20,753 research outputs found
Quantum Mechanics Lecture Notes. Selected Chapters
These are extended lecture notes of the quantum mechanics course which I am
teaching in the Weizmann Institute of Science graduate physics program. They
cover the topics listed below. The first four chapter are posted here. Their
content is detailed on the next page. The other chapters are planned to be
added in the coming months.
1. Motion in External Electromagnetic Field. Gauge Fields in Quantum
Mechanics.
2. Quantum Mechanics of Electromagnetic Field
3. Photon-Matter Interactions
4. Quantization of the Schr\"odinger Field (The Second Quantization)
5. Open Systems. Density Matrix
6. Adiabatic Theory. The Berry Phase. The Born-Oppenheimer Approximation
7. Mean Field Approaches for Many Body Systems -- Fermions and Boson
A family of total Lagrangian Petrov-Galerkin Cosserat rod finite element formulations
The standard in rod finite element formulations is the Bubnov-Galerkin
projection method, where the test functions arise from a consistent variation
of the ansatz functions. This approach becomes increasingly complex when highly
nonlinear ansatz functions are chosen to approximate the rod's centerline and
cross-section orientations. Using a Petrov-Galerkin projection method, we
propose a whole family of rod finite element formulations where the nodal
generalized virtual displacements and generalized velocities are interpolated
instead of using the consistent variations and time derivatives of the ansatz
functions. This approach leads to a significant simplification of the
expressions in the discrete virtual work functionals. In addition, independent
strategies can be chosen for interpolating the nodal centerline points and
cross-section orientations. We discuss three objective interpolation strategies
and give an in-depth analysis concerning locking and convergence behavior for
the whole family of rod finite element formulations.Comment: arXiv admin note: text overlap with arXiv:2301.0559
NSC++: Non-Standard Cosmologies in C++
We introduce NSC++, a header-only C++ library that simulates the evolution of
the plasma and a decaying fluid in the early Universe. NSC++ can be used in C++
programs or called directly from python scripts without significant overhead.
There is no special installation process or external dependencies. Furthermore,
there are example programs that can be modified to handle several cases.Comment: 18 pages; 3 figures; 4 tables; The stable version of the library can
be found at https://github.com/dkaramit/NSCpp/tree/stable. v2: Added
reference; Minor corrections to text. v3: Minor corrections, matches
published versio
Instruction of my personal computing library
This document is prepared to introduce and explain how to use the computing
library composed by T. Oishi. The library-01 TOSPEM solves, for the spherical
nucleus, (i) the Schroedinger equation for the single-nucleon states within the
Woods-Saxon potential, (ii-a) the electric or magnetic transition strength,
B(EJ) or B(MJ), between the arbitrary set of initial and final states of the
nucleus of interest, and (ii-b) Weisskopf estimate for comparison with results
in (ii-a). The library-02 RESONA is composed to solve the resonant eigenstates
of spherical Schroedinger equations. The final version is expected to be
published for educational and commercial purposes. Before the official
publication, under the agreement with publishers, I make the current,
preliminary version open for public. Two applications, GFORTRAN and GNUPLOT,
are necessary for full usage. Feedbacks and comments on products will be
appreciated. The source codes etc. are available in the GitHub repository [1].Comment: arXiv admin note: text overlap with arXiv:2303.1052
Full trajectory optimizing operator inference for reduced-order modeling using differentiable programming
Accurate and inexpensive Reduced Order Models (ROMs) for forecasting
turbulent flows can facilitate rapid design iterations and thus prove critical
for predictive control in engineering problems. Galerkin projection based
Reduced Order Models (GP-ROMs), derived by projecting the Navier-Stokes
equations on a truncated Proper Orthogonal Decomposition (POD) basis, are
popular because of their low computational costs and theoretical foundations.
