Nonequilibrium Distribution Function Research Articles

Analytical irreversible thermodynamics in conduction-radiative heat transfer of neutral gases through a lens of thermal radiation fields

An analytical investigation has been conducted on the steady-state conduction and a diluted gas exposed to a thermal radiation field experiences radiative heat transfer. Analytical solutions have been derived for the transport Boltzmann BGK partial differential equations system. The investigation delves into the irreversible thermodynamic behavior of the system through the utilization of the Liu-Lees model, which employs two-stream Maxwellian distribution functions. The moment approach is utilized to predict the behavior of macroscopic gas parameters like temperature, concentration, and Fourier heat flux. This process entails substituting these parameters into the two-stream Maxwellian distribution functions to analyze the system's non-equilibrium thermodynamic characteristics, which encompass gas particles and the heated plate. Through this analysis, kinetic coefficients, entropy, entropy flux, entropy production, and thermodynamic forces are determined. The investigation assesses the validation of Onsager's reciprocity relation, the Boltzmann H-theorem, the Le Chatelier principle, and the second law of thermodynamics within the system. Furthermore, the Gibbs formula is employed to estimate the ratios between various contributions to internal energy changes, which are determined by the total derivatives of extensive parameters. The non-equilibrium velocity distribution function was calculated for the first time, and its behavior was compared with that of the equilibrium one using 3D graphics. The behavior of the computed variables is predicted through graphical analysis and discussion of the results.

Magnetic field induced acceleration or deceleration of bubble rising inside ferrofluids: A fractional step lattice Boltzmann investigation

Bubble rising process inside a ferrofluid commonly occurs in industrial applications, but the interface deformation, the bubble motion, and their underlying mechanisms are not fully understood. In the present study, a magnetic field coupling fractional step lattice Boltzmann model is employed to simulate the complex interface behaviors and the magnetic flux density distribution during the bubble rising process. The magnetic field is solved by a self-correcting Poisson equation solver, and the flow properties and the interface are first predicted by the equilibrium distribution functions and then corrected by the nonequilibrium distribution functions and source terms. The acceleration and deceleration of the rising bubble under the combination effects of the hydrodynamic force, the surface tension force, and the magnetic force are investigated. This study aims to reveal the mechanisms of the magnetic effects, especially the magnetic field induced bubble interface deformation, and the acceleration and deceleration effects on the bubble rising process. Applying a vertical uniform magnetic field, the bubble rising velocity accelerates with the magnetic field strength, and various shape types are observed. However, when a horizontal uniform magnetic field is applied, the bubble rising velocity decelerates with an increase in the magnetic field strength. Meanwhile, the initial shape remains elongated only along the magnetic field direction. The numerical results from both the conditions show that the external magnetic field indirectly impacts on the bubble rising process through the squeezing effect generated by the surrounding magnetized ferrofluid. A shape regime map is presented for different bubble shapes during the rising process over wide range of the parameter space. Besides, by employing an appropriate combination of the surface tension and the magnetic field strength and direction, it is possible to either accelerate or decelerate the rising bubble. These findings not only provide valuable insights into the underlying mechanisms of the bubble deformation and the rising velocity, but also offer potential technical support for industrial applications.

A Parametric Kinetic Solver for Simulating Boundary-Dominated Turbulent Flow Phenomena

