Critical dynamics: A consequence of a random, stepwise growth of clusters?

Critical dynamics of correlated particles (here Glauber dynamics of singly flipping Ising spins) is explained by a random, stepwise growth and contraction of clusters, as follows. At equilibrium, a cluster of size s is described by its length l (a random walk-like path, connecting a sequence of neighbor spins). The length scales as l∼sρ, where ρ constitutes a new static critical exponent. We assume that, on the average, the random growth of a cluster from zero, to size s and length ls, requires a sequence of l2 spin flips. This gives for dynamic critical exponent, z=[2ρ(γ+β)−β]ν, where γ, β and ν are the usual static exponents. Exact results at dimension D=1 and 4, and simulation results at D=2 and 3, support the theory.