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arXiv:cond-mat/9510077 · Nuclear Physics B

Dynamical relaxation and universal short-time behavior in finite systems. The renormalization-group approach

U. Ritschel, H.W. Diehl

Published Apr 1, 1996• 26 citations
arXivHTMLPDF
DOI
10.1016/0550-3213(96)00012-0

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We study how the finite-sized n-component model A with periodic boundary conditions relaxes near its bulk critical point from an initial nonequilibrium state with short-range correlations. Particular attention is paid to the universal long-time traces that the initial condition leaves. An approach based on renormalization-group improved perturbation theory in 4−ǫ space dimensions and a nonperturbative treatment of the q = 0 mode of the fluctuating orderparameter field is developed. This leads to a renormalized effective stochastic equation for this mode in the background of the other, q = 0 modes; we explicitly derive it to one-loop order, show that it takes the expected finitesize scaling form at the fixed point, and solve it numerically. Our results confirm for general n that the amplitude of the magnetization density m(t) in the linear relaxation-time regime depends on the initial magnetization in the universal fashion originally found in our large-n analysis [J. Stat. Phys. 73 (1993) 1]. The anomalous short-time power-law increase of m(t) also is recovered. For n = 1, our results are in fair agreement with recent Monte Carlo simulations by Li, Ritschel, and Zheng [J. Phys. A 27 (1994) L837] for the three-dimensional Ising model.

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We study how the finite-sized n-component model A with periodic boundary conditions relaxes near its bulk critical point from an initial nonequilibrium state with short-range correlations. Particular attention is paid to the universal long-time traces that the initial condition leaves. An approach based on renormalization-group improved perturbation theory in 4−ǫ space dimensions and a nonperturbative treatment of the q = 0 mode of the fluctuating orderparameter field is developed. This leads to a renormalized effective stochastic equation for this mode in the background of the other, q = 0 modes; we explicitly derive it to one-loop order, show that it takes the expected finitesize scaling form at the fixed point, and solve it numerically. Our results confirm for general n that the amplitude of the magnetization density m(t) in the linear relaxation-time regime depends on the initial magnetization in the universal fashion originally found in our large-n analysis [J. Stat. Phys. 73 (1993) 1]. The anomalous short-time power-law increase of m(t) also is recovered. For n = 1, our results are in fair agreement with recent Monte Carlo simulations by Li, Ritschel, and Zheng [J. Phys. A 27 (1994) L837] for the three-dimensional Ising model.
We present a method to calculate short-time nonequilibrium universal exponents within the functional-renormalization-group scheme. As an example, we consider the classical critical dynamics of the relaxational model A after a quench of the temperature of the system and calculate the initial-slip exponent which characterizes the nonequilibrium universal short-time behavior of both the order parameter and correlation functions. The value of this exponent is found to be consistent with the result of a perturbative dimensional expansion and of Monte Carlo simulations in three spatial dimensions.
The renormalization group approach is applied to the study of the short-time critical behavior of the d-dimensional n-component spin systems with cubic anisotropy. First, the system is quenched from high temperature to the critical temperature and then relaxes to equilibrium within model A dynamics. The asymptotic scaling laws and the initial slip exponents ${\ensuremath{\theta}}^{\ensuremath{'}}$ and $\ensuremath{\theta}$ of the order parameter and the response function, respectively, are calculated to the second order in $\ensuremath{\epsilon}=4\ensuremath{-}d.$ Logarithmic corrections to short-time behaviors of both the autocorrelation and the order parameter are found in d=4d=4d=4 dimension.