Page 26 -

References
19
Fig. 1.7. Bust of Ludwig Boltzmann

to collisions leading to coalescence of the colliding drops. The time evolution of
the number density f of water drops in a cloud, as function of the drop mass
m> 0 is known to be described by the so-called stochastic coalescence equation,
whichhastheformofaspace-homogeneous(kinetic)Boltzmann-typeequation,
where the drop mass m plays the role of the independent variable. The equation
reads:
m
1
∂ t f (m, t) = K(m − m , m )f (m − m , t)f (m , t) dm

2
0
∞

− K(m, m )f (m , t)f (m, t) dm .
0
The quadratic operator on the right hand side models coalescing collisions of
drops.ThefunctionK(m, m )denotesthenon-negativecross-section.Fordetails

on the physics we refer to [18], mathematical results can be found in [19].
Acknowledgement The author is indebted to Benedikt Bica from the Institute
for Meteorology and Geophysics of the University of Vienna for providing the
cloud classification of the Images 1.2 to 1.6.
CommentonImage1.7 The bust of Ludwig Boltzmannat his grave at the Central
Cemetery of Vienna, Austria. The entropy formula is engraved. We acknowledge
9
the courtesy of Andrea Baczynski , who took this photograph.

References

[1] C. Bardos, F. Golse and D. Levermore, Fluid Dynamic Limits of Kinetic
Theory I: Formal Asymptotics Leading to Incompressible Hydrodynamics;J.
Stat. Phys. 63, pp. 323–344, 1991
[2] E.X. Berry, A Mathematical Framework for Cloud Models,Journal of the
Atmospheric Sciences, Vol. 26, No. 1, pp. 109–111, 1969
[3] L. Boltzmann, Weitere Studien über das Wärmegleichgewicht unter Gas-
molekülen, Sitzungsberichte der Akademie der Wissenschaften, Wien, Ber.
66, pp. 275–370, 1872
[4] C. Cercignani, The Boltzmann Equation and its Applications,Springer Ver-
lag, 1988
9
www.pbase.com/lacandonna