However, the accuracy of traditional GP-ROMs degrades over long time prediction
horizons. To address this issue, we extend the recently proposed Neural
Galerkin Projection (NeuralGP) data driven framework to
compressibility-dominated transonic flow, considering a prototypical problem of
a buffeting NACA0012 airfoil governed by the full Navier-Stokes equations. The
algorithm maintains the form of the ROM-ODE obtained from the Galerkin
projection; however coefficients are learned directly from the data using
gradient descent facilitated by differentiable programming. This blends the
strengths of the physics driven GP-ROM and purely data driven neural
network-based techniques, resulting in a computationally cheaper model that is
easier to interpret. We show that the NeuralGP method minimizes a more rigorous
full trajectory error norm compared to a linearized error definition optimized
by the calibration procedure. We also find that while both procedures stabilize
the ROM by displacing the eigenvalues of the linear dynamics matrix of the
ROM-ODE to the complex left half-plane, the NeuralGP algorithm adds more
dissipation to the trailing POD modes resulting in its better long-term
performance. The results presented highlight the superior accuracy of the
NeuralGP technique compared to the traditional calibrated GP-ROM method
Chiral active fluids: Odd viscosity, active turbulence, and directed flows of hydrodynamic microrotors
While the number of publications on rotating active matter has rapidly increased in recent years, studies on purely hydrodynamically interacting rotors on the microscale are still rare, especially from the perspective of particle based hydrodynamic simulations. The work presented here targets to fill this gap. By means of high-performance computer simulations, performed in a highly parallelised fashion on graphics processing units, the dynamics of ensembles of up to 70,000 rotating colloids immersed in an explicit mesoscopic solvent consisting out of up to 30 million fluid particles, are investigated. Some of the results presented in this thesis have been worked out in collaboration with experimentalists, such that the theoretical considerations developed in this thesis are supported by experiments, and vice versa. The studied system, modelled in order to resemble the essential physics of the experimentally realisable system, consists out of rotating magnetic colloidal particles, i.e., (micro-)rotors, rotating in sync to an externally applied magnetic field, where the rotors solely interact via hydrodynamic and steric interactions. Overall, the agreement between simulations and experiments is very good, proving that hydrodynamic interactions play a key role in this and related systems.
While already an isolated rotating colloid is driven out of equilibrium, only collections of two or more rotors have experimentally shown to be able to convert the rotational energy input into translational dynamics in an orbital rotating fashion. The rotating colloids inject circular flows into the fluid, such that detailed balance is broken, and it is not a priori known whether equilibrium properties of colloids can be extended to isolated rotating colloids. A joint theoretical and experimental analysis of isolated, pairs, and small groups of hydrodynamically interacting rotors is given in chapter 2. While the translational dynamics of isolated rotors effectively resemble the dynamics of non-rotating colloids, the orbital rotation of pairs of rotors can be described with leading order hydrodynamics and a two-dimensional analogy of Faxén’s law is derived.
In chapter 3, a homogeneously distributed ensemble of rotors (bulk) as a realisation of a chiral active fluid is studied and it is explicitly shown computationally and experimentally that it carries odd viscosity. The mutual orbital translation of rotors and an increase of the effective solvent viscosity with rotor density lead to a non-monotonous behaviour of the average translational velocity. Meanwhile, the rotor suspension bears a finite osmotic compressibility resulting from the long-ranged nature of hydrody- namic interactions such that rotational and odd stresses are transmitted through the solvent also at small and intermediate rotor densities. Consequently, density inhomogeneities predicted for chiral active fluids with odd viscosity can be found and allow for an explicit measurement of odd viscosity in simulations and experiments. At intermediate densities, the collective dynamics shows the emergence of multi-scale vortices and chaotic motion which is identified as active turbulence with a self-similar power-law decay in the energy spectrum, showing that the injected energy on the rotor scale is transported to larger scales, similar to the inverse energy cascade of clas- sical two-dimensional turbulence. While either odd viscosity or active turbulence have been reported in chiral active matter previously, the system studied here shows that the emergence of both simultaneously is possible resulting from the osmotic compressibility and hydrodynamic mediation of odd and active stresses. The collective dynamics of colloids rotating out of phase, i.e., where a constant torque instead of a constant angular velocity is applied, is shown to be qualitatively very similar. However, at smaller densities, local density inhomogeneities imply position dependent angular velocities of the rotors resulting from inter-rotor friction.
While the friction of a quasi-2D layer of active colloids with the substrate is often not easily modifiable in experiments, the incorporation of substrate friction into the simulation models typically implies a considerable increase in computational effort. In chapter 4, a very efficient way of incorporating the friction with a substrate into a two-dimensional multiparticle collision dynamics solvent is introduced, allowing for an explicit investigation of the influences of substrate on active dynamics. For the rotor fluid, it is explicitly shown that the influence of the substrate friction results in a cutoff of the hydrodynamic interaction length, such that the maximum size of the formed vortices is controlled by the substrate friction, also resulting in a cutoff in the energy spectrum, because energy is taken out of the system at the respective length. These findings are in agreement with the experiments.
Since active particles in confinement are known to organise in states of collective dynamics, ensembles of rotationally actuated colloids are studied in circular confinement and in the presence of periodic obstacle lattices in chapters 5 and 6, respectively. The results show that the chaotic active turbulent transport of rotors in suspension can be enhanced and guided resulting from edge flows generated at the boundaries, as has recently been reported for a related chiral active system. The consequent collective rotor dynamics can be regarded as a superposition of active turbulent and imposed flows, leading to on average stationary flows. In contrast to the bulk dynamics, the imposed flows inject additional energy into the system on the long length scales, and the same scaling behaviour of the energy spectrum as in bulk is only obtained if the energy injection scales, due to the mutual generation of rotor translational dynamics throughout the system and the edge flows, are well separated. The combination of edge flow and entropic layering at the boundaries leads to oscillating hydrodynamic stresses and consequently to an oscillating vorticity profile. In the presence of odd viscosity, this consequently leads to non-trivial steady-state density modulations at the boundary, resulting from a balance of osmotic pressure and odd stresses.
