Linearized Boltzmann Equation Research Articles

On the Derivation of the Linear Boltzmann Equation from the Nonideal Rayleigh Gas

On the Derivation of the Linear Boltzmann Equation from the Nonideal Rayleigh Gas

Lattice Boltzmann Simulation of Nonequilibrium Flows Using Spectral Multiple-Relaxation-Time Collision Model

Prediction of nonequilibrium flows is critical to space flight. The present work demonstrates that the recently developed spectral multiple-relaxation-time (SMRT) lattice Boltzmann (LB) model is theoretically equivalent to Grad’s eigensystem [“Principles of the Kinetic Theory of Gases,” Thermodynamik der Gase/Thermodynamics of Gases, Springer–Verlag, Berlin, 1958, pp. 205–294], where the eigenfunctions obtained by tensor decomposition of the Hermite polynomials are also those of the linearized Boltzmann equation. Numerical results of shock structure simulation using the Maxwell molecular model agree very well with those of a high-resolution fast spectral method code up to Mach 7, provided that the relaxation times of the irreducible tensor components match their theoretical values. If a reduced set of relaxation times is used, such as in the Shakhov model and lumped-sum relaxation of Hermite modes, non-negligible discrepancies start to occur as the Mach number is raised, indicating the necessity of the fine-grained relaxation model. Together with the proven advantages of LB, the LB-SMRT scheme offers a competitive alternative for nonequilibrium flow simulation.

Kinetic theory of tilted Dirac cone materials

We formulate the Boltzmann kinetic equations for interacting electrons with tilted Dirac cones in two space dimensions characterized by a tilt parameter 0≤ζ<1. By solving the linearized Boltzmann equation, we find that the broadening of the Drude pole is enhanced by κ(ζ)×(1−ζ2)−1/2, where the κ is interaction-induced enhancement factor. The intensity of the Drude pole is also anisotropically enhanced by (1−ζ2)−1. The ubiquitous “redshift” factors (1−ζ2)1/2 can be regarded as a manifestation of an underlying spacetime structure in such solids. The additional broadening κ that arises from the interactions cannot be obtained from a simple coordinate change and are more pronounced for electrons in a ζ-deformed Minkowski spacetime of tilted Dirac fermions.

Review of Fourth-Order Maximum Entropy Based Predictive Modeling and Illustrative Application to a Nuclear Reactor Benchmark: II. Best-Estimate Predicted Values and Uncertainties for Model Responses and Parameters

This work continues the review and illustrative application to energy systems of the “Fourth-Order Best-Estimate Results with Reduced Uncertainties Predictive Modeling” (4th-BERRU-PM) methodology. The 4th-BERRU-PM methodology uses the Maximum Entropy (MaxEnt) principle to incorporate fourth-order experimental and computational information, including fourth (and higher) order sensitivities of computed model responses with respect to model parameters. The 4th-BERRU-PM methodology yields the fourth-order MaxEnt posterior distribution of experimentally measured and computed model responses and parameters in the combined phase space of model responses and parameters. The 4th-BERRU-PM methodology encompasses fourth-order sensitivity analysis (SA) and uncertainty quantification (UQ), which were reviewed in the accompanying work (Part 1), as well as fourth-order data assimilation (DA) and model calibration (MC) capabilities, which will be reviewed and illustrated in this work (Part 2). The applicability of the 4th-BERRU-PM methodology to energy systems is illustrated by using the Polyethylene-Reflected Plutonium (acronym: PERP) OECD/NEA reactor physics benchmark, which is modeled using the linear neutron transport Boltzmann equation, involving 21,976 imprecisely known parameters. This benchmark is representative of “large-scale computations” such as those involved in the modeling of energy systems. The result (“response”) of interest for the PERP benchmark is the leakage of neutrons through the outer surface of this spherical benchmark, which can be computed numerically and measured experimentally. The impact of the high-order sensitivities of the response with respect to the PERP model parameters is quantified for “high-precision” parameters (2% standard deviations) and “typical-precision” parameters (5% standard deviations). Analyzing the best-estimate results with reduced uncertainties for the 1st—through 4th-order moments (mean values, covariance, skewness, and kurtosis) produced by the 4th-BERRU-PM methodology for the PERP benchmark indicates that, even for systems modeled by linear equations (e.g., the PERP benchmark), retaining only first-order sensitivities is insufficient for reliable predictive modeling (including SA, UQ, DA, and MC). At least second-order sensitivities should be retained in order to obtain reliable predictions.

