I am studying through the book Thermodynamics and Statistical Mechanics by Walter Greiner and I’ve got a couple of doubts when I was reading about the classical ensembles, specially the Canonical ensemble (chapter 7, pages 159-160, Springer, 2004).
In the case of the canonical ensemble it is considered that the system has a fixed temperature due to the contact with the termal reservoir and it can assume a large range of values of energy (I think, but I am not sure, that these different values of energy are attained by the system – in termal contact with the reservoir – until the final equilibrium state is attained.)
In the page 160, about the probability $p_{i}$ of finding the system $S$ in a microstate $i$ with an energy $E_{i}$, it is told:
“If $S$ is a closed system, $p_{i}$ will be proportional to the number of microstates ${\Omega}_{S}(E_{i})$. Analogously, $p_{i}$ is proprotional to the number of microstates in the total closed system for which $S$ lies in the microstate with the energy $E_{i}$. Obviously this is just equal to the number of microstates of the heat bath for the energy$E–E_{i}$, since $S$ assumes one microstate $i$ [...]”
What I don’t understand is: How “obviously” is that? Because to me it’s not obvious.
After that, I don’t understand too how to get the Equation (7.4), since we the expansion is “with respect to $E_{i}$”.
\begin{equation} k{\ln}{\Omega}_{R}(E-E_{i}){\approx}k{\ln}{\Omega}_{R}(E)-{\frac{\partial}{{\partial}E}}(k{\ln}{\Omega}_{R}(E))E_{i}+..., \end{equation} where $E$ is the total energy, $E_{R}$ is the energy of the reservoir, $E_{S}=E_{i}$ is the energy of the system in a given microstate and $E=E_{R}+E_{S}$.
If anybody could help, I really would appreciate.
