Hilbert Expansions Research Articles

QDT — A Matlab toolbox for the simulation of coupled quantum systems and coherent multidimensional spectroscopy

We present QDT (“quantum dynamics toolbox”), an open-source Matlab software package that enables users to simulate coupled quantum systems in the subsystem energy eigenbasis using modular functions. QDT requires no user knowledge of operator matrix assembly and automatically performs all necessary operator constructions and Hilbert space expansions. Density matrix propagation is performed by numerically solving the Liouville–von-Neumann equation. In order to simulate dissipation and decoherence effects, the Lindblad formalism is implemented. Furthermore, QDT supplies practical analysis and plotting functions, such as visualization of density matrix and expectation value dynamics, that facilitate the evaluation of simulation results. QDT further provides a module for the simulation of coherent multidimensional spectroscopy. Program summaryProgram Title: QDT — A Quantum Dynamics ToolboxCPC Library link to program files:https://doi.org/10.17632/786vb4rgy3.1Code Ocean capsule:https://codeocean.com/capsule/4608215Licensing provisions: MITProgramming language: Matlab R2018a (and above) with parallel computing toolbox (optional) [1]Supplementary material: Available on the QDT website [2]Nature of problem: Coupled quantum systems can be easily set up and discussed as level schemes in the subsystem energy eigenbasis. The simulation of the time evolution of these systems by numerically solving differential equations is not straightforward, however. Simulation code that is written from scratch is time-consuming to create and prone to errors during implementation. Furthermore, if code bases deviate drastically between research groups, reproducibility of results becomes challenging, and communication is hindered.Solution method: QDT addresses the above issues by providing a modular toolbox that contains, among others, elements for the simulation of multi-level quantum systems, quantum harmonic oscillators, their interaction with external electric fields, and dissipation. These elements can be combined by the user to build a specific system of interest. All operators are created and expanded automatically based on a single script file provided by the user. Simulation tasks are automated by a single function call, and parallelized in case of extensive tasks.

A stochastic hierarchical model for low grade glioma evolution

A stochastic hierarchical model for the evolution of low grade gliomas is proposed. Starting with the description of cell motion using a piecewise diffusion Markov process (PDifMP) at the cellular level, we derive an equation for the density of the transition probability of this Markov process based on the generalised Fokker–Planck equation. Then, a macroscopic model is derived via parabolic limit and Hilbert expansions in the moment equations. After setting up the model, we perform several numerical tests to study the role of the local characteristics and the extended generator of the PDifMP in the process of tumour progression. The main aim focuses on understanding how the variations of the jump rate function of this process at the microscopic scale and the diffusion coefficient at the macroscopic scale are related to the diffusive behaviour of the glioma cells and to the onset of malignancy, i.e., the transition from low-grade to high-grade gliomas.

Glioma invasion and its interplay with nervous tissue and therapy: A multiscale model

A multiscale mathematical model for glioma cell migration and proliferation is proposed, taking into account a possible therapeutic approach. Starting with the description of processes occurring at the subcellular level, the equation for the mesoscopic level is formulated and a macroscopic model is derived, via parabolic limit and Hilbert expansions in the moment equations. After the model set up and the study of the well-posedness of this macroscopic setting, we investigate the role of the fibers in the tumor dynamics. In particular, we focus on the fiber density function, with the aim of comparing some common choices present in the literature and understanding which differences arise in the description of the actual fiber density and orientation. Finally, some numerical simulations, based on real data, highlight the role of each modelled process in the evolution of the solution of the macroscopic equation.

Gyrokinetics from variational averaging: Existence and error bounds

The gyrokinetic paradigm in the long wavelength regime is reviewed from the perspective of variational averaging (VA). The VA-method represents a third pillar for averaging kinetic equations with highly oscillatory characteristics, besides classical averaging and Hilbert expansions. VA operates on the level of the Lagrangian function and preserves the Hamiltonian structure of the characteristics at all orders. We discuss the methodology of VA in detail by means of charged-particle motion in a strong magnetic field. The application of VA to a broader class of highly oscillatory problems can be envisioned. For the charged particle, we prove the existence of a coordinate map in phase space that leads to a gyrokinetic Lagrangian at any order of the expansion, for general external fields. We compute this map up to third order, independent of the electromagnetic gauge. Moreover, an error bound for the solution of the derived gyrokinetic equation with respect to the solution of the Vlasov equation is provided, allowing us to estimate the quality of the VA-approximation in this particular case.

