Boltzmann equation, linearized

in kinetic gas theory

A linear integro-differential equation which approximately describes the evolution of the one-particle distribution function of a sufficiently-rarefied gas without internal degrees of freedom for small deviations from equilibrium.

This equation,

$$ \frac{\partial \phi }{\partial t } + \langle c, \nabla _ {x} \phi \rangle = \ \frac{1} \epsilon L _ {0} ( \phi ), $$

is obtained from the Boltzmann equation

$$ \frac{\partial f }{\partial t } + \langle c, \nabla _ {x} f\rangle = \frac{1} \epsilon L (f, f) $$

by substituting

$$ f = \pi ^ {- 3/2 } e ^ { -c ^ {2} } + \mu e ^ {-c ^ {2} / 2 } \phi $$

and equating the terms in which the parameter $ \mu $ appears in the first degree. The operator $ L _ {0} $ is said to be the linearized collision operator. The linearized Boltzmann equation gives a satisfactory description of the evolution of the distribution function only if

$$ \sup _ { x,c,t } \ | f(x, c, t) - \pi ^ {- 3/2 } e ^ {-c ^ {2} } | \ll 1. $$

If certain very general assumptions are made, the operator $ L _ {0} $ is non-positive, $ \langle \phi , L _ {0} \phi \rangle \leq 0 $, and can be written as

$$ L _ {0} ( \phi ) = \ - \nu (c ) \phi + G \phi , $$

where $ \nu (c) $( which is sometimes called the collision frequency) is a multiplication operator acting on $ \phi $ and $ G $ is a completely-continuous integral operator. The function $ \nu (c) $ and the kernel of $ G $ have the following form for the hard-sphere model [1]:

$$ \nu (c) = \ 4 \pi ^ {2} \left [ { \frac{1}{2} } e ^ {- c ^ {2} /2 } + \left ( | c | + { \frac{1}{2 | c | } } \right ) \int\limits _ { 0 } ^ { {|c| } } e ^ {- \alpha ^ {2} } \ d \alpha \right ] , $$

$$ G (c, c ^ \prime ) = \frac{4 \pi }{| c - c ^ \prime | } \mathop{\rm exp} \left \{ \frac{- | c - c ^ \prime | ^ {2} }{4 } - \frac{| c | ^ {2} - | c ^ \prime | ^ {2} }{4 | c - c ^ \prime | ^ {2} } \right \} - $$

$$ - 2 \pi | c - c ^ \prime | \mathop{\rm exp} \left \{ - \frac{c ^ {2} + c ^ \prime2 }{2 } \right \} $$

The existence theorem for the solution of the Cauchy problem as $ t \rightarrow \infty $ and $ \epsilon \rightarrow 0 $ was proved for the linearized Boltzmann equation, and the dispersion equation was studied. The equation is principally employed in the molecular acoustics of ideal gases. The equation yields correct values for the transfer coefficients (viscosity, thermal conduction, velocity of sound) and the Stokes–Kirchhoff law of ultra-sound absorption.

References

[1]	T. Carleman, "Problèmes mathématiques dans la théorie kinétique des gaz" , Mittag-Leffler Inst. (1957)
[2]	A.A. Arsen'ev, "The Cauchy problem for the linearized Boltzmann equation" USSR Comput. Math. Math. Phys. , 5 : 5 (1965) pp. 116–136 Zh. Vychisl. Mat. i Mat. Fiz. , 5 : 5 (1965) pp. 864–882

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)