Boundary layer flow plays a very important role in shaping the entire flow feature near and behind obstacles inside fluids. Thus, boundary treatment methods are crucial for a physically consistent fluid simulation, especially when turbulence occurs at a high Reynolds number, in which accurately handling thin boundary layer becomes quite challenging. Traditional Navier-Stokes solvers usually construct multi-resolution body-fitted meshes to achieve high accuracy, often together with near-wall and sub-grid turbulence modeling. However, this could be time-consuming and computationally intensive even with GPU accelerations. An alternative and much faster approach is to switch to a kinetic solver, such as the lattice Boltzmann model, but boundary treatment has to be done in a cut-cell manner, sacrificing accuracy unless grid resolution is much increased. In this paper, we focus on simulating the boundary-dominated turbulent flow phenomena with an efficient kinetic solver. In order to significantly improve the cut-cell-based boundary treatment for higher accuracy without excessively increasing the simulation resolution, we propose a novel parametric boundary treatment model, including a semi-Lagrangian scheme at the wall for non-equilibrium distribution functions, together with a purely link-based near-wall analytical mesoscopic model by analogy with the macroscopic wall modeling approach, which is yet simple to compute. Such a new method is further extended to handle moving boundaries, showing increased accuracy. Comprehensive analyses are conducted, with a variety of simulation results that are both qualitatively and quantitatively validated with experiments and real life scenarios, and compared to existing methods, to indicate superiority of our method. We highlight that our method not only provides a more accurate way for boundary treatment, but also a valuable tool to control boundary layer behaviors. This has not been achieved and demonstrated before in computer graphics, which we believe will be very useful in practical engineering.

Full-band modeling of AM and FM interband cascade laser frequency combs

A numerical model for the simulation of frequency combs in interband cascade lasers is developed. The approach is based on slowly varying amplitude approximation for electromagnetic field and semiconductor Bloch equations for the electron gas. The electron gas nonequilibrium distribution function is computed explicitly. Effects of both radiative and nonradiative recombination are considered. The model is applied to both amplitude modulated and frequency modulated combs.

Electrical resistivity and the Hall effect in the doped Mott-Hubbard material with strong spin-charge coupling

The kinetic characteristics of the doped Mott-Hubbard material are considered within the realistic spin-fermion model which takes into account the strong spin-charge coupling. The kinetic equation constructed on the basis of the mechanism of carrier scattering on the spin fluctuations is solved using the multi-moment method, which allows one to analyze the temperature behavior of nonequilibrium distribution function in the problems of electrical resistivity ρ and the Hall coefficient R H . The calculated dependences ρ(T) and R H (T) for the underdoped and optimally doped regimes demonstrate good qualitative agreement with the experimental data. In particular, the Hall coefficient calculated for the underdoped regime reproduces the experimentally observed sharp drop and even a change in sign at low temperatures.

Energy magnetization and transport in systems with a non-zero Berry curvature in a magnetic field

We demonstrate that the well-known expression for the charge magnetization of a sample with a non-zero Berry curvature can be obtained by demanding that the Einstein relation holds for the electric transport current. We extend this formalism to the transport energy current and show that the energy magnetization must satisfy a particular condition. We provide a physical interpretation of this condition, and relate the energy magnetization to circulating energy currents in Chern insulators due to chiral edge states. We further recover the expression for the energy magnetization with this alternative formalism. We also solve the Boltzmann Transport Equation for the non-equilibrium distribution function in 2D for systems with a non-zero Berry curvature in a magnetic field. This distribution function can be used to obtain the regular Hall response in time-reversal invariant samples with a non-zero Berry curvature, for which there is no anomalous Hall response.

A discrete unified gas kinetic scheme for simulating transient hydrodynamics in porous media with fractures

Tight gas reservoirs have typical characteristics of well-developed natural fractures, extremely low permeability, and low porosity. The flow in such media can be described by a dual-porosity dual-permeability (DPDP) model with the consideration of gas slippage and real gas effects. Based on this model, in this study, a discrete unified gas kinetic scheme is developed to describe the transient hydrodynamic behaviors in porous media with fractures. With an interporosity term, two kinetic equations are employed to capture the pressure propagation in matrix and fracture sub-systems, respectively. The proposed DUGKS can correctly recover the DPDP model through the Chapman–Enskog analysis. Unlike the traditional numerical methods, the velocity in DUGKS can be calculated locally by the non-equilibrium distribution functions. The presented method is verified by three typical problems, including the flow in a fractured porous medium, the flow around a vertical production well, and the flow around a fractured horizontal production well. The numerical results are in good agreement with the reference data. In the test, the global relative errors in pressures are on the order of 10−3. Moreover, the transient hydrodynamics in porous media with fractures are evaluated. The simulation results show that the slippage mainly occurs in the matrix, and this effect may result in a 20% increase in permeability. With the consideration of slippage and real gas effects, the predicted production is higher than that predicted by Darcy’s law. In addition, the results indicate that hydraulic fractures have a great impact on production.

