Dynamical relaxation and universal short-time behavior of finite systems

Abstract

A system belonging to the dynamic universality class of model A is considered in a block (V=L ^d) geometry with periodic boundary conditions. The relaxation of the order parameterm(t) from an initial value m⁽ⁱ⁾ is investigated at the bulk critical temperature. We demonstrate that a proper scaling description of the problem involves two characteristic times,t _L∼L^z andt _L∼[m⁽ⁱ⁾]^−z/x, wherez is the familiar dynamic bulk exponent, while x_i is an independent new bulk exponent discovered recently. Previous analyses of the problem either were restricted tot ≫t _i, or tacitly used the incorrect assumption thatx _i=β/vν. Thus the short-time regimet≪t_i with universal dependence on m⁽ⁱ⁾ was missed. As a concrete example we study the exact solution in the large-n limit.