Why is b=1?

The exponent b of the log-linear frequency-magnitude relation for natural seismicity commonly takes values that are statistically indistinguishable from b=1.  There are some exceptions, notably with respect to focal mechanism and for volcanic and induced seismicity, but it is possible these could be explained at least in part by variability in the dynamic range of measurements between the minimum magnitude of complete reporting and the maximum magnitude, especially where the dynamic range of the statistical sample is small.  However, in laboratory experiments and in discrete element simulations a wide range of b-values for acoustic emissions are consistent (after accounting for systematic differences in the transducer response) with systematic variations in the range  as the stress intensity factor increases from its minimum to its maximum, critical value.  The question remains: why is  an attractor stationary state for large-scale seismicity?  Previous attempts to answer this question have relied on a simple geometric ‘tiling’ argument that is inconsistent with the spatial distribution of earthquake locations, or a hierarchical ‘triple-junction’ model that has not been validated by observation.  Here, we derive a closed analytical solution for the maximum entropy -value, conditional on the assumption that earthquake magnitude scales linearly with the logarithm of rupture area. In the limit of infinite dynamic range, the solution is .  The maximum entropy -value converges to this value asymptotically from above as dynamic range increases for large systems at steady state.  This is in contrast to a previous maximum entropy solution based on analysing the spectrum in ‘natural time’ of earthquake catalogues, where larger samples with greater dynamic range lead to a divergence from .  The new theory is consistent with the trend in b-value convergence from above towards an asymptotic limit of in b=1.027±0.015 at 95% confidence from the global CMT earthquake frequency-moment catalogue for data since 1990.