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