Microcanonical vs. Canonical Partition Functions: What's the Difference?

[soumobrata]

What is the difference between micro canonical Partition function and canonical Partition function?
Is the mathematical expression of the above two Partition function are same?
If it is then why??
[emoji29]

 

[stevendaryl]

As I understand it, the various types of ensembles differ in what is held constant, and what is allowed to change (by exchanging something with the environment--energy, or particles, or whatever). The microcanonical ensemble holds the following quantities fixed:

1. Total Energy
2. Volume
3. Number of particles (of each type)

For a canonical ensemble, instead of holding the total energy constant, the temperature is held constant.

The expressions for the partition function definitely are different in the two cases. The canonical partition function, $Z(N,V,T)$ is a function of T, V, and N, while the microcanonical partition function, $\Omega(N,V,E)$ is a function of E, V and N. The microcanonical partition function is usually called the "density of states". They are related to thermodynamic quantities:

$Z = e^{-A/kT}$ where $A$ is the Helmholtz free energy, $A = E - TS$.
$\Omega = e^{S/k}$ where $S$ is the entropy

 

[soumobrata]

Thanks i will show u a problem tomorrow...then my doubts wiil be clear!