The typical way in which differential equations based physics is introduced is through particle mechanics using Newton’s Second Law. In this type of physics the interactions between the particle and its environment and/or other particle either don’t happen at all (e.g. no friction), are modelled (e.g. linear drag), or taken to happen so quickly balances can be made between before collision and after collision (e.g. elastic and inelastic collisions). Solid state physics is about as far away from this world as possible. Its basic assumption is that particles interact strongly all the time. The prototype system one can think of is a crystal lattice of identical molecules. Systems like this exhibit a great deal of symmetry, and their inherent economic value has made the study of crystal symmetry an extremely well developed science.
Of course there are other solid systems which are just as interesting, but a bit more difficult to visualize on the microscopic level (like the jello pictured above).
Solid state physics has its own set of formalisms and at the introductory level these don’t have much overlap with continuum scale theories. Continuum scale theories begin by observing that in any finite chunk of material there is an astronomical number of particles (Avogadro’s number) and thus a macroscopic description that uses the continuum assumption (that we can treat quantities of interest like the temperature, T, as piecewise differentiable functions of the space variables) can be expected to give reasonably accurate results.
Solid State Physics
