A discrete time evolution model for fracture networks

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 R0 and R1, 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.