Boltzmann Inequality

Boltzmann 's inequality is an inequality that relates any distribution function satisfying the Boltzmann equation and the collision integral .

Content

Wording

For any distribution function $f$ ${\ displaystyle f}$ satisfying the Boltzmann equation, the inequality

\int (\ln f)Q(f,f){\frac {{\rm {d}}\mathbf {p} }{m}}\leqslant 0,

{\ displaystyle \ int (\ ln f) Q (f, f) {\ frac {{\ rm {d}} \ mathbf {p}} {m}} \ leqslant 0,}

Where $Q(f,f)$ ${\ displaystyle Q (f, f)}$ - collision integral, $\mathbf {p}$ ${\ displaystyle \ mathbf {p}}$ - impulse $m$ ${\ displaystyle m}$ - mass of particles. The equal sign is achieved if and only if $f(\mathbf {p} )=\exp \left({a+\left(\mathbf {b} ,{\frac {\mathbf {p} }{m}}\right)+c\,{\frac {p^{2}}{m^{2}}}}\right),$ ${\ displaystyle f (\ mathbf {p}) = \ exp \ left ({a + \ left (\ mathbf {b}, {\ frac {\ mathbf {p}} {m}} \ right) + c \, { \ frac {p ^ {2}} {m ^ {2}}}} \ right),}$ which corresponds to the Maxwell distribution (here $a$ ${\ displaystyle a}$ and $c$ ${\ displaystyle c}$ - scalar, and $\mathbf {b}$ ${\ displaystyle \ mathbf {b}}$ - vector constants; inner parentheses denote the scalar product of vectors) ^[1] .

Proof

The proof is in the famous book of ^[2] .

Notes

↑ Karniadakis G. M., Beskok A., Aluru N. Microflows and Nanoflows: Fundamentals and Simulation . - New York: Springer Science & Business Media , 2005 .-- xxi + 818 p. - (Interdisciplinary Applied Mathematics, vol. 29). - ISBN 978-0387-22197-7 . - P. 589.
↑ Churchignani, 1978 , p. 93.

Literature

Cherchinyani K. Theory and applications of the Boltzmann equation. - M .: Mir , 1978.- 495 p.

[1] Karniadakis G. M., Beskok A., Aluru N. Microflows and Nanoflows: Fundamentals and Simulation . - New York: Springer Science & Business Media , 2005 .-- xxi + 818 p. - (Interdisciplinary Applied Mathematics, vol. 29). - ISBN 978-0387-22197-7 . - P. 589.

[_897b18bbb959a144-2] Churchignani, 1978 , p. 93.