Inhomogeneous Term Research Articles

Thermodynamic limit and boundary energy of the spin-1 Heisenberg chain with non-diagonal boundary fields

The thermodynamic limit and boundary energy of the isotropic spin-1 Heisenberg chain with non-diagonal boundary fields are studied. The finite size scaling properties of the inhomogeneous term in the T-Q relation at the ground state are calculated by the density matrix renormalization group. Based on our findings, the boundary energy of the system in the thermodynamic limit can be obtained from Bethe ansatz equations of a related model with parallel boundary fields. These results can be generalized to the SU(2) symmetric high spin Heisenberg model directly.

Simulating the dynamics of electronic observables via reduced-dimensionality generalized quantum master equations.

We describe a general-purpose framework for formulating the dynamics of any subset of electronic reduced density matrix elements in terms of a formally exact generalized quantum master equation (GQME). Within this framework, the effect of coupling to the nuclear degrees of freedom, as well as to any projected-out electronic reduced density matrix elements, is captured by a memory kernel and an inhomogeneous term, whose dimensionalities are dictated by the number of electronic reduced density matrix elements included in the subset of interest. We show that the memory kernel and inhomogeneous term within such GQMEs can be calculated from projection-free inputs of the same dimensionality, which can be cast in terms of the corresponding subsets of overall system two-time correlation functions. The applicability and feasibility of such reduced-dimensionality GQMEs is demonstrated on the two-state spin-boson benchmark model. To this end, we compare and contrast the following four types of GQMEs: (1) a full density matrix GQME, (2) a single-population scalar GQME, (3) a populations-only GQME, and (4) a subset GQME for any combination of populations and coherences. Using a method based on the mapping Hamiltonian approach and linearized semiclassical approximation to calculate the projection-free inputs, we find that while single-population GQMEs and subset GQMEs containing only one population are less accurate, they can still produce reasonable results and that the accuracy of the results obtained via the populations-only GQME and a subset GQME containing both populations is comparable to that obtained via the full density matrix GQMEs.

Evolution of interface singularities in shallow water equations with variable bottom topography

AbstractWave front propagation with nontrivial bottom topography is studied within the formalism of hyperbolic long wave models. Evolution of nonsmooth initial data is examined, and, in particular, the splitting of singular points and their short time behavior is described. In the opposite limit of longer times, the local analysis of wave fronts is used to estimate the gradient catastrophe formation and how this is influenced by the topography. The limiting cases when the free surface intersects the bottom boundary, belonging to the so‐called “physical” and “nonphysical” vacuum classes, are examined. Solutions expressed by power series in the spatial variable lead to a hierarchy of ordinary differential equations for the time‐dependent series coefficients, which are shown to reveal basic differences between the two vacuum cases: for nonphysical vacuums, the equations of the hierarchy are recursive and linear past the first two pairs, whereas for physical vacuums, the hierarchy is nonrecursive, fully coupled, and nonlinear. The former case may admit solutions that are free of singularities for nonzero time intervals, whereas the latter is shown to develop nonstandard velocity shocks instantaneously. Polynomial bottom topographies simplify the hierarchy, as they contribute only a finite number of inhomogeneous forcing terms to the equations in the recursion relations. However, we show that truncation to finite‐dimensional systems and polynomial solutions is in general only possible for the case of a quadratic bottom profile. In this case, the system's evolution can reduce to, and is completely described by, a low‐dimensional dynamical system for the time‐dependent coefficients. This system encapsulates all the nonlinear properties of the solution for general power series initial data, and, in particular, governs the loss of regularity in finite times at the dry point. For the special case of parabolic bottom topographies, an exact, self‐similar solution class is introduced and studied to illustrate via closed‐form expressions the general results.

The renewal equation with unbounded inhomogeneous term

The renewal equation with unbounded inhomogeneous term

Feasibility Study of 3D-VMAT-Based GRID Therapy.

