Boltzmann equation

The Boltzmann equation or Boltzmann transport equation (after the physicist Ludwig Boltzmann ) is the basic integro-differential equation in the six-dimensional phase space of the kinetic theory of gases and non-equilibrium - Thermodynamics . It is an equation for the statistical distribution of particles in a medium.

The Boltzmann equation is used when the mean free path of the particles is large, i.e. when there are only a few gas particles in a given volume, so that the mean collision duration is small compared to the mean free time of flight and only two- particle collisions have to be considered. In a medium where this is not the case, i. H. in the limit of the small mean free path, the Boltzmann equation (under certain conditions) changes into the much simpler Navier-Stokes equation of continuum mechanics . In this sense, the Boltzmann equation is a mesoscopic one Equation that stands between the microscopic description of individual particles and the macroscopic description.

The Boltzmann equation finds an important application in the proof of the H-theorem with which Boltzmann was able to derive the 2nd law of thermodynamics from statistical assumptions. Current applications concern, for example, flows in a dilute gas. In practice this occurs e.g. B. when calculating phenomena in the outer atmosphere of the earth , for example when the space shuttle re- enters . The distribution of neutrons in a nuclear reactor or the intensity of heat radiation in a combustion chamber can also be described by the Boltzmann equation.

A numerical solution to the Boltzmann equation is provided by the Lattice-Boltzmann method .

equation

The Boltzmann equation describes the total time derivative of the distribution density (left side of the equation) as a collision integral (right side of the equation):

${\displaystyle \left({\frac {\partial }{\partial t}}+{\vec {v}}\cdot \nabla _{\vec {x}}+{\frac {\vec {F}}{m}}\cdot \nabla _{\vec {v}}\right)f({\vec {x}},{\vec {v}},t)=\left.{\frac {\partial f}{\partial t}}\right|_{\mathrm {Stoss} }}$

With

• the distribution density in the state space ${\displaystyle f({\vec {x}},{\vec {v}},t)}$
  • the place ${\displaystyle {\vec {x}}}$
  • the speed ${\displaystyle {\vec {v}}}$
  • the time ${\displaystyle t}$
• a given external force ${\displaystyle {\vec {F}}}$
• the mass of the particles. ${\displaystyle m}$

The second term is also called the transport term and the third term field term , since it describes the interaction with external fields. ${\displaystyle {\vec {v}}\cdot \nabla _{\vec {x}}f}$ ${\displaystyle {\tfrac {\vec {F}}{m}}\cdot \nabla _{\vec {v}}f}$

The distribution density can be interpreted in such a way that the value indicates the relative number of particles that are located in the spatial volume at the time and have speeds in the range . ${\displaystyle f({\vec {x}},{\vec {v}},t)\,{\text{d}}{\vec {x}}\,{\text{d}}{\vec {v}}}$ ${\displaystyle t}$ ${\displaystyle [{\vec {x}},{\vec {x}}+{\text{d}}{\vec {x}}]}$ ${\displaystyle [{\vec {v}},{\vec {v}}+{\text{d}}{\vec {v}}]}$

The collision integral is a multidimensional integral in which there is a non-linear link. It indicates that contribution to the equation that arises from the collision of the individual particles (if it were not there, the equation could be solved using the means of classical mechanics ). ${\displaystyle \left.{\tfrac {\partial f}{\partial t}}\right|_{\mathrm {Stoss} }}$ ${\displaystyle f}$

In a narrower sense, the Boltzmann equation means the above equation together with a special approach for the collision integral ( Boltzmann's impact number approach ):

${\displaystyle \left.{\frac {\partial f}{\partial t}}\right|_{\mathrm {Stoss} }=\int W({\vec {v}}_{1},{\vec {v}}_{2},{\vec {v}}_{3},{\vec {v}})\left\{f({\vec {x}},{\vec {v}}_{1},t)f({\vec {x}},{\vec {v}}_{2},t)-f({\vec {x}},{\vec {v}}_{3},t)f({\vec {x}},{\vec {v}},t)\right\}{\text{d}}{\vec {v}}_{1}{\text{d}}{\vec {v}}_{2}{\text{d}}{\vec {v}}_{3}}$

Here is the probability per unit time that in a collision between two particles before the collision speeds and possess after the collision, the velocities and , respectively. The exact shape of depends on the type and shape of the particles and must be determined from a microscopic theory (e.g. from quantum mechanics ). ${\displaystyle W({\vec {v}}_{1},{\vec {v}}_{2},{\vec {v}}_{3},{\vec {v}})}$ ${\displaystyle {\vec {v}}_{1}}$ ${\displaystyle {\vec {v}}_{2}}$ ${\displaystyle {\vec {v}}_{3}}$ ${\displaystyle {\vec {v}}}$ ${\displaystyle W}$

Both the theoretical and the numerical treatment of the Boltzmann equation is very complex. For a simple introduction see p. Müller-Kirsten.

In order to draw conclusions from the Boltzmann equation, one analyzes its speed moments. The nth speed moment is obtained by multiplying by the Boltzmann equation and then integrating over the speed space. From this, for example, the Maxwell stress tensor can be obtained. ${\displaystyle {\vec {v}}^{n}}$

literature

• Hartmut Haug: Statistical Physics - Equilibrium Theory and Kinetics. 2nd Edition. Springer 2006, ISBN 3-540-25629-6 .
• Hans Babovsky: The Boltzmann equation. Vieweg + Teubner Verlag, 1998, ISBN 3-519-02380-6 .

Individual evidence

1. ↑ Harald JW Müller-Kirsten: Basics of Statistical Physics. 2nd ed. (World Scientific, 2013), Chapter 13: The Boltzmann Transport Equation. ISBN 978-981-4449-53-3
2. ↑ George Schmidt: Physics of High Temperature Plasmas . ISBN 0-323-16176-6 , pp. 59 and following .
3. ↑ George Schmidt: Physics of High Temperature Plasmas . ISBN 0-323-16176-6 , pp. 63, equation 3-41 .