We study the critical relaxation of the two-dimensional Ising model from a fully ordered configuration by series expansion in time
t and by Monte Carlo simulation. Both the magnetization
(m) and energy series are obtained up to 12th order. An accurate estimate from series analysis for the dynamical critical exponent
z is difficult but compatible with 2.2. We also use Monte Carlo simulation to determine an effective exponent, ${z}_{\mathrm{eff}}(t)=\ensuremath{-}\frac{1}{8}d\mathrm{ln}t/d\mathrm{ln}m$, directly from a ratio of three-spin correlation to
m. Extrapolation to $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty}$ leads to an estimate $z=2.169\ifmmode\pm\else\textpm\fi{}0.003$.
We consider power spectra for fluctuating quantities at a critical point, P( omega ) varies as omega - phi , where phi is determined by the dynamic exponent z and static exponents specific to the quantity considered. From the magnetization spectra obtained with simulations, z is found to be 2.13+or-0.03 for the two-dimensional Glauber kinetic Ising model, hence in accord with recent Monte Carlo relaxational studies, but at odds with those obtained by series expansion and damage spreading methods. Excellent dynamic finite-size scaling for system sizes L=10 to 128 supports our estimate.
This thesis work deals with Monte Carlo (MC) simulations with the focus on quantum mechanical systems. I start with an introduction about Monte-Carlo methods in general and their application to the classical Ising model. The second, most important part deals with the problem in quantum mechanics, and presents a possible simulation method, the Stochastic Series Expansion (SSE). I give an introduction about the idea behind that method and give some results obtained for simulations for finite spin chains. All programming was performed in FORTRAN.