Relevant for the efficient dispersion and mixing of inert particles on the mesoscale by means of active turbulent mixing powered by rotors, a study of the dynamics of a binary mixture consisting out of rotors and passive particles is presented in chapter 7. Because the rotors are not self-propelled, but the translational dynamics is induced by the surrounding rotors, the passive particles, which do not inject further energy into the system, are transported according to the same mechanism as the rotors. The collective dynamics thus resembles the pure rotor bulk dynamics at the respective density of only rotors. However, since no odd stresses act between the passive particles, only mutual rotor interactions lead to odd stresses leading to the accumulation of rotors in the regions of positive vorticity. This density increase is associated with a pressure increase, which balances the odd stresses acting on the rotors. However, the passive particles are only subject to the accumulation induced pressure increase such that these particles are transported into the areas of low rotor concentration, i.e., the regions of negative vorticity. Under conditions of sustained vortex flow, this results in segregation of both particle types.
Since local symmetry breaking can convert injected rotational into translational energy, microswimmers can be constructed out of rotor materials when a suitable breaking of symmetry is kept in the vicinity of a rotor. One hypothetical realisation, i.e., a coupled rotor pair consisting out of two rotors of opposite angular velocity and of fixed distance, termed a birotor, are studied in chapter 8. The birotor pumps the fluid into one direction and consequently translates into the opposite direction, and creates a flow field reminiscent of a source doublet, or sliplet flow field. Fixed in space the birotor might be an interesting realisation of a microfluidic pump. The trans- lational dynamics of a birotor can be mapped onto the active Brownian particle model for single swimmers. However, due to the hydrodynamic interactions among the rotors, the birotor ensemble dynamics do not show the emergence of stable motility induced clustering. The reason for this is the flow created by birotor in small aggregates which effectively pushes further arriving birotors away from small aggregates, which eventually are all dispersed by thermal fluctuations
Open-source high-performance software packages for direct and inverse solving of horizontal capillary flow
This work introduces Fronts, a set of open-source numerical software packages for nonlinear horizontal capillary-driven flow problems in unsaturated porous media governed by the Richards equation. The software uses the Boltzmann transformation to solve such problems in semi-infinite domains. The scheme adopted by Fronts allows it to be faster and easier to use than other tools, and provide continuous functions for all involved fields. The software is capable of solving problems that appear in hydrology, but also in other particular domains of interest such as paper-based microfluidics. As the first known open-source implementation to adopt this approach, Fronts has been validated against analytical solutions as well as existing software achieving remarkable results in terms of computational costs and numerical precision, and is meant to aid the study and modeling of capillary flow. Fronts can be freely downloaded and installed, and offers a friendly environment for new users with its complete documentation and tutorial cases.Cited as: Gerlero, G. S., Berli, C. L. A., Kler, P. A. Open-source high-performance software packages for direct and inverse solving of horizontal capillary flow. Capillarity, 2023, 6(2): 31-40. https://doi.org/10.46690/capi.2023.02.0
Four Lectures on the Random Field Ising Model, Parisi-Sourlas Supersymmetry, and Dimensional Reduction
Numerical evidence suggests that the Random Field Ising Model loses
Parisi-Sourlas SUSY and the dimensional reduction property somewhere between 4
and 5 dimensions, while a related model of branched polymers retains these
features in any . These notes give a leisurely introduction to a recent
theory, developed jointly with A. Kaviraj and E. Trevisani, which aims to
explain these facts. Based on the lectures given in Cortona and at the IHES in
2022.Comment: 55 pages, 11 figures; v2 - minor changes, mentioned forthcoming work
by Fytas et a
Robust Exponential Runge-Kutta Embedded Pairs
Exponential integrators are explicit methods for solving ordinary
differential equations that treat linear behaviour exactly. The stiff-order
conditions for exponential integrators derived in a Banach space framework by
Hochbruck and Ostermann are solved symbolically by expressing the Runge--Kutta
weights as unknown linear combinations of phi functions. Of particular interest
are embedded exponential pairs that efficiently generate both a high- and
low-order estimate, allowing for dynamic adjustment of the time step. A key
requirement is that the pair be robust: if the nonlinear source function has
nonzero total time derivatives, the order of the low-order estimate should
never exceed its design value. Robust exponential Runge--Kutta (3,2) and (4,3)
embedded pairs that are well-suited to initial value problems with a dominant
linearity are constructed.Comment: 24 pages, 8 figures. The Mathematica scripts mentioned in the paper
can be found in: https://github.com/stiffode/expint
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