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\sect{Chemical equilibrium for elementary particles}

In a certain medium, there were at the beginning ${N}$ neutrons per
volume unit. Some of them decomposed according to
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\[n \Leftrightarrow  p + e^{-} + \bar{\nu}\]
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All of the particles are fermions with spin ${\frac{1}{2}}$.
Their masses are ${\mathsf{m_{n}}, \mathsf{m_{p}}, \mathsf{m_{e}}, 
and \mathsf{m_{\nu}}=0}$. Assume temperature ${T}$.

Denote by ${N'}$ density of the neutrons in a thermal equilibrium. 
Write the equation for ${N'}$ in four cases: 

(1) The particles are non-relativistic (except the nutrino)

(2) The particles are hyper relativistic (negligible mass).

(3) The temperature is zero.

(4) The temperature is high (Boltzmann approximation).

Define the conditions for the assumptions to be valid. 
Write the equations using the data only. 
There is no need to solve the equations.