A novel one-step simplified lattice Boltzmann method and its application to multiphase flows with large density ratio

Recently, a one-step simplified lattice Boltzmann method abandoning the original predictor–corrector scheme has been proposed for single-phase flows. In this method, the information of non-equilibrium distribution function (DF) is implicitly included in the difference of two equilibrium DFs at two different locations and time levels. Due to this treatment, the one-step method faces challenges such as extra virtual memory cost and additional boundary treatments. To overcome these drawbacks, a novel one-step simplified lattice Boltzmann method (NOSLBM) is developed by directly constructing the non-equilibrium DF with macroscopic variables. The NOSLBM preserves the merits of high computational efficiency and simple code programming in the original one-step method. Moreover, the present method is extended to multiphase flows. One NOSLBM for the solution of the Cahn–Hilliard equation is employed to capture the interface. Another one is adopted to solve the Navier–Stokes equations for the hydrodynamic fields. Numerical tests about interface capturing and single-phase flows indicate that the present method has a better performance on computational efficiency than that of the simplified multiphase lattice Boltzmann method (SMLBM), in which the predictor–corrector scheme is applied. Numerical tests about binary fluids with large density ratio imply the great accuracy and numerical stability of the present method.

Inferring equilibrium transition rates from nonequilibrium protocols

We develop a theory for inferring equilibrium transition rates from trajectories driven by a time-dependent force using results from stochastic thermodynamics. Applying the Kawasaki relation to approximate the nonequilibrium distribution function in terms of the equilibrium distribution function and the excess dissipation, we formulate a nonequilibrium transition state theory to estimate the rate enhancement over the equilibrium rate due to the nonequilibrium protocol. We demonstrate the utility of our theory in examples of pulling of harmonically trapped particles in one and two dimensions, as well as a semiflexible polymer with a reactive linker in three dimensions. We expect our purely thermodynamic approach will find use in both molecular simulation and force spectroscopy experiments.

Relaxation-time model for the post-Newtonian Boltzmann equation

The non-equilibrium contributions to the post-Newtonian hydrodynamic equations are determined from a relaxation-time model of the post-Newtonian Boltzmann equation. The Chapman- Enskog method is used to calculate the non-equilibrium distribution function. The components of the energy–momentum tensor are found from the knowledge of the non-equilibrium and the post-Newtonian equilibrium Maxwell–Jüttner distribution functions. The linearized field equations for the mass, momentum and internal energy densities coupled with the three Poisson equations of the post-Newtonian approximation are investigated by considering a plane wave representation of the fields. The constitutive equations for the viscous stress and heat flux vector are obtained and it is shown that the transport coefficients of shear viscosity and heat conductivity do depend on the Newtonian gravitational potential.

Generalized Einstein relation in disordered organic semiconductors: Influence of the acoustic phonons–charge carriers scattering

In this work, we analyze the generalized Einstein relation for disordered organic semiconductors with a non-equilibrium Druyvesteyn-type distribution function. The Druyvesteyn behavior of hot electrons in a solid state is associated with the acoustic phonons–charge carriers scattering. Such a case has been experimentally demonstrated in electroluminescent inorganic rare–earth–doped zinc chalcogenides. Therefore, we can assume that, in a part of organic materials used in organic light-emitting diodes (OLEDs), we can also find the Druyvesteyn-type distribution of charge carriers under external electric fields. It looks that the electric-field-dependent diffusion coefficient, which is observed in the phonon-induced hopping electron transport, plays a key role here. The theoretical analysis of the Einstein relation shows that for lower concentrations of charge carriers, the diffusivity–mobility ratio (D/μ) reaches a value lower than 1 (in kT/q units). The temperature dependencies of D/μ are similar to the tendencies reported for the equilibrium conditions. The obtained satisfactory agreement between experimental results and calculations based on the Druyvesteyn-type distribution function confirms the usefulness of the presented model.

