Boundary conditions for the field theory of dynamic critical behavior in semi-infinite systems with conserved order parameter.

Abstract

The problem of constructing a field theory that describes the dynamic critical behavior of semi-infinite systems whose dynamic bulk critical behavior is represented by model B and whose order parameter is conserved both in the bulk and at the surface is reconsidered. Particular attention is paid to the derivation of the boundary conditions satisfied by the order-parameter field (\ensuremath{\varphi}(x,t) and the associated response field \ensuremath{\varphi}\ifmmode \tilde{}\else \~{}\fi{}(x,t). It is shown that the extremely complicated boundary conditions for \ensuremath{\varphi} obtained recently by Binder and Frisch [Z. Phys. B 84, 403 (1991)] simplify considerably if all irrelevant surface contributions to the action are discarded. In particular, the boundary conditions for both \ensuremath{\varphi} and \ensuremath{\varphi}\ifmmode \tilde{}\else \~{}\fi{} do not involve time derivatives. Although power counting alone admits surface terms other than those anticipated in the paper by Dietrich and Diehl [Z. Phys. B 51, 343 (1983)], the requirements of detailed balance imply that these extra terms are either absent or redundant. As an application and test, the relaxation of the order-parameter profile from a spatially homogeneous initial nonequilibrium state into thermal equilibrium is investigated, using a zero-loop approximation. The results are in conformity with those of Binder and Frisch.