Temperature and pressure jump coefficients at a liquid–vapor interface

The temperature and pressure jump coefficients at a liquid–vapor interface are calculated from the solution of the Shakhov kinetic model for the linearized Boltzmann equation. Complete and partial evaporation/condensation at the vapor–liquid interface are assumed as the boundary condition. The discrete velocity method is used to solve the problem numerically. The jump coefficients are tabulated as functions of the evaporation/condensation coefficient. The profiles of the vapor temperature and pressure deviations from that values at the interface corresponding to the liquid temperature and saturation pressure are plotted, and the solutions obtained from kinetic theory and continuum approach are shown to underline the effect of the jumps at the interface. The obtained results have been compared to those given by other authors, who applied the linearized Boltzmann equation as well as the model proposed by Bhatnagar, Gross, and Krook to it, and it was found that the pressure and temperature jump coefficients are relatively insensitive to the collision laws.

Spectral asymptotics and metastability for the linear relaxation Boltzmann equation

We consider the linear relaxation Boltzmann equation in a semiclassical framework. We construct a family of sharp quasimodes for the associated operator which yields sharp spectral asymptotics for its small spectrum in the low temperature regime. We deduce some information on the long time behavior of the solutions with a sharp estimate on the return to equilibrium as well as a quantitative metastability result. The main novelty is that the collision operator is a pseudo-differential operator in the critical class S^{1/2} and that its action on the Gaussian quasimodes yields a superposition of exponentials.

Dissipation in hydrodynamics from micro- to macroscale: wisdom from Boltzmann and stochastic thermodynamics

We show that macroscopic irreversible thermodynamics for viscous fluids can be derived from exact information-theoretic thermodynamic identities valid at the microscale. Entropy production, in particular, is a measure of the loss of many-particle correlations in the same way in which it measures the loss of system-reservoirs correlations in stochastic thermodynamics (ST). More specifically, we first show that boundary conditions at the macroscopic level define a natural decomposition of the entropy production rate (EPR) in terms of thermodynamic forces multiplying their conjugate currents, as well as a change in suitable nonequilibrium potential that acts as a Lyapunov function in the absence of forces. Moving to the microscale, we identify the exact identities at the origin of these dissipative contributions for isolated Hamiltonian systems. We then show that the molecular chaos hypothesis, which gives rise to the Boltzmann equation at the mesoscale, leads to a positive rate of loss of many-particle correlations, which we identify with the Boltzmann EPR. By generalizing the Boltzmann equation to account for boundaries with nonuniform temperature and nonzero velocity, and resorting to the Chapman–Enskog expansion, we recover the macroscopic theory we started from. Finally, using a linearized Boltzmann equation we derive ST for dilute particles in a weakly out-of-equilibrium fluid and its corresponding macroscopic thermodynamics. Our work unambiguously demonstrates the information-theoretical origin of thermodynamic notions of entropy and dissipation in macroscale irreversible thermodynamics.

Control and analysis of floating ability of aeronautical cylindrical spiral groove gas seal based on F-K model

The slip flow phenomenon caused by the gas floating seal in ultra-thin gas films and high-altitude rarefied gas environments occurs frequently. This study represents the first attempt to calculate the gas film floating force by considering the coupling relationship among the slip flow effect, the surface micro-grooves, and the eccentricity using a high-precision eight-node finite difference method based on the linearized Boltzmann equation for a spiral-grooved cylindrical gas seal. Furthermore, the influence of slip flow on the operational and groove parameters of the spiral-grooved cylindrical gas seal is investigated and discussed. Results show that the velocity gradient of the lubrication gas is reduced and the effect of the fluid hydrodynamic pressure is weakened because of slip flow, particularly in high-speed, low-pressure, and high-eccentricity fields. However, increases in groove depth, number, and length improve the gas film floating force, which strengthens the slip flow response in the grooves. Therefore, the slip flow reduces the gas film floating force, but when the groove depth exceeds 32 μm and the groove length is greater than 45 mm, the slip flow may become negligible. The results presented here provide a theoretical basis to broaden the application scope of dynamic seals in the aerospace field.

