Infrared catastrophe in two-quasiparticle collision integral

Relaxation of a non-equilibrium state in a disordered metal with a spin-dependent electron energy distribution is considered. The collision integral due to the electron-electron interaction is computed within the approximation of a two-quasiparticle scattering. We show that the spin-flip scattering processes with a small energy transfer may lead to the divergence of the collision integral for a quasi one-dimensional wire. This divergence is present only for a spin-dependent electron energy distribution which corresponds to the total electron spin magnetization M=0 and only for non-zero interaction in the triplet channel. In this case a non-perturbative treatment of the electron-electron interaction is needed to provide an effective infrared cut-off.Comment: 6 pages, 3 figure

Landau Collision Integral Solver with Adaptive Mesh Refinement on Emerging Architectures

The Landau collision integral is an accurate model for the small-angle dominated Coulomb collisions in fusion plasmas. We investigate a high order accurate, fully conservative, finite element discretization of the nonlinear multi-species Landau integral with adaptive mesh refinement using the PETSc library (www.mcs.anl.gov/petsc). We develop algorithms and techniques to efficiently utilize emerging architectures with an approach that minimizes memory usage and movement and is suitable for vector processing. The Landau collision integral is vectorized with Intel AVX-512 intrinsics and the solver sustains as much as 22% of the theoretical peak flop rate of the Second Generation Intel Xeon Phi, Knights Landing, processor

Variance Reduction For A Discrete Velocity Gas

We extend a variance reduction technique developed by Baker and Hadjiconstantinou [1] to a discrete velocity gas. In our previous work, the collision integral was evaluated by importance sampling of collision partners [2]. Significant computational effort may be wasted by evaluating the collision integral in regions where the flow is in equilibrium. In the current approach, substantial computational savings are obtained by only solving for the deviations from equilibrium. In the near continuum regime, the deviations from equilibrium are small and low noise evaluation of the collision integral can be achieved with very coarse statistical sampling. Spatially homogenous relaxation of the Bobylev-Krook-Wu distribution [3,4], was used as a test case to verify that the method predicts the correct evolution of a highly non-equilibrium distribution to equilibrium. When variance reduction is not used, the noise causes the entropy to undershoot, but the method with variance reduction matches the analytic curve for the same number of collisions. We then extend the work to travelling shock waves and compare the accuracy and computational savings of the variance reduction method to DSMC over Mach numbers ranging from 1.2 to 10.Aerospace Engineering and Engineering Mechanic

Differential Form of the Collision Integral for a Relativistic Plasma

The differential formulation of the Landau-Fokker-Planck collision integral is developed for the case of relativistic electromagnetic interactions.Comment: Plain TeX, 5 page

Side-jumps in the spin-Hall effect: construction of the Boltzmann collision integral

We present a systematic derivation of the side-jump contribution to the spin-Hall current in systems without band structure spin-orbit interactions, focusing on the construction of the collision integral for the Boltzmann equation. Starting from the quantum Liouville equation for the density operator we derive an equation describing the dynamics of the density matrix in the first Born approximation and to first order in the driving electric field. Elastic scattering requires conservation of the total energy, including the spin-orbit interaction energy with the electric field: this results in a first correction to the customary collision integral found in the Born approximation. A second correction is due to the change in the carrier position during collisions. It stems from the part of the density matrix off-diagonal in wave vector. The two corrections to the collision integral add up and are responsible for the total side-jump contribution to the spin-Hall current. The spin-orbit-induced correction to the velocity operator also contains terms diagonal and off-diagonal in momentum space, which together involve the total force acting on the system. This force is explicitly shown to vanish (on the average) in the steady state: thus the total contribution to the spin-Hall current due to the additional terms in the velocity operator is zero.Comment: Added references, expanded discussion, revised introductio

Metriplectic Integrators for the Landau Collision Operator

We present a novel framework for addressing the nonlinear Landau collision integral in terms of finite element and other subspace projection methods. We employ the underlying metriplectic structure of the Landau collision integral and, using a Galerkin discretization for the velocity space, we transform the infinite-dimensional system into a finite-dimensional, time-continuous metriplectic system. Temporal discretization is accomplished using the concept of discrete gradients. The conservation of energy, momentum, and particle densities, as well as the production of entropy is demonstrated algebraically for the fully discrete system. Due to the generality of our approach, the conservation properties and the monotonic behavior of entropy are guaranteed for finite element discretizations in general, independently of the mesh configuration.Comment: 24 pages. Comments welcom

Small damping approach in Fermi-liquid theory

The validity of small damping approximation (SDA) for the quasi-classical description of the averaged properties of nuclei at high temperatures is studied within the framework of collisional kinetic theory. The isoscalar collective quadrupole vibrations in hot nuclei are considered. We show that the extension of the SDA, by accounting for the damping of the distribution function $\delta f$ in the collision integral reduces the rate of variation with temperature of the Fermi surface distortion effects. The damping of the $\delta f$ in the collision integral increases significantly the collisional width of the giant quadrupole resonance (GQR) for small enough values of the relaxation time. The temperature dependence of the eigenenergy of the GQR becomes much more weaker than in the corresponding SDA case.Comment: 11 pages, 3 figure