Diffusion process | EPFL Graph Search

In probability theory and statistics, diffusion processes are a class of continuous-time Markov process with almost surely continuous sample paths. Diffusion process is stochastic in nature and hence is used to model many real-life stochastic systems. Brownian motion, reflected Brownian motion and Ornstein–Uhlenbeck processes are examples of diffusion processes. It is used heavily in statistical physics, statistical analysis, information theory, data science, neural networks, finance and marketing. A sample path of a diffusion process models the trajectory of a particle embedded in a flowing fluid and subjected to random displacements due to collisions with other particles, which is called Brownian motion. The position of the particle is then random; its probability density function as a function of space and time is governed by an advection–diffusion equation. Mathematical definition A diffusion process is a Markov process with continuous sample paths for which the Kol

Etude par STM de la déposition d'agrégats d'or sur TiO2 et mesure de l'émission électronique secondaire induite par l'impact d'agrégats sur une surface

Pierre Convers

This thesis work focuses on impact and diffusion processes as well as equilibrium positions of a metal deposited on a surface. The metal is deposited in the form of clusters containing n atoms in a controlled way. Gold or silver clusters cations (Au+n and Ag+n ) are used. Their size (n ∈ [1, 9]), charge state and deposition energy (E = 10 - 4500 eV) are well defined. The surface being at room temperature during the deposition, can be annealed and transferred in a scanning tunnelling microscope (STM) after deposition. The variable temperature microscope was developed during this thesis. The approach of the tip is carried out by an axial motor, which renders the microscope very stable. The first part of this work treats of the evolution of gold deposited on a TiO2(110) surface. Position and size of gold islands created by cluster or atomic depositions are determined by STM. Compared to atomic deposition at the same energy (10 < E < 100 eV), the cluster deposition produces smaller islands with a large islands density after annealing at 800 K. Regardless the deposition conditions, gold atoms diffuse on the surface already at 300 K. The impact energy is the key parameter to reduce the diffusion by cluster or atomic implantation or defect creation. Upon a high energy deposition, part of the deposited gold is invisible to the STM. The fate of these gold atoms is unclear : they are either buried under the surface or ejected in vacuum at the impact. The control of the islands size should permit the creation of a stable system with an optimum catalytic activity (i. e. CO combustion). Research groups have shown that this activity is strongly influenced by the islands size. It is equal to zero if the diameter is greater than 5 nm. The second part of this work is devoted to the electron emission γ induced by the impact of silver clusters Ag+n on a Pt and HOPG surface. Measurements are made by varying the impact energy (and so the speed v) of size-selected clusters on a defined surface. Impact velocity ranges from 104 to 105 m/s, which is lower than the classical threshold. Nevertheless every cluster size produces an electron emission. A potential emission (γ(v=0) ≠ 0) is observed for the monomer on both surfaces, which is caused by excited ions in a metastable state. The γ(v) curves are increasing, and no mesurable oscillation are observed. This does not confirm earlier work by Meiwes-Broer et al. These authors found oscillations in γ on similar systems relating them to the electronic structure of the clusters and the surface. A model based on heating of the electron gas is developed. This model gives good agreements with the γ(v) curves. The electronic temperature is estimated to 3000-8000 K. Similar behavior on both surfaces (Pt and HOPG) depending of the cluster size is shown. This behavior is probably related to the geometrical structure of the clusters. Finally, a more pronounced molecular effect is observed for the HOPG : the number of electrons emitted by the impact of a Ag+n is up to 7 times higher than the number of electrons emitted by impacts of n independent atoms. This value is smaller than 2 for the Pt surface.

EPFL2005

Diffusion Process Modeling by Using Fractional-Order Models

Tomas Skovranek

This paper deals with a concept and description of a RC network as an electro-analog model of diffusion process which enables to simulate heat dissipation under different initial and boundary conditions. It is based on well-known analogy between heat and electrical conduction. In the paper are compared analytical solution together with numerical solution and experimentally measured data. For the first time a fractional order model of diffusion process and its modeling via lumped RC network has been used. Simple examples of simulations, measurements and their comparison are shown.

Elsevier2015

The power of adaptivity in source identification with time queries on the path

Victor Cyril L Lecomte, Gergely Odor, Patrick Thiran

We study the problem of identifying the source of a stochastic diffusion process spreading on a graph based on the arrival times of the diffusion at a few queried nodes. In a graph

G=(V,E)

, an unknown source node

v^* \in V

is drawn uniformly at random, and unknown edge weights

w(e)

for

e\in E

, representing the propagation delays along the edges, are drawn independently from a Gaussian distribution of mean

1

and variance

\sigma^2

. An algorithm then attempts to identify

v^*

by querying nodes

q \in V

and being told the length of the shortest path between

q

and

v^*

in graph

G

weighted by

w

. We consider two settings: \emph{non-adaptive}, in which all query nodes must be decided in advance, and \emph{adaptive}, in which each query can depend on the results of the previous ones. Both settings are motivated by an application of the problem to epidemic processes (where the source is called patient zero), which we discuss in detail. We characterize the query complexity when

G

is an

n

-node path. In the non-adaptive setting,

\Theta(n\sigma^2)

queries are needed for

\sigma^2 \leq 1

, and

\Theta(n)

for

\sigma^2 \geq 1

. In the adaptive setting, somewhat surprisingly, only

\Theta(\log\log_{1/\sigma}n)

are needed when

\sigma^2 \leq 1/2

, and

\Theta(\log \log n)+O_\sigma(1)

when

\sigma^2 \geq 1/2

. This is the first mathematical study of source identification with time queries in a non-deterministic diffusion process.

2022

Etude par STM de la déposition d'agrégats d'or sur TiO2 et mesure de l'émission électronique secondaire induite par l'impact d'agrégats sur une surface

Pierre Convers

EPFL2005

The power of adaptivity in source identification with time queries on the path

Victor Cyril L Lecomte, Gergely Odor, Patrick Thiran

We study the problem of identifying the source of a stochastic diffusion process spreading on a graph based on the arrival times of the diffusion at a few queried nodes. In a graph

G=(V,E)

, an unknown source node

v^* \in V

is drawn uniformly at random, and unknown edge weights

w(e)

for

e\in E

, representing the propagation delays along the edges, are drawn independently from a Gaussian distribution of mean

1

and variance

\sigma^2

. An algorithm then attempts to identify

v^*

by querying nodes

q \in V

and being told the length of the shortest path between

q

and

v^*

in graph

G

weighted by

w

G

is an

n

-node path. In the non-adaptive setting,

\Theta(n\sigma^2)

queries are needed for

\sigma^2 \leq 1

, and

\Theta(n)

for

\sigma^2 \geq 1

. In the adaptive setting, somewhat surprisingly, only

\Theta(\log\log_{1/\sigma}n)

are needed when

\sigma^2 \leq 1/2

, and

\Theta(\log \log n)+O_\sigma(1)

when

\sigma^2 \geq 1/2

. This is the first mathematical study of source identification with time queries in a non-deterministic diffusion process.

2022