BA24-CorDis

Code Base to the bachelor's thesis "Studies of ERM Models with Correlated Disorder". It holds the code to simulate the ERM models with correlated disorder using Born approximation, as well as the final version of the thesis as a PDF file. It corresponds to the print version of the thesis, which is available at the University of Constance. Its reference is:

    BA2024-StudiesOfERMmodelswithcorrelateddisorder Final.pdf

Each correction of mistakes results in the updated version:

    BA2024-StudiesOfERMmodelswithcorrelateddisorder.pdf

Table of Contents

1. Structure
2. Usage
3. Corrections
4. Known Mistakes

Structure

The main contents of the thesis are held in:

    Bachelorarbeit/Inhalt/Sektionen

It contains the text files for each section of the thesis and uses references to several other folders. I assume the folder titles to be self-explanatory. Numerical results are stored in:

    Bachelorarbeit/Inhalt/Numerik

Hereby Cor-S-SOS-Comp contains results for the Heaviside approach presented in section 4, Cor-S-exp-SOS-Comp stores the respective exponential approach. Since a cruical mistake has been done in the simulation, the corrected results have the attribute *Cor* in their name. They both contain numerical results for $k\mapsto S(k) = 1.0$ and $k\mapsto S(k) = 1.0 + \mathcal{F}(g - 1)(k)$. The naming convention is as follows:

    DENSITY - GRID SIZE - MAX k - SPRING FUNCTION APPROACH .csv

There are logs provided which contain information about the computation time and the parameters used for the Simulation/runsim.sh script, as well as norm values for every iteration of the fixed point solver involved in the Dyson equation.

Usage

The LaTeX project files requires a package containing several customizations I have made. It is not publicly available at the moment. Therefore the user is advised to use the provided PDF file.

The numerical results can reproduced by running the Simulation/runsim.sh script. Needed packages for the julia simulation can be checked and installed via Simulation/PackageCheck.jl. For the fixed point solver multithreading is recommended. It can be activated using the -t flag in the runsim.sh script.

Corrections

The final version of the thesis that was used for the print version might contain mistakes. Since the project formally is completed, any corrections made will be stored in the Bachelorarbeit folder. To access the original LaTeX code please see the commit on august 8th, 2024 with the hash

    2611f066b6d9c5a104089db123ca1d1eb520cbe4

Known Mistakes

• The calculation of the velocity of sound contains a second root, i.e. the mean over $q\mapsto ((D_{\rho_\ast}(q))^{0.5}/q)^{0.5}$ was calculated. Obviously this does not represent local derivatives, such that a corrected version should follow $c(\rho_\ast) := \underset{q\in G_q}{\text{mean}}\left(\sqrt{D_{\rho_\ast}(q)}/q\right)$, while faulty calculations were based on $c_{\text{faulty}}(\rho_\ast) := \underset{q\in G_q}{\text{mean}}\left(\left(\sqrt{D_{\rho_\ast}(q)}/q\right)^{0.5}\right)$.

  This mistake is present in the print version of the thesis. It is fixed in the Code Base within the file Bachelorarbeit/Inhalt/Numerik/Schallgeschwindigkeit_Modul_fixed.jl. Since $x\mapsto \sqrt{x}$ is a monotonically increasing function, the mistake only changes "the form" of the velocity's graphs around $1$, but keeps near $1$ behaviour.

  This directly affects figures 17 and 21 and is corrected for commits after august 12th. One can see the change in an example on figure 17:

  

  compared to the faulty calculation:

  

  So thankfully, general results stay the same, but velocity amplitudes change with $((D_{\rho_\ast}(q))^{0.5}/q)^{0.5} \geq \sqrt{D_{\rho_\ast}(q)}/q$ for $q < 1$ and reversed for $q > 1$.

  The new data can be seen in Bachelorarbeit/Inhalt/Numerik/Cor-S-SOS-Comp/VelocityOfSound-S-fixed.csv and compared to the old data Bachelorarbeit/Inhalt/Numerik/Cor-S-SOS-Comp/VelocityOfSound-S.csv.

• Figure 15 uses a wrong label on the x-axis. Although its purpose can be deduced from the legend, it should be corrected to $\text{min}(D_{\rho_\ast})$, where $\text{min}$ operates on the respective grid $G_q$ for wave vectors $q$.

• Figure 9 also uses wrong labels on both axis, which should be corrected to $||\mathbf{r}||_2$ on the x-axis and $g_0(\mathbf{r})$ on the y-axis.

Known Typos

• In Definition 2.22 (External Field Shifting) there is mistakenly $\Phi_\varphi$ used within the integrnal, which should be corrected to only $\Phi$, since it is the only introduced argment.

• In Postulate 2.4 (The Moment Generating Function) the random variable has both $r$ and $R$ as its name. To avoid confusion and match overall notation, it should be corrected to $R$.

• In the definition of the first Lagrange operator summand $\mathcal{L}_{f,\overline{\text{II}},\mathcal{F}}$ on page 29 right after Corollary 2.4 (Perturbative Basis of the MGF with Normalization) the signs in the second functional derivative should be inverted, i.e. $\delta/\delta \hat J(p_1 + p_2)$. Furthermore to every $J$ a hat should also be added.