这是一份warwick华威大学PX154的成功案例

$\mathbf{d p}=\mathbf{F} d t$
Now let us integrate with respect to time;
$$
\mathbf{J}=\int_{t_{1}}^{t_{2}} \mathbf{F} d t
$$
however;
$$
\mathbf{F}=\frac{d \mathbf{p}}{d t}
$$
Substituting for $\mathbf{F}$ from in the integral, we obtain:
$$
\mathbf{J}=\int_{t_{1}}^{t_{2}} \frac{d \mathbf{p}}{d t} d t=\int_{p_{1}}^{p_{2}} d \mathbf{p}
$$

PX154 COURSE NOTES ：

The distribution functions:
Boltzmann:
$$
\bar{n}{i}^{\mathrm{B}}=\frac{N{i}}{A_{i}}=\frac{1}{\alpha \exp \left(+\frac{\varepsilon_{i}}{k_{\mathrm{B}} T}\right)}
$$
Bose-Einstein:
$$
\bar{n}{i}^{\mathrm{BE}}=\frac{N{i}}{A_{i}}=\frac{1}{\alpha \exp \left(+\frac{\varepsilon_{i}}{k_{\mathrm{B}} T}\right)-1}
$$
Fermi-Dirac:
$$
\bar{n}{i}^{\mathrm{FD}}=\frac{N{i}}{A_{i}}=\frac{1}{\alpha \exp \left(+\frac{\varepsilon_{i}}{k_{\mathrm{B}} T}\right)+1}
$$