# 玻尔兹曼方程

（重定向自波茲曼方程式

## 概述

### 相空间与密度函数

${\displaystyle d^{3}\mathbf {r} \,d^{3}\mathbf {p} =dx\,dy\,dz\,dp_{x}\,dp_{y}\,dp_{z}\ }$

${\displaystyle dN=f(\mathbf {r} ,\mathbf {p} ,t)\,d^{3}\mathbf {r} \,d^{3}\mathbf {p} }$

${\displaystyle dN}$ 被定义为在时间 ${\displaystyle t}$，位于 ${\displaystyle (\mathbf {r} ,\mathbf {p} )}$ 的空间元 ${\displaystyle d^{3}\mathbf {r} \,d^{3}\mathbf {p} }$ 中的粒子总数[5]:61-62。对坐标空间与动量空间的一个区域积分即可得该区域内所有具有对应位置和动量的粒子的总数：

${\displaystyle N=\int \limits _{\mathrm {positions} }d^{3}\mathbf {r} \int \limits _{\mathrm {momenta} }d^{3}\mathbf {p} \,f(\mathbf {r} ,\mathbf {p} ,t)=\iiint \limits _{\mathrm {positions} }\quad \iiint \limits _{\mathrm {momenta} }f(x,y,z,p_{x},p_{y},p_{z},t)\,dx\,dy\,dz\,dp_{x}\,dp_{y}\,dp_{z}}$

### 一般形式

${\displaystyle {\frac {\partial f}{\partial t}}=\left({\frac {\partial f}{\partial t}}\right)_{\mathrm {force} }+\left({\frac {\partial f}{\partial t}}\right)_{\mathrm {diff} }+\left({\frac {\partial f}{\partial t}}\right)_{\mathrm {coll} }}$

## “force”项和“diff”项

${\displaystyle f\left(\mathbf {r} +{\frac {\mathbf {p} }{m}}\Delta t,\mathbf {p} +\mathbf {F} \Delta t,t+\Delta t\right)\,d^{3}\mathbf {r} \,d^{3}\mathbf {p} =f(\mathbf {r} ,\mathbf {p} ,t)\,d^{3}\mathbf {r} \,d^{3}\mathbf {p} }$

{\displaystyle {\begin{aligned}dN_{\mathrm {coll} }&=\left({\frac {\partial f}{\partial t}}\right)_{\mathrm {coll} }\Delta td^{3}\mathbf {r} d^{3}\mathbf {p} \\&=f\left(\mathbf {r} +{\frac {\mathbf {p} }{m}}\Delta t,\mathbf {p} +\mathbf {F} \Delta t,t+\Delta t\right)d^{3}\mathbf {r} d^{3}\mathbf {p} -f(\mathbf {r} ,\mathbf {p} ,t)d^{3}\mathbf {r} d^{3}\mathbf {p} \\&=\Delta fd^{3}\mathbf {r} d^{3}\mathbf {p} \end{aligned}}}

(1)

${\displaystyle {\frac {df}{dt}}=\left({\frac {\partial f}{\partial t}}\right)_{\mathrm {coll} }}$

(2)

${\displaystyle f}$的全微分：

{\displaystyle {\begin{aligned}df&={\frac {\partial f}{\partial t}}dt+\left({\frac {\partial f}{\partial x}}dx+{\frac {\partial f}{\partial y}}dy+{\frac {\partial f}{\partial z}}dz\right)+\left({\frac {\partial f}{\partial p_{x}}}dp_{x}+{\frac {\partial f}{\partial p_{y}}}dp_{y}+{\frac {\partial f}{\partial p_{z}}}dp_{z}\right)\\&={\frac {\partial f}{\partial t}}dt+\nabla f\cdot d\mathbf {r} +{\frac {\partial f}{\partial \mathbf {p} }}\cdot d\mathbf {p} \\&={\frac {\partial f}{\partial t}}dt+\nabla f\cdot {\frac {\mathbf {p} dt}{m}}+{\frac {\partial f}{\partial \mathbf {p} }}\cdot \mathbf {F} dt\end{aligned}}}

(3)

