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In this paper we consider a one-dimensional Mindlin model describing linear elastic behaviour of isotropic materials with micro-structural effects. After introducing the kinetic and the potential energy, we derive a system of equations of motion by means of the Euler-Lagrange equations. A class of exact solutions is obtained. They have a wave behaviour due to a good property of the potential energy. Numerical solutions are obtained by using a weighted essentially non-oscillatory finite difference scheme coupled by a total variation diminishing Runge-Kutta method. A comparison between exact and numerical solutions shows the robustness and the accuracy of the numerical scheme. A numerical example of solutions for an inhomogeneous material is also shown.

The Boltzmann equation describes the detailed microscopic behaviour of a dilute gas, and represents the basis of the kinetic theory of gases. In order to reduce the difficulties in solving the Boltzmann equation, simple expressions of a collision operator have been proposed to replace the true Boltzmann integral term. These new equations are called kinetic models. The most popular and widely used kinetic model is the Bhatnagar-Gross-Krook (BGK) model. In this work we propose and analyse a BGK model for charge transport in graphene.

In this work we consider the classical non-linear Boltzmann equation, where the unknown is the distribution function $f$, which depends on the time $t$, the vector $\mathbf{x}$ (the position of a molecule) and its velocity $\mathbf{\xi}$. From the Boltzmann equation we derive a semi-discrete model, which consists in a set of partial differential equations in time and space. The new unknowns are two moments of the distribution function $f$ and, according to a finite volume scheme, integrals of $f$ with respect to the velocity $\mathbf{\xi}$, over bounded and open sets. We do not introduce any approximation for the partial derivatives with respect to $\mathbf{x}$ and the time $t$. We also propose the use of truncated octahedra as bounded domains of integration. This reduces both the computing complexity and the number of the constant numerical coefficients arising from the collision operator. All the numerical parameters can be obtained by means of numerical quadrature.

The aim of this work is to simulate the charge transport in a monolayer graphene on different substrates. This requires the inclusion of the scatterings of the charge carriers with the impurities and the phonons of the substrate, besides the interaction mechanisms already present in the graphene layer. As physical model, the semiclassical Boltzmann equation is assumed and the results are based on Direct Simulation Monte Carlo (DSMC). A crucial point is the correct inclusion of the Pauli Exclusion Principle (PEP). Two different substrates are investigated: SiO$_2$ and hexagonal boron nitride (h-BN). In the adopted model for the charge-impurities scattering, a crucial parameter is the distance $d$ between the graphene layer and the impurities of the substrate. Usually $d$ is considered constant. Here we assume that $d$ is a random variable in order to take into account the roughness of the substrate and the randomness of the location of the impurities. We confirm, where only the low-field mobility has been investigated, that h-BN is one of the most promising substrate also for the high-field mobility on account of the reduced degradation of the velocity due to the remote impurities.

The purpose of this work is to incorporate numerically, in a discontinuous Galerkin (DG) solver of a Boltzmann-Poisson model for hot electron transport, an electronic conduction band whose values are obtained by the spherical averaging of the full band structure given by a local empirical pseudopotential method (EPM) around a local minimum of the conduction band for silicon, as a midpoint between a radial band model and an anisotropic full band, in order to provide a more accurate physical description of the electron group velocity and conduction energy band structure in a semiconductor. This gives a better quantitative description of the transport and collision phenomena that fundamentally define the behaviour of the Boltzmann - Poisson model for electron transport used in this work. The numerical values of the derivatives of this conduction energy band, needed for the description of the electron group velocity, are obtained by means of a cubic spline interpolation. The EPM-Boltzmann-Poisson transport with this spherically averaged EPM calculated energy surface is numerically simulated and compared to the output of traditional analytic band models such as the parabolic and Kane bands, numerically implemented too, for the case of 1D $n^+-n-n^+$ silicon diodes with 400nm and 50nm channels. Quantitative differences are observed in the kinetic moments related to the conduction energy band used, such as mean velocity, average energy, and electric current (momentum)., Comment: submission to CMAME (Computer Methods in Applied Mechanics and Engineering) Journal as a reply to the reviewers on February 2017

The present work is motivated by the development of a fast DG based deterministic solver for the extension of the BTE to a system of transport Boltzmann equations for full electronic multi-band transport with intra-band scattering mechanisms. Our proposed method allows to find scattering effects of high complexity, such as anisotropic electronic bands or full band computations, by simply using the standard routines of a suitable Monte Carlo approach only once. In this short paper, we restrict our presentation to the single band problem as it will be also valid in the multi-band system as well. We present preliminary numerical tests of this method using the Kane energy band model, for a 1-D 400nm $n^{+}-n-n^{+}$ silicon channel diode, showing moments at $t=0.5$ps and $t=3.0$ps., Comment: Appeared in Computational Electronics (IWCE), 2012 International Workshop on, 22-25 May 2012, IEEE Xplore Digital Library (6242847.pdf) (2012)

In this paper we consider a Vlasov or collisionless Boltzmann equation describing the dynamics of planetary rings. We propose a simple physical model, where the particles of the rings move under the gravitational Newtonian potential of two primary bodies. We neglect the gravitational forces between the particles. We use a perturbative technique, which allows to find explicit solutions at the first order and solutions at the second order, solving a set of two linear ordinary differential equations., Comment: 12 pages

We propose a new macroscopic model derived from the classical nonlinear Boltzmann equation. A set of partial differential equations is obtained easily. The unknowns depend on the time and space coordinates, and are related to the distribution function, which is the unknown of the Boltzmann equation. This new model guarantees the conservation of the mass, momentum and energy. We prove that the set of equations coincides with the system obtained applying the discontinuous Galerkin method to the Boltzmann equation (see A. Majorana, Kinetic and Related models, 4 (2011), 139--151)., Comment: Conference in Austin, Texas (USA), April 2011

We propose a new deterministic numerical scheme, based on the discontinuous Galerkin method, for solving the Boltzamnn equation for rarefied gases. The new scheme guarantees the conservation of the mass, momentum and energy. We avoid any stochastic procedures in the treatment of the collision operator of the Boltzmamn equation., Comment: Conference: Partial differential equations in kinetic theories: kinetic description of biological models. Nov 8, 2010 - Nov 12, 2010 Edinburgh (UK) The second version differs from the first only in the MSC class data

In this paper, we present results of a discontinuous Galerkin (DG) scheme applied to deterministic computations of the transients for the Boltzmann-Poisson system describing electron transport in semiconductor devices. The collisional term models optical-phonon interactions which become dominant under strong energetic conditions corresponding to nano-scale active regions under applied bias. The proposed numerical technique is a finite element method using discontinuous piecewise polynomials as basis functions on unstructured meshes. It is applied to simulate hot electron transport in bulk silicon, in a silicon $n^+$-$n$-$n^+$ diode and in a double gated 12nm MOSFET. Additionally, the obtained results are compared to those of a high order WENO scheme simulation and DSMC (Discrete Simulation Monte Carlo) solvers., Comment: 51 pages, 82 figures