Exponential decay

Summary

A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where N is the quantity and λ (lambda) is a positive rate called the exponential decay constant, disintegration constant, rate constant, or transformation constant: :\frac{dN}{dt} = -\lambda N. The solution to this equation (see derivation below) is: :N(t) = N_0 e^{-\lambda t}, where N(t) is the quantity at time t, N0 = N(0) is the initial quantity, that is, the quantity at time t = 0. Measuring rates of decay Mean lifetime If the decaying quantity, N(t), is the number of discrete elements in a certain set, it is possible to compute the average length of time that an element remains in the set. This is called the mean lifetime (or simply the lifetime), where the exponential time constant, \tau

Official source

https://en.wikipedia.org/wiki/Exponential_decay

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This thesis aims to find, for the first time, a direct relation between the size and morphology of small metallic nanostructures (gold in this case) supported on a metal-oxide surface to their catalytic activity. In this perspective, three main topics have been treated during this thesis. The first part concerns the design and realization of a Scanning Tunneling Microscope (STM) supposed to work over a wide temperature range (4K < T ≤ 300K). This new device replaces an existing solution. The coarse approach of the scanning tip towards the sample is realized by an axial motor based on a sapphire prism gliding on shear piezos in the stick&slip mode. The new STM is very rigid moving the resonance frequencies to higher values with respect to the existing solution. Operation of the motor down to T = 8K has been proven, however topographic imaging has been performed only in the temperature range 77K < T ≤ 300K. The second part of this work focuses on a study of the evolution of the morphology of gold nanoparticles on a TiO2(110) surface. Size-selected clusters Aun+ (n = 5, 7) are deposited at a well defined kinetic energy on the surface held at room temperature. Subsequent annealing of the sample has been performed stepwise. After each temperature increase, the morphology has been determined by STM. The evolution as a function of surface temperature has been studied for two different surface reconstructions, TiO2(110)-(1×1) and TiO2(110)-(2×1). The deposition process leads only to small fragmentation and the morphology is stable up to T = 400K. Further increase of the surface temperature leads to sintering of the particles by Ostwald ripening as shown by an exponential decrease of the island density with temperature. The main topic of this thesis, the correlation between morphology and catalytic activity, is described in the last part of this manuscript. For the first time we are able to relate the onset of CO2 production from CO and O2 to a clear change in the morphology. The catalytic activity of the particles strongly depends on their size and dimensionality. The relative activity per particle has been determined and we find a clear maximum for clusters containing 60 atoms and are 3 to 4 monolayers high. These results are discussed in contrast of literature data on the same but also on different metal-oxide surfaces.

EPFL2009

Novel corrector problems with exponential decay of the resonance error for numerical homogenization

Edoardo Paganoni

Multiscale problems, such as modelling flows through porous media or predicting the mechanical properties of composite materials, are of great interest in many scientific areas. Analytical models describing these phenomena are rarely available, and one must recur to numerical simulations. This represents a great computational challenge, because of the prohibitive computational cost of resolving the small scales. Multiscale numerical methods are therefore necessary to solve multiscale problems within reasonable computational time and resources. In particular, numerical homogenization techniques aim to capture the macroscopic behaviour with equations whose coefficients are computed numerically from the solutions of corrector problems at the microscale. A lack of knowledge of the coupling conditions between the micro- and the macro-scales brings in the so-called resonance error, which affects the accuracy of all multiscale methods. This source of error often dominates the numerical discretization errors and increasing its rate of decay is crucial for improving the accuracy of multiscale methods. In this work, we propose two novel upscaling schemes with arbitrarily high convergence rates of the resonance error to approximate the homogenized coefficients of scalar, linear second order elliptic differential equations. The first one is based on a parabolic equation, inspired by a model employed to compute the effective diffusive coefficients in stochastic diffusion processes. By using the approximation properties of smooth filtering functions, the homogenized coefficients can be approximated with arbitrary rates of accuracy. This claim is proved through an a priori convergence analysis, under the assumption of smooth periodic multiscale coefficients. Numerical experiments verify the expected convergence rates also under more general assumptions, such as discontinuous and random coefficients. The second method originates from the first by integrating the parabolic equation over a finite time interval. This method is referred to as the modified elliptic approach, because of the presence of a right-hand side which can be interpreted in terms of continuous semigroups and can be approximated numerically by Krylov subspace methods. The same convergence results as in the parabolic approach hold true. As a last step, a convergence analysis of the resonance error for the modified elliptic approach in the context of equations with random coefficients is performed. In this case, the resonance error is composed of a variance and a bias term, which can be bounded from above by a function decaying to zero. Numerical experiments reveal that the convergence rate of the resonance error for random coefficients is hampered, in comparison to the case of periodic coefficients, but the modified elliptic approach nevertheless outperforms standard methods.

EPFL2020

MATHICSE technical Report : A parabolic local problem with exponential decay of the resonance error for numerical homogenization

Doghonay Arjmand, Edoardo Paganoni

This paper aims at an accurate and ecient computation of eective quantities, e.g., the homogenized coecients for approximating the solu- tions to partial dierential equations with oscillatory coecients. Typical multiscale methods are based on a micro-macro coupling, where the macro model describes the coarse scale behaviour, and the micro model is solved only locally to upscale the eective quantities, which are missing in the macro model. The fact that the micro problems are solved over small domains within the entire macroscopic domain, implies imposing arti- cial boundary conditions on the boundary of the microscopic domains. A naive treatment of these articial boundary conditions leads to a rst order error in "=, where " < represents the characteristic length of the small scale oscillations and d is the size of micro domain. This er- ror dominates all other errors originating from the discretization of the macro and the micro problems, and its reduction is a main issue in to- day's engineering multiscale computations. The objective of the present work is to analyse a parabolic approach, rst announced in [A. Abdulle, D. Arjmand, E. Paganoni, C. R. Acad. Sci. Paris, Ser. I, 2019], for computing the homogenized coecients with arbitrarily high convergence rates in "=. The analysis covers the setting of periodic microstructure, and numerical simulations are provided to verify the theoretical ndings for more general settings, e.g. random stationary micro structures.

MATHICSE2020

EPFL2009

Novel corrector problems with exponential decay of the resonance error for numerical homogenization

Edoardo Paganoni

EPFL2020