For example, how does graphite transform into diamond as pressure increases at a certain (adequate) temperature? I'm not sure if this is a great example to look at the bigger picture, so another example would be pure iron. At atmospheric pressure, as liquid iron cools down it solidifies to $\delta\text{-}\ce{Fe}$, with a bcc crystal structure. Further cooling down brings it to $\gamma\text{-}\ce{Fe}$, which has a fcc. Then, it keeps cooling down and it solidifies to $\alpha\text{-}\ce{Fe}$, again with a bcc structure. I don't really see how do the atoms physically rearrange to form the different crystal structures, neither do I see why would they jump back and forth between these. To add to this, I believe the difference between $\alpha$ and $\delta$ are their paramagnetic properties.

I guess the general question is, how do the atoms in the solid phase rearrange themselves by changes in pressure or temperature? (in contrast to, for example, a liquid-gas phase transition). And of course, why? What drives this change?

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    $\begingroup$ Magnetic properties have nothing to do with all this and are not the cause of difference between $\alpha$ and $\delta$. Indeed, $\alpha$ iron may or may not be ferromagnetic (dependent on temperature), while its crystal structure remains the same. $\endgroup$ – Ivan Neretin Mar 15 '18 at 8:26
  • $\begingroup$ So they're both bcc, yet different? $\endgroup$ – ralk912 Mar 15 '18 at 8:30
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    $\begingroup$ In a way, they can be thought of as not different at all. Imagine one phase which is stable until certain temperature, then ceases to be stable, then becomes stable again, then melts. $\endgroup$ – Ivan Neretin Mar 15 '18 at 8:57
  • $\begingroup$ Then why call them differently? And so, why would this happen? $\endgroup$ – ralk912 Mar 15 '18 at 9:00
  • $\begingroup$ Because one exists in one range of temperatures, and another exist in a different range, and never the twain shall meet. As to why, that's a very broad question, about as broad as chemistry itself. Solid phase structures are notoriously hard to predict. $\endgroup$ – Ivan Neretin Mar 15 '18 at 9:47

Phase transitions in solids occur in the same manner as in liquids or gases, through nucleation and growth. Yes, those processes require atomic motion. Yet,atoms diffuse in solids all the time, through various mechanisms. While potentially slower, the thermodynamics of solid phase transitions are easily seen in calorimetry measurements for example.

Nucleation and growth of precipitates is a key part of many engineering alloys. On the other hand, some phase changes (like diamond to graphite) are very hard to make happen at near-human-tolerable temperatures.

The solid phase also sees the martensitic phase transitions, which are second-order diffusionless atomic rearrangements. The name comes from the martensite phase observed in steels. Shape memory alloys are another example of martensitic phases.

A classic materials science text would be 'Phase Transformations in Metals and Alloys", by D.A. Porter and K.E. Easterling. In my opinion, the second edition is vastly preferable to the more recent third edition where the changes mainly seemed to be reintroducing all the errors that were removed in the second edition (sigh).

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