Linear Transport Theory Research Articles

Atomistic spin dynamics simulations of magnonic spin Seebeck and spin Nernst effects in altermagnets

Magnon band structures in altermagnets are characterized by an energy splitting of modes with opposite chirality, even in the absence of applied external fields and relativistic effects, because of an anisotropy in the Heisenberg exchange interactions. We perform quantitative atomistic spin dynamics simulations based on electronic structure calculations on rutile RuO2, a prototypical “d-wave” altermagnet, to study magnon currents generated by thermal gradients. We report substantial spin Seebeck and spin Nernst effects, i.e., longitudinal or transverse spin currents, depending on the propagation direction of the magnons with respect to the crystal, together with a finite spin accumulation associated with nonlinearities in the temperature profile. Our findings are consistent with the altermagnetic spin-group symmetry, as well as predictions from linear spin-wave theory and semiclassical Boltzmann transport theory. Published by the American Physical Society 2024

Boundary-layer structures arising in linear transport theory.

We consider boundary-layer structures that arise in connection with the transport of neutral particles (e.g., photons or neutrons) through a participating medium. Such boundary-layer structures were previously identified by the authors in certain particular cases [Phys. Rev. E 104, L032801 (2021)2470-004510.1103/PhysRevE.104.L032801]. Extending the previous work to anisotropic scattering and general Fresnel boundary conditions, this contribution presents computational algorithms which (1) resolve the aforementioned layers as well as previously unreported boundary layers associated with Fresnel boundary transmission and reflection, and (2) yield accurate simulations at fixed computational cost for transport under phase functions with arbitrarily strong anisotropy. The present paper additionally includes (3) Mathematical proofs which justify the numerical methods proposed for resolution of boundary-layer structures. The impact of the new theory on algorithmic performance is demonstrated through a series of 1D computational benchmarks that emulate typical photon- and neutron-transport applications such as, e.g., optical tomography, and nuclear reactor analysis and design. Experimental results for transmission of photons through turbid media are presented, exhibiting close agreement between simulated and experimental data. As illustrated by means of a variety of numerical results, the proposed boundary-layer-based approach tackles transport problems with unprecedented accuracy and efficiency.

Adding and doubling solution to the 1D Fokker-Planck Equation

The 1D Fokker-Planck equation (FPE) plays a major role in the propagation of light in the universe. It specifically describes small angle scattering of photons (and electrons) as they travel in participating media. In particular, the differential scattering term representing the phase function scattering law enables the small angle scattering. This term also makes the FPE a challenge to solve in the discrete ordinate sense. Our approach utilizes adding and doubling, which has been successfully applied since the 1960s to solve the linear Boltzmann equation. With the help of Morel’s discrete ordinate equivalence of the angular Laplacian, the FPE becomes similar to the discrete ordinates equation of linear transport theory. We then take advantage of the similarity through adding and doubling for its solution.

Reaction kinetics in the system Y2O3/Al2O3 – Use of an external electric field to control the product phase formation in a system forming multiple product phases

This study investigates the influence of an external electric field on the kinetics of a heterogeneous solid state reaction between Al2O3 and Y2O3. The reaction couples were prepared by means of pulsed laser deposition (PLD) by growing Y2O3 films on single crystalline alumina substrates with an (0001) orientation. The solid state reaction was performed at a temperature of 1400° (1673 K). Utilising attached platinum electrodes, an electric field of 350 V/mm was applied. The superposed field led to an ionic current through the reacting sample and modifies the individual growth kinetics of the three product phases/layers, Y3Al5O12 (YAG), YAlO3 (YAP) and Y4Al2O9 (YAM) The cross-sections of the reacted samples were characterised by means of SEM and XRD. Depending on the direction of the ionic current, the kinetics of the YAP phase formation in particular was strongly influenced. The general kinetics of a solid state reaction forming multiple product phases was analysed using linear transport theory. The effect of an electric field for controlling the product phase formation to prefer or to kinetically suppress the formation of a distinct phase is demonstrated.

Low-resistance contact in MoSe2 -based solid-state thermionic devices

Solid-state thermionic structures made from layered van der Waals heterostructures have shown promising thermal-to-electrical energy conversion efficiencies theoretically. In this paper, we further study these structures using first-principles calculations combined with the Green's function method. By calculating the electron-phonon relaxation length, we confirm ballistic transport in these structures. We study the effect of the number of layers, the energy barrier, and the asymmetry of the contacts on the performance of ${\mathrm{MoSe}}_{2}$-based thermionic converters. We show that the key to high-performance thermionic diodes is to make a low-energy barrier and low-resistance metallic contacts, and we identify copper as the optimum metallic contact to ${\mathrm{MoSe}}_{2}$-based devices. We further show that, unlike the vacuum-based thermionic diodes, asymmetry does not result in improved performance within the linearized transport theory.

