  Numerical simulations have shown that certain driven nonlinear systems can be
characterized by mean-field statistical properties often associated with
ergodic dynamics [C.D. Ferguson, W. Klein, and J.B. Rundle, Phys. Rev. E 60,
1359 (1999); D. Egolf, Science 287, 101 (2000)]. These driven mean-field
threshold systems feature long-range interactions and can be treated as
equilibrium-like systems with dynamics that are statistically stationary over
long time intervals. Recently the equilibrium property of ergodicity was
identified in an earthquake fault system, a natural driven threshold system, by
means of the Thirumalai-Mountain (TM) fluctuation metric developed in the study
of diffusive systems [K.F. Tiampo, J.B. Rundle, W. Klein, J.S. Sa Martins, and
C. D. Ferguson, Phys. Rev. Lett. 91, 238501 (2003)]. In this work we analyze
the seismicity of three naturally-occurring earthquake faults networks from a
variety of tectonic settings in an attempt to investigate the range of
applicability of effective ergodicity, using the TM metric and other, related
statistics. Results suggest that, once variations in the catalog data resulting
from technical and network issues are accounted for, all of these natural
earthquake systems display stationary periods of metastable equilibrium and
effective ergodicity that are disrupted by large events. We conclude that a
constant rate of events is an important prerequisite for these periods of
punctuated ergodicity, and that while the level of temporal variability in the
spatial statistics is the controlling factor in the ergodic behavior of seismic
networks, no single statistic is sufficient to ensure quantification of
ergodicity. Specifically, we demonstrate that stationarity, while a necessary
condition, is not sufficient to ensure ergodicity in fault systems.