${\displaystyle {\frac {\partial f}{\partial \mathbf {p} }}=\mathbf {\hat {e}} _{x}{\frac {\partial f}{\partial p_{x}}}+\mathbf {\hat {e}} _{y}{\frac {\partial f}{\partial p_{y}}}+\mathbf {\hat {e}} _{z}{\frac {\partial f}{\partial p_{z}}}=\nabla _{\mathbf {p} }f}$

### 最终形式

${\displaystyle {\frac {\partial f}{\partial t}}+{\frac {\mathbf {p} }{m}}\cdot \nabla f+\mathbf {F} \cdot {\frac {\partial f}{\partial \mathbf {p} }}=\left({\frac {\partial f}{\partial t}}\right)_{\mathrm {coll} }}$

## 碰撞项（Stosszahlansatz）和分子混沌

${\displaystyle \left({\frac {\partial f}{\partial t}}\right)_{\mathrm {coll} }=\iint gI(g,\Omega )[f(\mathbf {p'} _{A},t)f(\mathbf {p'} _{B},t)-f(\mathbf {p} _{A},t)f(\mathbf {p} _{B},t)]\,d\Omega \,d^{3}\mathbf {p} _{A}\,d^{3}\mathbf {p} _{B}.}$

${\displaystyle g=|\mathbf {p} _{B}-\mathbf {p} _{A}|=|\mathbf {p'} _{B}-\mathbf {p'} _{A}|}$

## 对碰撞项的简化

${\displaystyle {\frac {\partial f}{\partial t}}+{\frac {\mathbf {p} }{m}}\cdot \nabla f+\mathbf {F} \cdot {\frac {\partial f}{\partial \mathbf {p} }}=\nu (f_{0}-f)}$

## 普适方程（对于混合物）

${\displaystyle {\frac {\partial f_{i}}{\partial t}}+{\frac {\mathbf {p} _{i}}{m_{i}}}\cdot \nabla f_{i}+\mathbf {F} \cdot {\frac {\partial f_{i}}{\partial \mathbf {p} _{i}}}=\left({\frac {\partial f_{i}}{\partial t}}\right)_{\mathrm {coll} }}$

${\displaystyle \left({\frac {\partial f_{i}}{\partial t}}\right)_{\mathrm {coll} }=\sum _{j=1}^{n}\iint g_{ij}I_{ij}(g_{ij},\Omega )[f'_{i}f'_{j}-f_{i}f_{j}]\,d\Omega \,d^{3}\mathbf {p'} .}$

${\displaystyle g_{ij}=|\mathbf {p} _{i}-\mathbf {p} _{j}|=|\mathbf {p'} _{i}-\mathbf {p'} _{j}|}$

Iij 是粒子i和粒子j之间的微分散射截面。此积分的和描述的是某一相空间元中，组分i粒子的进出。

## 应用和推广

### 守恒方程

${\displaystyle n=\int f\,d^{3}p}$

${\displaystyle \langle A\rangle ={\frac {1}{n}}\int Af\,d^{3}p}$

${\displaystyle \int g{\frac {\partial f}{\partial t}}\,d^{3}p={\frac {\partial }{\partial t}}(n\langle g\rangle )}$
${\displaystyle \int {\frac {p_{j}g}{m}}{\frac {\partial f}{\partial x_{j}}}\,d^{3}p={\frac {1}{m}}{\frac {\partial }{\partial x_{j}}}(n\langle gp_{j}\rangle )}$
${\displaystyle \int gF_{j}{\frac {\partial f}{\partial p_{j}}}\,d^{3}p=-nF_{j}\left\langle {\frac {\partial g}{\partial p_{j}}}\right\rangle }$
${\displaystyle \int g\left({\frac {\partial f}{\partial t}}\right)_{\mathrm {coll} }\,d^{3}p=0}$

${\displaystyle g=m}$，即粒子质量，积分后的玻尔兹曼方程化为质量守恒方程[9]:pp 12,168

${\displaystyle {\frac {\partial }{\partial t}}\rho +{\frac {\partial }{\partial x_{j}}}(\rho V_{j})=0}$

${\displaystyle \rho =mn}$ 为质量密度，${\displaystyle V_{i}=\langle w_{i}\rangle }$ 为平均流体速度。

