A New Approach to Dynamic Finite-Size Scaling

Abstract

In this work we have considered the Taylor series expansion of the dynamic scaling relation of the magnetization with respect to small initial magnetization values in order to study the dynamic scaling behavior of two- and three-dimensional Ising models. We have used the literature values of the critical exponents and of the new dynamic exponent x0 to observe the dynamic finite-size scaling behavior of the time evolution of the magnetization during early stages of the Monte Carlo simulation. For the three-dimensional Ising model we have also presented that this method opens the possibility of calculating z and x0 separately. Our results show good agreement with the literature values. Measurements done on lattices with different sizes seem to give very good scaling.

A study of dynamic finite size scaling behavior of the scaling functions—calculation of dynamic critical index of Wolff algorithm

“…Jansen, Schaub and Schmittmann [2] showed that for a dynamic relaxation process, in which a system is evolving according to a dynamics of Model A [3] and is quenched from a very high temperature to the critical temperature, a universal dynamic scaling behavior within the short-time regime exists [4,5,6]. The existence of finite size scaling even in the early stages of the Monte Carlo simulation has been tested for various spin systems [5,6,7,8,9,10,11,12], the dynamic critical behavior is well-studied and it has been shown that the dynamic finite size scaling relation holds for the magnetization and for the moments of the magnetization. For the k th moment of the magnetization of a spin system, dynamic finite size scaling relation can be written as [2] M (k) (t, ǫ, m 0 , L) = L (−kβ/ν) M (k) (t/τ, ǫL 1/ν , m 0 L x 0 ) (1) where L is the spatial size of the system, β and ν are the well-known critical exponents, t is the simulation time, ǫ = (T − T c )/T c is the reduced temperature and x 0 is an independent exponent which is the anomalous dimension of the initial magnetization (m 0 ).…”