Background: Spatially fractionated radiotherapy (GRID) could effectively de-bulk tumor volumes for shallow and deep-seated locally advanced tumors. A new treatment planning method using the three-dimensional-volumetric modulated arc therapy (VMAT) technique combined with a novel, software-generated, virtual GRID block (VGB) was developed which allows better conformity plans (VMAT-GRID) and maintain the GRID dosimetric characteristics. The dosimetric metrics calculated via the valley/peak ratio (Dmin/Dmax), D90/D10, gross tumor volume (GTV) mean dose (Dmean), GTV equivalent uniform dose (EUD), and normal tissue maximum dose. Methods: Twenty-five patients with tumor volumes ranging between 71.6 cc and 4683 cc at various tumor sites were retrospectively studied. The prescription was 20 Gy to the maximum point of GTV in a single fraction, and the VMAT-GRID plan was generated using 6 MV/10 MV flattening-filter-free beams. Results: The optimized VGB was designed with the median center-to-center distance of 27 mm, and 9 mm for the median diameter of the opening area in this study. These 2 values can be used to design any optimized VGB, the final VGB may be modified to generate a patient-specific VGB. The median GTV mean dose was 918 (877- 938) cGy, and the median GTV EUD dose was 818 (597-916) cGy. In terms of dose inhomogeneity, the median valley-to-peak dose ratio was 0.07 (0.02-0.26); and the median ratio of D90/D10 was 0.70 (0.38-0.94). For the organ-at-risk doses, there was a rapid dose drop-off in the normal tissue area immediately adjacent to the target, and the maximum global doses were all located inside the GTV. Conclusion: Our results indicated that the VMAT-GRID planning approach could successfully deliver dose with acceptable GRID dose metric while sparing the normal tissue especially in the region near the target due to the rapid dose drop-off and restricting maximum dose inside the target.

Design and Shimming Method of Low Length-to-Interdiameter Ratio Halbach Magnet

The conventional Halbach permanent magnet usually has a high length to inter diameter ratio to gain a higher magnetic field homogeneity but can cause an increase in volume and weight, limiting the application in miniaturization. This paper proposes a novel design and shimming method for the Halbach magnet to have not only a lower length to inter diameter ratio but also a better homogeneity. In this paper, a multi-layer magnetic ring structure was first designed through theoretical calculation and simulation optimization, which compensated for the magnetic field inhomogeneity. Then, a passive shimming method was applied which simulated the first-order and second-order terms of magnetic field inhomogeneity using small magnetic blocks whose magnetization direction was consistent with the main magnetic field. This method has high shimming efficiency and can avoid the demagnetization of the magnetic blocks caused by their magnetization directions against the direction of the main magnetic field, especially when the intensity of the magnetic field is close to or beyond the remanence of the magnetic blocks. The magnetic field intensity of the proposed magnet is 1.08 T and the homogeneity reaches close to the sub-ppm. The performance of the magnet was verified in the portable MRI system combined with a set of homemade low-power gradient coils. MRI images of phantoms and okra reached a resolution of 156 μm (pixel size 156 μm, matrix 128 × 128, FOV 20 × 20 mm2), which shows the potential for wide application of the proposed design and shimming method.

Subdynamics of fluctuations in a many-particle quantum system

It is shown that the relevant part of an equilibrium two-time correlation function of operators related to a subsystem of S < N particles, selected from the whole system of N ≫ 1 quantum particles, obeys the exact completely closed (homogeneous) evolution equation. This equation differs from the conventional Nakajima–Zwanzig equation by the absence of the undesirable inhomogeneous term comprising all interparticle correlations at the initial time moment. The equation defines the subdynamics in a subspace of S selected particles and is obtained without any use of approximation like the Boltzmann ‘molecular chaos’, factorized initial state or Bogoliubov’s principle of weakening of initial correlations. In the second order in a small inter-particle interaction, this equation reduces to the Markovian evolution equation, which for a one-particle correlation function (S = 1) takes the form of the linear generalized quantum Boltzmann-like equation for a relevant part of a correlation function (one particle momentum distribution function) with the additional terms conditioned by initial correlations taken into account. This equation makes it possible to obtain the solution for the corresponding one-particle correlation function, which is valid for all timescales and reveals the influence of initial correlations on the evolution in time process. On a large timescale t ≫ t cor ∼ ℏ/k B T (decoherence time), the initial correlations cease to influence the damping process and the evolution enters a kinetic stage. However, the factor related to initial correlations survives on the large timescale, and can contribute to the kinetic coefficients.

