We examine geological crack patterns using the mean field theory of convex mosaics.
We assign the pair \left({\overline{n } }^{*},{\overline{v } }^{*}\right) n ¯ ∗ ,
v ¯ ∗ of average corner degrees (Domokos et al. in A two-vertex theorem for normal
tilings. Aequat Math https://doi.org/10.1007/s00010-022-00888-0 , 2022) to each crack
pattern and we define two local, random evolutionary steps R 0 and R 1 , corresponding
to secondary fracture and rearrangement of cracks, respectively. Random sequences
of these steps result in trajectories on the \left({\overline{n } }^{*},{\overline{v
} }^{*}\right) n ¯ ∗ , v ¯ ∗ plane. We prove the existence of limit points for several
types of trajectories. Also, we prove that cell density \overline{\rho }= \frac{{\overline{v
} }^{*}}{{\overline{n } }^{*}} ρ ¯ = v ¯ ∗ n ¯ ∗ increases monotonically under any
admissible trajectory.
