Two-particle Equation Research Articles

Monte Carlo study of electron–electron scattering effects in FET channels

In this work, we study the effect of electron–electron scattering on the high-energy tail of the electron distribution function in FET channels. A new Monte Carlo algorithm for solving a two-particle kinetic equation in the presence of a position-dependent electric field is proposed. In stationary bulk simulations, no visible effect of electron–electron scattering on the distribution function and consequently on its moments is observed. In stationary device simulations, however, a clear effect on the high-energy tail can be seen. The enhancement of the tail relative to the thermal tail is found to scale with the electron concentration.

Pair density functional theory for excited states of Coulomb systems

Pair density functional theory is extended to excited states of Coulomb systems. It is shown that the pair density determines the Hamiltonian of the Coulomb system. A universal kinetic energy functional appropriate for the ground and all bound excited states is defined. The Euler equation can be rewritten as a two-particle auxiliary equation in which the unknown Pauli-like potential should be approximated.

Many-body theory of positronium scattering and pickoff annihilation in noble-gas atoms

The many-body-theory approach to positronium-atom interactions developed in [Phys. Rev. Lett. \textbf{120}, 183402 (2018)] is applied to the sequence of noble-gas atoms He-Xe. The Dyson equation is solved separately for an electron and positron moving in the field of the atom, with the entire system enclosed in a hard-wall spherical cavity. The two-particle Dyson equation is solved to give the energies and wave functions of the Ps eigenstates in the cavity. From these, we determine the scattering phase shifts and cross sections, and values of the pickoff annihilation parameter $^1Z_\text{eff}$ including short-range electron-positron correlations via vertex enhancement factors. Comparisons are made with available experimental data for elastic and momentum-transfer cross sections and $^1Z_\text{eff}$. Values of $^1Z_\text{eff}$ for He and Ne, previously reported in [Phys. Rev. Lett. \textbf{120}, 183402 (2018)], are found to be in near-perfect agreement with experiment, and for Ar, Kr, and Xe within a factor of 1.2.

Model two-particle kinetic equation for pairs of quasiparticles

Because the main process of interaction in the Boltzmann model of real gases is the binary collision of molecules, it is convenient to use the two-particle kinetic equation to describe the dynamics of a rarefied gas. This equation was written using the same physical assumptions as those used by Ludwig Boltzmann. The right-hand side of this equation contains the product of the linear scattering operator and chaos projector. The Boltzmann equation follows from this equation without any additional approximations after simple integration of the velocities and positions of the second particle. Using the divergence form of the scattering operator, this equation can be represented as the Liouville equation, which implies that real molecules can be replaced by quasiparticles whose distribution function is the same as that of real molecules but whose dynamics are completely different. Pairs of quasiparticles do not collide but move along continuous trajectories in the phase space. The relative velocities in pairs of quasiparticles slowly rotate with an angular velocity vector depending on the distribution function. We provide an explicit approximate expression for the angular velocity through the first few velocity moments, using a special covariant Grad expansion for the velocity distribution function, which reduces to the exact Bobylev–Kruk–Wu solution in the isotropic case. We simulated the relaxation of distribution function to equilibrium and compared results with the existing exact solutions. The described algorithm will be effective for modeling flow regions with low Knudsen numbers, where the standard Direct Monte Carlo Simulation (DSMC) method encounters significant difficulties.

Mechanisms and energetics in the early stages of solvent-assisted low-temperature sintering of ZnO

The Hydro/Solvothermal Sintering (HSS) is a technique enabling the low temperature sintering of ceramics and composites, by using a solvent and uniaxial pressure. This energy-efficient technique also opens new perspectives in designing new composites with tailored functional properties. While the solvent plays a key role in reducing the sintering temperature, the modeling and deep understanding of sintering mechanisms remain an open field of investigation. This study unveils the energetics and mechanisms involved in the first stage of HSS. The strategy highlighted in this paper includes: (i) the building of a hydrothermal sintering device equipped with a dilatometer to monitor the shrinkage in situ, (ii) numerical stress calculations at the contact between particles to show the suitability of the two-particle kinetic equation, and (iii) the use of anisothermal and non-conventional stepwise isothermal methodologies for the investigation of mechanisms and energetics. The model material, ZnO, was densified to high relative densities with acetic acid as a solvent, and a pressure and temperature of 320 MPa and 150 °C, respectively. With these experimental conditions, the kinetic analysis obtained from the two methodologies is consistent and implies a dissolution reaction of the material as the rate controlling mechanism with possible coupling to grain boundary sliding. The activation energy of 90 kJ.mol—1 is determined with the different analysis methodologies. The interest of the stepwise isothermal methodology in obtaining accurate activation energy is also discussed.

