This chapter “derives” the scaling laws found in the previous chapter for incompressible flocks, using a simple heuristic argument which gives some physical insight into the mechanism, and its essentially nonequilibrium nature.

I present a purely dynamical derivation of the Mermin–Wagner–Hohenberg theorem, and compare it with the standard equilibrium derivation. This also provides an opportunity to introduce diffusion equations and gradient expansions, both of which play a large role in what follows.

This chapter deals with incompressible flocks in spatial dimensions d > 2, for which the dynamical renormalization group yields exact scaling laws.

I introduce, and describe in detail, the dynamical renormalization group, using the KPZ equation as an example. In addition to spelling out the mechanics of the technique in great detail, I also emphasize its philosophical importance, as the answer to Einstein’s famous question “Why is the universe intelligible?” and its role as a guide to the formulation of hydrodynamic theories.

This chapter treats incompressible flocks in two dimensions, and shows that they map onto both equilibrium two-dimensional smectics, and our old friend the KPZ equation (albeit in one dimension), as well as a peculiar type of constrained magnet. Exact scaling laws are again found, this time by exploiting these mappings.

This chapter applies the dynamical renormalization group introduced in Chapter 4 to the flocking problem, and uses it to show that nonlinear terms in the dynamics are “relevant,” and change the dynamics in precisely the way needed to circumvent the Mermin–Wagner–Hohenberg theorem.

The final chapter treats “Malthusian” flocks; that is, flocks with birth and death. Here, a full dynamical renormalization group calculation must be done; specifically, it can only be done using a d = 4-epsilon expansion.