Having defined the phase space distribution function $f(\textbf{r},\textbf{p},t)$ in $\mu-$space, one can express the information that there are $N$ particles in the volume $V$ through the condition \begin{equation} \int f(\textbf{r},\textbf{p},t) d^3\textbf{r}d^3\textbf{p}=N \end{equation} as given in, for example, Kerson Huang, Statistical mechanics, second edition, Sec. 3.1, Eq. 3.4).

In equilibrium, the distribution function is independent of $t$ so that \begin{equation}\int f(\textbf{r},\textbf{p}) d^3\textbf{r}d^3\textbf{p}=N. \end{equation} If the particles are uniformly distributed in space, so that $f$ is independent of $\textbf{r}$, then the number density is given by \begin{equation} n=\frac{N}{V}=\int d^3\textbf{p}f(\textbf{p}) \end{equation}

1. The expression for number density as given in Cosmology book by Kolb and Turner, they have an extra factor of $\frac{1}{(2\pi)^3}$: \begin{equation} n=\frac{N}{V}=\frac{g}{(2\pi)^3}\int d^3\textbf{p}f(\textbf{p}) \end{equation} where $g$ counts the number of internal degrees of freedom. But where does the factor $\frac{1}{(2\pi)^3}$ come from? Why is this factor missing in Huang's expression for number density?

2. Kolb and Turner also writes an expression for the pressure for the relativistic particles satisfying $E^2=|\textbf{p}|^2+m^2$ (units in which $c=1$) in equilibrium as $$p=\frac{g}{(2\pi)^3}\int \frac{|\textbf{p}|^2}{3E}f(\textbf{p})d^3\textbf{p}.$$ What should the starting point in deriving this formula for pressure?

Having defined the phase space distribution function $f(\textbf{r},\textbf{p},t)$ in $\mu-$space, one can express the information that there are $N$ particles in the volume $V$ through the condition \begin{equation} \int f(\textbf{r},\textbf{p},t) d^3\textbf{r}d^3\textbf{p}=N \end{equation} as given in, for example, Kerson Huang, Statistical mechanics, second edition, Sec. 3.1, Eq. 3.4).

In equilibrium, the distribution function is independent of $t$ so that \begin{equation}\int f(\textbf{r},\textbf{p}) d^3\textbf{r}d^3\textbf{p}=N. \end{equation} If the particles are uniformly distributed in space, so that $f$ is independent of $\textbf{r}$, then the number density is given by \begin{equation} n=\frac{N}{V}=\int d^3\textbf{p}f(\textbf{p}) \end{equation}

1. The expression for number density as given in Cosmology book by Kolb and Turner, they have an extra factor of $\frac{1}{(2\pi)^3}$: \begin{equation} n=\frac{N}{V}=\frac{g}{(2\pi)^3}\int d^3\textbf{p}f(\textbf{p}) \end{equation} where $g$ counts the number of internal degrees of freedom. But where does the factor $\frac{1}{(2\pi)^3}$ come from? Why is this factor missing in Huang's expression for number density?

2. Kolb and Turner also writes an expression for the pressure for the relativistic particles satisfying $E^2=|\textbf{p}|^2+m^2$ (units in which $c=1$) in equilibrium as $$p=\frac{g}{(2\pi)^3}\int \frac{|\textbf{p}|^2}{3E}f(\textbf{p})d^3\textbf{p}.$$ What should the starting point in deriving this formula for pressure? In other words, what is the general formula for equilibrium pressure?

Having defined the phase space distribution function $f(\textbf{r},\textbf{p},t)$ in $\mu-$space, one can express the information that there are $N$ particles in the volume $V$ through the condition \begin{equation} \int f(\textbf{r},\textbf{p},t) d^3\textbf{r}d^3\textbf{p}=N \end{equation} as given in, for example, Kerson Huang, Statistical mechanics, second edition, Sec. 3.1, Eq. 3.4).

In equilibrium, the distribution function is independent of $t$ so that \begin{equation}\int f(\textbf{r},\textbf{p}) d^3\textbf{r}d^3\textbf{p}=N. \end{equation} If the particles are uniformly distributed in space, so that $f$ is independent of $\textbf{r}$, then the number density is given by \begin{equation} n=\frac{N}{V}=\int d^3\textbf{p}f(\textbf{p}) \end{equation}

The expression for number density as given in Cosmology book by Kolb and Turner, they have an extra factor of $\frac{1}{(2\pi)^3}$: \begin{equation} n=\frac{N}{V}=\frac{g}{(2\pi)^3}\int d^3\textbf{p}f(\textbf{p}) \end{equation} where $g$ counts the number of internal degrees of freedom. But where does the factor $\frac{1}{(2\pi)^3}$ come from? Why is this factor missing in Huang's expression for number density?

Having defined the phase space distribution function $f(\textbf{r},\textbf{p},t)$ in $\mu-$space, one can express the information that there are $N$ particles in the volume $V$ through the condition \begin{equation} \int f(\textbf{r},\textbf{p},t) d^3\textbf{r}d^3\textbf{p}=N \end{equation} as given in, for example, Kerson Huang, Statistical mechanics, second edition, Sec. 3.1, Eq. 3.4).

In equilibrium, the distribution function is independent of $t$ so that \begin{equation}\int f(\textbf{r},\textbf{p}) d^3\textbf{r}d^3\textbf{p}=N. \end{equation} If the particles are uniformly distributed in space, so that $f$ is independent of $\textbf{r}$, then the number density is given by \begin{equation} n=\frac{N}{V}=\int d^3\textbf{p}f(\textbf{p}) \end{equation}

1. The expression for number density as given in Cosmology book by Kolb and Turner, they have an extra factor of $\frac{1}{(2\pi)^3}$: \begin{equation} n=\frac{N}{V}=\frac{g}{(2\pi)^3}\int d^3\textbf{p}f(\textbf{p}) \end{equation} where $g$ counts the number of internal degrees of freedom. But where does the factor $\frac{1}{(2\pi)^3}$ come from? Why is this factor missing in Huang's expression for number density?

2. Kolb and Turner also writes an expression for the pressure for the relativistic particles satisfying $E^2=|\textbf{p}|^2+m^2$ (units in which $c=1$) in equilibrium as $$p=\frac{g}{(2\pi)^3}\int \frac{|\textbf{p}|^2}{3E}f(\textbf{p})d^3\textbf{p}.$$ What should the starting point in deriving this formula for pressure?