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

- Page ID
- 152769

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

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