Nonequilibrium dynamics in Ising-like models with biased initial condition

We investigate the dynamical fixed points of the zero temperature Glauber dynamics in Ising-like models. The stability analysis of the fixed points in the mean field calculation shows the existence of an exponent that depends on the coordination number $z$ in the Ising model. For the generalized voter model, a phase diagram is obtained based on this study. Numerical results for the Ising model for both the mean field case and short ranged models on lattices with different values of $z$ are also obtained. A related study is the behavior of the exit probability $E\left(x_0\right)$, defined as the probability that a configuration ends up with all spins up starting with $x_0$ fraction of up spins. An interesting result is $E\left(x_0\right)=x_0$ in the mean field approximation when $z=2$, which is consistent with the conserved magnetization in the system. For larger values of $z, E\left(x_0\right)$ shows the usual finite size dependent nonlinear behavior both in the mean field model and in the Ising model with nearest neighbor interaction on different two dimensional lattices. For such a behavior, a data collapse of $E\left(x_0\right)$ is obtained using $y=\frac{(x_0-x_c)}{x_c}{L}^{1/\nu }$ as the scaling variable and $f\left(y\right)=\frac{1+tanh\left(\lambda y\right)}{2}$ appears as the scaling function. The universality of the exponent and the scaling factor is investigated.

Figure 1

(a) Flow diagram for the mean field Ising model for $z>2$. $x=0$ and 1 are the stable fixed points and $x=0.5$ is unstable. (b) shows the flow diagram for the generalized voter model. $x_1=0.5$ is the unstable fixed point and $x_2=\frac{1}{2}+\frac{1}{2}\sqrt{\frac{2z_1+z_2-2}{2z_1-z_2}}$ and $x_3=\frac{1}{2}-\frac{1}{2}\sqrt{\frac{2z_1+z_2-2}{2z_1-z_2}}$ are stable fixed points provided that the exponent is positive.

Figure 2

Variation of $\delta \left(t\right)$ is shown with time for $z=4$ for different ${\delta }_0$ where mean field approach is used. This plot also shows the data for several system sizes. Data are fitted to the exponential function, mentioned in the key. Maximum number of configuration was 5000. $\delta \left(t\right)$ attains the saturation value faster for larger ${\delta }_0$ and the process is slower for smaller value of ${\delta }_0$.

Figure 3

Data collapse of $\delta \left(t\right)$ is shown with time for several ${\delta }_0$ where $z=4$ and data are fitted to an exponential form as mentioned in the key. These data are for a system of $2^{16}$ spins averaged over a maximum of 5000 realizations. Inset shows the unscaled data. In this simulation mean field approach is used.

Figure 4

Plots of $\delta \left(t\right)$ are shown against time for several system sizes where $z=6$. Data for several values of ${\delta }_0$ are also shown. Inset shows the data collapse for different ${\delta }_0$ for a system of $2^{16}$ spins. As ${\delta }_0$ increases, $\delta \left(t\right)$ rapidly saturates. Minimum number of different initial configuration was 2000 and mean field approximation is used for this simulation.

Figure 5

Variation of $\delta \left(t\right)$ with time is shown in square lattice nearest neighbor Ising model for several ${\delta }_0$. Data are fitted to the power law form; exponents are mentioned in the key. The power law exponent $\beta$ decreases as the value of ${\delta }_0$ increases. Inset shows the variation of $\beta$ with ${\delta }_0$. These data are for system size $L\times L=64\times 64$ averaged over 5000 realizations.

Figure 6

Variation of $\delta \left(t\right)$ with time is shown in triangular lattice for nearest neighbor interaction. Data are fitted to Eq. (10) and the exponents are mentioned in the key. These data are for system size $L\times L=64\times 64$ averaged over 5000 realizations.

Figure 7

Variation of $\delta \left(t\right)$ with time is shown in square lattice next nearest neighbor Ising model for several system sizes. Data are fitted to the power law form as mentioned in the key.

Figure 8

Variation of $\delta \left(t\right)$ with time is shown in cubic lattice nearest neighbor Ising model for different ${\delta }_0$. $\beta$ decreases as the value of ${\delta }_0$ increases. Data are fitted to the power law form; exponents are mentioned in the key. The power law exponent $\beta$ decreases as the value of ${\delta }_0$ increases. These data are for system size $L^3=8^3$ averaged over 5000 realizations.

Figure 9

Data collapse of $E\left(x_0\right)$ is shown for different system sizes in square lattice Ising model using $\nu =1.307$. Data are fitted to the form of Eq. (13) as mentioned in the key. Inset shows the unscaled data. Number of different initial configuration was 5000 for these simulations.

Figure 10

Variation of the least square error $\epsilon$ with $\lambda$ is shown for $\nu =1.204$ in triangular lattice. The minimum of the curve is at $\lambda =0.857$.

Figure 11

Data collapse of $E\left(x_0\right)$ is shown for different system sizes in triangular lattice using $\nu =1.204$ and data are fitted to Eq. (13). Number of different initial configuration was 5000 for these simulations.

Figure 12

$E\left(x_0\right)$ is shown against $x_0$ for $z=2$ where mean field approach is used. $E\left(x_0\right)$ shows a linear variation with $x_0$. Number of different initial configuration was 5000 for these simulations.

Figure 13

(a) Data collapse of $E\left(x_0\right)$ is shown for different system sizes using ${\nu }^{\prime }=2$ where $z=4$. (b) shows the data collapse of $E\left(x_0\right)$ for different system sizes with ${\nu }^{\prime }=2$ for $z=6$. Mean field approximation is used in these simulations where number of different initial configuration was 5000. Data are fitted to the functions as mentioned in the key.

Figure 14

Magnetization $\left|m\right|$ is shown as a function of $z_1$ and $z_2$.