Expansion of the Two-particle Dirac Equation in Powers of 1/c to Higher Orders

Using an extension of the Foldy–Wouthuysen transformation to two-particle wave equations, we consider the problem of expansion of the two-body Dirac Hamiltonian in powers of 1/c up to the fourth order. The transformed Hamiltonian in an even-even form toorder 1/c4 is obtained. It is shown that the extra terms which can be eliminated by additional unitary transformations appear in the expansion in higher orders. As an example, the Breit equation for Coulomb particles is considered, and all the terms of order 1/c4 in its reduction are calculated. The obtained results can be used for the expansion of relativistic and quasirelativistic two-particle wave equations to higher orders.

Two-particle kinetic equation: method and exact solution

In the paper (Saveliev, in: AIP conference proceedings, vol 1333, p 134, 2011), the kinetic equation for two-particle distribution function was obtained by making use of exactly the same physical assumptions as Ludwig Boltzmann did. Instead of the collision integral, there are the linear scattering operator and the chaos projector in the right part of this equation. The Boltzmann equation follows from this two-particle equation without any additional assumptions after a simple integration. The article presents the method of generalized functions and considers the properties of the obtained exact solution for the two-particle kinetic equation for Maxwell's molecules, which is an intermediate asymptotics for problems of spatial homogeneous relaxation. After reducing the two-particle distribution function to a one-particle distribution function, the solution is reduced to the well-known Bobylev-Krook-Wu (BKW) mode.

Three-body Faddeev equations in two-particle Alt-Grassberger-Sandhas form with distorted-wave-Born-approximation amplitudes as effective potentials

In this paper, I prove that the sub-Coulomb $A(d,p)B$ reaction amplitude, which is a solution of the three-body Faddeev equations in the Alt-Grassberger-Sandhas (AGS) form, is peripheral if a peripheral is the corresponding DWBA amplitude. Hence the Faddeev's reaction amplitude for the sub-Coulomb $A(d,p)B$ reactions can also be parametrized in terms of the ANC of the $(n\,A)$ bound state. First, I consider the original AGS equations with separable potentials and prove that such equations are peripheral at sub-Coulomb energies. After that, the two-particle AGS equations are derived for the general potentials for sub-Coulomb transfer reactions. The effective AGS potentials are expressed in terms of the DWBA amplitudes for the sub-Coulomb reactions. Again, I demonstrate that the amplitude of the $A(d,p)B$ transfer reaction obtained from the AGS equation is peripheral and can be parametrized in terms of the ANC for the $(nA)$ bound state because the corresponding DWBA amplitude is peripheral. Finally, the AGS equations are generalized by including the optical nuclear potentials in the same manner as it is done in the DWBA. The obtained two-particle AGS equations contain the DWBA effective potentials with distorted waves generated by the sum of the nuclear optical and the channel Coulomb potentials. The AGS equation for the $A(d,p)B$ reactions is analyzed above the Coulomb barrier and it is shown again that the reaction amplitude satisfying generalized AGS equation with optical potentials depends on the ANC if a peripheral is the DWBA amplitude. The two-body AGS equations are generalized by including the intermediate three-body continuum and more than one bound state in each channel.

Direct interaction along light cones at the quantum level

Here, we point out that interactions with time delay can be described at the quantum level using a multi-time wave function , i.e. a wave function depending on one spacetime variable per particle. In particular, such a wave function makes it possible to implement direct interaction along light cones (not mediated by fields), as in the Wheeler–Feynman formulation of electrodynamics. Our results are as follows. (1) We derive a covariant two-particle integral equation and discuss it in detail. (2) It is shown how this integral equation (or equivalently, a system of two integro-differential equations) can be understood as defining the time evolution of ψ in a consistent way. (3) We demonstrate that the equation has strong analogies with Wheeler–Feynman electrodynamics and therefore suggests a possible new quantization of that theory. (4) We propose two natural ways how the two-particle equation can be extended to N particles. It is shown that exactly one of them leads to the usual Schrödinger equation with Coulomb-type pair potentials if time delay effects are neglected.

