5.3: Entropy and Counting States

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\( \newcommand\VTheta{\vec\Theta}\)
\( \newcommand\VLambda{\vec\Lambda}\)
\( \newcommand\VXi{\vec\Xi}\)
\( \newcommand\VPi{\vec\Pi}\)
\( \newcommand\VSigma{\vec\Sigma}\)
\( \newcommand\VUps{\vec\Upsilon}\)
\( \newcommand\VPhi{\vec\Phi}\)
\( \newcommand\VPsi{\vec\Psi}\)
\( \newcommand\VOmega{\vec\Omega}\)
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\(\newcommand \gtwid{\approx}\)

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\( \newcommand\bsqcap{\mbox{\boldmath{$\sqcap$}}}\)

\( \def\pabc#1#2#3{\left({\pz #1\over\pz #2}\right)\ns_{\!\!#3}}\)
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\( \newcommand\longrightleftharpoons{ \mathop{\vcenter{\hbox{\ooalign{\raise1pt\hbox{$\longrightharpoonup\joinrel$}\crcr  \lower1pt\hbox{$\longleftharpoondown\joinrel$}}}}}}\)
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\( \newcommand\rok{\Br\ns_1\,,\,\ldots\,,\,\Br\ns_k}\)
\( \newcommand\xhiOZ{\xhi^\ssr{OZ}}\)
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Suppose we are to partition \(N\) particles among \(J\) possible distinct single particle states. How many ways \(\ROmega\) are there of accomplishing this task? The answer depends on the statistics of the particles. If the particles are fermions, the answer is easy: \(\ROmega\ns_\ssr{FD}={J\choose N}\). For bosons, the number of possible partitions can be evaluated via the following argument. Imagine that we line up all the \(N\) particles in a row, and we place \(J-1\) barriers among the particles, as shown below in Figure [BEcount]. The number of partitions is then the total number of ways of placing the \(N\) particles among these \(N+J-1\) objects (particles plus barriers), hence we have \(\ROmega\ns_\ssr{BE}={N+J-1\choose N}\). For Maxwell-Boltzmann statistics, we take \(\ROmega\ns_\ssr{MB}=J^N/N!\) Note that \(\ROmega\ns_\ssr{MB}\) is not necessarily an integer, so Maxwell-Boltzmann statistics does not represent any actual state counting. Rather, it manifests itself as a common limit of the Bose and Fermi distributions, as we have seen and shall see again shortly.

[BEcount] Partitioning \(N\) bosons into \(J\) possible states (\(N=14\) and \(J=5\) shown). The \(N\) black dots represent bosons, while the \(J-1\) white dots represent markers separating the different single particle populations. Here \(n_1=3\), \(n_2=1\), \(n_3=4\), \(n_4=2\), and \(n_5=4\).

The entropy in each case is simply \(S=\kB\ln\ROmega\). We assume \(N\gg 1\) and \(J\gg 1\), with \(n\equiv N/J\) finite. Then using Stirling’s approximation, \(\ln(K!)=K\ln K - K + \CO(\ln K)\), we have \[\begin{split} S\ns_\ssr{MB}&=-J\kB \, n\ln n \\ S\ns_\ssr{BE}&=-J\kB\big[ n\ln n - (1+n)\ln (1+n)\big] \bvph \\ S\ns_\ssr{FD}&=-J\kB\big[ n\ln n + (1-n)\ln (1-n)\big]\ . \end{split}\] In the Maxwell-Boltzmann limit, \(n\ll 1\), and all three expressions agree. Note thatR \[\begin{split} \pabc{S\ns_\ssr{MB}}{N}{J} &= -\kB \, \big( 1 + \ln n\big) \\ \pabc{S\ns_\ssr{BE}}{N}{J} &= \kB\ln\!\big(n^{-1}+1\big) \bvph \\ \pabc{S\ns_\ssr{FD}}{N}{J} &= \kB\ln\!\big(n^{-1}-1\big)\ . \end{split}\]

Now let’s imagine grouping the single particle spectrum into intervals of \(J\) consecutive energy states. If \(J\) is finite and the spectrum is continuous and we are in the thermodynamic limit, then these states will all be degenerate. Therefore, using \(\alpha\) as a label for the energies, we have that the grand potential \(\Omega=E-TS-\mu N\) is given in each case by \[\begin{split} \Omega\ns_\ssr{MB} &= J\sum_\alpha \Big[ (\ve\ns_\alpha-\mu)\,n\ns_\alpha+\kT\,n\ns_\alpha\ln n\ns_\alpha\Big] \\ \Omega\ns_\ssr{BE} &= J\sum_\alpha \Big[ (\ve\ns_\alpha-\mu)\,n\ns_\alpha+\kT\,n\ns_\alpha\ln n\ns_\alpha -\kT\,(1+n\ns_\alpha)\ln (1+n\ns_\alpha)\Big] \\ \Omega\ns_\ssr{FD} &= J\sum_\alpha \Big[ (\ve\ns_\alpha-\mu)\,n\ns_\alpha+\kT\,n\ns_\alpha \ln n\ns_\alpha +\kT\,(1-n\ns_\alpha)\ln (1-n\ns_\alpha)\Big] \ . \end{split}\] Now - lo and behold! - treating \(\Omega\) as a function of the distribution \(\{n\ns_\alpha\}\) and extremizing in each case, subject to the constraint of total particle number \(N=J\sum_\alpha n\ns_\alpha\), one obtains the Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac distributions, respectively: \[{\delta\over\delta n\ns_\alpha}\Big(\Omega-\lambda \, J\sum_{\alpha'} n\ns_{\alpha'}\Big) = 0 \quad\Rightarrow \quad \begin{cases} n^\ssr{MB}_\alpha=e^{(\mu-\ve\ns_\alpha)/k\ns_\RB T} \\ \\ n^\ssr{BE}_\alpha=\big[e^{(\ve\ns_\alpha-\mu)/k\ns_\RB T}-1\big]^{-1}\\ \\ n^\ssr{FD}_\alpha=\big[e^{(\ve\ns_\alpha-\mu)/k\ns_\RB T}+1\big]^{-1} \ . \end{cases}\] As long as \(J\) is finite, so the states in each block all remain at the same energy, the results are independent of \(J\).