The unconditional constants for Hilbert space frame expansions

The most fundamental notion in frame theory is the frame expansion of a vector. Although it is well known that these expansions are unconditionally convergent series, no characterizations of the unconditional constant were known. This has made it impossible to get accurate quantitative estimates for problems which require using subsequences of a frame. We will prove some new results in frame theory by showing that the unconditional constants of the frame expansion of a vector in a Hilbert space are bounded by BA, where A,B are the frame bounds of the frame. Tight frames thus have unconditional constant one, which we then generalize by showing that Bessel sequences have frame expansions with unconditional constant one if and only if the sequence is an orthogonal sum of tight frames. We give further results concerning frame expansions, in which we examine when BA is actually attained or not. We end by discussing the connections of this work to frame multipliers. These results hold in both real and complex Hilbert spaces.

Derivation of the Isotropic Diffusion Source Approximation (IDSA) for Supernova Neutrino Transport by Asymptotic Expansions

We present Chapman--Enskog and Hilbert expansions applied to the $\BigO(v/c)$ Boltzmann equation for the radiative transfer of neutrinos in core-collapse supernovae. Based on the Legendre expansion of the scattering kernel for the collision integral truncated after the second term, we derive the diffusion limit for the Boltzmann equation by truncation of Chapman--Enskog or Hilbert expansions with reaction and collision scaling. We also give asymptotically sharp results obtained by the use of an additional time scaling. The diffusion limit determines the diffusion source in the \emph{Isotropic Diffusion Source Approximation (IDSA)} of Boltzmann's equation for which the free streaming limit and the reaction limit serve as limiters. Here, we derive the reaction limit as well as the free streaming limit by truncation of Chapman--Enskog or Hilbert expansions using reaction and collision scaling as well as time scaling, respectively. Finally, we motivate why limiters are a good choice for the definition of the source term in the IDSA.

Hilbert Fluid Dynamics Equations Expressed in Chapman-Enskog Pressure Tensor and Heat Current

The connection between the Chapman-Enskog and Hilbert expansions is investigated in detail. In particular, the fluid dynamics equations of any order in the Hilbert expansion are given in terms of the pressure tensor and heat current of the Chapman-Enskog expansion.

DIFFUSION APPROXIMATION FOR TRANSPORT PROCESSES WITH GENERAL REFLECTION BOUNDARY CONDITIONS

Diffusion approximations are obtained for space inhomogeneous linear transport models with reflection boundary conditions. The collision kernel is not required to satisfy any balance condition and the scattering kernel on the boundary is general enough to include all examples of boundary conditions known to the authors (with conservation of the number of particles) and, in addition, to model the Debye sheath. The mathematical approach does not rely on Hilbert expansions, but rather on martingale and stochastic averaging techniques.

Phase segregation and interface dynamics in kinetic systems

We consider a kinetic model of two species of particles interacting with a reservoir at fixed temperature, described by two coupled Vlasov–Fokker–Planck equations. We prove that in the diffusive limit the evolution is described by a macroscopic equation in the form of the gradient flux of the macroscopic free energy functional. Moreover, we study the sharp interface limit and find by formal Hilbert expansions that the interface motion is given in terms of a quasi-stationary problem for the chemical potentials. The velocity of the interface is the sum of two contributions: the velocity of the Mullins–Sekerka motion for the difference of the chemical potentials and the velocity of a Hele–Shaw motion for a linear combination of the two potentials. These equations are identical to the ones in Otto and E (1997 J. Chem. Phys. 107 10177–84) modelling the motion of a sharp interface for a polymer blend.

High field approximations of the spherical harmonics expansion model for semiconductors

We present an asymptotic analysis (with the scaled mean free path as small parameter) of the spherical-harmonics expansion (SHE-) model for semiconductors in the case of a large electric field. The Hilbert and Chapman-Enskog expansions are performed and the dependence of macroscopic parameter-functions such as the mobility and the diffusivity on the details of the considered elastic and inelastic scattering processes are investigated.¶For example, we verify so called velocity-saturation mobility models, so far obtained by heuristic considerations, by means of an asymptotic analysis for certain scattering processes.

