Linear and nonlinear critical slowing down in the kinetic Ising model: High-temperature series

The difference between the critical exponents (${\Delta }^{\left(\mathrm{l}\right)}$ and ${\Delta }^{\left(\mathrm{nl}\right)}$) of the linear and nonlinear relaxation times of the order parameter (${\tau }^{\left(\mathrm{l}\right)}$ and ${\tau }^{\left(\mathrm{nl}\right)}$) is investigated in the two-dimensional one-spin-flip kinetic Ising model. We have calculated the high-temperature series for ${\tau }^{\left(\mathrm{nl}\right)}$ up to ninth order and made use of the known series for ${\tau }^{\left(\mathrm{l}\right)}$ up to twelfth order. The series are analyzed by the ratio and the Padé-approximant methods. The correlation between the critical-point and critical-exponent estimates in the Padé approximants allows an improvement in the determination of ${\Delta }^{\left(\mathrm{l}\right)}$. The result, ${\Delta }^{\left(\mathrm{l}\right)}=2.125\mathrm{}\pm 0.01$, is higher than previous estimates of 2.0 ± 0.05. The estimate of ${\Delta }^{\left(\mathrm{nl}\right)}$ is less precise but the result ${\Delta }^{\left(\mathrm{nl}\right)}=1.95\mathrm{}\pm 0.15$ leads to the conclusion that ${\Delta }^{\left(\mathrm{l}\right)}\ne {\Delta }^{\left(\mathrm{nl}\right)}$ and that the difference between the two is of order $\beta$ (critical index of the order parameter). This is in accord with the scaling prediction.