Effect of material inhomogeneity under creep and plastic to creep transition of cracks

The driving force of a crack in a non-linear elastic material is described by the conventional J-integral, J (Rice 1968). For elastic−plastic materials, J does not describe the crack driving force. However, it describes the intensity of the crack tip field and can be, therefore, used to quantify the fracture toughness (Rice 1968). For the high-temperature behaviour of materials where creep deformation dominated, analogous parameter to J-integral were deduced, the C*-integral and a similar parameter, the Ct-integral (Landes and Begley 1976). C* has been proven to have a good correlation with the creep crack growth rate for many materials, mostly under extensive creep conditions. However, it does not describe the crack driving force.Based on the concept of configurational forces, a J -integral for elastic–plastic materials, Jep, was derived, which is able to quantify the true crack driving force in accordance with incremental theory of plasticity [4]. Hereby, the plastic strain is treated as an eigen-strain. Jep can be applied also in cases of non-proportional loading, e.g. for a growing crack (Simha et al. 2008) or for cyclic loading conditions (Ochensberger and Kolednik 2015). In this work, the concept of configurational force is applied to evaluate the crack driving force under small, medium and extensive creep conditions. Hereby, the creep strain is treated as a time dependent eigenstrain. A case study is performed where the path dependences of the elastic–plastic J-integral, Jep, and the conventional parameters Ct and C* are studied for materials which undergo elastic+creep deformation, with material inhomogeneity.It has been confirmed in the recent studies (Kolednik et al. 2014, Tiwari et al. 2020) that the material inhomogeneity term affects the crack depending on the nature of material inhomogeneity. This effect is unknown for creep deformation and the same has been studies on idealized fracture specimen under creep deformation using configurational forces and finite element method simulating real situations of dissimilar metals used at high temperatures in nuclear power plants and jet engines.

Dynamic fracture analysis using a high-accuracy manifold element modelling scheme

In this paper, a high-accuracy manifold element modelling scheme for dynamic fracture analysis of two-dimensional elastic cracked bodies is presented. The displacement discontinuity across crack surfaces is naturally simulated attributing to the dual cover systems in the numerical manifold method. The singularity of field gradients at crack tips is appropriately captured through the use of asymptotic basis functions. The discrete equations for dynamic analysis are solved by a new fully explicit direct time integration algorithm based on precise time step integration method, in which Taylor expansion theorem and downscaling time step integration method are used to calculate the exponential transfer matrix with high accuracy. To eliminate the sensitivity of accuracy of results to time-step size, Fourier expansion with trigonometric basis functions is utilized to obtain the approximate expression for different time-dependent loadings, the exponential transfer matrix is also further utilized to calculate the response matrix of inhomogeneous term. A matrix-based spectral analysis is used to prove the stability and convergence of proposed scheme, the results demonstrate that the present scheme is unconditionally stable and convergent. Finally, the applicability and accuracy of the proposed scheme for calculating dynamic stress intensity factors are examined by two numerical examples with different crack configurations and time-dependent loadings. By comparing with the analytical/reference solutions, it is proved that the present scheme has significant advantages in accuracy and efficiency than Newmark scheme for dynamic fracture analysis.

The Lamm–Rivière system I: $$L^p$$ regularity theory

Motivated by the heat flow and bubble analysis of biharmonic mappings, we study further regularity issues of the fourth order Lamm–Rivière system $$\begin{aligned} \Delta ^{2}u=\Delta (V\cdot \nabla u)+\mathrm{div}(w\nabla u)+(\nabla \omega +F)\cdot \nabla u+f \end{aligned}$$in dimension four, with an inhomogeneous term f which belongs to some natural function space. We obtain optimal higher order regularity and sharp Hölder continuity of weak solutions. Among several applications, we derive weak compactness for sequences of weak solutions with uniformly bounded energy, which generalizes the weak convergence theory of approximate biharmonic mappings.

Solution of Inhomogeneous Fractional Differential Equations with Polynomial Coefficients in Terms of the Green’s Function, in Nonstandard Analysis

Discussions are presented by Morita and Sato in Mathematics 2017; 5, 62: 1–24, on the problem of obtaining the particular solution of an inhomogeneous ordinary differential equation with polynomial coefficients in terms of the Green’s function, in the framework of distribution theory. In the present paper, a compact recipe in nonstandard analysis is presented, which is applicable to an inhomogeneous ordinary and also fractional differential equation with polynomial coefficients. The recipe consists of three theorems, each of which provides the particular solution of a differential equation for an inhomogeneous term, satisfying one of three conditions. The detailed derivation of the applications of these theorems is given for a simple fractional differential equation and an ordinary differential equation.

