Edit analytical expressions and add calculation
This commit is contained in:
Janita Willumsen
parent
cfef3c7366
commit
7d402f4ded
@ -10,6 +10,9 @@
|
||||
% Allows special characters (including æøå)
|
||||
\usepackage[utf8]{inputenc}
|
||||
\usepackage{fontawesome}
|
||||
\usepackage{multirow}
|
||||
\usepackage[table]{xcolor}
|
||||
% \rowcolors{2}{gray!15}{white}
|
||||
% \usepackage{tabularx}
|
||||
% \usepackage[english]{babel}
|
||||
|
||||
@ -23,12 +26,12 @@
|
||||
\usepackage{xcolor} % set colors
|
||||
\usepackage{hyperref} % automagic cross-referencing
|
||||
\usepackage{listings} % display code
|
||||
\usepackage{subfigure} % imports a lot of cool and useful figure commands
|
||||
% \usepackage{subfigure} % imports a lot of cool and useful figure commands
|
||||
\usepackage{subcaption}
|
||||
% \usepackage{float}
|
||||
%\usepackage[section]{placeins}
|
||||
\usepackage{algorithm}
|
||||
\usepackage[noend]{algpseudocode}
|
||||
\usepackage{subfigure}
|
||||
\usepackage{tikz}
|
||||
% \usetikzlibrary{quantikz}
|
||||
% defines the color of hyperref objects
|
||||
@ -73,6 +76,8 @@
|
||||
% Conclusion
|
||||
\subfile{sections/conclusion}
|
||||
|
||||
\clearpage
|
||||
\newpage
|
||||
% Appendix
|
||||
\subfile{sections/appendices}
|
||||
|
||||
|
||||
@ -67,3 +67,20 @@
|
||||
}
|
||||
|
||||
# Miscellaneous things
|
||||
@misc{obermeyer:2020:ising,
|
||||
author = {Johannes Obermeyer},
|
||||
title = {The Ising Model in One and Two Dimensions},
|
||||
year = {2020},
|
||||
url = {https://www.thphys.uni-heidelberg.de/~wolschin/statsem20_3s.pdf},
|
||||
urldate = {2023-11-13},
|
||||
pages = {16},
|
||||
note = {Seminar on Statistical Physics at the University of Heidelberg, summary}
|
||||
}
|
||||
|
||||
@book{springer:2012:modernstat,
|
||||
author = {Jay L. Devore and Kenneth N. Berk},
|
||||
title = {Modern Mathematical Statistics with Application},
|
||||
publisher = {Cham: Springer International Publishing AG},
|
||||
year = {2021},
|
||||
edition = {3}
|
||||
}
|
||||
@ -3,6 +3,24 @@
|
||||
\begin{document}
|
||||
\appendix
|
||||
\section{Ising model system states}\label{sec:system_states}
|
||||
Units
|
||||
\begin{table}[H]
|
||||
\centering
|
||||
\begin{tabular}[c]{cc} % @{\extracolsep{\fill}}
|
||||
\hline
|
||||
Value & Unit \\
|
||||
\hline
|
||||
$[ E ]$ & $J$ \\
|
||||
$[ T ]$ & $J / k_{\text{B}}$ \\
|
||||
$[ M ]$ & $\dots$ \\
|
||||
$[ C_{\text{V}} ]$ & $k_{\text{B}}$ \\
|
||||
$[ \chi ]$ & $1 / J$ \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\caption{Units, given by the coupling constant $J$ and the Boltzmann constant $k_{\text{B}}$.}
|
||||
\label{tab:units}
|
||||
\end{table}
|
||||
|
||||
To avoid counting duplicates, we used
|
||||
\begin{figure}\label{fig:tikz_counting}
|
||||
\centering
|
||||
@ -30,14 +48,196 @@ To avoid counting duplicates, we used
|
||||
\caption{Rules for multiplying spin pairs.}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}\label{fig:tikz_neighbor}
|
||||
\centering
|
||||
\begin{subfigure}{0.4\linewidth}
|
||||
\begin{tikzpicture}
|
||||
\draw (0, 0) grid (2, 2);
|
||||
\node (s1) at (0.5, 1.5) {$\uparrow$};
|
||||
\node (s2) at (1.5, 1.5) {$\uparrow$};
|
||||
