Multiplicity and Inverse of probability distribution, what do they mean?

[silverwhale]

Hello Everybody.

I have a rather simple question, which still kept me thinking for two hours without any result.

If we want to determine the multiplicity in the microcanonical ensemble we just divide the volume of the shell containing the accessible microstates over the volume of one microstate; let us write the multplicity as:
$$\Gamma = \frac{\omega }{\omega_{0}}.$$
Now, we can write the RHS as:
$$\frac{\omega }{\omega_{0}} = \frac{\int' d^{3N}q d^{3N}p}{d^{3N}q d^{3N}p}.$$
Next multiplying by $$\rho_{0},$$ where $$\rho_0$$ is the constant density distridubtiion of the ensemble members in a microcanonical ensemble. we get:
$$\frac{\omega }{\omega_{0}} = \frac{\int' d^{3N}q d^{3N}p \, \rho_0}{d^{3N}q d^{3N}p \, \rho_0};$$
Now this would mean that the mutliplicity $$\Gamma$$ is given by:
$$\Gamma = \frac{N}{n_0};$$
where N is the total number of ensemble members and n_0 is the number of ensemble elements in a given microstate.

First question, is this correct? Is this ok to write?

Next, let's suppose everything is fine. What would this expression then mean?

$$\frac{\int d^{3N}q d^{3N}p \, \rho(q,p)}{d^{3N}q d^{3N}p \, \rho(q,p)} = \frac{N}{n(q,p)};$$

where $$\rho(q,p)$$ is the density distribution again but this time it is not constant anymore, it varies from microstate to microstate. We would get something like $$\Gamma (q,p).$$ What does it mean? Can anyone help? I can't firugre out what this "multiplcity", that is (q,p) dependent, means. Incidently its inverse is similar to the probability distribution of the canonical ensemble and grand canonical ensemble.

Any help would be greatly appreciated.

 

[eljose79]

Hello!

It seems like you are on the right track with your calculations. The expression you have derived for multiplicity, \Gamma = \frac{N}{n_0}, is indeed correct. This means that the multiplicity is equal to the total number of ensemble members divided by the number of ensemble elements in a given microstate. In other words, it represents the number of ways in which a particular microstate can be realized in the ensemble.

Now, in the second part of your question, you have introduced a density distribution that varies from microstate to microstate. This is a more general expression for the multiplicity, where it takes into account the varying density of ensemble members in different microstates. The inverse of this expression, \Gamma (q,p), can be interpreted as a probability distribution for the microstates in the ensemble. This means that the probability of a particular microstate occurring in the ensemble is proportional to its multiplicity.

In summary, the multiplicity is a measure of the number of ways in which a particular microstate can be realized in the ensemble, and it can be expressed in terms of a constant density distribution or a varying one. The inverse of the multiplicity can be interpreted as a probability distribution for the microstates in the ensemble. I hope this helps clarify your understanding. Let me know if you have any further questions.