Generation of Pairwise Potentials Using Multidimensional Data Mining.

The rapid development of molecular structural databases provides the chemistry community access to an enormous array of experimental data that can be used to build and validate computational models. Using radial distribution functions collected from experimentally available X-ray and NMR structures, a number of so-called statistical potentials have been developed over the years using the structural data mining strategy. These potentials have been developed within the context of the two-particle Kirkwood equation by extending its original use for isotropic monatomic systems to anisotropic biomolecular systems. However, the accuracy and the unclear physical meaning of statistical potentials have long formed the central arguments against such methods. In this work, we present a new approach to generate molecular energy functions using structural data mining. Instead of employing the Kirkwood equation and introducing the "reference state" approximation, we model the multidimensional probability distributions of the molecular system using graphical models and generate the target pairwise Boltzmann probabilities using the Bayesian field theory. Different from the current statistical potentials that mimic the "knowledge-based" PMF based on the 2-particle Kirkwood equation, the graphical-model-based structure-derived potential developed in this study focuses on the generation of lower-dimensional Boltzmann distributions of atoms through reduction of dimensionality. We have named this new scoring function GARF, and in this work we focus on the mathematical derivation of our novel approach followed by validation studies on its ability to predict protein-ligand interactions.

Goldstonic pseudoscalar mesons in Bethe–Salpeter-inspired setting

For a two-particle bound-state equation closer to its Bethe–Salpeter origins than Salpeter’s equation, with effective interaction kernel deliberately forged such as to ensure, in the limit of zero mass of the bound-state constituents, the vanishing of the arising bound-state mass, we scrutinize the emerging features of the lightest pseudoscalar mesons for their agreement with the behavior predicted by a generalization of the Gell-Mann–Oakes–Renner relation.

Solution of Relativistic Two-Particle Equations with Arbitrary Orbital Angular Momentum

Bound states of relativistic systems of two particles are considered in the special case of an interaction operator which admits reduction of partial integral equations to differential equations directly in the momentum representation. Exact analytical solutions of these equations are obtained for some values of the energy (mass) of a system of two particles. Equations with other energies are solved numerically. Dependences of the mass spectra of two-particle systems and their wave functions on the parameters are found.

Inversion of Zeeman splitting of exciton states in InGaAs quantum wells

Zeeman splitting of quantum-confined states of excitons in InGaAs quantum wells (QWs) is experimentally found to depend strongly on quantization energy. Moreover, it changes sign when the quantization energy increases with a decrease in the QW width. In the 87-nm QW, the sign change is observed for the excited quantum-confined states, which are above the ground state only by a few meV. A two-step approach for the numerical solution of the two-particle Schroedinger equation, taking into account the Coulomb interaction and valence-band coupling, is used for a theoretical justification of the observed phenomenon. The calculated variation of the g-factor convincingly follows the dependencies obtained in the experiments.

Removal of self-interactions in the Dirac-Maxwell equations in one spatial dimension

We propose a theoretical framework that permits us to eliminate the unphysical self-repulsion that occurs if a spatially localized charged particle interacts with its own electric field. As an example of this framework, we study the time-resolved interaction between an electronic and positronic wave packet by solving the coupled set of two-particle Dirac-Maxwell equations. The restriction of the dynamics to only one spatial dimension permits us to neglect the magnetic field and therefore any effects due to retardation are absent. Here the unwanted self-repulsion can be removed by separating the total electric field into two portions, each of which is generated by only one particle and is evolved independently of the other. For example, the Maxwell equation for the electronic field has only the electronic charge density as a source term and only this field is coupled to the positron in the two-particle Dirac equation.

Minimal model for charge transfer excitons at the dielectric interface

Theoretical description of the charge transfer (CT) exciton across the donor-acceptor interface without the use of a completely localized hole (or electron) is a challenge in the field of organic solar cells. We calculate the total wavefunction of the CT exciton by solving an effective two-particle Schrodinger equation for the inhomogeneous dielectric interface. We formulate the magnitude of the CT and construct a minimal model of the CT exciton under the breakdown of inversion symmetry. We demonstrate that both a light hole mass and a hole localization along the normal to the dielectric interface are crucial to yield the CT exciton.

