Local Equilibrium Distribution Function Research Articles

Hydrodynamic Navier-Stokes equations in two-dimensional systems with Rashba spin-orbit coupling

We study a two-dimensional (2D) electron system with a linear spectrum in the presence of Rashba spin-orbit (RSO) coupling in the hydrodynamic regime. We derive a semiclassical Boltzmann equation with a collision integral due to Coulomb interactions on the basis of the eigenstates of the system with RSO coupling. Using the local equilibrium distribution functions, we obtain a generalized hydrodynamic Navier–Stokes equation for electronic systems with RSO coupling. In particular, we discuss the influence of the spin-orbit coupling on the viscosity and the enthalpy of the system and present some of its observable effects in hydrodynamic transport.

Solving 2D damped Kuramoto-Sivashinsky with multiple relaxation time lattice Boltzmann method

Lattice Boltzmann method (LBM) is a powerful fluid flow solver. Using this method to solve other PDEs might be a difficult task. The first challenge is to find a suitable local equilibrium distribution function (EDF) capable of recovering the desired PDE. The next difficulty arises from the explicit nature of LBM. The conditional stability of the LBM algorithm affects the numerical solution accuracy. Damped Kuramoto–Sivashinsky (DKS) equation is a fourth–order PDE that recently challenged many numerical methods' abilities. This equation is highly sensitive to a parameter that causes three states of solutions in a small interval of freedom. In this paper, we challenged LBM to solve the two–dimensional DKS equation by finding EDF using the Chapman–Enskog analysis up to the fourth–order. Also, the von Neumann analysis and a simple genetic algorithm are applied to find reliable values for the free parameters. Furthermore, a modification on image-based ghost node method is proposed for implementation of the boundary conditions in the complex geometries.

Gas‐kinetic unified algorithm for two‐dimensional planar and axisymmetric nozzle flows

SummaryFor high‐altitude nozzle or micronozzle flows and other gas flows in the slip or transition flow regime, how to solve it reliably has always been a difficult problem. In this paper, a unified gas kinetic expression is presented to describe the two‐dimensional planar and the axisymmetric nozzle flows, and the computable modeling of the Boltzmann equation is developed at the first time for the nozzle flows by introducing the gas molecular collision relaxing parameter and the local equilibrium distribution function integrated in the unified expression with the flow state controlling parameter, including the macroscopic flow variables, the gas viscosity transport coefficient, the thermodynamic effect, the molecular power law, and molecular models, covering a full spectrum of flow regimes. The boundary conditions involved in the nozzle flow have been mathematically expressed at the level of the gas molecular velocity distribution function, and the model equations have been solved uniformly by the gas‐kinetic unified algorithm (GKUA) for rarefied transition to continuum flows. The presented simulated results from the test cases show promising of simulating gas flow from transitional flow to continuum flow in the same flow domain, especially for those flow involving low‐speed gas flow in rarefied regime, which is otherwise practically difficult using direct simulation Monte Carlo (DSMC).

Numerical solutions to the Sharma–Tasso–Olver equation using Lattice Boltzmann method

AbstractA novel numerical scheme depending on Lattice Boltzmann method is proposed to solve the Sharma–Tasso–Olver equation in one, two and three dimension. The local equilibrium distribution functions and modified functions are obtained via Taylor expansion and Chapman–Enskog multiscale expansion techniques. The macro equation is recovered accurately based on the above functions, and the stability conditions of the equations are deduced. In addition, through simulating numerically some of the initial‐boundary value problems of multidimensional STO equations, the results demonstrate that the numerical solutions and the exact solutions tend to be identical. It also indicates that the model is feasible within a specific range and provides a reliable the approach for solving the multidimensional STO equations.

General Propagation Lattice Boltzmann Model for the Boussinesq Equation.

A general propagation lattice Boltzmann model is used to solve Boussinesq equations. Different local equilibrium distribution functions are selected, and the macroscopic equation is recovered with second order accuracy by means of the Chapman–Enskog multi-scale analysis and the Taylor expansion technique. To verify the effectiveness of the present model, some Boussinesq equations with initial boundary value problems are simulated. It is shown that our model can remain stable and accurate, which is an effective algorithm worthy of promotion and application.

Thermal spin–orbit torque in spintronics

Based on the spinor Boltzmann equation (SBE) formalism, we present a theory of temperature-dependent thermal spin–orbit torque for a system in the presence of Rashba spin–orbit interaction. Under the local equilibrium assumption, we can expand the distribution function of spinor Boltzmann equation around local equilibrium distribution; then, the spin diffusion equation can be derived from SBE, where the spin transfer torque, spin orbit torque as well as thermal spin–orbit torque can be naturally obtained. It is shown that this thermal spin–orbit torque originates from the temperature gradient of local equilibrium distribution function, which is explicit and straightforward than previous works. Finally, we illustrate them by an example of spin-polarized transport through a ferromagnet with Rashba spin–orbit coupling, in which those torques driven whatever by temperature gradient or bias are manifested quantitatively.Graphic abstractWe proposed a new expression for the thermal spin–orbit torque, which can be unified with the usual spin orbit torque as a generalized spin orbit torque.

