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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

New results from short-time dynamics

H. J. Luo, Lothar Schülke, Bo Zheng

Published Jan 25, 2000• 0 citations

Abstract

In 1989 Janssen, Schaub and Schmittmann have shown that universality and scaling hold already in the early stage of the dynamical evolution of statistical systems, if the system, initially at a very high temperature, is suddenly quenched to the critical temperature and then released to the dynamic evolution according to model A. This allows for a measurement of all the static and dynamic critical exponents and even for the critical point already in the short-time regime, i.e., far from equilibrium. Since the correlation length is still small here, the simulations do not suffer from critical slowing down, a problem encountered in the usual measurements in equilibrium. The concept has been successfully applied to a variety of statistical systems. We will report about recent results for the fully frustrated XY model, where the short-time approach is particularly efficient, since the standard cluster algorithm does not apply because of the frustration.

DOI: 10.1117/12.375410

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In 1989 Janssen, Schaub and Schmittmann have shown that universality and scaling hold already in the early stage of the dynamical evolution of statistical systems, if the system, initially at a very high temperature, is suddenly quenched to the critical temperature and then released to the dynamic evolution according to model A. This allows for a measurement of all the static and dynamic critical exponents and even for the critical point already in the short-time regime, i.e., far from equilibrium. Since the correlation length is still small here, the simulations do not suffer from critical slowing down, a problem encountered in the usual measurements in equilibrium. The concept has been successfully applied to a variety of statistical systems. We will report about recent results for the fully frustrated XY model, where the short-time approach is particularly efficient, since the standard cluster algorithm does not apply because of the frustration.

An introductory review to short-time critical dynamics is given. From the scaling relation valid already in the early stage of the evolution of a system at or near the critical point, one derives power law behaviour for various quantities. By a numerical simulation of the system one can measure the critical exponents and, by searching for the best power law behaviour, one can determine the critical point. Critical slowing down as well as finite size corrections are nearly absent, since the correlation length is still small for times far before equilibrium is reached. By measuring the (pseudo) critical points it is also possible to distinguish (weak) first-order from second-order phase transitions.

Monte Carlo simulations of the short-time critical dynamics are reviewed. The short-time universal scaling behavior of the dynamic Ising model and Potts model are discussed in detail, while extension and application to more complex systems as the XY model, the fully frustrated XY model and other dynamic systems are also presented. The investigation of the universal behavior of the short-time dynamics not only enlarges the fundamental knowledge on critical phenomena but also, more interestingly, provides possible new ways to determine not only the new critical exponents θ and θ1, but also the traditional dynamic critical exponent z as well as all static critical exponents.

With Monte Carlo methods we investigate the dynamic relaxation of the fully frustrated

\mathrm{XY}

model in two dimensions below and at the Kosterlitz-Thouless phase transition temperature. Special attention is given to the sublattice structure of the dynamic evolution. The short-time scaling behavior is found and universality confirmed. The critical exponent $\ensuremath{\theta}$ is measured for different temperatures and with different algorithms.

Dynamic relaxation of the XY-model quenched from a high temperature state to the critical temperature or below is investigated with Monte Carlo methods. When a non-zero initial magnetization is given, in the short-time regime of the dynamic evolution the critical initial increase of the magnetization is observed. The dynamic exponent is directly determined. The results show that the exponent varies with respect to the temperature. Furthermore, it is demonstrated that this initial increase of the magnetization is universal, i.e. independent of the microscopic details of the initial configurations and the algorithms.