Here we study the (complicated but well-defined) exact universal map from density to pair density. We survey how many common functionals, including the most basic version of the local density approximation (Dirac exchange with no correlation contribution), arise from particular approximations of this map. We develop an algorithm to compute the map numerically and apply it to one-parameter families $\{\alpha\rho(\alpha x)\}_{\alpha>0}$ of one-dimensional homogeneous and inhomogeneous single-particle densities. We observe that the pair density develops remarkable multiscale patterns which strongly depend on both the particle number and the “width” $\alpha^{-1}$ of the single-particle density. The simulation results are confirmed by rigorous asymptotic results in the limiting regimes $\alpha\gg$ and $\alpha\ll1$. For one-dimensional homogeneous systems, we show that the whole spectrum of patterns is reproduced surprisingly well by a simple asymptotics-based ansatz which slowly smoothens out the “strictly correlated” $\alpha=0$ pair density while slowly turning on the $\alpha=\infty$ “exchange” terms as $\alpha$ increases. Our findings lend theoretical support to the celebrated semi-empirical idea [A. D. Becke, J. Chem. Phys., 98 (1993), pp. 5648--5652] to mix in a fractional amount of exchange, albeit not to assuming the mixing to be additive and taking the fraction to be a system-independent constant.

Nonlinear dynamic response of glass-forming liquids to random pinning.

We use large scale computer simulations of a glass-forming liquid in which a fraction c of the particles has been permanently pinned. We find that the relaxation dynamics shows an exponential dependence on c. This result can be rationalized by assuming that the configurational entropy of the pinned liquid decreases linearly upon increasing of c. This behavior is discussed in the context of thermodynamic theories for the glass transition, notably the Adam-Gibbs picture and the random first order transition theory. For intermediate and low temperatures we find that the slowing down of the dynamics due to the pinning saturates and that the cooperativity decreases with increasing c, results which indicate that in glass-forming liquids there is a dynamic crossover at which the shape of the relaxing entities changes.

Importance of asymptotic freedom for the pseudocritical temperature in magnetized quark matter

Although asymptotic freedom is an essential feature of QCD, it is absent in effective chiral quark models like the Nambu--Jona-Lasinio and linear sigma models. In this work we advocate that asymptotic freedom plays a key role in the recently observed discrepancies between results of lattice QCD simulations and quark models regarding the behavior of the pseudocritical temperature ${T}_{pc}$ for chiral-symmetry restoration in the presence of a magnetic field $B$. We show that the lattice predictions that ${T}_{pc}$ decreases with $B$ can be reproduced within the Nambu--Jona-Lasinio model if the coupling constant $G$ of the model decreases with $B$ and the temperature. Without aiming at numerical precision, we support our claim by considering a simple ansatz for $G$ that mimics the asymptotic-freedom behavior of the QCD coupling constant $1/{\ensuremath{\alpha}}_{s}\ensuremath{\sim}ln(eB/{\ensuremath{\Lambda}}_{\text{QCD}}^{2})$ for large values of $B$.

Schrödinger method asN-body double and UV completion of dust

We investigate large-scale structure formation of collisionless dark matter in the phase space description based on the Vlasov equation whose nonlinearity is induced solely by gravitational interaction according to the Poisson equation. Determining the time-evolution of density and peculiar velocity demands solving the full Vlasov hierarchy for the moments of the phase space distribution function. In the presence of long-range interaction no consistent truncation of the hierarchy is known apart from the pressure- less fluid (dust) model which is incapable of describing virialization due to the occurrence of shell-crossing singularities and the inability to generate vorticity and higher cumulants like velocity dispersion. Our goal is to find a simple ansatz for the phase space distribution function that approximates the full Vlasov distribution function without pathologies in a controlled way and therefore can serve as theoretical N-body double and as a replacement for the dust model. We argue that the coarse-grained Wigner probability distribution obtained from a wave function fulfilling the Schr\"odinger-Poisson equation (SPE) is the sought-after function. We show that its evolution equation approximates the Vlasov equation and therefore also the dust equations before shell-crossing, but cures the shell-crossing singularities and is able to describe regions of multi-streaming and virialization. This feature was already employed in simulations by Widrow & Kaiser (1993). The coarse-grained Wigner ansatz allows to calculate all higher moments from density and velocity analytically, thereby incorporating nonzero higher cumulants in a self-consistent manner. On this basis we are able to show that the Schr\"odinger method (ScM) automatically closes the corresponding hierarchy such that it suffices to solve the SPE in order to directly determine density and velocity and all higher cumulants.

Static and Dynamical Correlation in Diradical Molecules by Quantum Monte Carlo Using the Jastrow Antisymmetrized Geminal Power Ansatz.