A Non-Equilibrium Interpolation Scheme for IB-LBM Optimized by Approximate Force

A non-equilibrium scheme and an optimized approximate force are proposed for the immersed boundary–lattice Boltzmann method (IB-LBM) to solve the fluid–structure interaction (FSI) equations. This new IB-LBM uses the discrete velocity distribution function and non-equilibrium distribution function to establish the interpolation operator and the spread operator at the mesoscopic scale. In the interpolation operator, we use the force model of LBM to derive a direct force with a simple form. In the spread operator, we give a theoretical proof with local second-order accuracy of the spread process using the non-equilibrium theory from the LBM. A non-iterative explicit force approximation scheme optimizes the direct force in that the streamlines have no penetration phenomenon, and the no-slip condition is strictly satisfied. Different from other schemes for the IB-LBM, we try to apply the non-equilibrium theory from the LBM to the IB-LBM and obtain good results. The explicit force obtained using the non-equilibrium scheme and then optimized via the non-iterative streamline correction equation simplifies the explicit direct force scheme and the original implicit scheme previously proposed but obtains a similar streamline correction result compared with the implicit method. Numerical tests prove the applicability and accuracy of this method in the simulation of complex conditions such as moving rigid bodies and deforming flexible bodies.

Kinetic Nonequilibrium Signatures in the Distribution Function of Earth-Escaping Hydrogen Atoms

The lightest atmospheric gas constituents, like H- and He- atoms, are known to escape from planetary gravitational fields to open space. Hereby it counts that not only the uppermost atmospheric layer, the so-called exobase, contributes to this planetary gas escape, but the layers below do contribute as well, especially since from below a nonthermal character of the final particle outflow is induced.We consider the outflow of H - atoms from lower levels of a planetary oxygen-dominated upper atmosphere, stratified by the planet‘s gravitational field - as given in case of the Earth. The terrestrial H-atom outflow is locally modified by elastic collisions of upwards flying H-atoms with the heavy major atmospheric background constituent. This causes a collision–induced velocity-modulation of the upwards directed H-atom flow and induces nonthermal kinetic signatures of the local H ˗ atom distribution function. An important, hitherto unrespected point hereby is that angle-integrated elastic O ˗ H- collision cross sections are velocity-dependent, falling off with increasing velocity v like (1/v). Consequently this modulation influences low velocity H-atoms stronger than high-velocity ones, which changes the kinetic profile of the escaping H-atoms and causes deviations from the classic Jeans escape. Deeply down in the lower thermosphere the local H-atoms, like as well the O-atoms, indeed are in a thermodynamical equilibrium characterized by Maxwell distributions with a common temperature TH = TO. Nevertheless, at the upper exobase border of the atmosphere the resulting H-atom escape flow turns out to be a, non-equilibrium flow with non-thermal escape-relevant properties. Here we describe this collisional modification of the H-escape flow and quantify the upcome of kinetic non-equilibrium features like power laws in the wings of the H-distribution function. This collisional modulation via velocity-dependent collision cross-sections acts as a typical process to convert equilibrium distributions into non-equilibrium kappa-like distribution functions. On the basis of this theoretical approach and stabilizing the upper atmosphere by the type of so-called "iso-baric Kappa functions" which all represent the same H ˗ atom- pressure we can calculate the effective escape flux of H- atoms to open space and can quantify its difference with respect to the classical Jeans escape value. While finding again that the classical Jeans formula slightly overestimates the actual escape flux, we now for the first time taking into account the nonthermal influence of collisional modulations can show that the actual escape with respect to the actual Jeans value is even enhanced by factors 2 to 3.