A localized collocation scheme with fundamental solutions for long-time anomalous heat conduction analysis in functionally graded materials

This paper makes the first attempt to develop a novel localized collocation scheme based on fundamental solutions for long-time anomalous heat conduction analysis in functionally graded materials. In the present numerical framework, the Laplace transformation and Fixed Talbot numerical inverse Laplace transformation are employed to deal with the time fractional derivative in anomalous time fractional heat conduction model. The localized collocation scheme based on fundamental solutions is applied to solve Laplace-transformed time-independent partial differential equations (PDEs), and the generalized reciprocity method (GRM) coupled with the derived problem-dependent generalized fundamental solution is proposed to deal with the inhomogeneous term in Laplace-transformed time-independent PDEs. Numerical tests demonstrate that the developed computational strategy can be considered as one of the competitive alternatives in the simulation of anomalous heat conduction behavior of functionally graded materials. Furthermore, the effect of time fractional derivative order on the behavior of anomalous heat conduction is numerically investigated.

Numerical Solution of Fractional Elliptic Problems with Inhomogeneous Boundary Conditions

The numerical solution of fractional-order elliptic problems is investigated in bounded domains. According to real-life situations, we assumed inhomogeneous boundary terms, while the underlying equations contain the full-space fractional Laplacian operator. The basis of the convergence analysis for a lower-order boundary element approximation is the theory for the corresponding continuous problem. In particular, we need continuity results for Riesz potentials and the fractional-order extension of the theory for boundary integral equations with the Laplacian operator. Accordingly, the convergence is stated in fractional-order Sobolev norms. The results were confirmed in a numerical experiment.

Uniform stability for a spatially discrete, subdiffusive Fokker–Planck equation

We prove stability estimates for the spatially discrete, Galerkin solution of a fractional Fokker-Planck equation, improving on previous results in several respects. Our main goal is to establish that the stability constants are bounded uniformly in the fractional diffusion exponent $\alpha\in(0,1]$. In addition, we account for the presence of an inhomogeneous term and show a stability estimate for the gradient of the Galerkin solution. As a by-product, the proofs of error bounds for a standard finite element approximation are simplified.

Anisotropic Cahn-Hilliard free energy and interfacial energies for binary alloys with pairwise interactions

The original Cahn-Hilliard derivation of the contribution of compositional inhomogeneity to the free energy of a binary alloy with pairwise interactions is extended to include higher-order inhomogeneity terms. For alloys on a cubic lattice, the coefficient of the first inhomogeneity is a second-rank tensor and reduces to a scalar, but it is shown that the second order and the third order inhomogeneity terms are weighted by fourth-rank and sixth-rank tensors, thus resulting in anisotropic contributions. Furthermore, each interaction shell generates a unique set of inhomogeneity coefficients that is determined by the set of vectors connecting an atom to its neighbors on that shell. These coefficients are calculated for fcc and bcc alloys with interactions up fourth nearest neighbors. Phase field simulations based on these extended Cahn-Hilliard free energies are performed to measure interface free energies along specific crystallographic directions as a function of temperature, and to obtain the equilibrium shape of precipitates. Interface free energies, and the resulting anisotropies, are compared to those obtained by discrete models and Monte Carlo simulations.

Lessons fromTμμon inflation models: Two-loop renormalization ofηin the scalar QED

A non-minimal coupling $\eta$ has been attracting growing interest particularly in the context of inflation models, though its quantum nature is not clear yet. We study the renormalization of a non-minimal coupling in the scalar quantum electrodynamics (QED). We find ${\it no}$ inhomogeneous term of the renormalization group equation (RGE) at the two-loop level. This is similar to other theories, where an inhomogeneous term of the RGE appears only at a higher-loop order: e.g., four-loop order in $\lambda \phi^{4}$ theory.

