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\sect{The extensive property of the grand energy}

Explain why ${F, \Omega}$ are extensive functions in the
thermodynamic limit, so that
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\[F\left(\beta; \lambda V, \lambda N\right)=\lambda F\left(\beta;
V,N\right)\]
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\[\Omega\left(\beta\mu; \lambda V\right)=\lambda \Omega
\left(\beta\mu; V\right)\]
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Guideline: Note that if you split the system, then in neglecting
"surface" interaction, the functions ${Z,Z}$ will be factorized.
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\[Z\approx Z^{A}+Z^{B},F\approx F^{A}+F^{B}\]
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Result: therefore,
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\[\Omega\left(\beta\mu;V\right)=V\Omega \left(\beta\mu;1\right)\]
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Prove that from here, we can conclude that
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\[\Omega\left(\beta\mu;V\right)=-V*{p}\left(\beta\mu\right)\]
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remark: Generalization of considerations such these were written by
Euler.

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