Average energy per mode classical point of view.
2024.11.28 19:43 Stock-Nerve4558 Average energy per mode classical point of view.
This was how my proffessor's notes explained how the classical theory calculated the average energy per mode in a black body radiation from the classical point of of view. But I neither understand what the integral says nor understand where did it come from.
\begin{itemize}
\item From Kinetic Theory, we know that the probability that an entity has an energy when in equilibrium with the surroundings at temperature $T$ is proportional to the Boltzmann factor:
\[
e^{-\alpha \beta}, \quad \beta = \frac{1}{kT}.
\]
\end{itemize}
The average energy per mode:
Classically, when all energies above a certain energy (minimum) are allowed, the expectation energy is
\[
\langle E \rangle = \frac{\int_{0}^{\infty} \alpha e^{-\alpha \beta} \, d\alpha}{\int_{0}^{\infty} e^{-\alpha \beta} \, d\alpha}.
\]
\[
\rightarrow \text{(17)}.
\]
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