Symbolic Computation Applied to Cauchy Type Singular Integrals

The development of operator theory is stimulated by the need to solve problems emerging from several fields in mathematics and physics. At the present time, this theory has wide applications in the study of non-linear differential equations, in linear transport theory, in the theory of diffraction of acoustic and electromagnetic waves, in the theory of scattering and of inverse scattering, among others. In our work, we use the computer algebra system Mathematica to implement, for the first time on a computer, analytical algorithms developed by us and others within operator theory. The main goal of this paper is to present new operator theory algorithms related to Cauchy type singular integrals, defined in the unit circle. The design of these algorithms was focused on the possibility of implementing on a computer all the extensive symbolic and numeric calculations present in the algorithms. Several nontrivial examples computed with the algorithms are presented. The corresponding source code of the algorithms has been made available as a supplement to the online edition of this article.

Reaction kinetics in the system Y2O3/Al2O3 – A solid state reaction forming multiple product phases investigated by using thin film techniques

The kinetics of the heterogeneous solid state reaction between Al2O3 and Y2O3 is investigated by using thin film techniques. Y2O3 films are grown by means of pulsed laser deposition (PLD) on single crystalline alumina substrates with a (0001) orientation. The solid state reactions are performed at a temperature of 1400 °C (1673 K). The cross sections of the reacted samples were investigated by means of SEM and XRD and exhibit a sequence of three product layers, Y3Al5O12 (YAG), YAlO3 (YAP) and Y4Al2O9 (YAM). The simultaneous growth of the product layers is controlled by a diffusional kinetics and the thickness increases are coupled to each other. According to Wagner and Schmalzried, one has to distinguish between rate constants of the first kind (“practical” Tammann constant), in the case of simultanous and coupled growth of multiple product phases, and rate constants of the second kind (“true” Tammann constant), in the case of the uncoupled and only growth of one product phase in equilibrium with the adjacent phases. The growth kinetics for a solid reaction forming three product phases (layer) is analysed in detail using linear transport theory. For the formation of YAG, YAP and YAM the rate constants of the second kind are determined from the experimental data and those of the first kind are calculated and compared to the available literature data. Based on these considerations, a detailed overview about the phase formation kinetics in the temperature range of 1200–1400 °C can be given for the first time. The (Nernst-Planck coupled) cation conductivities are calculated.

Linear Transport in Porous Media

The linear transport theory is developed to describe the time dependence of the number density of tracer particles in porous media. The advection is taken into account. The transport equation is numerically solved by the analytical discrete ordinates method. For the inverse Laplace transform, the double-exponential formula is employed. In this paper, we consider the travel distance of tracer particles whereas the half-space geometry was assumed in our previous paper [Amagai et al. (2020). Trans. Porous Media 132:311–331].

Nonisothermal Rarefied Gas Flow through a Long Cylindrical Channel under Arbitrary Pressure and Temperature Drops

The problem of rarefied gas flow through a long cylindrical channel as a function of the pressure and the temperature maintained at the channel ends is considered on the basis of the S-model of the Boltzmann kinetic equation. The pressure and temperature drops between the channel ends vary from small values at which the linear transport theory is valid to large values at which the gas molecule mean free path ceases to be constant along the channel. The solution to the model kinetic equation is found by means of the collocation method using the Chebyshev polynomials and rational functions. The mass flow and the pressure in the channel are obtained. Isobaric and isothermal flows are investigated.

A Reciprocal Formulation of Nonexponential Radiative Transfer. 2: Monte Carlo Estimation and Diffusion Approximation

When lifting the assumption of spatially-independent scattering centers in classical linear transport theory, collision rate is no longer proportional to angular flux/radiance because the macroscopic cross-section depends on the distance s to the previous collision or boundary. This creates a nonlocal relationship between collision rate and flux and requires revising a number of familiar deterministic and Monte Carlo methods. We generalize collision and track-length estimators to support unbiased estimation of either flux integrals or collision rates in generalized radiative transfer (GRT). To provide benchmark solutions for the Monte Carlo estimators, we derive the four Green’s functions for the isotropic point source in infinite media with isotropic scattering. Additionally, new moment-preserving diffusion approximations for these Green’s functions are derived, which reduce to algebraic expressions involving the first four moments of the free-path lengths between collisions.

