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- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Stellar_Atmospheres_(Tatum)/08%3A_Boltzmann's_and_Saha's_Equations/8.04%3A_Boltzmann's_EquationA dynamic equilibrium between collisional excitations and radiative de-excitations leads to a certain distribution of the atoms among their various energy levels. Most of the atoms will be in low-lyin...A dynamic equilibrium between collisional excitations and radiative de-excitations leads to a certain distribution of the atoms among their various energy levels. Most of the atoms will be in low-lying levels; the number of atoms in higher levels will decrease exponentially with energy level. The lower the temperature, the faster will be the population drop at the higher levels. Only at very high temperatures will high-lying energy levels be occupied by an appreciable number of atoms.
- https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book%3A_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04%3A_Statistical_Ensembles/4.04%3A_Ordinary_Canonical_Ensemble_(OCE)\[\begin{split} D\ns_{\ssr{W}}(E\ns_{\ssr{U}}-E\ns_n)\,\RDelta E &\to e^{S\ns_{\ssr{W}}(E\ns_{\ssr{U}}-E\ns_n\,,\,\RDelta E)} \equiv \mathop{\textsf{Tra}}_{\ssr{W}} \hskip-0.7cm\int\limits_{E\ns_{\ssr...\[\begin{split} D\ns_{\ssr{W}}(E\ns_{\ssr{U}}-E\ns_n)\,\RDelta E &\to e^{S\ns_{\ssr{W}}(E\ns_{\ssr{U}}-E\ns_n\,,\,\RDelta E)} \equiv \mathop{\textsf{Tra}}_{\ssr{W}} \hskip-0.7cm\int\limits_{E\ns_{\ssr{U}}-E\ns_n}^{E\ns_{\ssr{U}}-E\ns_n+\RDelta E}\hskip-0.7cm dE\>\delta(E-\HH\ns_{\ssr{W}})\\ D\ns_{\ssr{U}}(E\ns_{\ssr{U}})\,\RDelta E &\to e^{S\ns_{\ssr{U}}(E\ns_{\ssr{U}}\,,\,\RDelta E)} \equiv \mathop{\textsf{Tra}}_{\ssr{U}} \hskip-0.4cm\int\limits_{E\ns_{\ssr{U}}}^{E\ns_{\ssr{U}}+\RDelta E}\hski…
- https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book%3A_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04%3A_Statistical_Ensembles/4.01%3A_Microcanonical_Ensemble_(CE)Note that D(E) has dimensions of inverse energy, so one might ask how we are to take the logarithm of a dimensionful quantity in Equation ???. We must introduce an energy scale, such as ...Note that D(E) has dimensions of inverse energy, so one might ask how we are to take the logarithm of a dimensionful quantity in Equation ???. We must introduce an energy scale, such as \RDeltaE in Equation ???, and define ˜D(E;\RDeltaE)=D(E)\RDeltaE and S(E;\RDeltaE)≡\kBln˜D(E;\RDeltaE).