\node (s3) at (0.5, 0.5) {$\downarrow$};
|
||||
\node (s4) at (1.5, 0.5) {$\downarrow$};
|
||||
\end{tikzpicture}
|
||||
\caption{}
|
||||
\label{fig:sub_tikz_neighbor_a}
|
||||
\end{subfigure}
|
||||
\
|
||||
\begin{subfigure}{0.4\linewidth}
|
||||
\begin{tikzpicture}
|
||||
\draw (0, 0) grid (2, 2);
|
||||
\node (s1) at (0.5, 1.5) {$\uparrow$};
|
||||
\node (s2) at (1.5, 1.5) {$\downarrow$};
|
||||
\node (s3) at (0.5, 0.5) {$\downarrow$};
|
||||
\node (s4) at (1.5, 0.5) {$\uparrow$};
|
||||
\end{tikzpicture}
|
||||
\caption{}
|
||||
\label{fig:sub_tikz_neighbor_b}
|
||||
\end{subfigure}
|
||||
\caption{Possible spin configurations for two spins up.}
|
||||
\end{figure}
|
||||
|
||||
|
||||
\section{Analytical expressions}\label{sec:analytical_expressions}
|
||||
The Boltzmann distribution is normalized using a partition function $Z$ given by
|
||||
The Boltzmann distribution is normalized using a partition function $Z$, given by
|
||||
\begin{equation}\label{eq:partition}
|
||||
Z = \sum_{\text{all possible } \mathbf{s}}^{N}
|
||||
\end{equation}
|
||||
We use the values estimated for the $2 \times 2$ case, found in \ref{tab:lattice_config},
|
||||
and find
|
||||
\begin{align*}
|
||||
Z &= \sum_{\text{all possible } s_{i}}^{N} e^{-\beta E(\mathbf{s})} \\
|
||||
&= \dots \\
|
||||
&= 4 \cosh (8 \beta J) + 12 \\
|
||||
Z &= 1 \cdot e^{-\beta (-8J)} + 4 \cdot e^{-\beta (0)} + 4 \cdot e^{-\beta (0)} + 2 \cdot e^{-\beta (8J)} \\
|
||||
& \quad + 4 \cdot e^{-\beta (0)} 1 \cdot e^{-\beta (-8J)} \\
|
||||
&= 2e^{8 \beta J} + 2e^{-8 \beta J} + 12.
|
||||
\end{align*}
|
||||
We rewrite the expression using $\cosh(8 \beta J) = 1/2 \big( e^{8 \beta J} + e^{-8 \beta J})$, and get
|
||||
\begin{align*}
|
||||
z &= 4 \cosh (8 \beta J) + 12
|
||||
\end{align*}
|
||||
The Boltzmann distribution is given by
|
||||
\begin{equation}\label{eq:boltzmann}
|
||||
p(\mathbf{s} \ | \ T) = \frac{1}{Z} e^{-\beta E(\mathbf{s})},
|
||||
\end{equation}
|
||||
for a given temperature $T$. With our expression for the partition function, we
|
||||
get the probability distribution
|
||||
\begin{align*}
|
||||
p(\mathbf{s} \ | \ T) &= \frac{1}{4 \cosh (8 \beta J) + 12} e^{-\beta E(\mathbf{s})}
|
||||
\end{align*}
|
||||
|
||||
For discrete random variables $X$, with a known probability distribution, the
|
||||
expected value of $x$ is given by
|
||||
\begin{align*}
|
||||
\mathbb{E}(x) &= \sum_{x \in D} x \cdot p(x) & \text{\cite[p. 127]{springer:2012:modernstat}}.
|
||||
\end{align*}
|
||||
For a function of a stochastic random variable, the expected value of $x$ is
|
||||
\begin{align*}
|
||||
\mathbb{E}(h(X)) &= \sum_{x \in D} h(x) \cdot p(x)
|
||||
\end{align*}
|
||||
And in the case of a linear function we have
|
||||
\begin{align*}
|
||||
\mathbb{E}(aX + b) &= a \cdot \mathbb{E}(X) + b & \text{\cite[p. 131]{springer:2012:modernstat}}
|
||||
\end{align*}
|
||||
In our case the discrete random variable is the spin configuration, and we want
|
||||
to find the expected value of the function $E(\mathbf{s})$. In addition, we use
|
||||
$\langle E \rangle$ as notation for expexted value of a given variable, since $\mathbf{s}$ is the stochastic
|
||||
random variable.
