The grand potential and total Helmholtz free energy

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SUMMARY

The discussion focuses on the relationship between the grand potential \( J = F - \mu N \) and the total Helmholtz free energy \( F^{tot} \) as presented in K. Sekimoto's "Stochastic Energetics." Key points include the limits of the total free energy and particle number, specifically \( J = \lim[F_{tot} - \mu N_{tot}] \) and \( \lim[F^c] = \lim[\mu(N_{tot} - N)] \). Additionally, the formula on page 311 is critiqued for potentially missing a crucial factor \( \frac{N_{tot}^{N_{tot}}}{N_{tot}!} \), which becomes infinite as \( N_{tot} \) approaches infinity. The discussion raises questions about the accuracy of formulas A.75 and the first formula on page 311.

PREREQUISITES
  • Understanding of grand thermodynamic potentials, specifically grand potential and Helmholtz free energy.
  • Familiarity with statistical mechanics concepts as outlined in K. Sekimoto's "Stochastic Energetics."
  • Knowledge of limits and asymptotic behavior in mathematical expressions.
  • Basic comprehension of particle number \( N \) and its implications in thermodynamic equations.
NEXT STEPS
  • Review K. Sekimoto's "Stochastic Energetics" focusing on pages 182, 310, and 311 for detailed context.
  • Study the derivation and implications of grand potential in thermodynamics.
  • Investigate the mathematical treatment of limits in thermodynamic equations.
  • Explore the significance of factorial terms in statistical mechanics, particularly in relation to large particle numbers.
USEFUL FOR

Physicists, thermodynamicists, and researchers in statistical mechanics who are analyzing the relationships between thermodynamic potentials and their implications in various systems.

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what's the relation of the grand potential J=F-\mu N and total Helmholtz free energy of "system and particle environment" F^{tot}?
In K. Sekimoto's book "Stochastic Energetics" P182, P310 and P311 (see screenshots in the link):
Does it mean the followings:J=lim[F_{tot}-\mu N_{tot}], lim[F^c]=lim[\mu(N_{tot}-N)], lim[\Omega^cf^c]=lim[\mu N_{tot}],
And the right term of the formula above A.77 seems to be lack of a factor \frac{N_{tot}^{N_{tot}}}{N_{tot}!},which is infinite when N_{tot}->\infty.
 
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Nobody?:L:L
Google book of Stochastic Energetics: link
Is there any problem in the formula A.75(p310) and the first formula in p311?
 

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