Effective ergodicity in single-spin-flip dynamics

A quantitative measure of convergence to effective ergodicity, the Thirumalai-Mountain (TM) metric, is applied to Metropolis and Glauber single-spin-flip dynamics. In computing this measure, finite lattice ensemble averages are obtained using the exact solution for a one dimensional Ising model, whereas the time averages are computed with Monte Carlo simulations. The time evolution of the effective ergodic convergence of Ising magnetization is monitored. By this approach, diffusion regimes of the effective ergodic convergence of magnetization are identified for different lattice sizes, nonzero temperature, and nonzero external field values. Results show that caution should be taken when using the TM metric at system parameters that give rise to strong correlations.

Figure 1

Inverse effective ergodic convergence in Glauber dynamics with different system sizes with fixed temperature $\beta =1.0$ and external fields $H=\{0.5,1.0\}$ at (a) and (b), respectively.

Figure 2

Inverse effective ergodic convergence in Metropolis dynamics with different system sizes with fixed temperature $\beta =1.0$ and external fields $H=\{0.5,1.0\}$ at (a) and (b), respectively.

Figure 3

Effective ergodic convergence in Glauber dynamics with different external field values with fixed size $N=512$ and temperature $\beta =\{0.5,1.0\}$ at (a) and (b), respectively.

Figure 4

Effective ergodic convergence in Glauber dynamics with different temperatures with fixed size $N=512$ and external fields $H=\{0.5,1.0\}$ at (a) and (b), respectively.