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The probability of finding a particle at a particular energy level when energy is considered discrete is according to Boltzmann:

$$P(E_j) = \frac{g_j\cdot e^{-\beta E_j}}{\sum_{j=1}^\infty g_j \cdot e^{-\beta E_j}}$$

Where $g_j$ is the degeneracy of energy level $Ej$ and is derived by calculating the surface of 1/8th of a sphere in n-dimensions. When energy is considered continuous, it is said that the probability density $P′(E)$ is proportional to: $$P'(E) \propto D(E) \cdot e^{-\beta E}$$ Here $D(E)$ is the Density of states formula and is the derivative of 1/8th of the volume of the sphere in n-dimensions.

I have a hard time understanding the derivation of this proportionality in the continuous approach. How is it concluded from the discrete formula and/or by other means?

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  • $\begingroup$ This is a general rule. The "derivation" comes from probability theory (proper counting of states that fit the energy constraint). Probability of energy $E$ is the probability of a microstate with energy $E$ (which is $\exp(-\beta E)/Z$ for discrete and continuous cases) times the number of states that have this specific energy (actual number in discrete systems, and in the continuous limits it is density of states, $D(E)$, since Integration of D(E) over some $E$ range gives the number of energy states in this region). $\endgroup$ – Alexander Mar 18 at 6:30
  • $\begingroup$ @Alexander What surprises me is that $exp(-\beta E) / Z$ is also valid for the continuous approach while one can not fill in a specific value for $E$, neither in the partition function $Z$ or in $exp(-\beta E)$ because energy is continuous. $\endgroup$ – JohnnyGui Mar 18 at 7:04
  • $\begingroup$ @Alexander Would you be able to answer my previous question? I am quite curious abou this. $\endgroup$ – JohnnyGui Mar 25 at 18:38

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