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what's the relation of the grand potential [itex]J=F-\mu N[/itex] and total Helmholtz free energy of "system and particle environment" [itex]F^{tot}[/itex]?

In K. Sekimoto's book "Stochastic Energetics" P182, P310 and P311 (see screenshots in the link):

Does it mean the followings:[itex]J=lim[F_{tot}-\mu N_{tot}], lim[F^c]=lim[\mu(N_{tot}-N)], lim[\Omega^cf^c]=lim[\mu N_{tot}][/itex],

And the right term of the formula above A.77 seems to be lack of a factor [itex]\frac{N_{tot}^{N_{tot}}}{N_{tot}!}[/itex],which is infinite when [itex]N_{tot}->\infty[/itex].

In K. Sekimoto's book "Stochastic Energetics" P182, P310 and P311 (see screenshots in the link):

Does it mean the followings:[itex]J=lim[F_{tot}-\mu N_{tot}], lim[F^c]=lim[\mu(N_{tot}-N)], lim[\Omega^cf^c]=lim[\mu N_{tot}][/itex],

And the right term of the formula above A.77 seems to be lack of a factor [itex]\frac{N_{tot}^{N_{tot}}}{N_{tot}!}[/itex],which is infinite when [itex]N_{tot}->\infty[/itex].

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