Weak exchange interactions in biradicals: A pseudopotential for unpaired electrons and an asymptotic methods for calculating the exchange integral

A theory of weak exchange interactions in biradicals is developed. The basic idea is to describe the motion of two weakly bound electrons on paramagnetic centers by a two-particle Schrobinger equation that takes into account the interaction of these electrons with the closed electron shells of the biradical, described by a model local potential, and their mutual Coulomb repulsion. The model potential can be constructed using the pseudopotential method, based on results of ab initio quantum chemical calculations. The exchange interaction between the unpaired electrons on the paramagnetic centers is calculated by a generalized asymptotic method that takes into account the quasi-classical nature of the subbarrier motion of the electrons in the region between the paramagnetic centers. The developed theory makes it possible to estimate the magnitude of the exchange interaction and to determine how this interaction depends on the distance between the paramagnetic centers and their relative orientation.

Excited-state pair-density-functional theory

The theory of pair density is extended to excited states. The theory of a single excited state is generalized for the pair density. A two-particle equation is derived for the square root of the pair density of the given excited state. An expression for the effective potential of this equation is presented.

Stacking effects on the electronic and optical properties of bilayer transition metal dichalcogenidesMoS2,MoSe2,WS2, andWSe2

Employing the random phase approximation we investigate the binding energy and Van der Waals (vdW) interlayer spacing between the two layers of bilayer transition metal dichalcogenides ${\mathrm{MoS}}_{2}$, ${\mathrm{MoSe}}_{2}$, ${\mathrm{WS}}_{2}$, and ${\mathrm{WSe}}_{2}$ for five different stacking patterns, and examine the stacking-induced modifications on the electronic and optical/excitonic properties within the GW approximation with a priori inclusion of spin-orbit coupling and by solving the two-particle Bethe-Salpeter equation. Our results show that for all cases, the most stable stacking order is the high symmetry $A{A}^{\ensuremath{'}}$ type, distinctive of the bulklike $2H$ symmetry, followed by the $AB$ stacking fault, typical of the $3R$ polytypism, which is by only 5 meV/formula unit less stable. The conduction band minimum is always located in the midpoint between K and $\ensuremath{\Gamma}$, regardless of the stacking and chemical composition. All $M{X}_{2}$ undergo an direct-to-indirect optical gap transition going from the monolayer to the bilayer regime. The stacking and the characteristic vdW interlayer distance mainly influence the valence band splitting at K and its relative energy with respect to $\ensuremath{\Gamma}$, as well as, the electron-hole binding energy and the values of the optical excitations.

Translational and rotational dynamics of colloidal particles in suspension: Effect of shear

We report a generalization of a nonequilibrium thermodynamic theory for the mesoscopic dynamics of radially symmetric interacting particles to anisotropic pairwise interactions and attain the one- and two-particle Fokker-Planck kinetics equations at a low-density limit that provides the translational-rotational coupling of their motion due to hydrodynamic interactions, from which we derived the balance equations of linear, angular momentum, and energy dissipation due to particle interactions and energy interchange with heat bath. In this low-density approximation, an already-known virial expression for the long-time translational collective diffusion coefficient of an orientational isotropic suspension in terms of the fluid equilibrium microstructure is recovered. An external shear flow induces, in the diffusive regime, vorticity effects into the rotational diffusion property of the colloidal particles. They manifest in the appearance of the particle's rotational viscosity due to vortex flow. The Smoluchowski equation that governs the dynamical relaxation of colloid microstructure due to particle's Brownian motion under stationary flow is provided.

Explicit form of Pauli potential for direct derivation of pair density from a two-particle differential equation for the quintet state of four electrons with harmonic interparticle interactions

Recently, an analytic two-particle density matrix (2DM) has been derived for the quintet state of four electrons interacting via two-body harmonic forces. Here we use this 2DM to extract the exact pair density \(\Gamma (\mathbf{r}_1,\mathbf{r}_2)\). This is then employed in the known two-particle partial differential equation for the pair density amplitude to extract the Pauli potential \(v_P(\mathbf{r}_1,\mathbf{r}_2)\) for this quintet state.