Center-manifold renormalization in dynamic critical phenomena for dissipative spin systems

We consider a purely dissipative spin system operating far from thermodynamic equilibrium. We prove that a coarse graining for short-wavelength modes is obtained near criticality by performing a center-manifold reduction of fast-relaxing modes. The spin systems considered are realizations of a time-dependent Ginzburg-Landau model. Agreement with renormalization-group derivations is obtained. Neutron scattering cross sections for a magnetic material in the dynamic critical case are calculated making use of the center-manifold reduction. Experimental measurements of static spin autocorrelations are used to obtain the distribution of fast modes about the center manifold. Making use of this information, predictions are made on neutron scattering for magnetic materials coupled to a heat bath near a dynamic critical point. The existence of a center manifold determines scaling relations between the strength of the statistical forcing field and the spin-correlation length. Thus, the relevance of a time-dependent experiment under far-from-equilibrium conditions becomes apparent inasmuch as such an experiment can lead to the elucidation of the nature and strength of the statistical noise.