Is it possible to derive the canonical ensemble without the microcanonical one?

Question

I'm trying to understand if the microcanonical entropy $S_{mc}=\log(\Omega)$ with $\Omega=\int\frac{d\vec{z}}{h^{3N}N!}\theta(E-H(\vec{z}))$ is somewhat more "fundamental" than the canonical entropy $S_{c}=\beta E-\log(Z)$ with $Z=\beta\mathcal{L}[\Omega]=\int\frac{d\vec{z}}{h^{3N}N!}e^{-\beta H(\vec{z})}$. My guess is that it is, because to define a canonical ensemble system one would need a thermal bath, and the total system (thermal bath + canonical system) is to be regarded as a microcanonical one. Is this reasoning right? If not, what are other ways to derive the canonical ensemble that don't require a microcanonical one?

Answer 1

The idea that microcanonical ensemble would be "more fundamental" than other ensembles is strongly related to a mechanics-based approach to statistical mechanics, where the clean starting point would be a Hamiltonian isolated system of N particles. Therefore, a system at constant energy.

However, that this is not the only possible point of view was already clear to Boltzmann and physicists of the early stages of statistical mechanics. When Paul and Tatiana Ehrenfest wrote their contribution on "The Conceptual Foundations of the Statistical Approach in Mechanis" for the German Encyclopedia of Mathematical Sciences", they presented as "Modern Formulation" the one based on the so-called method of the most probable probability distribution. I.e. on the search for the maximum of the probability of finding $a_1$ systems in the state of energy $\epsilon_1$, $a_2$ in the state of energy $\epsilon_2$ and so on, subject to the constraint on the total numer of systems ($\sum a_k = N$), and possible additional constraints.

Such a more general method does not imply a special role of the microcanonical ensemble and, dependiong on the additional constraints, can provide on the same foot micro-canonical, canonical and grand-canonical ensembles.

More appeal on such an approach comes after realizing that it is equivalent to the application of Shannon's definition of entropy of a generic probability distribution to the cases of the equilibrium probability distributions in phase space. Whatever is the chosen route (Shannon's entropy or Boltzmann's most probable distribution) canonical ensemble does not require a previous introduction of a microcanonical ensemble, thus eliminating any need of assuming the existence of a larger, encompassing isolated system.

Another consideration which I feel important is that the "most fundamental character" of an ensemble may be related not to its formal derivation, but to how easy is to justify the link with thermodynamics.

In this respect, it would be the Grand-canonical ensemble playing the role of the most fundamental. Indeed, while in the case of micro-canonical or canonical ensemble it is not easy to show the relation between chemical potential and derivative of the relevant thermodynamic potential with respect to number of particles, in the Grand-canonical ensemble it is trivial to show that the first derivatives of the logarithm of the grand-partition function with respect to $\beta \mu$ is directly related to the average number of particles.

Summarizing, I would conclude that in general, there is not a more fundamental ensemble. This is precisely what makes Statistical Mechanics fully consistent with Thermodynamics where all fundamental equations are related by Legendre transforms, but there is not one more fundamental than the others.