We prove exponential concentration estimates and a strong law of large numbers for a particle system that is the simplest representative of a general class of models for 2D grain boundary coarsening introduced by the first author (2014). The system consists of
n
n
particles in
(
0
,
∞
)
(0,\infty )
that move at unit speed to the left. Each time a particle hits the boundary point
0
0
, it is removed from the system along with a second particle chosen uniformly from the particles in
(
0
,
∞
)
(0,\infty )
. Under the assumption that the initial empirical measure of the particle system converges weakly to a measure with density
f
0
(
x
)
∈
L
+
1
(
0
,
∞
)
f_0(x) \in L^1_+(0,\infty )
, the empirical measure of the particle system at time
t
t
is shown to converge to the measure with density
f
(
x
,
t
)
f(x,t)
, where
f
f
is the unique solution to the kinetic equation with nonlinear boundary coupling
∂
t
f
(
x
,
t
)
−
∂
x
f
(
x
,
t
)
=
−
f
(
0
,
t
)
∫
0
∞
f
(
y
,
t
)
d
y
f
(
x
,
t
)
,
0
>
x
>
∞
,
\begin{equation*} \partial _t f (x,t) - \partial _x f(x,t) = -\frac {f(0,t)}{\int _0^\infty f(y,t)\, dy} f(x,t), \quad 0>x > \infty , \end{equation*}
and initial condition
f
(
x
,
0
)
=
f
0
(
x
)
f(x,0)=f_0(x)
.
The proof relies on a concentration inequality for an urn model studied by Pittel, and Maurey’s concentration inequality for Lipschitz functions on the permutation group.