The microcanonical ensemble is just a specialization of the canonical ensemble: We only sum up micro states \(p_i\) with energies \(U_i = U\). The remaining \(p_i\) are zero; thus we find
| \begin{equation*} 1 = \sum_{i=1}^W p_i = \frac{1}{Z} \exp\left( - \frac{U}{k T} \right) \sum_{i=1}^W 1 = \frac{1}{Z} \exp\left( - \frac{U}{k T} \right) W \qquad . \end{equation*} | (5.60) |
and
| \begin{equation*} S = k \ln(Z) + \frac{1}{T} U = k \ln(W) \qquad . \end{equation*} | (5.61) |
\(W\): Number of the micro states of a system with energies \(U\).
The isolated
system is therefor characterized by \(U = const.\) and \(S = const.\)
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© J. Carstensen (TD Kin II)