Diradical molecules are essential species involved in many organic and inorganic chemical reactions. The computational study of their electronic structure is often challenging, because a reliable description of the correlation, and in particular of the static one, requires multireference techniques. The Jastrow correlated antisymmetrized geminal power (JAGP) is a compact and efficient wave function ansatz, based on the valence-bond representation, which can be used within quantum Monte Carlo (QMC) approaches. The AGP part can be rewritten in terms of molecular orbitals, obtaining a multideterminant expansion with zero-seniority number. In the present work we demonstrate the capability of the JAGP ansatz to correctly describe the electronic structure of two diradical prototypes: the orthogonally twisted ethylene, C2H4, and the methylene, CH2, representing respectively a homosymmetric and heterosymmetric system. In the orthogonally twisted ethylene, we find a degeneracy of π and π* molecular orbitals, as correctly predicted by multireference procedures, and our best estimates of the twisting barrier, using respectively the variational Monte Carlo (VMC) and the lattice regularized diffusion Monte Carlo (LRDMC) methods, are 71.9(1) and 70.2(2) kcal/mol, in very good agreement with the high-level MR-CISD+Q value, 69.2 kcal/mol. In the methylene we estimate an adiabatic triplet-singlet (X̃(3)B1-ã(1)A1) energy gap of 8.32(7) and 8.64(6) kcal/mol, using respectively VMC and LRDMC, consistently with the experimental-derived finding for Te, 9.363 kcal/mol. On the other hand, we show that the simple ansatz of a Jastrow correlated single determinant (JSD) wave function is unable to provide an accurate description of the electronic structure in these diradical molecules, both at variational level (VMC torsional barrier of C2H4 of 99.3(2) kcal/mol, triplet-singlet energy gap of CH2 of 13.45(10) kcal/mol) and, more remarkably, in the fixed-nodes projection schemes (LRDMC torsional barrier of 97.5(2) kcal/mol, triplet-singlet energy gap of 13.36(8) kcal/mol) showing that a poor description of the static correlation yields an inaccurate nodal surface. The suitability of JAGP to correctly describe diradicals with a computational cost comparable with that of a JSD calculation, in combination with a favorable scalability of QMC algorithms with the system size, opens new perspectives in the ab initio study of large diradical systems, like the transition states in cycloaddition reactions and the thermal isomerization of biological chromophores.

Disrupted coarsening in complex Cahn-Hilliard dynamics

Predicting the pattern formation in a system maintained far from equilibrium is a complex task. For a given dynamics governed by the evolution of a conservative order parameter, recent investigations have demonstrated that the knowledge of the long time expression of the order parameter is sufficient to predict the existence of disrupted coarsening, i.e., the pinning of the inhomogeneities wavelength to a well defined value. However, there exists some dynamics for which the asymptotic form of the order parameter remains unknown. The Cahn-Hilliard-like equation used to describe the stability of solids under irradiation belongs to this class of equations. In this paper, we present an alternative to predict the patterning induced by this equation. Based on a simple ansatz, we calculated the form factor and proved that a disrupted coarsening takes place in such dynamics. This disrupted coarsening results from the bifurcation of the implicit equation linking the characteristic length of the dynamics (k_{m}^{∞})^{-1} to a control parameter describing the irradiation. This analysis is supported by direct simulations. From this paper, it clearly appears that the bifurcation of k_{m}^{∞} is a criterion for disrupted coarsening.

Kinetics of particle wrapping by a vesicle.

We present theoretical results on kinetics for the passive wrapping of a single, rigid particle by a flexible membrane. Using a simple geometric ansatz for the shape of the membrane/particle complex we first compute free energy profiles as a function of the particle size, attraction strength between the particle and vesicle, and material properties of the vesicle--bending stiffness and stretching modulus. The free energy profiles thus computed are taken as input to a stochastic model of the wrapping process, described by a Fokker-Planck equation. We compute average uptake rates of the particle into the vesicle. We find that the rate of particle uptake falls to zero outside of a thermodynamically allowed range of particle sizes. Within the thermodynamically allowed range of particle size, the rate of uptake is variable and we compute the optimal particle size and maximal uptake rate as a function of the attraction strength, the vesicle size, and vesicle material properties.

