Statistical Mechanics and Elasticity

What is Elasticity theory? Given the strain on a solid, i.e., for a given macroscopic deformation, it tells you what the stresses in the solid are. There, as always, are two levels at which one studies the theory of elasticity. Suppose the constitutive relationship between the stress and strain is known, for example Hooke's law is a constitutive relation that is linear. Then, the problem is one of solving a set of differential equations to obtain the stress. When the theory is linear, as in the above case, this is simple exercise. The other level of study is associated with deriving this constitutive relation. This is the issue addressed in the rest of this rambling.

First, let us consider the context of an ordered crystalline solid. Further, let us assume that the solid is a defect free single crystal. The energy required to deform such a solid is set by the bonding energy that is primarily electrostatic. So, the thermal energy is much smaller than the typical energy scale in the problem. Therefore, a very good model for the solid would be a lattice of balls interconnected by stiff springs. In this case, the elasticity is linear up to very large applied stresses and the problem of getting a constitutive relation reduces to a Newtonian N particle problem that can be readily solved by going to generalized coordinates that are the normal modes of the system. Next, we ask, if we superpose thermal fluctuations on this answer, how is it changed? The result? It is not changed at all. This result is surprising at first sight, but is a rather obvious artifact of the harmonic nature of the interaction. One can readily verify this by writing down the partition function. The conclusion then is that for a perfect crystalline solid, statistical mechanics is irrelevant for understanding the elastic response of the system! You can hardly ever say this for a finite temperature N particle problem!

So, when does stat mech play a role in understanding elasticity of a crystalline solid? When there are defects in the solid. What happens is that as you increase the applied force on the solid, well before you probe the limits of the harmonic approximation made on the interaction, the defect gives. So, the response of this system is governed by what the defects do, rather than what the background perfect crystal is doing. The dynamics of defects in a solid is an elegantly formulated problem. A well developed theory exists a la E & M for understanding this. I don't know much about this so I will stop by saying that the problem of defect dynamics and defect interactions is a nicely formulated one that one can look up. The point of interest here is that the defects will now have a statistics associated with the temperature of the system. Therefore, in order to understand the elasticity of this system, one needs to take into account the fluctuations of the defects in the system. I don't know much about this either. But I mean to look it up and will tell you when I do.

So much for crystalline solids. There is of course a whole class of amorphous solids whose entire elasticity is statistical in origin, rubber being the standing prototype of this class. They are the reason I started thinking about elasticity and stat mech in the first place. But, I guess I must take another idle morning to sort this stuff out for myself.