1.4.4.3 Some computations on particle distribution
At equilibrium, that is, under the invariant law, the 
are uniform on 
and hence, the 
for 
are i.i.d. uniform on 
. The strong law of large numbers and the central limit theorem then yield that
Hence, as 
goes to infinity, the instantaneous proportion of molecules in each compartment converges to 
with fluctuations of order 
. For instance,
and as a numerical illustration, as , the choice and yields that and hence
For an arbitrary initial law, ...
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