Nonequilibrium temperature, entropy production, and bulk viscosity of one-dimensional normal shock waves.

The nonequilibrium temperature in the kinetic theory of gases is reexamined and an alternative definition of the temperature in terms of the local equilibrium distribution function is proposed. The alternative definition introduces a new physical quantity, 'exoenergy,' which represents the nonequilibrium nature of thermodynamic systems. The internal energy equation is split into two equations, the temperature equation and the exoenergy equation. In order to rationalize the equation splitting, the nonequilibrium thermodynamics is considered introducing the nonequilibrium entropy phenomenologically. The proposed temperature equation resolves the overshooting anomaly of temperature profiles of the Monte Carlo data for one-dimensional normal shock waves. The exoenergy equation makes the theory self-consistent and gives the entropy production of shock waves in closed form. The theory gives a general form of the shock wave equation and the general relation of the bulk viscosity to the shear viscosity and the heat conductivity of dilute monatomic gases.

Lattice Boltzmann method for shallow water flow with wave radiation stress

The protection of the marine environment is of growing concern, and the development of hydrodynamic models is required to study ocean processes. In this work, a two-dimensional hydrodynamic lattice Boltzmann model was developed to study wave–current coupling problems that combine wave radiation, bed shear stress and wind shear stress. The model is based on nonlinear shallow water equations and two different forms are developed to model wave radiation stress. The first form uses an additional external force term, including a weight factor in the lattice Boltzmann equation at the streaming step. The second form modifies a local equilibrium distribution function at the collision step. Von Neumann stability method is used to study the two schemes. This model was verified by three classic cases: waves diffracted by a straight vertical wall, waves diffracted by two breakwaters, and waves in a circular channel. The numerical results demonstrate the validity of the proposed model, indicating that it can accurately describe the characteristics of water flow with interactions between waves and currents.

Angular mode expansion of the Boltzmann equation in the small-angle approximation

We use an expansion in angular mode functions in order to solve the Boltzmann equation for a gluon plasma undergoing longitudinal expansion. By comparing with the exact solution obtained numerically by other means we show that the expansion in mode functions converges rapidly for all cases of practical interest, and represents a substantial gain in numerical effort as compared to more standard methods. We contrast the cases of a non expanding plasma and of longitudinally expanding plasmas, and follow in both cases the evolutions towards thermalization. In the latter case, we observe that, although the spherical mode function appears to be well reproduced after some time by a local equilibrium distribution function depending on slowly varying temperature and chemical potential, thereby suggesting thermalization of the system, the longitudinal and transverse pressures take more time to equilibrate. This is because the expansion hinders the relaxation of the first angular mode function. This feature was also observed in a simpler context where the Boltzmann equation is solved in terms of special moments within the relaxation time approximation, and attributed there to the particular coupling between the first two moments of the distribution function. The present analysis confirms this observation in a more realistic setting.

Mesoscopic Simulation of the (2 + 1)-Dimensional Wave Equation with Nonlinear Damping and Source Terms Using the Lattice Boltzmann BGK Model.

In this work, we develop a mesoscopic lattice Boltzmann Bhatnagar-Gross-Krook (BGK) model to solve (2 + 1)-dimensional wave equation with the nonlinear damping and source terms. Through the Chapman-Enskog multiscale expansion, the macroscopic governing evolution equation can be obtained accurately by choosing appropriate local equilibrium distribution functions. We validate the present mesoscopic model by some related issues where the exact solution is known. It turned out that the numerical solution is in very good agreement with exact one, which shows that the present mesoscopic model is pretty valid, and can be used to solve more similar nonlinear wave equations with nonlinear damping and source terms, and predict and enrich the internal mechanism of nonlinearity and complexity in nonlinear dynamic phenomenon.

Nonlinear responses of chiral fluids from kinetic theory

The second-order nonlinear responses of inviscid chiral fluids near local equilibrium are investigated by applying the chiral kinetic theory (CKT) incorporating side-jump effects. It is shown that the local equilibrium distribution function can be non-trivially introduced in a co-moving frame with respect to the fluid velocity when the quantum corrections in collisions are involved. For the study of anomalous transport, contributions from both quantum corrections in anomalous hydrodynamic equations of motion and those from the CKT and Wigner functions are considered under the relaxation-time (RT) approximation, which result in anomalous charge Hall currents propagating along the cross product of the background electric field and the temperature (or chemical-potential) gradient and of the temperature and chemical-potential gradients. On the other hand, the nonlinear quantum correction on the charge density vanishes in the classical RT approximation, which in fact satisfies the matching condition given by the anomalous equation obtained from the CKT.