Modeling the collective relaxation time of glass-forming polymers at intermediate length scales: Application to polyisobutylene

In a recent paper [V. N. Novikov, K. S. Schweizer, and A. P. Sokolov, J. Chem. Phys. 138, 164508 (2013)] a simple analytical ansatz has been proposed to describe the momentum transfer (Q) dependence of the collective relaxation time of glass-forming systems in a wide Q-range covering the region of the first maximum of the static structure factor S(Q) and the so-called intermediate length scale regime. In this work we have generalized this model in order to deal with glass-forming systems where the atomic diffusive processes are sub-linear in nature. This is for instance the case of glass-forming polymers. The generalized expression considers a sub-linear jump-diffusion model and reduces to the expression previously proposed for normal diffusion. The generalized ansatz has been applied to the experimental results of the Q- and temperature-dependence of polyisobutylene (PIB), which were previously published. To reduce the number of free parameters of the model to only one, we have taken advantage of atomistic molecular dynamics simulations of PIB properly validated by neutron scattering results. The model perfectly describes the experimental results capturing both, Q- and temperature-dependences. Moreover, the model also reproduces the experimental Q-dependence of the effective activation energy of the collective relaxation time in the temperature range of observation. This non-trivial result gives additional support to the way the crossover between two different relaxation mechanisms of density fluctuations is formulated in the model.

“Massive” perturbative QCD, Regular in the IR limit

The goal of the research is to devise a modification of the perturbative QCD (pQCD) that should be regular in the low-energy region and could serve as a practical means for the analysis of data below 1 GeV down to the IR limit. Recent observation of the four-loop pQCD series “blow-up” in the region below 1 GeV for the Bjorken Sum Rule gave an impetus to this attempt. The proposed “massive” analytic pQCD has two sources. It can be treated as the common logarithmic pQCD with only one parameter added, the effective “glueball mass” m gl ≲ 1 GeV, serving as an IR regulator. At the same time, it looks like a modification of Analytic Perturbation Theory (APT) comprising nonpower perturbative expansion that makes it compatible with linear integral transformations. Figuratively (with minor reservations), the proposed MAPT differs from the minimal APT by simple ansatz Q 2 → Q 2 + m g1 2 .

Beyond classical theory: Predicting the free energy barrier of bubble nucleation in polymer foaming

Based on the Gibbs‐Tolman‐Koenig formalism, we considered the Tolman correction to the free energy barrier of bubble nucleation in polymer‐gas binary mixtures. For this class of systems, the correction may be estimated with a reasonable accuracy using experimentally accessible macroscopic thermodynamic quantities only. Although the Tolman correction is applicable only in the low supersaturation regime, a simple ansatz regarding the supersaturation dependence of the Tolman length can be made to extend the usefulness of this approach and to yield the free energy barrier that vanishes at the mean‐field spinodal as demanded by thermodynamic considerations. © 2013 American Institute of Chemical Engineers AIChE J, 59: 3042–3053, 2013

N-electron Slater determinants from nonunitary canonical transformations of fermion operators

Mean-field methods such as Hartree-Fock (HF) and Hartree-Fock-Bogoliubov (HFB) constitute the building blocks upon which more elaborate many-body theories are based. The HF and HFB wave functions are built out of independent quasiparticles resulting from a unitary linear canonical transformation of the elementary fermion operators. Here, we discuss the possibility of allowing the HF transformation to become nonunitary. The properties of such HF vacua are discussed, as well as the evaluation of matrix elements among such states. We use a simple ansatz to demonstrate that a nonunitary transformation brings additional flexibility that can be exploited in variational approximations to many-fermion wave functions. The action of projection operators on nonunitary-based HF states is also discussed and applied, in a variation-after-projection approach, to the one-dimensional Hubbard model with periodic boundary conditions.

Viscoelastic convection: Few-modes model and numerical simulations of field equations for Maxwellian fluids

The structure formation of convection rolls in Maxwellian fluids that are heated from below in a Rayleigh-Bénard setup is investigated close to onset with a simple few-modes ansatz and by solving the hydrodynamic field equations with a finite-difference method. Depending on the magnitude of the viscoelastic relaxation time one can have besides stationary convection also oscillatory patterns in the form of standing or traveling waves. The existence and stability regions of these convection structures are determined. The convection behavior of the model is compared with the results of full numerical simulations. Furthermore, the effect of modulating the heating periodically in time on the stability of the quiescent conductive state of the fluid and on its convection behavior is investigated as a function of the fluid's viscoelasticity.

Can Diffusions Propagate?

The diffusion of relativistic particle in a fluid at equilibrium is investigated through an analytical and numerical study of the Relativistic Ornstein-Uhlenbeck process (ROUP). Contrary to expectations, the ROUP exhibits short-time propagation in physical space and only displays typical diffuse behavior at asymptotic times. The short-time propagation is understood through an analytical computation and the density profile is fitted at all times by a simple Ansatz. A generalization of Fick’s law is finally obtained, in which the diffusion coefficient is replaced by a time-dependent metric. These results connect relativistic diffusion with gravitational horizons and geometrical flows.

Chromoelectric flux tubes and coherence length in QCD

The transverse profile of the chromoelectric flux tubes in SU(2) and SU(3) pure gauge theories is analyzed by a simple variational ansatz using a strict analogy with ordinary superconductivity. Our method allows to extract the penetration length and the coherence length of the flux tube.

