In a lattice, what, technically, are random (thermal) atomic motions? I'm familiar with normal modes and their periodic nature. And I know that the term 'independent mode' is used to describe how each individual atom in a lattice can move (degrees of freedom).
But, what confuses me is the term 'random atomic motion'. Wikipedia discusses this in their Thermodynamics section (https://en.wikipedia.org/wiki/Phonon). Are random (thermal) atomic motions in a lattice (which move about their mean position) the same thing as 'independent modes'? If not, what are random/thermal atomic motions? 
I ask because somewhere I've heard the random/thermal motions of atoms described as 'wiggling' and that would seem to define something different than the bending and stretching of independent modes.
I would more than appreciate clarification on this.
 A: There are no random motions in solids. All motions are highly correlated, you are just adding up a lot of modes at different frequencies, which looks like random motion if you are only looking at a single atom. 
It's not totally wrong to look at single particles being in random motion, though, since the Fourier transform of a lot of frequencies at arbitrary phases is a pseudorandom signal and heat in a thermodynamic sense is also based on the notion of statistical independence (correlation is the enemy of simplifying assumptions about averages). 
From an experimental point of view phonons are only visible at low temperatures, where the number of excited modes is not too large, at room temperature the "random" approximation should be more than satisfied in most cases, with the caveat that things are getting complicated even for high temperatures when we get close to phase transitions like the Curie temperature, where ferromagnets loose their magnetization. In that case we can see fluctuations of the size of the entire solid near the phase transition temperature, i.e. the assumption that individual atoms are not correlated breaks down, again, because hard to analyze macroscopic collective phenomena are at work.  
