\input{preamble.tex}

\begin{document}
    \section{Omega and Z}
    \newcommand{\ThE}{thermal equilibrium}

        In this section, we want to answer why $Z$ is a more efficient technology than the good 'ol $\Omega$.
        \begin{align*}
            &P(\mu_i) = e^{-\beta E_i}
            \left( \frac{Z_{N-1}(\beta)}{Z_N(\beta)}\right)\\
            &P(\mu_1,\mu_2,\mu_3,\ldots) = \frac{e^{-\beta E_1} e^{-\beta E_2} e^{-\beta E_3} \cdots}{Z(\beta)}
                              = \frac{e^{-\beta E} }{Z(\beta)}\\
        \end{align*}

        %&=& &=& &=& &=& &=& &=& &=& &=& &=& =
        Let's zoom into the system to see what's really going in there.
        As usual, in \ThE, the probability of a microstate only depends on its energy.

            \[P(\mu_i) = \frac1{\Omega_1(E_{\mu_i})} \]
        \begin{align}
            P(\mu_1,\mu_2,\mu_3,\ldots) &= \frac1{\Omega_1(E_1)}
                                \frac1{\Omega_1(E_2)}
                                \frac1{\Omega_1(E_3)}
                                \ldots \\
            &= \frac1{\Omega_N(E)}
        \end{align}

        Notice that these equations already represent thermal equilibrium,
        since they are characterized only by energy.
        Thermal equilibrium is our one and only approximation\footnote{this includes the assumption that $\beta$ is constant in the range $E\pm\Delta E$},
        hence, one shouldn't expect to do more approximation in order to arrive at Z.\\
        There are three ways to exploit the relation between (1) and (2), but first I need to introduce several notations:

        %Preliminary
        \newcommand{\aE}{\epsilon}
        \newcommand{\soi}{\sum_i}
        \newcommand{\Stothei}[1] {{\hat\pdc}^{#1} S_1(\aE)}
        Equation (1) to (2) may look like a tiny step in \LaTeX,
        but it is actually a giant step.
        Rewrite $\Omega_1$ as $e^{ S_1}$, hence the relation becomes
        \begin{equation}
            \label{eq:mainassumption}
            \soi S_1(E_i) = S_N(E) = S_N\Big(\soi E_i\Big) \\
        \end{equation}


            \paragraph{1. principle of least assumption
            \footnote{a more mediocre and tedious derivation is given in the appendix}}
            For \eqref{eq:mainassumption} to be true,
            $S_N(E)$ must not depend on the spread of the $E_i$ distribution,
            where mathematically you can freely rearrange your energy distribution as you wish
            \[S_1(E_1) + S_1(E_2) = S_1(E_1+E_2) + S_1(0) \]
            From here we know that $S_1$ must be a linear function
            \[ S_1(x) = S_1(0) + \beta x\]
            where $\beta = \pdc S_1 = \mathrm{constant}$.


            Interpretations:
            \begin{itemize}
                \item Hey, the entropy is automatically maximized, too
                    (which means, it is redundant
                    to further assume that entropy is maximized).\\
                \item We can now write the probability as
                    \[ P(\mu_i) = e^{-S_1(0) -\beta E_i} \]
                \item any visual reasons on why the entropy increases linearly?
            \end{itemize}


            \paragraph{2. vary N}
            Back to one particle, we are now authorized to use this relation
            \begin{align*}
                &p(\mu_i) = \frac{\Omega_{N -1}(E-E_{\mu_i})}{\Omega_N(E)} \\
                &= \exp(\log\Omega_{N-1}(E-E_{\mu_i}) - \log \Omega_N(E))\\
                & = \frac{\Omega_{N-1}( E )}{\Omega_N(E)}
                    \exp( -\pd{}{E}\log \Omega_{N-1}E_i
                          - \frac12\pddc \log\Omega_{N-1}E_i^2
                          - \cdots
                          )\\
                & = \frac{\Omega_{N-1}( E )}{\Omega_N(E)}
                    \exp(-\beta E_i
                         -\frac12 \pd{\beta}{E} E_i^2
                         - \cdots  )\\
                & = \frac{\exp(-\beta E_i)}{Z_1(\beta,\mathcal{O}(E_i^2))}\\ 
            \end{align*}
            There are two completely different arguments
            to arrive at Boltzmann distribution:
            \begin{itemize}
                \item the multiplicity is maximized
                \item the state only depends on energy
            \end{itemize}

            Another way to look at thermal equilibrium:
            we randomly sample a small element of our system,
            and yet it will still look exactly the same
            as its parent system (redundancy of information, rescaling, whatever).

            Possible mathematical relations:
            \[Z_{1R}(\beta,E_i)  = \frac{\Omega_N(E)}{ \Omega_{N-1}(E)}
                             \frac1{\exp(+ \frac{\beta}{2C_v T}E_i^2
                                         - \cdots  )}\]
            Now we begin to see why we need log. All of our arguments
            \[\log Z_1 = \pd{}{N}\log \Omega_N(E)\]
            The difference between the two description is that
            the former gives us the real-life particle description,
            why the latter only tells us about the smoothed out representation of the system.

            In the Z description,
            we are instead viewing the system as a field (``N is not fixed'', but we don't care).\\
            %topic: http://ajp.aapt.org/resource/1/ajpias/v71/i11/p1136_s1


        \input{statmech2appendix.tex}

\end{document}
