61 lines
1.5 KiB
TeX
61 lines
1.5 KiB
TeX
\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 3.15}
|
|
\subtitle{Entropy maximum and temperature}
|
|
\author{\begin{telugu}హృదయ్ దీపక్ మల్లుభొట్ల\end{telugu}}
|
|
% want empty date
|
|
\predate{}
|
|
\date{}
|
|
\postdate{}
|
|
|
|
% !TeX spellcheck = en_GB
|
|
\begin{document}
|
|
\maketitle
|
|
|
|
Explain in words why, for two weakly coupled systems
|
|
\begin{equation}
|
|
\rho(E_1) = \flatfrac{\Omega_1(E_1) \Omega_2(E - E_1)}{\Omega(E)}
|
|
\end{equation}
|
|
is intuitive for a system where all states of energy $E$ have equal probability density.
|
|
Using $S = \kb \log(\Omega)$, show in one step that maximising the probability of $E_1$ makes the two temperatures $\frac{1}{T} = \pdv{S}{E}$ the same, and hence that maximising $\rho(E_1)$ maximises the total entropy.
|
|
|
|
\section{Solution} \label{sec:solution}
|
|
|
|
Basically just a probability calculation of independent stuff, so it's intuitive.
|
|
|
|
Derivative is easy enough.
|
|
|
|
\newpage
|
|
\listoftodos
|
|
|
|
\end{document}
|