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Prison Architect 2.7.0.2 MULTILANGUAGE Game Download [2021]

Prison Architect 2.7.0.2 MULTILANGUAGE Game Download [2021]





 
 
 
 
 
 
 

Prison Architect 2.7.0.2 MULTILANGUAGE Game Download

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How to calculate the particle number in a box?

How to calculate the particle number in a box? The calculation of particle number is said to be based on the particle number density, which is calculated as the magnitude of the gradient of the energy function – $ \hbar \omega _n $ divided by $V$.
Do we have some nice way to calculate this particle number density? I have no idea how to evaluate this magnitude of gradient in the context of a quantum field theory.

A:

I suppose that we have a finite-temperature quantum field theory in a finite-size box and the vacuum state has a non-zero occupation number $n$, so that the free energy $F$ is zero. The free energy in a finite-size box is
$$F=-\frac{1}{\beta}\ln Z$$
where $Z$ is the partition function
$$Z=\int \mathcal{D}\Phi\,e^{ -S[\Phi]}.$$
(where the action $S$ is really the Euclidean action, but the integral is over the time-continuum, so that the periodic boundary condition is automatically satisfied. See the end of my answer for a technical point.)
So $F=0$ is the equation for the equation of state, and we have
$$\frac{\partial F}{\partial \beta} = -\frac{1}{\beta^2}\frac{\partial

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