Stripe Excitation of High Gain Media With Disorder

We discuss the spatial and spectral intensity distribution of emission in optical gain media under stripe excitation in the case where the intensity reaches saturation level. It is found that modes propagating along the stripe in different directions are spatially separated if they affect each other due to saturation. The investigation includes the effects of wavelength-dependent inhomogenities, such as localized losses and reflective perturbations. Even relatively small distortions of this kind are found to cause considerable spatial and spectral redistribution of the intensity compared to an ideal disorder-free medium. Our results indicate that a simple ansatz may describe some mechanisms that lead to the formation of random laser action that is commonly observed in high gain media, such as organic semiconductors. Furthermore, consequential difficulties of gain measurements in such media using stripe excitation experiments are highlighted.

Recent neutrino data and a realistic tribimaximal-like neutrino mixing matrix

In light of the recent neutrino experimental results from the Daya Bay and RENO Collaborations, we construct a realistic tribimaximal-like Pontecorvo–Maki–Nakagawa–Sakata (PMNS) leptonic mixing matrix. Motivated by the Qin–Ma (QM) parametrization for the quark mixing matrix in which the CP-odd phase is approximately maximal, we propose a simple ansatz for the charged lepton mixing matrix, namely, it has the QM-like parametrization, and assume the tribimaximal mixing (TBM) pattern for the neutrino mixing matrix. The deviation of the leptonic mixing matrix from the TBM one is then systematically studied. While the deviation of the solar and atmospheric neutrino mixing angles from the corresponding TBM values, i.e. sin2θ12=1/3 and sin2θ23=1/2, is fairly small, we find a non-vanishing reactor mixing angle given by sinθ13≈λ/2 (λ≈0.22 being the Cabibbo angle). Specifically, we obtain θ13≃9.2° and δCP≃δQM≃O(90°). Furthermore, we show that the leptonic CP violation characterized by the Jarlskog invariant is |JCPℓ|≃λ/6, which could be tested in the future experiments such as the upcoming long baseline neutrino oscillation ones.

SU(3) gauge family symmetry and prediction for the lepton-flavor mixing and neutrino masses with maximal spontaneous CP violation

A model for the lepton-flavor mixing and CP violation is proposed based on the SUF(3) gauge family symmetry and the Majorana feature of neutrinos. A consistent prediction for the lepton-flavor mixing and masses is shown to be resulted from the appropriate vacuum structure of SUF(3) gauge symmetry breaking. By choosing the SUF(3) gauge fixing condition to possess a residual Z2 symmetry and requiring the vacuum structure of spontaneous symmetry breaking to have approximate global U(1) family symmetries, we obtain naturally the tri-bimaximal mixing matrix and largely degenerate neutrino masses in the neutrino sector and the small mixing matrix in the charged-lepton sector. With a simple ansatz that all the smallness due to the approximate global U(1) family symmetries is characterized by a single Wolfenstein parameter λ≃0.22, and the charged-lepton mixing matrix has a similar hierarchy structure as the CKM quark mixing matrix, we arrive at a consistent prediction for the MNSP lepton-flavor mixing with a maximal spontaneous CP violation: δ=π/2, sin2θ13≃12λ2≃0.024 (sin22θ13≃0.094), sin2θ12≃13(1−2λ3)≃0.326 and sin2θ23≃12(1−λ2)≃0.48, which agree well with the current experimental data. The CP-violating Jarlskog-invariant is obtained to be JCP≃16λ(1−λ2/2−λ3)sinδ≃0.035, which is detectable in next generation neutrino experiment. The small masses of the neutrinos and charged leptons are simply attributed to the standard seesaw mechanism. The largely degenerate neutrino masses with the normal hierarchy and inverse hierarchy are discussed and found to be at the order mνi≃O(λ2)≃0.04–0.06 eV with a total mass ∑mν∼0.15 eV, which is testable in future precision astrophysics and cosmology.

Flavor mixing democracy and minimal CP violation

We point out that there is a unique parametrization of quark flavor mixing in which every angle is close to the Cabibbo angle θC≃13° with the CP-violating phase ϕq around 1°, implying that they might all be related to the strong hierarchy among quark masses. Applying the same parametrization to lepton flavor mixing, we find that all three mixing angles are comparably large (around π/4) and the Dirac CP-violating phase ϕl is also minimal as compared with its values in the other eight possible parametrizations. In this spirit, we propose a simple neutrino mixing ansatz which is equivalent to the tri-bimaximal flavor mixing pattern in the ϕl→0 limit and predicts sinθ13=1/2sin(ϕl/2) for reactor antineutrino oscillations. Hence the Jarlskog invariant of leptonic CP violation Jl=(sinϕl)/12 can reach a few percent if θ13 lies in the range 7°⩽θ13⩽10°.