一类含源Boussinesq系统解的数值分析及仿真

For a class of initial value problems of the high-order Boussinesq systems with source terms, a D1Q5 nonstandard lattice Boltzmann model (LBM) with correction functions and source terms was proposed. Different local equilibrium distribution functions and correction functions were selected, and the nonlinear wave equation was recovered by means of the Chapman-Enskog multi-scale analysis and the Taylor expansion technique. Some initial boundary value problems of the Boussinesq systems with analytical solutions were simulated to verify the effectiveness of the LBM. The results show that the numerical solutions agree well with the analytical solutions and the norm errors obtained with the LBM are smaller than those with the modified finite difference method (MFDM). Furthermore, some problems without analytical solutions were numerically studied with the present method and the MFDM. The comparison shows that the numerical solutions from the LBM are in good agreement with those from the MFDM, which validates the effectiveness and stability of the proposed model.

LATTICE BOLTZMANN MODEL FOR TWO-DIMENSIONAL GENERALIZED SINE-GORDON EQUATION

The nonlinear sine-Gordon equation arises in various problems in science and engineering. In this paper, we propose a numerical model based on lattice Boltmann method to obtain the numerical solutions of two-dimensional generalized sine-Gordon equation, including damped and undamped sine-Gordon equation. By choosing properly the conservation condition between the macroscopic quantity <i>u</i>_<i>t</i> and the distribution functions and applying the Chapman-Enskog expansion, the governing equation is recovered correctly from the lattice Boltzmann equation. Moreover, the local equilibrium distribution function is obtained. The numerical results of the first three examples agree well with the analytic solutions, which indicates the lattice Boltzmann model is satisfactory and efficient. Numerical solutions for cases involving the most known from the bibliography line and ring solitons are given. Numerical experiments also show that the present scheme has a good long-time numerical behavior for the generalized sine-Gordon equation. Moreover, the model can also be applied to other two-dimensional nonlinear wave equations, such as nonlinear hyperbolic telegraph equation and Klein-Gordon equation.

Numerical Simulation of a Class of Nonlinear Wave Equations by Lattice Boltzmann Method

In this paper, we develop a lattice Boltzmann model for a class of one-dimensional nonlinear wave equations, including the second-order hyperbolic telegraph equation, the nonlinear Klein–Gordon equation, the damped and undamped sine-Gordon equation and double sine-Gordon equation. By choosing properly the conservation condition between the macroscopic quantity \(u_t\) and the distribution functions and applying the Chapman–Enskog expansion, the governing equation is recovered correctly from the lattice Boltzmann equation. Moreover, the local equilibrium distribution function is obtained. The results of numerical examples have been compared with the analytical solutions to confirm the good accuracy and the applicability of our scheme.

Decoupled scheme based on the Hermite expansion to construct lattice Boltzmann models for the compressible Navier-Stokes equations with arbitrary specific heat ratio.

A decoupled scheme based on the Hermite expansion to construct lattice Boltzmann models for the compressible Navier-Stokes equations with arbitrary specific heat ratio is proposed. The local equilibrium distribution function including the rotational velocity of particle is decoupled into two parts, i.e., the local equilibrium distribution function of the translational velocity of particle and that of the rotational velocity of particle. From these two local equilibrium functions, two lattice Boltzmann models are derived via the Hermite expansion, namely one is in relation to the translational velocity and the other is connected with the rotational velocity. Accordingly, the distribution function is also decoupled. After this, the evolution equation is decoupled into the evolution equation of the translational velocity and that of the rotational velocity. The two evolution equations evolve separately. The lattice Boltzmann models used in the scheme proposed by this work are constructed via the Hermite expansion, so it is easy to construct new schemes of higher-order accuracy. To validate the proposed scheme, a one-dimensional shock tube simulation is performed. The numerical results agree with the analytical solutions very well.

Numerical simulation of a class of FitzHugh-Nagumo systems based on the lattice Boltzmann method