Coherent propagation and incoherent diffusion of elastic waves in a two dimensional continuum with a random distribution of edge dislocations

We study the coherent propagation and incoherent diffusion of in-plane elastic waves in a two dimensional continuum populated by many, randomly placed and oriented, edge dislocations. Because of the Peierls–Nabarro force the dislocations can oscillate around an equilibrium position with frequency ω0 . The coupling between waves and dislocations is given by the Peach–Koehler force. This leads to a wave equation with an inhomogeneous term that involves a differential operator. In the coherent case, a Dyson equation for a mass operator is set up and solved to all orders in perturbation theory in independent scattering approximation (ISA). As a result, a complex index of refraction is obtained, from which an effective wave velocity and attenuation can be read off, for both longitudinal and transverse waves. In the incoherent case a Bethe–Salpeter equation is set up, and solved to leading order in perturbation theory in the limit of low frequency and wave number. A diffusion equation is obtained and the (frequency-dependent) diffusion coefficient is explicitly calculated. It reduces to the value obtained with energy transfer arguments at low frequency. An important intermediate step is the obtention of a Ward–Takahashi identity (WTI) for a wave equation that involves a differential operator, which is shown to be compatible with the ISA.

Global Solutions to Isothermal System with Source

In this short note, we are concerned with the global existence of solutions to the isothermal system with source, where the inhomogeneous terms f(x,t,ρ,u)=b(x,t)ρ+(a′(x)/a(x))*ρu^2+α(x,t)ρu|u| are appeared in the momentum equation. Our work extended the results in the previous papers “Resonance for the Isothermal System of Isentropic Gas Dynamics” (Proc. A.M.S.139(2011),2821-2826), “Global Existence and Stability to the Polytropic Gas Dynamics with an Outer Force” (Appl. Math. Let-ters, 95(2019), 35-40) and “Existence of Global Solutions for Isentropic GasFlow with Friction” (Nonlinearity, 33(2020), 3940-3969), where the global solution was obtained for the source f(x,t,ρ,u)=(a′(x)/a(x))*ρu^2, f(x,t,ρ,u)=b(x,t)ρ, f(x,t,ρ,u)=α(x,t)ρu|u| respectively.

The smooth forcing extension method: A high-order technique for solving elliptic equations on complex domains

High-order numerical methods for solving elliptic equations over arbitrary domains typically require specialized machinery, such as high-quality conforming grids for finite elements method, and quadrature rules for boundary integral methods. These tools make it difficult to apply these techniques to higher dimensions. In contrast, fixed Cartesian grid methods, such as the immersed boundary (IB) method, are easy to apply and generalize, but typically are low-order accurate. In this study, we introduce the Smooth Forcing Extension (SFE) method, a fixed Cartesian grid technique that builds on the insights of the IB method, and allows one to obtain arbitrary orders of accuracy. Our approach relies on a novel Fourier continuation method to compute extensions of the inhomogeneous terms to any desired regularity. This is combined with the highly accurate Non-Uniform Fast Fourier Transform for interpolation operations to yield a fast and robust method. Numerical tests confirm that the technique performs precisely as expected on one-dimensional test problems. In higher dimensions, the performance is even better, in some cases yielding sub-geometric convergence. We also demonstrate how this technique can be applied to solving parabolic problems and for computing the eigenvalues of elliptic operators on general domains, in the process illustrating its stability and amenability to generalization.

Field dependence of magnetic entropy change in Mn5Ge3 near room temperature

We present the investigation of the magnetic field dependence of magneto-caloric properties in Mn5Ge3 with a near-room temperature second order phase transition (SOPT) at Tc ≃ 297.5 K. The isothermal entropy change (∣ΔSM∣) exhibits a high-field (H ≥ 3 T) power-law behavior with exponent value of 0.58 (near TC), in corroboration with critical exponents study. The maximum entropy change (∣ΔSM∣max∣) exhibits a H2∕3 dependence with addition of spatial inhomogeneity and higher order magnetization (up to M6) terms in Landau′s theory of SOPT. The nearly identical magnetic field evolution of magnetocaloric effect and a reasonably high refrigerant capacity of 398 J/kg (at 5 T) for inexpensive and abundant Mn5Ge3 make it a promising alternative to rare-earth Gadolinium for near-room temperature magnetic refrigeration technology.