On a hierarchy of macroscopic models for semiconductors

This paper shows that various models of electron transport in semiconductors that have been previously proposed in the literature can be connected one with each other by the diffusion approximation methodology. We first investigate the diffusion limit of the semiconductor Boltzmann equation towards the so-called ‘‘spherical harmonic expansion model,’’ under the assumption of dominant elastic scattering. Then, this model is again connected, either to the energy-transport model or to a ‘‘periodic spherical harmonic expansion model’’ through a diffusion approximation, respectively making electron–electron or phonon scattering large. We provide the mathematical background which makes the Hilbert expansions associated with these various diffusion limits rigorous.

Convergence of Convective–Diffusive Lattice Boltzmann Methods

Lattice Boltzmann methods are numerical schemes derived as a kinetic approximation of an underlying lattice gas. A numerical convergence theory for nonlinear convective–diffusive lattice Boltzmann methods is established. Convergence, consistency, and stability are defined through truncated Hilbert expansions. In this setting it is shown that consistency and stability imply convergence. Monotone lattice Boltzmann methods are defined and shown to be stable, and hence convergent when consistent. Examples of diffusive and convective–diffusive lattice Boltzmann methods that are both consistent and monotone are presented.

Asymptotic expansions of solutions for the Boltzmann equation

Abstract The Hilbert and Chapman-Enskog expansions approximate solutions of the Boltzmann equation, but each has some disadvantages: The Hilbert expansion, which results in nonlinear and linearized Euler equations, is invalid for weak shocks, weak boundary layers and long time asymptotics. The Chapman-Enskog expansion results in nonlinear Euler then Navier-Stokes then Burnett and super-Burnett equations. Navier-Stokes is correct for weak shocks, weak boundary layers and long time asymptotics, but the Burnett equations have spurious high order dispersive effects. In this paper a modified expansion is developed, which combines the best features of the two expansions. It results in nonlinear and linearized Navier-Stokes equations only and is valid in the above-mentioned regimes.

Stokes Paradox in Kinetic Theory

The flow past a solid body is analyzed by means of the Boltzmann equation. It is shown that for small Mach numbers a formal linearization is affected by the same difficulties as the Stokes linearization in continuum fluid mechanics. In particular, it is proved that the linearized Boltzmann equation does not provide bounded solutions for the flow past an axisymmetric body. As a consequence, an inner-outer expansion must be constructed, and the linearization is to be reinterpreted as the first step of the inner expansion; the outer one being very similar to the Hilbert and Chapman-Enskog expansions.

Asymptotic Theory of the Boltzmann Equation

The precise mathematical relation that the Hilbert and Chapman-Enskog expansions bear to the manifold of solutions of the Boltzmann equation is described. These expansions yield inherently imprecise descriptions of a gas in terms of macroscopic fluid variables instead of a molecular distribution function. It is shown that these expansions are asymptotic to a very special class of solutions of the Boltzmann equation for sufficiently small mean free path. Next, a generalization of the Hilbert and Chapman-Enskog expansions is described in terms of extended sets of macroscopic state variables, each governed by partial differential equations similar to those found in fluid dynamics, but sufficiently general to approximate an arbitrary distribution function. The generalized expansions are shown to be asymptotic to quite arbitrary solutions of the Boltzmann equation. It is then shown that the ordinary Hilbert and Chapman-Enskog expansions can also be made asymptotic to very general solutions of the Boltzmann equation by reinterpreting the variables that enter these expansions as certain well-defined replacements for the actual fluid state of the gas. In this way the scope of the Euler, Navier-Stokes, Burnett equations, etc., is greatly extended by interpreting them as governing the artificial variables. Not only are general solutions of the Boltzmann equation shown to be approximated by fluid dynamics (in the limit of small mean free path), but the rapid decay of an arbitrary initial distribution function to a special Hilbert distribution function is also governed by sets of partial differential equations similar to those found in fluid dynamics.