A Conservative Discretization of a Hamiltonian System
(2018) FYTK02 20181Computational Biology and Biological Physics - Undergoing reorganization
Department of Astronomy and Theoretical Physics - Undergoing reorganization
- Abstract
- The reaction–diffusion equation (RDE) is a natural way of describing a system where there is not only diffusion but also interaction with its surroundings. The RDE has been the topic of interest in previous papers for its usefulness of explaining pattern formations in nature. The static RDE has, in one dimension, the form of a Newton equation. We study a discretized version of this equation, with applications in biology (where a cell can be seen a natural discretization), economics, and computer simulations.
In particular we consider a class of discretized Newton equations that allows for a conserved quantity. Using this we manage to find a conservative discretization of the $\Phi^4$ system. - Popular Abstract
- Almost anywhere you look in nature, you will see some sort of pattern. In the artichokes from your local supermarkets vegetable aisle, in the horns of wild deer or in the snowflakes falling on your face during winter, patterns arise everywhere. But what exactly is a pattern? Most humans have an instinctual affinity for patterns, we find them beautiful. But disregarding aesthetics, why do physicists find patterns so useful?
Patterns are a discernible regularities. These regularity can be very complex and to this day scientists and mathematicians haven't been able to explain exactly how patterns arise in nature. It was as late as the 80's when we understood how snowflakes pattern get their shapes, even though they had been a topic of... (More) - Almost anywhere you look in nature, you will see some sort of pattern. In the artichokes from your local supermarkets vegetable aisle, in the horns of wild deer or in the snowflakes falling on your face during winter, patterns arise everywhere. But what exactly is a pattern? Most humans have an instinctual affinity for patterns, we find them beautiful. But disregarding aesthetics, why do physicists find patterns so useful?
Patterns are a discernible regularities. These regularity can be very complex and to this day scientists and mathematicians haven't been able to explain exactly how patterns arise in nature. It was as late as the 80's when we understood how snowflakes pattern get their shapes, even though they had been a topic of interest for several hundred years.
The formation of a pattern is just one small step away from total chaos. When working with patterns, it feels like a small miracle that they even form in nature, considering how hard they are to reproduce mathematically. The difficulty of describing patterns in nature can be overcome by using so-called discrete time, since computers must use discrete time. What is meant by this, is that time is not continuous and you jump from one "time box" to the other, like you do when playing hopscotch. To illustrate this point let us look at how populations are studied. Populations, of course, change all the time but you can only measure them discretely. So it's only natural to describe population using a discrete time model. It can also be used in economics where the same situation as population applies. Finding and understanding patterns in models like this can be the difference between a recession and a economic boom, or how growth regulators for plants diffuses in cells.
One would believe that for patterns to arise there should be some sort of conserved quantity. Usually in physics the energy or momentum is conserved. And the conserved quantity for the discrete time case could be used to approximate the continuous time case, and therefore bridging the discrete time case of the computer to the continuous time world that we live in. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/student-papers/record/8964311
- author
- Emanuelsson, Robin LU
- supervisor
-
- Bo Söderberg LU
- organization
- course
- FYTK02 20181
- year
- 2018
- type
- M2 - Bachelor Degree
- subject
- language
- English
- id
- 8964311
- date added to LUP
- 2019-02-27 11:13:19
- date last changed
- 2019-02-27 11:13:19
@misc{8964311,
abstract = {{The reaction–diffusion equation (RDE) is a natural way of describing a system where there is not only diffusion but also interaction with its surroundings. The RDE has been the topic of interest in previous papers for its usefulness of explaining pattern formations in nature. The static RDE has, in one dimension, the form of a Newton equation. We study a discretized version of this equation, with applications in biology (where a cell can be seen a natural discretization), economics, and computer simulations.
In particular we consider a class of discretized Newton equations that allows for a conserved quantity. Using this we manage to find a conservative discretization of the $\Phi^4$ system.}},
author = {{Emanuelsson, Robin}},
language = {{eng}},
note = {{Student Paper}},
title = {{A Conservative Discretization of a Hamiltonian System}},
year = {{2018}},
}