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An equation in kinetic gas theory, proposed by L. Boltzmann for the determination of the single-particle distribution function of an ideal mono-atomic gas [[#References|[1]]]. In dimensionless variables the equation has the form:
 
An equation in kinetic gas theory, proposed by L. Boltzmann for the determination of the single-particle distribution function of an ideal mono-atomic gas [[#References|[1]]]. In dimensionless variables the equation has the form:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016820/b0168201.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$\frac{\partial f}{\partial t}+(v,\nabla_xf)+(F,\nabla_vf)=\frac1\epsilon L(f,f).\label{*}\tag{*}$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016820/b0168202.png" /> is the density of the distribution function of the number of particles in the phase space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016820/b0168203.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016820/b0168204.png" /> is the three-dimensional space coordinate, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016820/b0168205.png" /> is the velocity, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016820/b0168206.png" /> is the time, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016820/b0168207.png" /> is the field strength of the external forces, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016820/b0168208.png" /> is a dimensionless parameter (which is proportional to the ratio between the average distance covered by the particles between successive collisions and the typical scale of the phenomena under consideration). In the simplest case the collision operator has the form:
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Here $f(x,v,t)$ is the density of the distribution function of the number of particles in the phase space $x\otimes v$, $x$ is the three-dimensional space coordinate, $v$ is the velocity, $t$ is the time, $F$ is the field strength of the external forces, and $\epsilon$ is a dimensionless parameter (which is proportional to the ratio between the average distance covered by the particles between successive collisions and the typical scale of the phenomena under consideration). In the simplest case the collision operator has the form:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016820/b0168209.png" /></td> </tr></table>
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$$L(f,f)=\int[f(v')f(v_1')-f(v)f(v_1)]|v-v_1|\,d\omega\,dv_1,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016820/b01682010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016820/b01682011.png" /> are the velocities of the molecules before collision, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016820/b01682012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016820/b01682013.png" /> are the velocities of the molecules after collision, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016820/b01682014.png" /> is the solid angle element in the direction of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016820/b01682015.png" />.
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where $v_1$ and $v$ are the velocities of the molecules before collision, $v_1'$ and $v'$ are the velocities of the molecules after collision, and $d\omega$ is the solid angle element in the direction of the vector $v_1-v$.
  
In deriving the Boltzmann equation it is assumed that the evolution of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016820/b01682016.png" /> is determined by its value at a given moment of time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016820/b01682017.png" /> and by the pairwise collisions between the gas molecules, and that the time of interaction between two gas molecules during collision is much shorter than the time during which they move independently of each other. From the mathematical point of view the derivation of the Boltzmann equation is based on a certain algorithm which constructs the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016820/b01682018.png" /> in accordance with the well-known laws of motion of two gas molecules which collide with one another.
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In deriving the Boltzmann equation it is assumed that the evolution of the function $f(x,v,t)$ is determined by its value at a given moment of time $t$ and by the pairwise collisions between the gas molecules, and that the time of interaction between two gas molecules during collision is much shorter than the time during which they move independently of each other. From the mathematical point of view the derivation of the Boltzmann equation is based on a certain algorithm which constructs the operator $L$ in accordance with the well-known laws of motion of two gas molecules which collide with one another.
  
In equation (*) the range of variation of the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016820/b01682019.png" /> is the half-line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016820/b01682020.png" />; the range of variation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016820/b01682021.png" /> is the entire space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016820/b01682022.png" />; and the range of variation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016820/b01682023.png" /> is a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016820/b01682024.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016820/b01682025.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016820/b01682026.png" /> may coincide with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016820/b01682027.png" />). In accordance with its physical meaning, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016820/b01682028.png" /> should be non-negative and such that
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In equation \eqref{*} the range of variation of the variable $t$ is the half-line $t\geq0$; the range of variation of $v$ is the entire space $\mathbf R^3$; and the range of variation of $x$ is a subspace $\Omega$ in $\mathbf R^3$ ($\Omega$ may coincide with $\mathbf R^3$). In accordance with its physical meaning, the function $f(x,v,t)$ should be non-negative and such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016820/b01682029.png" /></td> </tr></table>
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$$\int f(x,v,t)v^2\,dv<\infty.$$
  
