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If one asks forcefully, what is the partition function for microc. ensemble, is it ok to say Ω ( the no. of microstates)?

What I can think of is this. Because partition function in canonical and grandcanonical is defined as the normalization factor for the probability distribution function, and in microc. , Probability distribution function is 1/Ω . So in a sense, Ω is the normalization factor and hence the partition function.

Shikhar Chamoli
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  • The simple answer is that there isn’t one. There is, however, a probability distribution with the trivial probabilities $p_j = \frac{1}{\Omega}$. – Matt Hanson Oct 13 '23 at 02:13

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The distribution function in microcanonical ensemble is the energy shell. E.g., for ideal gas with $$H=\sum_{i=1}^N\frac{\mathbf{p}_i^2}{2m}$$ we have the distribution function $$ W(\mathbf{p}_1,...\mathbf{p}_N|E)=\frac{1}{\Omega(E)}\delta\left(E-\sum_{i=1}^N\frac{\mathbf{p}_i^2}{2m}\right).$$ We can evaluate the normalization factor and use it as a partition function (although it is less useful than the temperature and parameter dependent partition functions for canonical and grand canonical ensembles.) E.g., The entropy can be defined as $$ S=\log\Omega(E). $$

See this thread regarding different definitions of entropy.

Roger V.
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