|
||||
|
||||
The expression for total energy and total magnetization is given in eq. \eqref{eq:energy} and \eqref{eq:magnetization}.
|
||||
The expected values for these is given by
|
||||
\begin{equation*}
|
||||
\langle E(\mathbf{s}) \rangle = \sum_{i=1}^{N} E(s_{i}) p(s_{i} \ | \ T)
|
||||
\end{equation*}
|
||||
|
||||
\begin{equation*}
|
||||
\langle |M(\mathbf{s})| \rangle = \sum_{i=1}^{N} |M(s_{i})| p(s_{i} \ | \ T)
|
||||
\end{equation*}
|
||||
|
||||
Since we want to compare expected values for different lattice sizes, we have to
|
||||
find the expected values per spin. We normalize the total expressions for total
|
||||
energy \eqref{eq:energy} and magnetizaation \eqref{eq:magnetization} by the
|
||||
number of spins to get
|
||||
\begin{equation}\label{eq:spin_energy}
|
||||
\epsilon(\mathbf{s}) = \frac{E(\mathbf{s})}{N}
|
||||
\end{equation}
|
||||
|
||||
\begin{equation}\label{eq:spin_magnetization}
|
||||
m(\mathbf{s}) = \frac{M(\mathbf{s})}{N}
|
||||
\end{equation}
|
||||
|
||||
Both energy per spin and magnetization per spin are functions of $\mathbf{s}$,
|
||||
we will use the short hand notation $\langle \epsilon \rangle$ and $\langle |m| \rangle$.
|
||||
In addition, the number of spins are given as a constant for each lattice. We can
|
||||
rewrite and use this when we find the expectation values per spin, for energy per spin
|
||||
\begin{align*}
|
||||
\langle \epsilon \rangle &= \sum_{i=1}^{N} \epsilon(s_{i}) p(s_{i} \ | \ T) \\
|
||||
&= \sum_{i=1}^{N} \frac{E(\mathbf{s})}{N} p(s_{i} \ | \ T) \\
|
||||
&= \frac{1}{N} \sum_{i=1}^{N} E(\mathbf{s}) p(s_{i} \ | \ T)
|
||||
\end{align*}
|
||||
The same applies for magnetization per spin
|
||||
\begin{align*}
|
||||
\langle |m| \rangle = \frac{1}{N} \sum_{i=1}^{N} |M(s_{i})| p(s_{i} \ | \ T)
|
||||
\end{align*}
|
||||
Continuing with the expectation values for a $2 \times 2$ lattice, excluding the terms which give zero we get
|
||||
\begin{align*}
|
||||
\langle E \rangle &= (-8J) \cdot \frac{1}{Z} e^{8 \beta J} + 2 \cdot (8J) \cdot \frac{1}{Z} e^{-8 \beta J} + (-8J) \cdot \frac{1}{Z} e^{8 \beta J} \\
|
||||
&= \frac{16J}{Z} \big(e^{-8 \beta J} - e^{8 \beta J}) \\
|
||||
&= -\frac{32J \sinh(8 \beta J)}{4(\cosh(8 \beta J) + 3)} \\
|
||||
&= -\frac{8J \sinh(8 \beta J)}{\cosh(8 \beta J) + 3},
|
||||
\end{align*}
|
||||
and
|
||||
\begin{align*}
|
||||
\langle |M| \rangle &= 4 \cdot \frac{1}{Z} \cdot e^{8 \beta J} + 4 \cdot 2 \cdot \frac{1}{Z} \cdot e^{0} \\
|
||||
& \quad + 4 \cdot | -2| \cdot \frac{1}{Z} \cdot e^{0} + | -4| \cdot e^{8 \beta J} \\
|
||||
&= \frac{8 e^{8 \beta J} + 16}{Z} \\
|
||||
&= \frac{4 (2e^{8 \beta J} + 4)}{4(\cosh(8 \beta J) + 3)} \\
|
||||
&= \frac{2(e^{8 \beta J} + 2)}{\cosh(8 \beta J) + 3}
|
||||
\end{align*}