The lattice Boltzmann method (LBM) was proposed as a novel mesoscopic numerical method, and is widely used to simulate complex nonlinear fluid systems. In this paper, we develop a lattice Boltzmann model with amending function and source term to solve a class of initial value problems of the FitzHugh Nagumo systems, which arises in the periodic oscillations of neuronal action potential under constant current stimulation higher than the threshold value. Firstly, we construct a non-standard lattice Boltzmann model with the proper amending function and source term. For different evolution equations, local equilibrium distribution functions and amending function are selected, and the nonlinear FitzHugh Nagumo systems can be recovered correctly by using the Chapman Enskog multi-scale analysis. Secondly, through the integral technique, we obtain a new method on how to construct the amending function. In order to guarantee the stability of the present model, the L stability of the lattice Boltzmann model is analyzed by using the extremum principle, and we get a sufficient condition for the stability that is the initial value u0(x) must satisfy |u0(x)|1 and the parameters must satisfy i-(1+)(t)/(x), (i=1-4). Thirdly, based on the results of the grid independent analysis and numerical simulation, it can be concluded that the present model is convergent with two order space accuracy. Finally, some initial boundary value problems with analytical solutions are simulated to verify the effectiveness of the present model. The results are compared with the analytical solutions and numerical solutions obtained by the modified finite difference method (MFDM). It is shown that the numerical solutions agree well with the analytical solutions and the global relative errors obtained by the present model are smaller than the MFDM. Furthermore, some test problems without analytical solutions are numerically studied by the present model and the MFDM. The results show that the numerical solutions obtained by the present model are in good agreement with those obtained by the MFDM, which can validate the effectiveness and stability of the LBM. In conclusion, our model not only can enrich the applications of the lattice Boltzmann model in simulating nonlinear partial difference equations, but also help to provide valuable references for solving more complicated nonlinear partial difference systems. Therefore, this research has important theoretical significance and application value.

Coherent flavour oscillation and CP violating parameter in thermal resonant leptogenesis

Solving the Kadanoff-Baym (KB) equations in a different method from our previous analysis, we obtain the CP violating parameter $\varepsilon$ in the thermal resonant leptogenesis without assuming smallness of the off-diagonal Yukawa couplings. For that purpose, we first derive a kinetic equation for density matrix of RH neutrinos with almost degenerate masses $M_i \ (i=1,2) \sim M$. If the deviation from thermal equilibrium is small, the differential equation is reduced to a linear algebraic equation and the density matrix can be solved explicitly in terms of the time variation of (local) equilibrium distribution function. The obtained CP-violating parameter $\varepsilon_i$ is proportional to an enhancement factor $(M^{2}_{i}-M^{2}_{j}) M_i \Gamma_j/ ((M^{2}_{i}-M^{2}_{j})^2 +R_{ij}^2)$ with a regulator $R_{ij}=M (\Gamma_i + \Gamma_j)$, consistent with the previous analysis. The decay width can be determined systematically by the 1PI self-energy of the RH neutrinos in the 2PI formalism.

Determination of bed elevation in the enhanced lattice Boltzmann method for the shallow-water equations

The bed slope in the shallow-water equations reflects the bed topography. It is not a flow variable and cannot be determined in the solution to the flow equations. An immovable nonflat bed affects a flow as a force term, but the flow has no effect on it. Only when the bed term is correctly represented in a numerical method can it generate an accurate solution. In the enhanced lattice Boltzmann method for the shallow-water equations (eLABSWE), using an individual Chapman-Enskog analysis, it is found that such a correct representation can be achieved by retaining C(α)=2λ(α), in which C(α) is the coefficient for bed elevation in the lattice Boltzmann equation and λ(α) is that for the water depth in the local equilibrium distribution function. The finding has been validated through simulations of a water at rest in a dish-shaped lake, a wind-induced shallow flow in the same lake, and a steady flow over a two-dimensional bed hump.

Convergence proof of the DSMC method and the Gas-Kinetic Unified Algorithm for the Boltzmann equation

This paper investigates the convergence proof of the Direct Simulation Monte Carlo (DSMC) method and the Gas-Kinetic Unified Algorithm in simulating the Boltzmann equation. It can be shown that the particle velocity distribution function obtained by the DSMC method converges to a modified form of the Boltzmann equation, which is the equation of the gas-kinetic unified algorithm to directly solve the molecular velocity distribution function. Their convergence is derived through mathematical treatment. The collision frequency is presented using various molecular models and the local equilibrium distribution function is obtained by Enskog expansion using the converged equation of the DSMC method. These two expressions agree with those used in the unified algorithm. Numerical validation of the converging consistency between these two approaches is illustrated by simulating the pressure driven Poiseuille flow in the slip transition flow regime and the two-dimensional and three-dimensional flows around a circular cylinder and spherical-cone reentry body covering the whole flow regimes from low speed micro-channel flow to high speed non-equilibrium aerothermodynamics.

On the effect of superfluid flows on the interaction of microwaves with He II

The paper proposes a possible mechanism of interaction of microwaves with superfluid helium that results in an experimentally observed narrow peak of microwave absorption on the frequencies by the order of the roton frequency. The obtained microwave photon absorption coefficient depends on the local equilibrium distribution function which is established due to fast roton-roton and roton-phonon interactions. With the availability of superfluid flows, the local equilibrium distribution function depends on their velocity. The critical velocity of the flows, at which the absorption of microwaves is replaced by their radiation, is found.