2.12: The Degeneracy of an Isolated System and Its Entropy
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In Section 20.9 , we find that the sum of the probabilities of the population sets of an isolated system is
\[1=\sum_{\left\{N_i\right\},E}{W\left(N_i,g_i\right){\rho }_{MS,N,E}}. \nonumber \]
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
\[1={\rho }_{MS,N,E}\sum_{\left\{N_i\right\},E}{W\left(N_i,g_i\right)} \nonumber \]
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,
\[\mathit{\Omega}_E=\sum_{\left\{N_i\right\},E}{W\left(N_i,g_i\right)} \nonumber \]
The sum of the probabilities of the population sets of an isolated system becomes
\[1={\rho }_{MS,N,E}{\mathit{\Omega}}_E \nonumber \]
In Section 20.9 , we infer that
\[\rho_{MS,N,E}=\prod^{\infty }_{i=1}{\rho \left({\epsilon }_i\right)^{N_i}} \nonumber \]
so we have
\[1={\mathit{\Omega}}_E\prod^{\infty }_{i=1}\rho \left(\epsilon_i\right)^{N_i} \nonumber \]