FYS3150/Project-4
3
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

Finished first review, and added a few comments.

Browse Source
This commit is contained in:
Janita Willumsen
2023-12-04 19:31:43 +01:00
parent cb4b35b811
commit df699e0ea0
7 changed files with 156 additions and 136 deletions
Download Patch File Download Diff File Expand all files Collapse all files

36
latex/sections/results.tex
View File

@@ -13,14 +13,14 @@ per spin for $T_{1}$ in Figure \ref{fig:burn_in_energy_1_0}, and $T_{2}$ in Figu
per spin for $T_{1}$ in Figure \ref{fig:burn_in_magnetization_1_0}, and for $T_{2}$
in Figure \ref{fig:burn_in_energy_2_4}.
% Not sure about the relevance of the paragraph below
We used random numbers to set the initial state and the index of spin to flip, and
observed different graphs ... However, when we set the seed of the random number
engine, we could reproduce the results at every run. It is possible to determine
the burn-in time analytically, using the correlation time given by $\tau \approx L^{d + z}$.
Where $d$ is the dimensionality of the system and $z = 2.1665 \pm 0.0012$
\footnote{This value was determined by Nightingale and Blöte for the Metropolis algorithm.}.
However, when we increased number of Monte Carlo cycles the sampled generated during
the burn-in time did not affect the mean value (...). % Should add result showing this
% We used random numbers to set the initial state and the index of spin to flip, and
% observed different graphs ... However, when we set the seed of the random number
% engine, we could reproduce the results at every run. It is possible to determine
% the burn-in time analytically, using the correlation time given by $\tau \approx L^{d + z}$.
% Where $d$ is the dimensionality of the system and $z = 2.1665 \pm 0.0012$
% \footnote{This value was determined by Nightingale and Blöte for the Metropolis algorithm.}.
% However, when we increased number of Monte Carlo cycles the sampled generated during
% the burn-in time did not affect the mean value (...). % Should add result showing this
The lattice was initialized in an ordered and an unordered state, for both temperatures. We observed
no change in expectation value of energy or magnetization for $T_{1}$, when we
@@ -86,7 +86,7 @@ centered around the expectation value $\langle \epsilon \rangle = -1.2370$. %
\label{fig:histogram_2_4}
\end{figure} %
However, we observed a higher variance of Var$(\epsilon) = 0.0203$. When the temperature
increased, the system moved from an ordered to an onordered state. The change in
increased, the system moved from an ordered to an unordered state. The change in
system state, or phase transition, indicates the temperature is close to a critical
point.
@@ -97,8 +97,8 @@ We continued investigating the behavior of the system around the critical temper
First, we generated $10$ million samples of spin configurations for lattices of
size $L \in \{20, 40, 60, 80, 100\}$, and temperatures $T \in [2.1, 2.4]$. We divided the
temperature range into $40$ steps, with an equal step size of $0.0075$. The samples
were generated in parallel, where we allocated $4$ sequential temperatures to $10$
MPI processes. Each process was set to spawn $10$ thread, resulting in a total of
were generated in parallel, where the program allocated $4$ sequential temperatures to $10$
MPI processes. Each process was set to spawn $10$ threads, resulting in a total of
$100$ threads working in parallel. We include results for $1$ million MC cycles
in Appendix \ref{sec:additional_results}
@@ -113,9 +113,9 @@ increase in $\langle \epsilon \rangle$ in the temperature range $T \in [2.25, 2.
\caption{$\langle \epsilon \rangle$ for $T \in [2.1, 2.4]$, $10^7$ MC cycles.}
\label{fig:phase_energy_10M}
\end{figure} %
We observe a deacrese in $\langle |m| \rangle$ for the same temperature range in
Figure \ref{fig:phase_magnetization_10M}, suggesting the system moves from an ordered
magnetized state to a state of no net magnetization. The system energy increase,
We observe a decrease in $\langle |m| \rangle$ for the same temperature range in
Figure \ref{fig:phase_magnetization_10M}, suggesting that the system moves from an ordered
magnetized state to a state of no net magnetization. The system energy increases,
however, there is a loss of magnetization close to the critical temperature.
\begin{figure}[H]
\centering
@@ -124,16 +124,16 @@ however, there is a loss of magnetization close to the critical temperature.
\label{fig:phase_magnetization_10M}
\end{figure} %
In Figure \ref{fig:phase_heat_10M}, we observe an increase in heat capacity in the
temperature range $T \in [2.25, 2.35]$. In addition, we observed a sharper peak
value of heat capacity the lattice size increase.
temperature range $T \in [2.25, 2.35]$. In addition, we observe a sharper peak
value of heat capacity when the lattice size increase.
\begin{figure}[H]
\centering
\includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/heat_capacity.pdf}
\caption{$C_{V}$ for $T \in [2.1, 2.4]$, $10^7$ MC cycles.}
\label{fig:phase_heat_10M}
\end{figure} %
The magnetic susceptibility in Figure \ref{fig:phase_susceptibility_10M}, show
the sharp peak in the same temperature range as that of the heat capacity. Since
The magnetic susceptibility in Figure \ref{fig:phase_susceptibility_10M}, shows
the sharp peak in the same temperature range as that of the heat capacity. Since the
shape of the curve for both heat capacity and the magnetic susceptibility become
sharper when we increase lattice size, we are moving closer to the critical temperature.
\begin{figure}[H]