Violent relaxation

The term violent relaxation ( engl. For violent relaxation ) describes a mechanism for the relaxation of a gas , d. H. to redistribute the kinetic energy so that a thermal equilibrium can be established. This process is particularly important in astrophysics , where collision-free gases (such as stars in galaxies , galaxies in galaxy clusters ) would otherwise not be able to create a thermal equilibrium. The relaxation takes place within the time scale of the free fall , whereby the redistribution of the kinetic energy is "forced" by temporal changes in the gravitational potential .

Mathematical description

The change in energy (kinetic plus potential) of a test mass over time is ( chain rule ): ${\displaystyle E(t)={\frac {m}{2}}v(t)^{2}+\Phi ({\vec {r}}(t),t)}$

${\displaystyle {\frac {\mathrm {d} E}{\mathrm {d} t}}={\frac {1}{2}}\cdot m\cdot {\frac {\mathrm {d} v^{2}}{\mathrm {d} t}}+{\frac {\partial \Phi }{\partial t}}+{\vec {v}}\cdot {\vec {\nabla }}\Phi }$

Here are:

• ${\displaystyle m}$ the test mass,
• ${\displaystyle {\vec {v}}}$ the speed and
• ${\displaystyle \Phi ({\vec {r}},t)}$ the gravitational potential.

The following applies now:

${\displaystyle {\frac {1}{2}}\cdot m\cdot {\frac {\mathrm {d} v^{2}}{\mathrm {d} t}}={\vec {v}}\cdot \left(m\cdot {\frac {\mathrm {d} {\vec {v}}}{\mathrm {d} t}}\right)={\vec {v}}\cdot {\vec {F}}}$

where the force is. If one uses now , two of the terms cancel each other out and one obtains for the time change of the energy of a test mass: ${\displaystyle {\vec {F}}}$ ${\displaystyle {\vec {F}}=-{\vec {\nabla }}\Phi }$

${\displaystyle {\frac {\mathrm {d} E}{\mathrm {d} t}}={\frac {\partial \Phi }{\partial t}}}$.

literature

• Peter Schneider: Introduction to extragalactic astronomy and cosmology . 1st edition. Springer, Berlin Heidelberg 2006, ISBN 3-540-25832-9 , pp. 238, 290–291 ( limited preview in Google Book Search - corrected reprint 2008).
• Donald Lynden-Bell: Statistical mechanics of violent relaxation in stellar systems . In: Monthly Notices of the Royal Astronomical Society . tape 136 , 1967, pp. 101-121 , bibcode : 1967MNRAS.136..101L .