Well, I know that a microcanonical ensemble is a certain statistical ensemble, i.e., a set of all possible physical realizations of the microstate of a system that is consistent with a given macrostate. And I know that the postulate of equal probabilities is a priori principle used in statistics physics.

But in the literature, this two concepts are introduced usually in the same time in a non clear way, in wich turns to be difficult to undertand their relationship. It gives me the sensation that a microcanonical ensemble is just an ensemble that follows the postulate as if they were the same thing.

Can someone explain with more clarity their fundamental distinctions and their relationship with other kinds of statistical ensembles?

  • $\begingroup$ As far as I know, the postulate of equal probabilities (Tolman) gives the condition: $\rho(H) = c$. In the microcanonical ensemble the energy is constant (isolated system) therefore we can apply Tolman's postulate and get $\rho_{mic}(H) = A \cdot \delta[E-H]$. The point is (I'm not sure about this) that there are other density functions that follow Tolman's postulate and don't have necessarily the form of $\rho_{mic}(H)$. As I said, I'm not sure about this. $\endgroup$
    – David
    Mar 30 '21 at 13:40
  • $\begingroup$ In physics, the words "microcanonical ensemble" imply that we've assigned equal probabilities to all of the members of the ensemble. That might be an abuse of the word "ensemble" (my own first sentence uses that word inconsistently!), but that's what physicists mean by "microcanonical ensemble." Does acknowledging that this is not a consistent way to use the word "ensemble" help answer the question? $\endgroup$ Mar 31 '21 at 1:13

Equal probability is a base concept for the microcanonical ensemble: all microstates have eqaul accessing probability.

The microcanonical ensemble is based on this concept, but also the Boltzmann hypothesis of entropy formula:

$$ S(E, V, N) = K_B \ln \Gamma(E, V, N). $$ Where $\Gamma$ is the total number of microscopic state with the given condition of $E$, $V$, and $N$.

This formula serves as a bridge connecting to the thermal dynamics through the Maxwell relations:

\begin{align} \frac{1}{T} =& \left( \frac{\partial S}{\partial E}\right)_{V,N};\\ \frac{P}{T} =& \left( \frac{\partial S}{\partial V}\right)_{E,N};\\ \frac{\mu}{T} =& -\left( \frac{\partial S}{\partial N}\right)_{E,V};\\ \end{align}

This structure providing a systematic development from statistical mechanics to thermodynamics is called microcanonical ensemble.


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