Rate of reaction- Arrhenius equation
Clear distinction has to be made between Arrhenius-equation and transition state theory. The formulas for TST can be derived, as opposed to Arrhenius equation, which by itself, is not based on a solid physical derivation. Basically any process can be modeled by an Arrhenius equation that has the formula:

$$ r \propto \exp{(-E_a/RT)} $$

Any molecular process can be reasonably approximated by this formula if it is a statistical transition between states with different energy. The reason for this is that the statistical probability of a state if proportional to the Boltzmann factor $g \exp{(-E/RT)} $, which is readily Arrhenius-like.

So what are some examples for Arrhenius-type behaviors? For example, if you look up the term Arrhenius-type equation on ScienceDirect you can see that research papers have been published on solid strain rates, membrane permeabilities, solubility rates and corrosion. 

So, as you can see, an Arrhenius-type equation can describe a huge number of physical/chemical processes, so it is quite reasonable to use it for solid state reactions. The question is only how to find the energy of activation and the pre-exponent factor. For this, there is no general recipe. The best thing to do is to use them as empirical model parameters to be measured by experiments.