Boltzmann equation | EPFL Graph Search

The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium; it was devised by Ludwig Boltzmann in 1872. The classic example of such a system is a fluid with temperature gradients in space causing heat to flow from hotter regions to colder ones, by the random but biased transport of the particles making up that fluid. In the modern literature the term Boltzmann equation is often used in a more general sense, referring to any kinetic equation that describes the change of a macroscopic quantity in a thermodynamic system, such as energy, charge or particle number. The equation arises not by analyzing the individual positions and momenta of each particle in the fluid but rather by considering a probability distribution for the position and momentum of a typical particle—that is, the probability that the particle occupies a given very small region of space (mathematically the volume element

Evolution of open many-electron systems

Aric Gliesche

Starting from the quantum statistical master equation derived in [1] we show how the connection to the semi-classical Boltzmann equation (SCBE) can be established and how irreversibility is related to the problem of separability of quantum mechanics. Our principle goal is to find a sound theoretical basis for the description of the evolution of an electron gas in the intermediate regime between pure classical behavior and pure quantum behavior. We investigate the evolution of one-particle properties in a weakly interacting N-electron system confined to a finite spatial region in a near-equilibrium situation that is weakly coupled to a statistical environment. The equations for the reduced n-particle density matrices, with n < N are hierarchically coupled through two-particle interactions. In order to elucidate the role of this type of coupling and of the inter-particle correlations generated by the interaction, we examine first the particular situation where energy transfer between the N-electron system and the statistical environment is negligible, but where the system has a finite memory. We then formulate the general master equation that describes the evolution of the coarse grained one-particle density matrix of an interacting confined electron gas including energy transfer with one or more bath subsystems, which is called the quantum Boltzmann equation (QBE). The connection with phase space is established by expressing the one-particle states in terms of the overcomplete basis of coherent states, which are localized in phase space. In this way we obtain the QBE in phase space. After performing an additional coarse-graining procedure in phase space, and assuming that the interaction of the electron gas and the bath subsystems is local in real space, we obtain the semi-classical Boltzmann equation. The validity range of the classical description, which introduces local dynamics in phase space is discussed.

EPFL2007

Gaussian Process Regression for Maximum Entropy Distribution

Mohsen Sadr

Maximum-Entropy Distributions offer an attractive family of probability densities suitable for moment closure problems. Yet finding the Lagrange multipliers which parametrize these distributions, turns out to be a computational bottleneck for practical closure settings. Motivated by recent success of Gaussian processes, we investigate the suitability of Gaussian priors to approximate the Lagrange multipliers as a map of a given set of moments. Examining various kernel functions, the hyperparameters are optimized by maximizing the log-likelihood. The performance of the devised data-driven Maximum-Entropy closure is studied for couple of test cases including relaxation of non-equilibrium distributions governed by Bhatnagar-Gross-Krook and Boltzmann kinetic equations.

2020

Entropic Fokker-Planck kinetic model

Mohammadhossein Gorji

The diffusion limit of kinetic systems has been subject of numerous studies since prominent works of Lebowitz et al. [1] and van Kampen [2]. More recently, the topic has seen a fresh interest from the rarefied gas simulation perspective. In particular, Fokker Planck based kinetic models provide novel approximations of the Boltzmann equation, where the relaxation induced by binary collisions is modeled via continuous stochastic processes. Hence in contrast to direct simulation Monte-Carlo, computational particles follow seemingly independent stochastic paths. As a result, a significant computational gain at small/vanishing Knudsen numbers can be obtained, where the dynamics of particles is overwhelmed by collisions. The cubic Fokker-Planck equation derived by [3] gives rise to the correct viscosity and Prandtl number for monatomic gases in the hydrodynamic limit, and further accurate behavior at moderate Knudsen numbers. Yet the model lacks a rigorous structure and more crucially does not admit the H-theorem. The latter underpins its accuracy e.g. in predicting shock wave profiles. This study addresses bridging the gap between diffusion processes and the Boltzmann equation by introducing the Entropic-Fokker-Planck kinetic model. The drift-diffusion closures derived for the model, allow for an H-theorem besides honoring consistent relaxation of moments. The devised model is validated with respect to direct simulation Monte-Carlo for high-Mach as well as Couette flows. Good performance of the model together with its easy to compute coefficients, makes the Entropic-Fokker-Planck framework attractive for computational investigation of gases beyond equilibrium. (C) 2021 The Authors. Published by Elsevier Inc.

2021

Evolution of open many-electron systems

Aric Gliesche

EPFL2007