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Thermalization of a particle with dissipative collisions
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Thermalization of a particle by dissipative collisions
Philippe A. Martin
Institut de Physique The?orique, Ecole Polytechnique Fe?de?rale de Lausanne, CH-1015,
Lausanne, Switzerland
Jaros law Piasecki
Institute of Theoretical Physics, University of Warsaw, Hoz?a 69, 00 681 Warsaw,
Poland
(February 1, 2008)
One considers the motion of a test particle in an homogeneous
fluid in equilibrium at temperature T , undergoing dissipative colli-
sions with the fluid particles. It is shown that the corresponding lin-
ear Boltzmann equation still posseses a stationary Maxwellian veloc-
ity distribution, with an effective temperature smaller than T . This
effective temperature is explicitly given in terms of the restitution
parameter and the masses.
PACS numbers: 51.10.+y, 44.90.+c
The search for stationary states of granular matter has recently been a subject
of interest for both experimental and theoretical reasons [1,2]. Granular matter can
be modelized by spherical particles that partially dissipate their kinetic energy at
collisions. If (u,v) and (u?, v?) denote the velocities of two spheres of mass m and
M before and after the collision, they are related by
mu? +M v? = mu+Mv (1)
σ? · (v? ? u?) = ?ασ? · (v ? u), 0 α ≤ 1 (2)
σ?
⊥ · (v? ? u?) = σ?⊥ · (v ? u)
where σ? is a unit vector normal to the surface of the spheres at the point of
impact, and σ?⊥ points in the tangent direction: σ?⊥ · σ? = 0. The first relation is
the conservation of the center of mass momentum, whereas the second one says
that the normal component of the relative velocity reverses its direction with a
magnitude reduced by the factor α, the so called restitution parameter (the tangent
component remains unchanged). Solving (1), (2) for u? and v? gives
u? = u+ (1? μ)(1 + α)(σ? · (v ? u))σ?, v? = v ? μ(1 + α)(σ? · (v ? u))σ? (3)
with μ = m/(m+M). The inverse relation is obtained by exchanging the
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