The simplest boundary condition on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016820/b01682030.png" /> has the form
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The simplest boundary condition on $\partial\Omega$ has the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016820/b01682031.png" /></td> </tr></table>
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$$f(v-2n(n,v),x,t)=f(v,x,t),\quad x\in\partial\Omega,\quad v\in\mathbf R^3,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016820/b01682032.png" /> is the normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016820/b01682033.png" />. There exist a number of rigorous statements of the Cauchy problem for equation (*).
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where $n$ is the normal to $\partial\Omega$. There exist a number of rigorous statements of the Cauchy problem for equation \eqref{*}.
  
 
====References====
 
====References====

Latest revision as of 15:08, 14 February 2020

An equation in kinetic gas theory, proposed by L. Boltzmann for the determination of the single-particle distribution function of an ideal mono-atomic gas [1]. In dimensionless variables the equation has the form:

$$\frac{\partial f}{\partial t}+(v,\nabla_xf)+(F,\nabla_vf)=\frac1\epsilon L(f,f).\label{*}\tag{*}$$

Here $f(x,v,t)$ is the density of the distribution function of the number of particles in the phase space $x\otimes v$, $x$ is the three-dimensional space coordinate, $v$ is the velocity, $t$ is the time, $F$ is the field strength of the external forces, and $\epsilon$ is a dimensionless parameter (which is proportional to the ratio between the average distance covered by the particles between successive collisions and the typical scale of the phenomena under consideration). In the simplest case the collision operator has the form:

$$L(f,f)=\int[f(v')f(v_1')-f(v)f(v_1)]|v-v_1|\,d\omega\,dv_1,$$

where $v_1$ and $v$ are the velocities of the molecules before collision, $v_1'$ and $v'$ are the velocities of the molecules after collision, and $d\omega$ is the solid angle element in the direction of the vector $v_1-v$.

In deriving the Boltzmann equation it is assumed that the evolution of the function $f(x,v,t)$ is determined by its value at a given moment of time $t$ and by the pairwise collisions between the gas molecules, and that the time of interaction between two gas molecules during collision is much shorter than the time during which they move independently of each other. From the mathematical point of view the derivation of the Boltzmann equation is based on a certain algorithm which constructs the operator $L$ in accordance with the well-known laws of motion of two gas molecules which collide with one another.

In equation \eqref{*} the range of variation of the variable $t$ is the half-line $t\geq0$; the range of variation of $v$ is the entire space $\mathbf R^3$; and the range of variation of $x$ is a subspace $\Omega$ in $\mathbf R^3$ ($\Omega$ may coincide with $\mathbf R^3$). In accordance with its physical meaning, the function $f(x,v,t)$ should be non-negative and such that

$$\int f(x,v,t)v^2\,dv<\infty.$$

The simplest boundary condition on $\partial\Omega$ has the form

$$f(v-2n(n,v),x,t)=f(v,x,t),\quad x\in\partial\Omega,\quad v\in\mathbf R^3,$$

where $n$ is the normal to $\partial\Omega$. There exist a number of rigorous statements of the Cauchy problem for equation \eqref{*}.

References

[1] L. Boltzmann, "Lectures on gas theory" , Univ. California Press , Berkeley (1964)
[2] N.N. Bogolyubov, "Selected works" , 2 , Kiev (1970) (In Russian)
[3] S. Chapman, T.G. Cowling, "The mathematical theory of non-uniform gases" , Cambridge Univ. Press (1939)


Comments

References

[a1] C. Cercignani, "Theory and application of the Boltzmann equation" , Scottish Acad. Press (1975)
[a2] C. Cercignani (ed.) , Kinetic theories and the Boltzmann equation , Springer (1984)
How to Cite This Entry:
Boltzmann equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boltzmann_equation&oldid=12483
This article was adapted from an original article by A.A. Arsen'ev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article