Step 1: Define terms.
- \(n_i\): Number of particles at energy level \(i\).
- \(g_i\): Degeneracy of energy level \(i\), i.e., number of available states.
- \(\frac{n_i}{g_i}\): Occupation index (average particles per state).
Step 2: Differentiate quantum and classical statistics.
- Quantum Statistics (Fermi-Dirac, Bose-Einstein): Applicable when the occupation index is not small. Particle distinguishability and occupation limits (Pauli exclusion principle for fermions) are crucial, indicating high particle density.
- Classical Statistics (Maxwell-Boltzmann): Valid for a small occupation index. Particles are treated as distinguishable with no occupation limits, applicable when particles are sparsely distributed across many states.
Step 3: State the condition for the classical limit.
The classical limit is reached when the occupation index is much less than 1:
\[ \frac{n_i}{g_i} \ll 1 \]
This means the number of available states \(g_i\) significantly exceeds the number of particles \(n_i\).
\[ g_i \gg n_i \implies \frac{g_i}{n_i} \gg 1 \]
This scenario describes a low-density, high-temperature gas where quantum effects are negligible.