|
||||
The squared function
|
||||
\begin{align*}
|
||||
\langle E^{2} \rangle &= (-8J)^{2} \cdot \frac{1}{Z} e^{8 \beta J} + 2 \cdot (8J)^{2} \cdot \frac{1}{Z} e^{-8 \beta J} + (-8J)^{2} \cdot \frac{1}{Z} e^{8 \beta J} \\
|
||||
&= \frac{128J^{2}}{Z} \big(e^{8 \beta J} + e^{-8 \beta J} \big) \\
|
||||
&= \frac{128J^{2} \cosh(8 \beta J)}{4(\cosh(8 \beta J) + 3)} \\
|
||||
&= \frac{64J^{2} \cosh(8 \beta J)}{\cosh(8 \beta J) + 3},
|
||||
\end{align*}
|
||||
and
|
||||
\begin{align*}
|
||||
\langle M^{2} \rangle &= 4^{2} \cdot \frac{1}{Z} \cdot e^{8 \beta J} + 4 \cdot 2^{2} \cdot \frac{1}{Z} \cdot e^{0} \\
|
||||
& \quad + 4 \cdot (-2)^{2} \cdot \frac{1}{Z} \cdot e^{0} + (-4)^{2}\cdot e^{8 \beta J} \\
|
||||
&= \frac{32e^{8 \beta J} + 32}{Z} \\
|
||||
&= \frac{4 (8e^{8 \beta J} + 8)}{4(\cosh(8 \beta J) + 3)} \\
|
||||
&= \frac{8e^{8 \beta J} + 8}{\cosh(8 \beta J) + 3}
|
||||
\end{align*}
|
||||
The squared expectation value is given by
|
||||
\begin{align*}
|
||||
\langle E \rangle^{2} &= \bigg(-\frac{8J \sinh(8 \beta J)}{\cosh(8 \beta J) + 3} \bigg)^{2} \\
|
||||
&= \frac{64J^{2} \sinh^{2}(8 \beta J)}{(\cosh(8 \beta J) + 3)^{2}},
|
||||
\end{align*}
|
||||
and
|
||||
\begin{align*}
|
||||
\langle |M| \rangle^{2} &= \Big( \frac{2(e^{8 \beta J} + 2)}{\cosh(8 \beta J) + 3} \Big)^{2} \\
|
||||
&= \frac{4(e^{8 \beta J} + 2)^{2}}{(\cosh(8 \beta J) + 3)^{2}}
|
||||
\end{align*}
|
||||
Calculating the heat capacity and susceptibility, we need the variance of both total
|
||||
energy and total magnetizaation. We obtain this using the definition
|
||||
\begin{align*}
|
||||
\mathbb{V}(X) &= \sum_{x \in D} [(x - \mathbb{E}(x))^2 \cdot p(x)] & \text{\cite[p. 132]{springer:2012:modernstat}}. \\
|
||||
&= \mathbb{E}(X^{2}) - [\mathbb{E}(X)]^{2}
|
||||
\end{align*}
|
||||
The variance of total energy is given by
|
||||
\begin{align*}
|
||||
\mathbb{V}(E) &= \mathbb{E}(E^{2}) - [\mathbb{E}(E)]^{2} \\
|
||||
&= \frac{64J^{2} \cosh(8 \beta J)}{\cosh(8 \beta J) + 3} - \frac{64J^{2} \sinh^{2}(8 \beta J)}{(\cosh(8 \beta J) + 3)^{2}} \\
|
||||
&= 64J^{2} \bigg( \frac{\cosh(8 \beta J)}{\cosh(8 \beta J) + 3} - \frac{\sinh^{2}(8 \beta J)}{(\cosh(8 \beta J) + 3)^{2}} \bigg) \\
|
||||
&= 64J^{2} \bigg( \frac{(\cosh(8 \beta J)) \cdot (\cosh(8 \beta J) + 3)}{(\cosh(8 \beta J) + 3)^{2}} \\
|
||||
& \quad - \frac{\sinh^{2}(8 \beta J)}{(\cosh(8 \beta J) + 3)^{2}} \bigg) \\
|
||||
&= 64J^{2} \bigg( \frac{\cosh^{2}(8 \beta J) + 3\cosh(8 \beta J) - \sinh^{2}(8 \beta J)}{(\cosh(8 \beta J) + 3)^{2}} \bigg) \\
|
||||
&= 64J^{2} \bigg( \frac{3\cosh(8 \beta J) + 1}{(\cosh(8 \beta J) + 3)^{2}} \bigg)
|
||||
\end{align*}
|
||||
\begin{align*}
|
||||
\mathbb{V}(M) &= \mathbb{E}(M^{2}) - [\mathbb{E}(|M|)]^{2} \\
|
||||
