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arXiv:cond-mat/0302571 · International Journal of Modern Physics C

A NEW APPROACH TO DYNAMIC FINITE-SIZE SCALING

MEHMET DİLAVER, SEMRA GÜNDÜÇ, MERAL AYDIN, YİĞİT GÜNDÜÇ

Published Sep 1, 2003• 1 citation
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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.

DOI
10.1142/s012918310300508x

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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 behaviour of 2- and 3-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 behaviour of the time evolution of the magnetization during early stages of the Monte Carlo simulation. For 3-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.
In this work we have calculated the dynamic critical exponent zzz for 2-, 3- and 4-dimensional Ising models using the Wolff's algorithm through dynamic finite size scaling. We have studied time evolution of the average cluster size, the magnetization and higher moments of the magnetization. It is observed that dynamic scaling is independent of the algorithm. In this sense, universality is established for a wide range of algorithms with their own dynamic critical exponents. For scaling, we have used the literature values of critical exponents to observe the dynamic finite size scaling and to obtain the value of zzz. From the simulation data a very good scaling is observed leading to vanishingly small zzz values for all three dimensions.
We simulate the critical relaxation process of the two-dimensional Ising model with the initial state both completely disordered or completely ordered. Results of a method to measure both the dynamic and static critical exponents are reported, based on the finite-size scaling for the dynamics at the early time. From the time-dependent Binder cumulant, the dynamical exponent z is extracted independently, while the static exponents \ensuremath{\beta}/\ensuremath{\nu} and \ensuremath{\nu} are obtained from the time evolution of the magnetization and its higher moments. \textcopyright{} 1996 The American Physical Society.
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We report the results of a Monte Carlo investigation of the (equilibrium) time-displaced correlation functions for the magnetization and energy of a simple cubic Ising model as a function of time, temperature, and lattice size. The simulations were carried out on a CDC CYBER 205 supercomputer employing a high-speed, vectorized multispin coding program and using a total of 5\ifmmode\times\else\texttimes\fi{}1012{10}^{12}1012 Monte Carlo spin-flip trials. We used L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L lattices with periodic boundary conditions and L as large as 96. The short-time and long-time behaviors of the correlation functions are analyzed by fits to a sum of exponential decays, and the critical exponent z for the largest relaxation time is extracted using a finite-size-scaling analysis. Our estimate z=2.04\ifmmode\pm\else\textpm\fi{}0.03 resolves an intriguing contradiction in the literature; it satisfies the theoretical lower bound and is in agreement with the prediction obtained by \ensuremath{\epsilon} expansion. We also consider various small systematic errors that typically occur in the analysis of relaxation functions and show how they can lead to spurious results if sufficient care is not exercised.