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→The force and diffusion terms “force”项与“diff”项
dN是在t时刻，关于（r，p）的微体积元<math> d^3\bf{r}</math>和微动量元<math> \mathrm{d}^3\bf{p}</math>内的分子数目。在位置空间和动量空间的一个区域上积分，得出在该区域中具有位置和动量的粒子总数：
dN是在t时刻，关于（r，p）的微体积元<math> d^3\bf{r}</math>和微动量元<math> \mathrm{d}^3\bf{p}</math>内的分子数目。在位置空间和动量空间的一个区域上积分，得出在该区域中具有位置和动量的粒子总数：
<math>N=\int d^{3}\mathbf{p}\int d^{3}\mathbf{r}f(\mathbf{r},\mathbf{p},t)=\iiint\iiint f(x,y,z,p_{x},p_{y},p_{z},t) \text{d}x\,\text{d}y\,\text{d}z\,\text{d}p_x\,\text{d}p_y\,\text{d}p_z</math>
<math>N=\underset{momenta}\int d^{3}\mathbf{p}\underset{positions}\int d^{3}\mathbf{r}f(\mathbf{r},\mathbf{p},t)=\underset{momenta}\iiint\; \; \; \underset{positions}\iiint f(x,y,z,p_{x},p_{y},p_{z},t) \text{d}x\,\text{d}y\,\text{d}z\,\text{d}p_x\,\text{d}p_y\,\text{d}p_z</math>
which is a [[multiple integral|6-fold integral]]. While ''f'' is associated with a number of particles, the phase space is for one-particle (not all of them, which is usually the case with [[deterministic]] [[many body problem|many-body]] systems), since only one '''r''' and '''p''' is in question. It is not part of the analysis to use '''r'''<sub>1</sub>, '''p'''<sub>1</sub> for particle 1, '''r'''<sub>2</sub>, '''p'''<sub>2</sub> for particle 2, etc. up to '''r'''<sub>''N''</sub>, '''p'''<sub>''N''</sub> for particle ''N''.
which is a [[multiple integral|6-fold integral]]. While ''f'' is associated with a number of particles, the phase space is for one-particle (not all of them, which is usually the case with [[deterministic]] [[many body problem|many-body]] systems), since only one '''r''' and '''p''' is in question. It is not part of the analysis to use '''r'''<sub>1</sub>, '''p'''<sub>1</sub> for particle 1, '''r'''<sub>2</sub>, '''p'''<sub>2</sub> for particle 2, etc. up to '''r'''<sub>''N''</sub>, '''p'''<sub>''N''</sub> for particle ''N''.
& = f\left ( \textbf{r}+\frac{\textbf{p}}{m}\Delta t,\textbf{p}+\textbf{F}\Delta t,t+\Delta t \right )\, d^{3}\textbf{r}\, d^{3}\textbf{p}- f(\textbf{r},\textbf{p},t)\, d^{3}\textbf{r}\, d^{3}\textbf{p}\\[5pt]
& = f\left ( \textbf{r}+\frac{\textbf{p}}{m}\Delta t,\textbf{p}+\textbf{F}\Delta t,t+\Delta t \right )\, d^{3}\textbf{r}\, d^{3}\textbf{p}- f(\textbf{r},\textbf{p},t)\, d^{3}\textbf{r}\, d^{3}\textbf{p}\\[5pt]
& =\Delta f d^{3}\textbf{r}\, d^{3}\textbf{p}
& =\Delta f d^{3}\textbf{r}\, d^{3}\textbf{p}
\end{align}</math>|3={{EquationRef|1}}|：}}mass
\end{align}</math>|3={{EquationRef|1}}|：}}where Δ''f'' is the ''total'' change in ''f''. Dividing ({{EquationNote|1}}) by <math> d^3\bf{r}</math> <math> d^3\bf{p}</math> Δ''t'' and taking the limits Δ''t'' → 0 and Δ''f'' → 0, we have{{NumBlk|2=<math>\frac{d f}{d t} = \left(\frac{\partial f}{\partial t} \right)_\mathrm{coll}</math>|3={{EquationRef|2}}|：}}
where Δ''f'' is the ''total'' change in ''f''. Dividing ({{EquationNote|1}}) by <math> d^3\bf{r}</math> <math> d^3\bf{p}</math> Δ''t'' and taking the limits Δ''t'' → 0 and Δ''f'' → 0, we have{{NumBlk|2=<math>\frac{d f}{d t} = \left(\frac{\partial f}{\partial t} \right)_\mathrm{coll}</math>|3={{EquationRef|2}}|：}}
The total [[differential of a function|differential]] of ''f'' is:
The total [[differential of a function|differential]] of ''f'' is: