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第39行： − − :<math> \text{d}^3\mathbf{r}\,\text{d}^3\mathbf{p} = \text{d}x\,\text{d}y\,\text{d}z\,\text{d}p_x\,\text{d}p_y\,\text{d}p_z. </math> − − < math > text { d } n = f (mathbf { r } ，mathbf { p } ，t) ，text { d } ^ 3 mathbf { r } ，text { d } ^ 3 mathbf { p } </math > − − 第59行：
第53行： − + − 《数学》+ − + − + − + − \begin{align}+ − + − 开始{ align }+ − + − is the number of molecules which ''all'' have positions lying within a volume element <math> d^3\bf{r}</math> about '''r''' and momenta lying within a [[momentum space]] element <math> \mathrm{d}^3\bf{p}</math> about '''p''', at time ''t''.<ref>{{Cite book |last=Huang |first=Kerson |year=1987 |title=Statistical Mechanics |url=https://archive.org/details/statisticalmecha00huan_475 |url-access=limited |location=New York |publisher=Wiley |isbn=978-0-471-81518-1 |page=[https://archive.org/details/statisticalmecha00huan_475/page/n65 53] |edition=Second }}</ref> [[Integration (calculus)|Integrating]] over a region of position space and momentum space gives the total number of particles which have positions and momenta in that region:+
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The set of all possible positions '''r''' and momenta '''p''' is called the [[phase space]] of the system; in other words a set of three [[coordinates]] for each position coordinate ''x, y, z'', and three more for each momentum component ''p<sub>x</sub>, p<sub>y</sub>, p<sub>z</sub>''. The entire space is 6-[[dimension]]al: a point in this space is ('''r''', '''p''') = (''x, y, z, p<sub>x</sub>, p<sub>y</sub>, p<sub>z</sub>''), and each coordinate is [[Parametric equation|parameterized]] by time ''t''. The small volume ("differential [[volume element]]") is written
The set of all possible positions '''r''' and momenta '''p''' is called the [[phase space]] of the system; in other words a set of three [[coordinates]] for each position coordinate ''x, y, z'', and three more for each momentum component ''p<sub>x</sub>, p<sub>y</sub>, p<sub>z</sub>''. The entire space is 6-[[dimension]]al: a point in this space is ('''r''', '''p''') = (''x, y, z, p<sub>x</sub>, p<sub>y</sub>, p<sub>z</sub>''), and each coordinate is [[Parametric equation|parameterized]] by time ''t''. The small volume ("differential [[volume element]]") is written
<math>\text{d}N = f (\mathbf{r},\mathbf{p},t)\,\text{d}^3\mathbf{r}\,\text{d}^3\mathbf{p}</math>
<math>\text{d}N = f (\mathbf{r},\mathbf{p},t)\,\text{d}^3\mathbf{r}\,\text{d}^3\mathbf{p}</math>
Since the probability of ''N'' molecules which ''all'' have '''r''' and '''p''' within <math> \mathrm{d}^3\bf{r}</math> <math> \mathrm{d}^3\bf{p}</math> is in question, at the heart of the equation is a quantity ''f'' which gives this probability per unit phase-space volume, or probability per unit length cubed per unit momentum cubed, at an instant of time ''t''. This is a [[probability density function]]: ''f''('''r''', '''p''', ''t''), defined so that,
Since the probability of ''N'' molecules which ''all'' have '''r''' and '''p''' within <math> \mathrm{d}^3\bf{r}</math> <math> \mathrm{d}^3\bf{p}</math> is in question, at the heart of the equation is a quantity ''f'' which gives this probability per unit phase-space volume, or probability per unit length cubed per unit momentum cubed, at an instant of time ''t''. This is a [[probability density function]]: ''f''('''r''', '''p''', ''t''), defined so that,
<math>
<math>
《数学》
\begin{align}
开始{ align }
is the number of molecules which ''all'' have positions lying within a volume element <math> d^3\bf{r}</math> about '''r''' and momenta lying within a [[momentum space]] element <math> \mathrm{d}^3\bf{p}</math> about '''p''', at time ''t''.<ref>{{Cite book |last=Huang |first=Kerson |year=1987 |title=Statistical Mechanics |url=https://archive.org/details/statisticalmecha00huan_475 |url-access=limited |location=New York |publisher=Wiley |isbn=978-0-471-81518-1 |page=[https://archive.org/details/statisticalmecha00huan_475/page/n65 53] |edition=Second }}</ref> [[Integration (calculus)|Integrating]] over a region of position space and momentum space gives the total number of particles which have positions and momenta in that region:
N & = \int\limits_\mathrm{momenta} \text{d}^3\mathbf{p} \int\limits_\mathrm{positions} \text{d}^3\mathbf{r}\,f (\mathbf{r},\mathbf{p},t) \\[5pt]
N & = \int\limits_\mathrm{momenta} \text{d}^3\mathbf{p} \int\limits_\mathrm{positions} \text{d}^3\mathbf{r}\,f (\mathbf{r},\mathbf{p},t) \\[5pt]