&= \frac{8e^{8 \beta J} + 8}{\cosh(8 \beta J) + 3} - \frac{4(e^{8 \beta J} + 2)^{2}}{(\cosh(8 \beta J) + 3)^{2}} \\
|
||||
&= \frac{(8(e^{8 \beta J} + 1)) \cdot (\cosh(8 \beta J) + 3) - 4(e^{8 \beta J} + 2)^{2}}{(\cosh(8 \beta J) + 3)^{2}} \\
|
||||
&= \frac{4(e^{8 \beta J} + 1) \cdot (e^{8 \beta J} + e^{-8 \beta J}) + 24(e^{8 \beta J} + 1) - 4(e^{8 \beta J} + 1)^{2}}{(\cosh(8 \beta J) + 3)^{2}} \\
|
||||
&= \frac{4e^{2(8 \beta J)} + 4e^{8 \beta J} 4e^{0} + 4e^{-8 \beta J} 24e^{8 \beta J} + 24 - 4e^{2(8 \beta J)} - 16e^{8 \beta J} - 16}{(\cosh(8 \beta J) + 3)^{2}} \\
|
||||
&= \frac{4(3e^{8 \beta J} + e^{-8 \beta J} + 3)}{(\cosh(8 \beta J) + 3)^{2}}
|
||||
\end{align*}
|
||||
|
||||
\begin{align*}
|
||||
C_{\text{V}} &= \frac{1}{N} \frac{1}{k_{\text{B} T^{2}}} (\mathbb{E}(E^{2}) - [\mathbb{E}(E)]^{2}) \\
|
||||
&= \frac{1}{N k_{\text{B} T^{2}}} \mathbb{V}(E) \\
|
||||
&= \frac{64J^{2} }{N k_{\text{B}} T^{2}} \bigg( \frac{3\cosh(8 \beta J) + 1}{(\cosh(8 \beta J) + 3)^{2}} \bigg)
|
||||
\end{align*}
|
||||
|
||||
\begin{align*}
|
||||
\chi &= \frac{1}{N} \frac{1}{k_{\text{B} T}} (\mathbb{E}(M^{2}) - [\mathbb{E}(M)]^{2}) \\
|
||||
&= \frac{1}{N k_{\text{B} T}} \mathbb{V}(M) \\
|
||||
&= \frac{4}{N k_{\text{B} T}} \bigg( \frac{3e^{8 \beta J} + e^{-8 \beta J} + 3}{(\cosh(8 \beta J) + 3)^{2}} \bigg)
|
||||
\end{align*}
|
||||
\begin{align*}
|
||||
\langle \epsilon^{2} \rangle &= \frac{1}{N^{2}} \sum_{i=1}^{N} E(\mathbf{s})^{2} p(s_{i} \ | \ T) \\
|
||||
&=
|
||||
\end{align*}
|
||||
The same applies for magnetization per spin
|
||||
\begin{align*}
|
||||
\langle |m| \rangle = \frac{1}{N} \sum_{i=1}^{N} |M(s_{i})| p(s_{i} \ | \ T)
|
||||
\end{align*}
|
||||
\end{document}
|
||||
@ -1,13 +1,75 @@
|
||||
\documentclass[../ising_model.tex]{subfiles}
|
||||
\documentclass[../ising_model.tex]{subfiles}
|
||||
|
||||
\begin{document}
|
||||
\begin{document}
|
||||
\section{Methods}\label{sec:methods}
|
||||
\subsection{The Ising model}\label{sec:ising_model}
|
||||
% Add definitions
|
||||
The Ising model correspond to a lattice with a number of spin sites, where
|
||||
each spin has the possible state up $\uparrow$ or down $\downarrow$. We will study the
|
||||
two-dimensional Ising model for ferromagnets, however, the model is not resticted
|
||||
to this dimentionality~\cite[p. 3]{obermeyer:2020:ising}. We will assume a square lattice, where the length $L$ of
|
||||
the lattice give the number of spin sites $N = L^{2}$. When we
|
||||
consider the entire 2D lattice we can study the system spin configuration, or the
|
||||
microstate, given by $\mathbf{s} = (s_{1}, s_{2}, \dots, s_{N})$. The value of
|
||||
each spin $s_{i}$, where $i \in [1, N]$, is given by the direction of the spin
|
||||
$\uparrow = +1$ and $\downarrow = -1$.
|
||||
|
||||
We find the total energy of the system using
|
||||
\begin{equation}\label{eq:energy}
|
||||
E(\mathbf{s}) = -J \sum_{\langle k l \rangle}^{N} s_{k} s_{l}
|
||||
\end{equation}
|
||||
where $\langle k l \rangle$ denotes the neighboring spins.
|
||||
|
||||
The total magnetization of the system is given by
|
||||
\begin{equation}\label{eq:magnetization}
|
||||
M(\mathbf{s}) = \sum_{i}^{N} s_{i},
|
||||
\end{equation}
|
||||
with degeneracy denoting how many states have the same values. We will consider
|
||||
a 2D lattice with $L = 2$, with periodic boundary conditions, following the
|
||||
counting pattern shown in figure~\ref{fig:tikz_counting}. The resulting values for
|
||||
the $2 \times 2$ lattice can be found in table \ref{tab:lattice_config}.
|
||||
|
||||
\begin{table}[H]
|
||||
\centering
|
||||
\begin{tabular}{cccc} % @{\extracolsep{\fill}}
|
||||
\hline
|
||||
Spins up & $E(\mathbf{s})$ & $M(\mathbf{s})$ & Degeneracy \\
|
||||
\hline
|
||||
4 & -8J & 4 & 1 \\
|
||||
3 & 0 & 2 & 4 \\
|
||||
\multirow{2}*{2} & 0 & 0 & 4 \\
|
||||
& 8J & 0 & 2 \\
|
||||
1 & 0 & -2 & 4 \\
|
||||
0 & -8J & -4 & 1 \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\caption{Values}
|
||||
\label{tab:lattice_config}
|
||||
\end{table}
|
||||
|
||||
Probability stuff, for a system with a given temperature $T$. We use the Boltzmann
|
||||
distribution in the analytical expression, to compare with our results. This is
|
||||
a probability distribution from the family of exponential
|
||||
\begin{equation}\label{eq:boltzmann_distribution}
|
||||
p(\mathbf{s} \ | \ T) = \frac{1}{Z} \exp^{-\beta E(\mathbf{s})},
|
||||
\end{equation}
|
||||
where $Z$ is the partition function $\beta$ is the related to the inverse of the
|
||||
temperature as
|
||||
\begin{equation}\label{eq:beta}
|
||||
\beta = \frac{1}{k_{\text{B}} T}.
|
||||
\end{equation}
|
||||
We derive an analytical expression for $Z$ in appendix \ref{sec:analytical_expressions}
|
||||
|
||||
We will assume a $2 \times 2$ lattice to study the possible system states, and
|
||||
to find analytical expressions necessary in the Markov Chain Monte Carlo method.
|
||||
These results are used to test our code during implementation.
|
||||
|
||||
We study the 2D lattice and find the total energy and the total magnetization of
|
||||
of the system, in addition to the degeneracy. In addition, we want to work with
|
||||
unitless spins. To obtain this, we introduce the base unit for energy $J$, and
|
||||
with the Boltzmann constant we derive other units necessary. The resulting units
|
||||
can be found in table \ref{tab:units}.
|
||||
|
||||
% Problem 1
|
||||
|
||||
\begin{equation}\label{eq:partition_function}
|
||||
@ -31,42 +93,42 @@ These results are used to test our code during implementation.