Linear Boltzmann equation solver for voxel-level dosimetry in radiopharmaceutical therapy: Comparison with Monte Carlo and kernel convolution.

With recent interest in patient-specific dosimetry for radiopharmaceutical therapy (RPT) and selective internal radiation therapy (SIRT), an increasing number of voxel-based algorithms are being evaluated. Monte Carlo (MC) radiation transport, generally considered to be the most accurate among different methods for voxel-level absorbed dose estimation, can be computationally inefficient for routine clinical use. This work demonstrates a recently implemented grid-based linear Boltzmann transport equation (LBTE) solver for fast and accurate voxel-based dosimetry in RPT and SIRT and benchmarks it against MC. A deterministic LBTE solver (Acuros MRT) was implemented within a commercial RPT dosimetry package (Velocity 4.1). The LBTE is directly discretized using an adaptive mesh refined grid and then the coupled photon-electron radiation transport is iteratively solved inside specified volumes to estimate radiation doses from both photons and charged particles in heterogeneous media. To evaluate the performance of the LBTE solver for RPT and SIRT applications, 177Lu SPECT/CT, 90Y PET/CT, and 131I SPECT/CT images of phantoms and patients were used. Multiple lesions (2-1052mL) and normal organs were delineated for each study. Voxel dosimetry was performed with the LBTE solver, dose voxel kernel (DVK) convolution with density correction, and a validated in-house MC code using the same time-integrated activity and density maps as input to the different dose engines. The resulting dose maps, difference maps, and dose-volume-histogram (DVH) metrics were compared, to assess the voxel-level agreement. Evaluation of mean absorbed dose included comparison with structure-level estimates from OLINDA. In the phantom inserts/compartments, the LBTE solver versus MC and DVK convolution demonstrated good agreement with mean absorbed dose and DVH metrics agreeing to within 5% except for the D90 and D70 metrics of a very low activity concentration insert of 90Y where the agreement was within 15%. In the patient studies (five patients imaged after 177Lu DOTATATE RPT, five after 90Y SIRT, and two after 131I radioimmunotherapy), in general, there was better agreement between the LBTE solver and MC than between LBTE solver and DVK convolution for mean absorbed dose and voxel-level evaluations. Across all patients for all three radionuclides, for soft tissue structures (kidney, liver, lesions), the mean absorbed dose estimates from the LBTE solver were in good agreement with those from MC (median difference<1%, maximum 9%) and those from DVK (median difference<5%, maximum 9%). The LBTE and OLINDA estimates for mean absorbed dose in kidneys and liver agreed to within 10%, but differences for lesions were larger with a maximum 14% for 177Lu, 23% for 90Y, and 26% for 131I. For bone regions, the agreement in mean absorbed doses between LBTE and both MC and DVK were similar (median<11%, max 11%) while for lung the agreement between LBTE and MC (median<1%, max 8%) was substantially better than between LBTE and DVK (median<16%, max 33%). Voxel level estimates for soft tissue structures also showed good agreement between the LBTE solver and both MC and DVK with a median difference<5% (maximum<13%) for the DVH metrics with all three radionuclides. The largest difference in DVH metrics was for the D90 and D70 metric in lung and bone where the uptake was low. Here, the difference between LBTE and MC had a median value<14% (maximum 23%) for bone and<4% (maximum 37%) for lung, while the corresponding differences between LBTE and DVK were<23% (maximum 31%) and<67% (maximum 313%), respectively. For a typical patient with a matrix size of 166×166×129 (voxel size 3×3×3 mm3), voxel dosimetry using the LBTE solver was as fast as ∼2 min on a desktop computer. Having established good agreement between the LBTE solver and MC for RPT and SIRT applications, the LBTE solver is a viable option for voxel dosimetry that can be faster than MC. Further analysis is being performed to encompass the broad range of radionuclides and conditions encountered clinically.