Gas Discharge Properties of Inverse-Brush-Electrodes Driven by Alternating Current

The inverse-brush-electrodes (IBE) structure with a large area is designed to investigate the discharge mechanism driven by alternating current (ac). It is found that the IBE structure increases the number of secondary electrons and reduces the breakdown voltage by comparing the discharge processes of IBE and plate electrodes. Meanwhile, the breakdown mechanism of gas discharge driven by ac is analyzed in detail. Additionally, the designed IBE structure has a negative differential conductivity enhancement effect than that of plate electrodes when plasmas are generated in the heavy noble gas. Finally, the negative differential conductivity enhancement phenomenon of the IBE structure is analyzed by experiments and simulations. This effect originates from the non-equilibrium electron energy distribution function (EEDF), which is due to the more suprathermal electrons generated by the IBE structure.

Topological Hall effect in three-dimensional centrosymmetric magnetic skyrmion crystals

Centrosymmetric skyrmions attract much attention from the research community because of their small sizes and high concentrations. These features can be useful for applications. Such skyrmions are originated due to the Ruderman-Kittel-Kasuya-Yosida and Kondo interactions rather than the Dzyaloshinski-Moria interaction. We study a topological Hall effect in such systems using the Boltzmann equation for a nonequilibrium distribution function. For the relaxation, we choose the electron-acoustic phonon and electron-skyrmion interactions. We find that the topological Hall resistivity exhibits nonlinear behavior depending on chemical potential. Because of noncubic lattice symmetry, we investigate the dependence of the resistivity tensor components along the $x$ direction (parallel to an applied electric field), ${\ensuremath{\rho}}_{xx}$; ${\ensuremath{\rho}}_{zz}$ (the component along the $z$ axis); and ${\ensuremath{\rho}}_{xy}$ (a topological Hall component) with respect to the effective mass ratio, ${m}_{z}/{m}_{x}$. We assume that the skyrmions are spaced on the $\mathit{xy}$ plane and stretched out along the $z$ direction. The temperature dependence of the resistivity tensor reveals the monotonic growth for all components. There is some concern in the interpretation of experiments. Sometimes it can be very difficult to measure the topological Hall resistivity. Indeed, we find that ${\ensuremath{\rho}}_{xy}$ is one to two orders of magnitude less than ${\ensuremath{\rho}}_{xx}$ and ${\ensuremath{\rho}}_{zz}$. Additionally, there is another important factor, which complicates the problem. The $z$ axis and an applied electric field are not exactly perpendicular because of experimental conditions. Thus, the perpendicular to the electric field resistivity contains a linear combination of ${\ensuremath{\rho}}_{xy}$ and $|{\ensuremath{\rho}}_{xx}\ensuremath{-}{\ensuremath{\rho}}_{zz}|$. To determine ${\ensuremath{\rho}}_{xy}$ from an experiment, we propose an experimental setup to measure the topological Hall effect and provide the equations that allow us to determine ${\ensuremath{\rho}}_{xy}$.

Unification of kinetic and hydrodynamic approaches in the theory of dense gases and liquids far from equilibrium

A system of non-Markovian transport equations is obtained for the non-equilibrium one-particle distribution function of particles and the non-equilibrium average value of the density of the potential energy of the interaction of the system particles far from the equilibrium state. Expressions for entropy, the partition function of the non-equilibrium state of the system, as well as non-equilibrium thermodynamic relations were obtained. The generalized structure of transfer nuclei is revealed in detail with the selection of short-range and long-range contributions of interactions between particles. The connection of transport nuclei with generalized diffusion coefficients, friction in the space of coordinates and momentum and the potential part of the thermal conductivity coefficient is established.

A magnetic field coupling fractional step lattice Boltzmann model for the complex interfacial behavior in magnetic multiphase flows