Constructing solutions to two-way diffusion problems

A variety of boundary value problems in linear transport theory are expressed as a diffusion equation of the two-way, or forward–backward, type. In such problems boundary data are specified only on part of the boundary, which introduces several technical challenges. Existence and uniqueness theorems have been established in the literature under various assumptions; however, calculating solutions in practice has proven difficult. Here we present one possible means of practical calculation. By formulating the problem in terms of projection operators, we derive a formal sum for the solution whose terms are readily calculated. We demonstrate the validity of this approach for a variety of physical problems, with focus on a periodic problem from the field of active matter.

On the modeling of equatorial shallow-water waves with the Coriolis effect

In the present study a simplified phenomenological model of shallow-water wave propagating mainly in the equatorial ocean regions with the Coriolis effect caused by the Earth’s rotation is formally derived. The model equation which is analogous to the Green–Naghdi equations with the second-order approximation of the Camassa–Holm scaling captures stronger nonlinear effects than the classical dispersive integrable equations like the Korteweg–de Vries and two-component Camassa–Holm system. The local well-posedness of the Cauchy problem is then established by the linear transport theory and wave-breaking phenomenon is investigated based on the method of characteristics and the Riccati-type differential inequality. Finally, the condition of permanent waves is demonstrated by analyzing competition between the slope of average of horizontal velocity component and the free surface component.

Singular Eigenfunction Expansion Solution for Time-Dependent Neutron Transport with Delayed Neutrons

We develop an analytic solution for time-dependent neutron transport with delayed neutrons using the singular eigenfunction expansion method. Our approach is based on a technique for solving time-dependent neutron-transport problems without delayed neutrons (Case and Zweifel, Linear Transport Theory, Addison-Wesley, 1967), which we effectively generalize to include the presence of delayed-neutron precursors. In particular, we obtain eigenfunctions composed of two parts: one corresponding to the neutron angular flux and one corresponding to the delayed-neutron precursor concentration. We further demonstrate that these eigenfunctions are complete. We also provide numerical results for an example problem.

The Solvability of the Inverse Problem for the Evolution Equation with a Superstable Semigroup

For the evolution equation in a Banach space, the linear inverse source problem is studied. It is required to recover an unknown nonhomogeneous term by means of an additional nonlocal condition written in the form of a Riemann-Stieltjes integral. The main assumption is related to the superstability (quasinilpotency) of the evolution semigroup. More precisely, it is assumed that the evolutionary semigroup associated with the abstract differential equation has an infinite negative exponential type. Without other restrictions, a theorem on the solvability of the inverse problem is obtained. It is shown that the solution can be represented by a convergent Neumann series. Exact conditions under which an infinite series becomes a finite sum are found. Here, the algorithm for calculating the solution becomes finite. Model examples are considered, including an important example of the inverse problem with final overdetermination. The above results can be applicated in special parts of mathematical physics related to the theory of elasticity and the linear transport theory. As is customary, our research takes place in the general case with a choice of the complex scalar field, but the main facts are also true in the real case. The created theory allows transfer to nonlocal problems for evolution equations, when instead of the traditional initial condition special time averaging is used to find the solution.

Port-Hamiltonian modeling and reduction of a burning plasma system

In this contribution, we develop a structured port-Hamiltonian model for a class of irreversible distributed parameter systems and exploit the obtained formulation for model reduction from dimension 3 to dimension 1 in space. The proposed methodology is motivated by the control of burning plasma profiles in Tokamak reactors. The burning plasma is viewed as a multi-physics system built on Maxwell equations and total mass, species, momentum, energies, and entropy balance equations. Moreover, the system presents nonlinear couplings, especially through transport coefficients, and its dynamic evolves over multiple time scales. The main couplings considered here are the Joule effect, the Lorentz forces, and the fusion reaction kinetics. The port-based modeling formulation and reduction rely on the use of the Gibbs relation, Onsager linear transport theory, Stokes–Dirac structures, and energy preserving geometric reduction.

