Amplitude de probabilité | EPFL Graph Search

vignette|Une fonction d'onde pour un seul électron dans l'orbite atomique 5d d'un atome d'hydrogène . La forme montre les endroits où la densité de probabilité de l'électron est supérieure à une certaine valeur, celle-ci est calculée avec l'amplitude de probabilité. La teinte sur la surface colorée montre la phase complexe de la fonction d'onde. En mécanique quantique, une amplitude de probabilité est un nombre complexe utilisé pour décrire le comportement d'un système. Le carré de son module donne la probabilité (ou la densité de probabilité) pour que le système soit mesuré dans un état donné. Un exemple basique Dans le cas très simple d'un photon pouvant être polarisé verticalement ou horizontalement, son état peut être écrit ainsi : :| \psi \rangle = \alpha |H \rangle + \beta |V\rangle où :

| \psi \rangle représente un état possible du photon. *|H \rangle et |V\rangle représentent les deux états de base (Horizontal et

Calculation of geological uncertainties associated with 3-D geological models

Ian Pomian-Srzednicki

3-D geological models are built with data collected in the field such as boreholes, geophysical measurements, pilot shafts or geological mapping. Unfortunately, these data are always limited in number. It implies that geological information is sparse and subsurface models are thus always built of both subjective interpretation and mathematical interpolation/extrapolation techniques. These models are therefore uncertain and this uncertainty is rarely pointed out in a geological prognosis. Our study proposes to bring a new methodology for the evaluation of geological uncertainties related to 3-D subsurface models and to test its potential use. The methodology we propose is based on the 3-D subsurface model, which is here considered as the most probable prediction (notion of best guess). The various geological interfaces that compose the subsurface model are handled individually as Gaussian random fields. At each location of an interface, the random function Z(u) describing the position of this interface is composed of a deterministic part m(u) which represents the expected position, and a random part σ(u)ε(u) which describes fluctuations around the predicted position. Then, a model of spatial variability (a variogram function γ(h)) is proposed in order to condition the random field according to available observations. Several structural constraints, such as the shape of folds and the thickness of layers can also be accounted for in this model. At this point, we are able to estimate the local variance all over the study area by the application of the kriging technique. Finally, the variability is converted into three-dimensional information by calculating probabilities, this describes the occurrence of the various rock masses that are present in the study area. The probabilities are calculated according to intersection rules that govern the stratigraphic sequence of the subsurface model, and they allow us to build a probabilistic model of subsurface structures in the form of a three-dimensional probability field. All of this has been incorporated in a computer program.

EPFL2002

Rotation measure and synchrotron emission signatures in simulations of magnetized galactic discs

Yoan Rappaz

We analyse observational signatures of magnetic fields for simulations of a Milky Way-like disc with supernova-driven interstellar turbulence and self-consistent chemical processes. In particular, we post-process two simulations data sets of the SILCC Project for two initial amplitudes of the magnetic field, B-0 = 3 and 6 mu G, to study the evolution of Faraday rotation measures (RM) and synchrotron luminosity. For calculating the RM, three different models of the electron density n(e) are considered. A constant electron density, and two estimations based on the density of ionized species and the fraction of the total gas, respectively. Our results show that the RM profiles are extremely sensitive to the n(e) models, which assesses the importance of accurate electron distribution observations/estimations for the magnetic fields to be probed using Faraday RMs. As a second observable of the magnetic field, we estimate the synchrotron luminosity in the simulations using a semi-analytical cosmic ray model. We find that the synchrotron luminosity decreases over time, which is connected to the decay of magnetic energy in the simulations. The ratios between the magnetic, the cosmic ray, and the thermal energy density indicate that the assumption of equipartition does not hold for most regions of the ISM. In particular, for the ratio of the cosmic ray to the magnetic field energy the assumption of equipartition could lead to a wrong interpretation of the observed synchrotron emission.

OXFORD UNIV PRESS2022

Local excitations in amorphous solids

Wencheng Ji

Amorphous solids are structurally disordered. They are very common and include glasses, colloids, and granular materials, but are far less understood than crystalline solids. Key aspects of these materials are controlled by the presence of excitations in which a group of particles rearranges. This motion can be triggered by (a) quantum fluctuations associated with two-level systems (TLS), which dominate the low temperature properties of conventional glasses and have practical importance on superconducting qubits; by (b) thermal fluctuations associated with activations, which are related to the famous and challenging

glass transition'' problem; or by (c) exerting an external stress or strain associated with shear transformations, which control the plasticity. Hence, it is important to understand how temperature and system preparation determines the density and geometry of these excitations. The possible unification of these excitations into a common description is also a fundamental problem.  These local excitations are thought to have a close relationship with

Quasi-localised modes (QLMs)'' which are present in the low-frequency vibrational spectrum in amorphous solids. Understanding the properties of QLMs and clarifying the relation between QLMs and these local excitations are important to the study of the latter. In this thesis: (1) we provide a theory for the QLMs, D_L(omega) ~ omega^alpha, that establishes the link between QLMs and shear transformations for systems under quasi-static loading. It predicts two regimes depending on the density of shear transformations P(x)~ x^theta (with x the additional stress needed to trigger a shear transformation). If theta>1/4, alpha=4 and a finite fraction of quasi-localised modes form shear transformations, whose amplitudes vanish at low frequencies. If theta

EPFL2021

Calculation of geological uncertainties associated with 3-D geological models

Ian Pomian-Srzednicki

EPFL2002