${\displaystyle g=mw_{i}}$，即粒子动量，积分后的玻尔兹曼方程化为动量守恒方程[9]:pp 15,169

${\displaystyle {\frac {\partial }{\partial t}}(\rho V_{i})+{\frac {\partial }{\partial x_{j}}}(\rho V_{i}V_{j}+P_{ij})-nF_{i}=0}$

${\displaystyle P_{ij}=\rho \langle (w_{i}-V_{i})(w_{j}-V_{j})\rangle }$ 为压强张量（加上流体静力学压强）。

${\displaystyle g={\tfrac {1}{2}}mw_{i}w_{i}}$，即粒子动能，积分后的玻尔兹曼方程化为能量守恒方程[9]:pp 19,169

${\displaystyle {\frac {\partial }{\partial t}}(u+{\tfrac {1}{2}}\rho V_{i}V_{i})+{\frac {\partial }{\partial x_{j}}}(uV_{j}+{\tfrac {1}{2}}\rho V_{i}V_{i}V_{j}+J_{qj}+P_{ij}V_{i})-nF_{i}V_{i}=0}$

${\displaystyle u={\tfrac {1}{2}}\rho \langle (w_{i}-V_{i})(w_{i}-V_{i})\rangle }$ 为动力热能密度（kinetic thermal energy density），${\displaystyle J_{qi}={\tfrac {1}{2}}\rho \langle (w_{i}-V_{i})(w_{k}-V_{k})(w_{k}-V_{k})\rangle }$ 热通量矢量。

### 哈密顿力学

${\displaystyle {\hat {\mathbf {L} }}[f]=\mathbf {C} [f],\,}$

${\displaystyle {\hat {\mathbf {L} }}_{\mathrm {NR} }={\frac {\partial }{\partial t}}+{\frac {\mathbf {p} }{m}}\cdot \nabla +\mathbf {F} \cdot {\frac {\partial }{\partial \mathbf {p} }}\,.}$

## 注释

1. ^ N. Gorban, Alexander; V. Karlin, Ilya. Introduction. Invariant Manifolds for Physical and Chemical Kinetics (Springer Berlin Heidelberg). 2005: 1–19. doi:10.1007/978-3-540-31531-5_1 （英语）.
2. Encyclopedia of physics 2nd. VCH. ISBN 3-527-26954-1.
3. ^ DiPerna, R. J.; Lions, P. L. Ordinary differential equations, transport theory and Sobolev spaces. Inventiones Mathematicae. 1989-10, 98 (3): 511–547. doi:10.1007/BF01393835.
4. ^ Gressman, Philip T.; Strain, Robert M. Global classical solutions of the Boltzmann equation without angular cut-off. Journal of the American Mathematical Society. 2011-09-01, 24 (3): 771–771. doi:10.1090/S0894-0347-2011-00697-8.
5. ^ Kerson Huang. Statistical mechanics. Wiley. 1987. ISBN 978-0-471-81518-1.
6. McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, ISBN 0-07-051400-3
7. ^ Bhatnagar, P. L.; Gross, E. P.; Krook, M. (1954-05-01).
8. ^ Grosso 2014，第501页.
9. de Groot, S.R.; Mazur, P. Non-Equilibrium Thermodynamics. New York: Dover Publications Inc. 1984 [2013-01-31]. ISBN 0-486-64741-2.
10. ^ Philip T. Gressman and Robert M. Strain (2010).
11. ^ "Mathematicians Solve 140-Year-Old Boltzmann Equation". news.upenn.edu.
12. ^ Evans, Ben; Morgan, Ken; Hassan, Oubay (2011-03-01).
13. ^ Pareschi, L.; Russo, G. (2000-01-01).

## 参考资料

• （英文）Harris, Stewart (1971). An introduction to the theory of the Boltzmann equation. Dover Books. p. 221. ISBN 978-0-486-43831-3. 此书的推导始于刘维尔定理BBGKY英语BBGKY hierarchy，介绍了玻尔兹曼方程在现代物理框架下的地位。其他大部分统计力学的教科书，例如黄克孙的《Statistical Mechanics》，甚至原样照搬玻尔兹曼最初的演算过程。为了简化论述，这些教科书运用启发式的解释，避而不谈玻尔兹曼方程的应用范围以及方程中的某些假设，而这些假设正是玻尔兹曼方程与其他输运方程例如福克-普朗克方程或朗道方程组的不同之处。