Description of critical dynamics by static geometry of clusters

Abstract

Critical slowing-down of relaxation (exponent z ) is modelled by clusters, which grow and recede at the end of connected sequences of particles. Consequently the relaxation time of a cluster of size s is proportional to the number of back and forth steps needed to travel its typical connected length l s . This and an assumption on mutual screening of ends, lead to z =[(2.5 ϱ − 0.25)( γ + β )− β ] ν , where ϱ is a static geometric exponent in l s ∼ s ϱ . Ising exponents ϱ measured (elsewhere), with the help of simulation, give z ϱ,β,ν . In addition, a Flory-like approximation is derived here; it relates ϱ to β and ν analytically, giving z βν . In D = 2, 3 and 4 − ɛ , respectively, we findz β,ν = 2.14, 2.05 and 2 + ɛ 2 75.1, while z ϱ,β,ν = 2.17(4) and 2.04(4); both sets of values are in fair agreement with the literature.