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20.12: The Degeneracy of an Isolated System and Its Entropy

  • Page ID
    152769
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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

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

    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

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


    This page titled 20.12: The Degeneracy of an Isolated System and Its Entropy is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.