Word Count : 450

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We start with the expression for the thermodynamic probability for fermions: [ W = \prod_{i=1}^{N} \frac{g_i!}{N_i! (g_i - N_i)!} ] where (g_i) is the degeneracy of the (i)-th energy level, and (N_i) is the number of particles in that level. The equilibrium distribution maximizes (\ln W) subject to the constraints of fixed total particle number (N = \sum_i N_i) and total energy (E = \sum_i N_i \epsilon_i). Using Stirling’s approximation for large numbers ((\ln n! \approx n \ln n - n)): [ \ln W = \sum_i \left[ g_i \ln g_i - g_i - N_i \ln N_i + N_i - (g_i - N_i) \ln (g_i - N_i) + (g_i - N_i) \right] ] __________ ______ ________ ________ ____ ________.
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