  We study the dynamic critical exponent from effective and microscopic
theories. We employ a simple TDGL model, or model A in the classification of
Hohenberg and Halperin, as an effective theory and the imaginary time formalism
of the finite-temperature filed theory as a microscopic theory. Taking an O(N)
scalar model as an example and carrying out the 1/N expansion up to the NLO in
the 1PI and 2PI effective actions, we compare the low-energy and low-momentum
behavior of the response function in the effective theory and of the retarded
Green's function in the microscopic theory. At the NLO of the 1PI 1/N expansion
the low-energy and low-momentum behavior of the two-point function is very much
different in the microscopic and effective theories: in the field theory it is
dominated by the propagating mode while in model A it is dominated by the
diffusive mode. Also, in the microscopic theory the dynamic critical exponent,
z, depends on whether the kinematics is relativistic or nonrelativistic. In
contrast, at the NLO of the 2PI 1/N expansion the microscopic and effective
theories are equivalent. They satisfy exactly the same Kadanoff-Baym equation.
Also, whether the kinematics is relativistic or nonrelativistic in the
microscopic theory becomes irrelevant. This implies that the diffusive mode
with z = 2 + O(1/N) is dominant at low energies and momenta even in the
microscopic theory at the NLO of the 2PI 1/N expansion, though we do not
explicitly solve the Kadanoff-Baym equation. We also try to improve the
calculation of the dynamic critical exponent of model A by incorporating the
static 2PI NLO correlations. The obtained critical exponent is slightly smaller
than the previous result and its N dependence is also milder than the previous
one.
