# 2.12: The Degeneracy of an Isolated System and Its Entropy

In Section 20.9, we find that the sum of the probabilities of the population sets of an isolated system is

$1=\sum_{\left\{N_i\right\},E}{W\left(N_i,g_i\right){\rho }_{MS,N,E}}.$

By the principle of equal a priori probabilities, $${\rho }_{MS,N,E}$$ is a constant, and it can be factored out of the sum. We have

$1={\rho }_{MS,N,E}\sum_{\left\{N_i\right\},E}{W\left(N_i,g_i\right)}$

Moreover, the sum of the thermodynamic probabilities over all allowed population sets is just the number of microstates that have energy $$E$$. This sum is just the degeneracy of the system energy, $$E$$. The symbol $$\mathit{\Omega}_E$$ is often given to this system-energy degeneracy. That is,

$\mathit{\Omega}_E=\sum_{\left\{N_i\right\},E}{W\left(N_i,g_i\right)}$

The sum of the probabilities of the population sets of an isolated system becomes

$1={\rho }_{MS,N,E}{\mathit{\Omega}}_E$

In Section 20.9, we infer that

$\rho_{MS,N,E}=\prod^{\infty }_{i=1}{\rho \left({\epsilon }_i\right)^{N_i}}$

so we have

$1={\mathit{\Omega}}_E\prod^{\infty }_{i=1}\rho \left(\epsilon_i\right)^{N_i}$