Kinetic Theory for the Low-Density Lorentz Gas

The Lorentz gas is one of the simplest and most widely-studied models for particle transport in matter. It describes a cloud of non-interacting gas particles in an infinitely extended array of identical spherical scatterers. The model was introduced by Lorentz in 1905 who, following the pioneering ideas of Maxwell and Boltzmann, postulated that in the limit of low scatterer density, the macroscopic transport properties of the model should be governed by a linear Boltzmann equation. The linear Boltzmann equation has since proved a useful tool in the description of various phenomena, including semiconductor physics and radiative transfer. A rigorous derivation of the linear Boltzmann equation from the underlying particle dynamics was given, for random scatterer configurations, in three seminal papers by Gallavotti, Spohn and Boldrighini-Bunimovich-Sinai. The objective of the present study is to develop an approach for a large class of deterministic scatterer configurations, including various types of quasicrystals. We prove the convergence of the particle dynamics to transport processes that are in general (depending on the scatterer configuration) not described by the linear Boltzmann equation. This was previously understood only in the case of the periodic Lorentz gas through work of Caglioti-Golse and Marklof-Strömbergsson. Our results extend beyond the classical Lorentz gas with hard sphere scatterers, and in particular hold for general classes of spherically symmetric finite-range potentials. We employ a rescaling technique that randomises the point configuration given by the scatterers’ centers. The limiting transport process is then expressed in terms of a point process that arises as the limit of the randomised point configuration under a certain volume-preserving one-parameter linear group action.

Spectrum of the Boltzmann collision operator for λϕ4 theory in the classical regime

We analytically determine all the eigenvalues and eigenfunctions of the linearized Boltzmann collision operator (computed using Boltzmann statistics) in massless scalar λϕ4 theory. This is used to determine this system's shear viscosity and particle diffusion transport coefficients in analytical form. The corresponding relaxation time approximation for this linearized Boltzmann equation is also derived.

Are offline ART decisions for NSCLC impacted by the type of dose calculation algorithm?

IntroductionDecisions for plan-adaptations may be impacted by a transitioning from one dose-calculation algorithm to another. This study examines the impact on dosimetric-triggered offline adaptation in LA-NSCLC in the context of a transition from superposition/convolution dose calculation algorithm (Type-B) to linear Boltzmann equation solver dose calculation algorithms (Type-C). Materials & MethodsTwo dosimetric-triggered offline adaptive treatment workflows are compared in a retrospective planning study on 30 LA-NSCLC patients. One workflow uses a Type-B dose calculation algorithm and the other uses Type-C. Treatment plans were re-calculated on the anatomy of a mid-treatment synthetic-CT utilizing the same algorithm utilized for pre-treatment planning. Assessment for plan-adaptation was evaluated through a decision model based on target coverage and OAR constraint violation. The impact of algorithm during treatment planning was controlled for by recalculating the Type-B plan with Type-C. ResultsIn the Type-B approach, 13 patients required adaptation due to OAR-constraint violations, while 15 patients required adaptation in the Type-C approach. For 8 out of 30 cases, the decision to adapt was opposite in both approaches. None of the patients in our dataset encountered CTV-target underdosage that necessitated plan-adaptation. Upon recalculating the Type-B approach with the Type-C algorithm, it was shown that 10 of the original Type-B plans revealed clinically relevant dose reductions (≥3%) on the CTV in their original plans. This re-calculation identified 21 plans in total that required ART. DiscussionIn our study, nearly one-third of the cases would have a different decision for plan-adaption when utilizing Type-C instead of Type-B. There was no substantial increase in the total number of plan-adaptations for LA-NSCLC. However, Type-C is more sensitive to altered anatomy during treatment compared to Type-B. Recalculating Type-B plans with the Type-C algorithm revealed an increase from 13 to 21 cases triggering ART.