Flow harmonics from self-consistent particlization of a viscous fluid

The quantitative extraction of quark-gluon plasma (QGP) properties from heavy-ion data, such as its specific shear viscosity $\ensuremath{\eta}/s$, typically requires comparison to viscous hydrodynamic or ``hybrid'' hydrodynamics + transport simulations. In either case, one has to convert the fluid to hadrons, yet without additional theory input the conversion is ambiguous for dissipative fluids. Here, shear viscous phase-space corrections calculated using linearized transport theory are applied in Cooper-Frye freeze-out to quantify the effects on anisotropic flow coefficients ${v}_{n}({p}_{T})$ at the energies available at both the BNL Relativistic Heavy Ion Collider and the CERN Large Hadron Collider. Expanding upon our previous flow harmonics studies [D. Molnar and Z. Wolff, Phys. Rev. C 95, 024903 (2017); Z. Wolff and D. Molnar, J. Phys.: Conf. Ser. 535, 012020 (2014)], we calculate pion and proton ${v}_{2}({p}_{T})$, ${v}_{4}({p}_{T})$, and ${v}_{6}({p}_{T})$, but here we incorporate a hadron gas that is chemically frozen below a temperature of 175 MeV and use hypersurfaces from realistic viscous hydrodynamic simulations. For additive quark model cross sections and relative phase-space corrections with ${p}^{3/2}$ momentum dependence rather than the quadratic Grad form, we find at moderately high transverse momentum noticeably higher ${v}_{4}({p}_{T})$ and ${v}_{6}({p}_{T})$ for protons than for pions. In addition, the value of $\ensuremath{\eta}/s$ deduced from elliptic flow data differs by nearly 50% from the value extracted using the naive ``democratic Grad'' form of freeze-out distributions. To facilitate the use of the self-consistent viscous corrections calculated here in hydrodynamic and hybrid calculations, we also present convenient parametrizations of the corrections for the various hadron species.

Roles of nonlocal conductivity on spin Hall angle measurement

Spin Hall angle characterizes the rate of spin-charge current conversion and it has become one of the most important material parameters for spintronics physics and device application. A long-standing controversy is that the spin Hall angles for a given material measured by spin pumping and by spin Hall torque experiments are inconsistent and they could differ by as much as an order of magnitude. By using the linear response spin transport theory, we explicitly formulate the relation between the spin Hall angle and measured variables in different experiments. We find that the nonlocal conductivity inherited in the layered structure plays a key role to resolve conflicting values of the spin Hall angle. We provide a generalized scheme for extracting spin transport coefficients from experimental data.

Geometric formulation of the Cauchy invariants for incompressible Euler flow in flat and curved spaces

Cauchy invariants are now viewed as a powerful tool for investigating the Lagrangian structure of three-dimensional (3D) ideal flow (Frisch & Zheligovsky,Commun. Math. Phys., vol. 326, 2014, pp. 499–505; Podviginaet al.,J. Comput. Phys., vol. 306, 2016, pp. 320–342). Looking at such invariants with the modern tools of differential geometry and of geodesic flow on the space SDiff of volume-preserving transformations (Arnold,Ann. Inst. Fourier, vol. 16, 1966, pp. 319–361), all manners of generalisations are here derived. The Cauchy invariants equation and the Cauchy formula, relating the vorticity and the Jacobian of the Lagrangian map, are shown to be two expressions of this Lie-advection invariance, which are duals of each other (specifically, Hodge dual). Actually, this is shown to be an instance of a general result which holds for flow both in flat (Euclidean) space and in a curved Riemannian space: any Lie-advection invariant$p$-form which is exact (i.e. is a differential of a$(p-1)$-form) has an associated Cauchy invariants equation and a Cauchy formula. This constitutes a new fundamental result in linear transport theory, providing a Lagrangian formulation of Lie advection for some classes of differential forms. The result has a broad applicability: examples include the magnetohydrodynamics (MHD) equations and various extensions thereof, discussed by Lingamet al.(Phys. Lett. A, vol. 380, 2016, pp. 2400–2406), and include also the equations of Tao (2016,arXiv:1606.08481 [math.AP]), Euler equations with modified Biot–Savart law, displaying finite-time blow-up. Our main result is also used for new derivations, and several new results, concerning local helicity-type invariants for fluids and MHD flow in flat or curved spaces of arbitrary dimension.

Statistical thermodynamics of thermal diffusion factors in binary hydrocarbon mixtures – an application

ABSTRACTThermal diffusion factors play key roles in the analysis of composition variation of the mixture components in hydrocarbon reservoirs and to ensure reliable formation fluid evaluation. This paper presents the statistical thermodynamics of thermal diffusion factors of binary hydrocarbon systems using the modified theory of linear transport. The theory is revised to include the effect of pressure and unlike interaction parameters in calculating the unlike collision integrals and account for the explicit effects of the intermolecular interactions, molecular mass, energy and size parameters. The dependence of the diffusion factors on temperature, pressure and composition of the mixtures is investigated in a systematic way. Comparisons of calculated results with experimental data show that the revised theory could predict the thermal diffusion factors of binary nonhydrocarbon gas mixtures and liquid hydrocarbon mixtures very well over a range of temperature and pressure including the reservoir conditions.

How to Construct Three-Dimensional Transport Theory Using Rotated Reference Frames

ABSTRACTIn linear transport theory, 3D equations reduce to 1D equations by means of rotated reference frames. In this paper, we illustrate how the technique works and 3D transport theories are obtained.