2. Recall that the Helmholtz free energy is given by $F = -k_B T \ln Z$, where $Z$ is the canonical partition function. Show that the Landau potential as defined in problem 1 is given by $\Phi = -k_B T \ln Z$, where $Z$ is the grand partition function.
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The Landau potential is defined as Ω = -kBT ln Z, where Z is the grand partition function. Show more…
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In Section 6.5 I derived the useful relation $F=-k T \ln Z$ between the Helmholtz free energy and the ordinary partition function. Use an analogous argument to prove that $$\Phi=-k T \ln \mathcal{Z}$$ where $\mathcal{Z}$ is the grand partition function and $\Phi$ is the grand free energy introduced in Problem 5.23.
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