Macroscopic estimate of the linear Boltzmann and Landau equations with Specular reflection boundary

Macroscopic estimate of the linear Boltzmann and Landau equations with Specular reflection boundary

Quantitative rates of convergence to equilibrium for the degenerate linear Boltzmann equation on the torus

AbstractWe study the linear relaxation Boltzmann equation on the torus with a spatially varying jump rate which can be zero on large sections of the domain. In Bernard and Salvarani (Arch. Ration. Mech. Anal. 208 (2013), no. 3, 977–984), Bernard and Salvarani showed that this equation converges exponentially fast to equilibrium if and only if the jump rate satisfies the geometric control condition of Bardos, Lebeau and Rauch. Han‐Kwan and Léautaud showed a more general result for linear Boltzmann equations under the action of potentials in different geometric contexts, including the case of unbounded velocities. In this paper, we obtain quantitative rates of convergence to equilibrium when the geometric control condition is satisfied, using a probabilistic approach based on Doeblin's theorem from Markov chains.

Viscous and thermal velocity slip coefficients via the linearized Boltzmann equation with ab initio potential

The viscous and thermal velocity slip coefficients of various monatomic gases are computed via the linearized classical Boltzmann equation, with ab initio potential, subject to Maxwell and Cercignani–Lampis boundary conditions. Both classical and quantum interatomic interactions are considered. Comparisons with hard sphere and Lennard–Jones potentials, as well as the linearized Shakhov model are performed. The produced database is dense, covers the whole range of the accommodation coefficients and is of high accuracy. Using symbolic regression, very accurate closed form expressions of the slip coefficients, easily implemented in the future computational and experimental works, are deduced. The thermal slip coefficient depends, much more than the viscous one, on the intermolecular potential. For example, in the case of diffuse scattering, the relative differences in the viscous slip coefficient data between HS and AI potentials are less than 4%, whilst the corresponding ones in the thermal slip coefficient data are about 6% for He, reaching 15% for Xe. Quantum effects are considered for He, at temperatures 1–104 K to deduce that deviations from the classical behaviour are not important in the viscous slip coefficient, but they become important in the thermal slip coefficient, where the differences between the classical and quantum approaches reach 15% at 1 K. The computational effort of solving the linearized Boltzmann equation with ab initio and Lennard–Jones potentials is the same. Since ab initio potentials do not contain any adjustable parameters, it is recommended to use them at any temperature.

Drag on a sphere in a slow flow of a binary mixture of rarefied gases

The drag force on a sphere in an isothermal binary mixture of rarefied gases is calculated on the basis of the McCormack model for the linearized Boltzmann equation. The diffuse scattering of the gas–surface interaction law is assumed as the boundary condition. The rigid-spheres model of interatomic interaction potential is used as well as ab initio potentials for helium, argon, and krypton gases. The force is obtained in a wide range of the gas rarefaction, which covers the free molecular, transition, and slip regimes of the gas flow. In the free molecular and slip flow regimes, the problem is solved analytically, while in the transition regime, the system of kinetic equations is solved numerically via the discrete velocity method optimized to overcome the problem of discontinuity of the distribution function of molecular velocities on the convex surface. The calculations are carried out for the mixtures helium–argon and helium–krypton at 300 K. In the slip flow regime, the data available in the literature for the viscous slip coefficient of the helium–argon mixture are used, while for the mixture helium–krypton, it is calculated. The influence of the interatomic interaction potential, molar fraction, and ratio of atomic mass of species in the mixture on the drag force is analyzed.

The Lorentz Gas with a Nearly Periodic Distribution of Scatterers

We consider the Lorentz gas in a distribution of scatterers which microscopically converges to a periodic distribution, and prove that the Lorentz gas in the low density limit satisfies a linear Boltzmann equation. This is in contrast with the periodic Lorentz gas, which does not satisfy the Boltzmann equation in the limit.

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.

Long‐time correlations for a hard‐sphere gas at equilibrium

AbstractIt has been known since Lanford that the dynamics of a hard‐sphere gas is described in the low density limit by the Boltzmann equation, at least for short times. The classical strategy of proof fails for longer times, even close to equilibrium. In this paper, we introduce a weak convergence method coupled with a sampling argument to prove that the covariance of the fluctuation field around equilibrium is governed by the linearized Boltzmann equation globally in time (including in diffusive regimes). This method is much more robust and simpler than the one devised in Bodineau et al which was specific to the 2D case.

Metastability Results for a Class of Linear Boltzmann Equations

Metastability Results for a Class of Linear Boltzmann Equations