IGNOU MPH 11 2026 - Using the relation of thermodynamic probability Obtain

Question

Using the relation of thermodynamic probability

$$W = \prod_{i=1}^N \frac{g_i}{N_i!(g_i - N_i)!}$
Obtain an expression of Fermi Dirac distribution function. Show a plot of $f(\epsilon)$ (the occupation index of a state corresponding to energy $\epsilon_i$ ) versus $\epsilon$ for different temperatures and discuss it. What is the physical interpretation of Fermi energy?

19 Jan 2026

Answer :

Word Count : 450

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] ] Simplifying: [ \ln W = \sum_i \left[ g_i \ln g_i - N_i \ln N_i - (g_i - N_i) \ln (g_i - N_i) \right] ] To maximize (\ln W)

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