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This problem was supposed to be an estimation task, but many contestants overlooked that and made more complex calculations than needed.
Some of the students considered drag force due to the fact that while in the frame of the sphere,
the black body radiation emitted by the sphere is isotropic, Doppler shift makes it anisotropic in the lab frame.
Similar, Doppler shift happens with the background radiation: it is isotropic in the lab frame, but anisotropic in the
sphere's frame. Because of that, there is a drag force acting on the sphere, proportional to its speed.
Hence, a fast-moving sphere will slow down exponentially in time. However, if we consider a
short time period - shorter than the correlation time τ (we can define this as the time during which the sphere's velocity turns by 90 degrees) then the average force due to random impacts from randomly emitted/absorbed photons
becomes larger than the drag force. If it were the other way around, the particle would not be performing a Brownian motion, but its speed would be decaying in a regular exponential manner (the drag force can become of the order of force fluctuations, though, if we average the forces over a time period which is of the order of τ or longer).
To sum up, it is not wrong to consider the Doppler-shift-caused drag force here, but it is not an essential component of the solution of this problem.
The best solution award is distributed between the five correct solutions as follows. One third of it goes to the shortes and simplest solution: Dolteanu Stefan. One third of the award goes to a mathematically rigorous and fancy solution Cai Zixing. Don't get discouraged if you don't understand the math here. Note that this solution gives only the correlation time τ, but the mean free path is from here only one short step away. The final third of is distributed between: Navneel Singhal and Tóbiás Marozsák The fifth solution has gaps (considers essentially only the drag force and does not consider the diffusion process in the velocity space), and hence is not provided here. And here are the results. Number of fully correct solutions: 5.
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