If we consider an atom as a system then what is its temperature? An atom can be considered a SYSTEM of particles (electrons and nucleons) that are structured in a particular way and moving with some velocity(electrons). So can it be assigned a temperature? What kind of mathematical treatment can be applied?
EDIT: just for fun can someone please apply statistical mechanics to a "model" atom  and tell us the results.
 A: Temperature is defined as an average energy per atom/molecule/particle. Thus, it is a statistical property of systems with many atoms/molecules/particles. Defining a temperature for a single atom is thus statistically unfounded. Such attempts are indeed regularly made and result in sensational article headlines and "paradoxes"... because the temperature does not behave as it should :)
A: If we consider the kinetic theory of gases, then atoms/molecules are effectively assumed to be small hard spheres with no description of what is inside them. They are very clearly particles and there is no question of an associated wave motion (classically) to each atom. In order to conduct experiments or describe real world effects of temperatures, this is an acceptable approximation of an atom (since an atom is not actually a hard sphere). 
However, for subatomic particles, their quantum mechanical properties begin to play a role that cannot as easily be ignored. Thus, as is mentioned in a comment, the understanding behind calling elementary particles as particles is different from calling atoms or molecules as particles. 
I am not aware of analyses that continue to do this at the subatomic level anyway, however this could be a possible starting point to sort of observe the unusual paradoxes that come up with newer definitions of temperature. 
https://www.sciencedaily.com/releases/2013/01/130104143516.htm
A: As other answers already pointed out, temperature is meaningless for a single atom. This can be defined mathematicaly as follows.
Temperature is defined as proportional to the average CoM kinetic energy of the system (as defined by König's theorem). But what matters is the standard deviation around this average value: if it's too large, the average value doesn't represent the system faithfully, as particles have a high probability to have a kinetic energy very different from the average.
Typically, standard deviation for the kinetic energy for a system of $N$ atoms is proportional to $1/\sqrt{N}$. Therefore:

*

*For a single particle, this standard deviation is very large, making the average value meaningless: you can compute it, but it doesn't represent anything relevent.

*For a large collection of particles, it goes to zero, so almost all particles have a kinetic energy very close to the average value.

Only in the second case can you make anything physically useful with the average kinetic energy, for example define:
$$\langle K\rangle=\frac{3}{2}\,kT$$
which is a way to define absolute temperature.
