I'm studying the ideal Fermi gas from "Statistical Mechanics", by R. K. Pathria. In particular, the following formula, which can be found on page 237: \begin{equation} \mu=\left(\frac{3N}{4 \pi g V}\right)^{\frac{2}{3}}\frac{h^2}{2m} \end{equation} describes the chemical potential in the grand canonical ensemble as a function of the number of particles $N$ and the volume $V$. However, on page 242 he uses this formula for the chemical potential in studying the canonical ensemble. There is a reason for which that formula should hold in both cases?
4
-
1$\begingroup$ I couldn't find this formula in my book. Which edition is yours? Also, how could you define a chemical potential for a canonical ensemble? $\endgroup$– QuantumBrickCommented Mar 3, 2016 at 17:11
-
$\begingroup$ I've got the third edition. To define the chemical potential in the canonical ensemble, one just studies the dependence of the energy from the number of particle, and than computes the derivate of $E$ with respect to $N$. $\endgroup$– OnTheHighlandsCommented Mar 3, 2016 at 17:26
-
$\begingroup$ But in the canonical ensemble the number of particles is fixed. $\endgroup$– QuantumBrickCommented Mar 3, 2016 at 17:28
-
$\begingroup$ Of course your system exchange only energy with the reservoir, but you can consider a problem in which the number of particles is $N$ and another in which the number of particles is $M \neq N$, and than consider the dependence of $E$ from the number of particles. You do the same when you define the pressure as the derivate of the energy with respect to the volume. Morover, also in the microcanonical ensamble all these extensive parameter are fixed, but you can define pressure, chemical potential and temperature. $\endgroup$– OnTheHighlandsCommented Mar 3, 2016 at 17:42
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
1
In statistical mechanics, we always consider systems at thermodynamic limit. By thermodynamic limit, we mean that the volume or the number of particles of a system tends to infinity. It can be shown that the difference between different ensembles vanish at this limit. See, for instance, J. E. Mayer and M. G. Mayer, Statistical Mechanics, (John Wiley, New York, 1940).
-
$\begingroup$ Just for the sake of conciseness: the thermodynamic limit is approached when both, the volume AND the particle number tend to infinity while maintaining a constant density: $N \rightarrow\infty, V\rightarrow\infty, N/V = \text{const}$ (see also: en.wikipedia.org/wiki/Thermodynamic_limit) $\endgroup$ Commented Jun 21, 2018 at 11:32