|
||||
\end{equation}
|
||||
%
|
||||
\begin{equation}\label{eq:energy_total_first}
|
||||
\langle E \rangle = \frac{-8 J \sinh(8 \beta J)}{\cosh(8 \beta J) + 3}
|
||||
\langle E \rangle = -\frac{8J \sinh(8 \beta J)}{\cosh(8 \beta J) + 3}
|
||||
\end{equation}
|
||||
%
|
||||
\begin{equation}\label{eq:energy_total_first_squared}
|
||||
\langle E \rangle^{2} = \frac{64 J^{2} \sinh(8 \beta J)}{(\cosh(8 \beta J) + 3)^{2}}
|
||||
\langle E \rangle^{2} = \frac{64J^{2} \sinh^{2}(8 \beta J)}{(\cosh(8 \beta J) + 3)^{2}}
|
||||
\end{equation}
|
||||
%
|
||||
\begin{equation}\label{eq:energy_total_second}
|
||||
\langle E^{2} \rangle = \frac{64 J^{2} \cosh(8 \beta J)}{\cosh(8 \beta J) + 3}
|
||||
\langle E^{2} \rangle = \frac{64J^{2} \cosh(8 \beta J)}{\cosh(8 \beta J) + 3}
|
||||
\end{equation}
|
||||
%
|
||||
\begin{equation}\label{eq:magnetization_total_first}
|
||||
\langle |M| \rangle = \frac{2 e^{8 \beta J}}{\cosh(8 \beta J) + 3}
|
||||
\langle |M| \rangle = \frac{2(e^{8 \beta J} + 2)}{\cosh(8 \beta J) + 3}
|
||||
\end{equation}
|
||||
%
|
||||
\begin{equation}\label{eq:magnetization_total_first_squared}
|
||||
\langle |M| \rangle^{2} = \frac{4e^{16 \beta J} + 16e^{8 \beta J} + 16}{(\cosh(8 \beta J) + 3)^{2}}
|
||||
\langle |M| \rangle^{2} = \frac{4(e^{8 \beta J} + 2)^{2}}{(\cosh(8 \beta J) + 3)^{2}}
|
||||
\end{equation}
|
||||
%
|
||||
\begin{equation}\label{eq:magnetization_total_second}
|
||||
\langle M^{2} \rangle = \frac{8e^{8 \beta J} + 8}{\cosh(8 \beta J) + 3}
|
||||
\langle M^{2} \rangle = frac{8(e^{8 \beta J} + 1)}{\cosh(8 \beta J) + 3}
|
||||
\end{equation}
|
||||
%
|
||||
\begin{equation}\label{eq:specific_heat_capacity}
|
||||
C_{V} = \frac{64 J^{2}}{N k_{\text{B}} T^{2}} \Big( \frac{3 \cosh(8 \beta J) + \cosh^{2}(8 \beta J) - \sinh^{2}(8 \beta J)}{(\cosh(8 \beta J) + 3)^{2}} \Big)
|
||||
C_{V} = \frac{64J^{2} }{N k_{\text{B}} T^{2}} \frac{(3\cosh(8 \beta J) + 1)}{(\cosh(8 \beta J) + 3)^{2}}
|
||||
\end{equation}
|
||||
%
|
||||
\begin{equation}\label{eq:sesceptibility}
|
||||
\chi = \frac{1}{N k_{\text{B}} T} \Big( \frac{12e^{8 \beta J} + 4 e^{-8 \beta J} + 12}{(\cosh(8 \beta J) + 3)^{2}} \Big)
|
||||
\chi = \frac{4}{N k_{\text{B} T}} \frac{(3e^{8 \beta J} + e^{-8 \beta J} + 3)}{(\cosh(8 \beta J) + 3)^{2}}
|
||||
\end{equation}
|
||||
The derivation of analytical expressions can be found in appendix \ref{sec:analytical_expressions}.
|
||||
|
||||
\subsection{Markov Chain Monte Carlo methods}\label{sec:mcmc_methods}
|
||||
|
||||
|
||||
\subsection{Implementation}\label{sec:implementation}
|
||||
\subsection{Algorithm and implementation}\label{sec:algo_implementation}
|
||||
|
||||
|
||||
\subsection{Tools}\label{sec:tools}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user