feat: adds 3.19

tex/3.19.tex

@@ -0,0 +1,71 @@
+\documentclass{article}
+
+% set up telugu
+\usepackage{fontspec}
+\newfontfamily\telugufont{Potti Sreeramulu}[Script = Telugu]
+\usepackage{polyglossia}
+\setdefaultlanguage{english}
+\setotherlanguage{telugu}
+
+%other packages
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{physics}
+\usepackage{siunitx}
+\usepackage{todonotes}
+\usepackage{luacode}
+\usepackage{titling}
+\usepackage{enumerate}
+
+% custom deepak packages
+\usepackage{luatrivially}
+\usepackage{subtitling}
+
+\usepackage{cleveref}
+
+\begin{luacode*}
+	math.randomseed(31415926)
+\end{luacode*}
+
+\newcommand{\kb}{k_{\mathrm{B}}}
+
+\title{Problem 3.19}
+\subtitle{Random energy model}
+\author{\begin{telugu}హృదయ్ దీపక్ మల్లుభొట్ల\end{telugu}}
+% want empty date
+\predate{}
+\date{}
+\postdate{}
+
+% !TeX spellcheck = en_GB
+\begin{document}
+	\maketitle
+	
+	Given $N$ spins (so $M = 2^N$ possible states), assign a random energy to each.
+	Given $j \in \left{1, 2, \ldots, 2^N \right}$, assume each energy $E_j$ is selected with probability $P(E) = \frac{1}{\sqrt{\pi N}} e^{\flatfrac{-E^2}{N}}$.
+	Gaussian with mean $0$ and standard deviation $\sqrt{\flatfrac{N}{2}}$
+
+	\subsubsection*{(a) Microcanonical ensemble}
+	Consider the states in a small range $E < E_j < E+\deltaE$.
+	Let the number of such states in this range be $\Omega(E) \delta E$.
+	
+	Calculate the average
+	\begin{equation}
+		\left< \Omega(N\epsilon) \right>_{REM}
+	\end{equation}
+	over the ensemble of REM systems, in terms of energy per particle $\epsilon$.
+	For energies per particle near zero, show that this average density of states grows exponentially as the system size $N$ grows.
+	In contrast, show that $\left< \Omega(N\epsilon) \right>_{REM}$ decreases exponentially for $\epsilon = \flatfrac{E}{N} < -\epsilon_\ast$ and for $\epsilon > \epsilon_\ast$, where the limiting energy per particle
+	\begin{equation}
+		\epsilon_\ast = \sqrt{\log 2}.
+	\end{equation}
+	(Hint: as $N$ grows, the probability density $P(N \epsilon)$ decreases exponentially, which the total number of states $2^N$ grows exponentially.
+	Which one wins?)
+
+	\section{Solution} \label{sec:solution}
+
+	
+	\newpage
+	\listoftodos
+
+\end{document}