Finite size effects in critical dynamics and the renormalization group

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Systems representable as a time-dependent Ginzburg-Landau model with nonconserved order parameter are considered in a block (V=L d) geometry with periodic boundary conditions, both for space dimensionalitiesd≧4 andd=4−e. A systematic approach for studying finite size effects on dynamic critical behavior is developed. The method consists in constructing an effective reduced dynamics for the lowest-energy (q=0) mode by integrating out the remaining degrees of freedom, and generalizes recent analytic approaches for studying static finite size effects to dynamics. Above four dimensions, the coupling to the other (q≠0) modes is irrelevant and the probability densityP(Φ,t) for the normalized order parameterΦ=∫dd xϕ(x,t)/V satisfies a Fokker-Planck equation. The dynamics is equivalently described by the Langevin equation for a particle moving in a |Φ|4 potential or by a supersymmetric quantum mechanical Hamiltonian. Dynamic finite size scaling is found to be broken, e.g. the order parameter relaxation rate varies at the bulk critical temperatureT c,∞ as ωυ(T c,∞ L)∼L −d/2 asL→∞. By contrast, ford<4, the coupling to the other (q≠0) modes cannot be ignored and dynamic finite size scaling is valid. The asymptotic behavior of correlation and response functions can be studied within the framework of an expansion in powers of ɛ1/2. The scaling function associated with ωυ is computed to one-loop order. Finally, the many component (n→∞) limit is briefly considered.