# The grand potential and total Helmholtz free energy

• xibeisiber

#### xibeisiber

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 $N_{tot}->\infty$.

Last edited:
Nobody?:L:L
Is there any problem in the formula A.75(p310) and the first formula in p311?

## What is the grand potential?

The grand potential is a thermodynamic potential that describes the equilibrium state of a system with varying particle number, temperature, and volume. It is denoted by Ω and is related to the total Helmholtz free energy through the following equation: Ω = F - μN, where F is the Helmholtz free energy, μ is the chemical potential, and N is the number of particles.

## How is the grand potential related to the total Helmholtz free energy?

The grand potential and the total Helmholtz free energy are related through the following equation: Ω = F - μN, where F is the Helmholtz free energy, μ is the chemical potential, and N is the number of particles. The grand potential is a thermodynamic potential that takes into account the varying particle number, temperature, and volume of a system, while the total Helmholtz free energy only considers the particle number and temperature.

## What is the significance of the total Helmholtz free energy?

The total Helmholtz free energy is a fundamental thermodynamic quantity that represents the maximum amount of work that can be extracted from a system at constant temperature and volume. It is also a measure of the system's stability, with a lower free energy indicating a more stable state.

## How is the grand potential used in statistical mechanics?

In statistical mechanics, the grand potential is used to calculate the thermodynamic properties of a system in equilibrium. It is particularly useful in systems with varying particle number, such as in the study of phase transitions. The grand potential can also be used to derive other thermodynamic potentials, such as the Gibbs free energy.

## What are the applications of the grand potential and total Helmholtz free energy?

The grand potential and total Helmholtz free energy have numerous applications in physics, chemistry, and engineering. They are used to study equilibrium states of systems, predict phase transitions, and calculate thermodynamic properties. They are also essential in the design and optimization of processes and materials in various industries, such as energy production and materials science.