In the present study, a magnetic field coupling fractional step lattice Boltzmann (FSLB) model is proposed to simulate the complex interfacial behaviors in magnetic multiphase flows with severe interface deformation and large density ratio, which can be recovered to the corresponding macroscopic governing equations with fully second-order accuracy. Since the equilibrium and the non-equilibrium terms are obtained by the Chapman-Enskog (C-E) expansion analysis, the basic idea of the present method is to predict the intermediate variables by updating the various moments of equilibrium distribution functions first, and then correct the physical variables by updating the various moments of non-equilibrium distribution functions. Besides, the present magnetic field coupling FSLB method does not need the evolution of distribution functions compared to the existing LB methods. The present method inherits the excellent performance from the conventional LB method and the relatively better numerical stability from the matured fractional-step method, which is more stable than the original LB method and more efficiency and easy-to-implement than the LB flux solver. Owing to those good properties, we successfully perform the first rigorous numerical simulation on the Rosensweig instability, which accurately captures the spikes forming at the surface of a ferrofluid under an applied magnetic field. Moreover, for the first time we numerically reproduce the falling ferrofluid droplet impacted on the solid surface. Intriguingly, our simulation results reveal that the falling velocity of the ferrofluid droplet is slightly accelerated by the elongation, although the direction of the magnetic field is opposite to that of the gravity. All the results for the interfacial behavior and the magnetic interaction highlights the comparative capability, stability, and accuracy of the present magnetic field coupling FSLB method for simulating the problems of complex fluid dynamics associated with a severe interface deformation and large density ratio in magnetic multiphase flows.

Quadrature-based moment methods for kinetic plasma simulations

Quadrature-based moment methods (QBMM) are applied to rarefied gas flow and plasma physics problems represented by the Boltzmann equation or the Vlasov-Poisson system of equations. QBMM are a computationally advantageous alternative to both direct Eulerian and Lagrangian particle solvers, able to provide a noise-free solution to the Boltzmann equation and capture non-equilibrium velocity distribution functions (VDF) without excessive computational cost. To provide a closure of the moment equations, the VDF is assumed to be represented by the sum of weighted kernel functions that are placed at given velocity abscissas. Three QBMM approaches, QMOM (Dirac delta kernels), HyQMOM (hyperbolic QMOM), and EQMOM (Gaussian kernels), are applied to canonical one-dimensional rarefied gas flow and plasma physics problems. A new algorithm for EQMOM that is able to treat pathological VDFs and provide control over large abscissas for more than two nodes is developed. QMOM and EQMOM are suitable for problems where the flow does not have shock-like features (for QMOM) or a discontinuity in the VDF (for EQMOM) and where the VDF is far from non-equilibrium and features strong phase-mixing. HyQMOM is generally applicable to all test problems and yields good representations of both the moments and the VDF. HyQMOM also provides significant wall-time improvement (factor of 10–100) compared to other direct Eulerian solvers.

The Mott-Smith solution to the regular shock reflection problem

The classical Mott-Smith solution for one-dimensional normal shock wave structure is extended to the two-dimensional regular shock reflection problem. The solution for the non-equilibrium molecular velocity distribution function along the symmetry-plane streamline is obtained as a weighted sum of four Maxwellians. An analysis of applicability of the solution has been performed using the results of direct simulation Monte Carlo calculations for a range of incident shock wave intensities. Accuracy of the solution improves with increasing$Ma_n$, the Mach number normal to the shock front, so that the solution becomes rather accurate for strong shocks with$Ma_n>8$.

Self-consistent theory of current injection into d and d + is superconductors

We present results for the steady-state nonlinear response of a superconducting film connected to normal-metal reservoirs under voltage bias, allowing for a subdominant s-wave component appearing near the interfaces. Our investigation is based on a current-conserving theory that self-consistently includes the non-equilibrium distribution functions, charge imbalance, and the voltage-dependencies of order parameters and scalar impurity self-energies. For a pure d-wave superconductor with [110] orientation of the interfaces to the contacts, the conductance contains a zero-bias peak reflecting the large density of zero-energy interface Andreev bound states. Including a subdominant s-wave pairing channel, it is in equilibrium energetically favorable for an s-wave order parameter component to appear near the interfaces in the time-reversal symmetry breaking combination d + is. The Andreev states then shift to finite energies in the density of states. Under voltage bias, we find that the non-equilibrium distribution in the contact area causes a rapid suppression of the s-wave component to zero as the voltage . The resulting spectral rearrangements and voltage-dependent scattering amplitudes lead to a pronounced non-thermally broadened split of the zero-bias conductance peak that is not seen in a non-selfconsistent Landauer–Büttiker scattering approach.