The quantities \( \tau_i \) are the correlation times for the system. They control also the auto-correlation function discussed above. The longest correlation time is obviously given by the second largest eigenvalue \( \tau_1 \), which normally defines the correlation time discussed above. For large times, this is the only correlation time that survives. If higher eigenvalues of the transition matrix are well separated from \( \lambda_1 \) and we simulate long enough, \( \tau_1 \) may well define the correlation time. In other cases we may not be able to extract a reliable result for \( \tau_1 \). Coming back to the time correlation function \( \phi(t) \) we can present a more general definition in terms of the mean magnetizations $ \langle \mathbf{M}(t) \rangle$. Recalling that the mean value is equal to $ \langle \mathbf{M}(\infty) \rangle$ we arrive at the expectation values

$$ \phi(t) =\langle \mathbf{M}(0)-\mathbf{M}(\infty)\rangle \langle \mathbf{M}(t)-\mathbf{M}(\infty)\rangle, $$

resulting in

$$ \phi(t) =\sum_{i,j\ne 0}m_i\alpha_im_j\alpha_je^{-t/\tau_i}, $$

which is appropriate for all times.