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 the 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, it was already clear to Boltzmann and other physicists in the early stages of statistical mechanics that this is not the only possible point of view. When Paul and Tatiana Ehrenfest wrote their contribution to "The Conceptual Foundations of the Statistical Approach in Mechanics" 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 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 number of systems ($\sum a_k = N$), and possible additional constraints.

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

Such an approach becomes more appealing after realizing that it is equivalent to applying Shannon's definition of the entropy of a generic probability distribution to the cases of equilibrium probability distributions in phase space. Whatever the chosen route (Shannon's entropy or Boltzmann's most probable distribution), a canonical ensemble does not require a previous introduction of a microcanonical ensemble, thus eliminating any need to assume the existence of a more extensive, encompassing isolated system.

Another important consideration is that the "most fundamental character" of an ensemble may be related not to its formal derivation but to how easily the link with thermodynamics can be justified.

The grand canonical ensemble would play the most fundamental role in this respect. 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 the number of particles, in the Grand-canonical ensemble, it is trivial to show that the first derivative of the logarithm of the grand-partition function with respect to $\beta \mu$ is directly related to the average number of particles.

In summary, 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 no one is more fundamental than the others.