Thermodynamics of a Double-Couple Fault Plane Model by Spin-Lattice Montecarlo Simulations

The lithosphere is the exterior boundary-layer of the solid earth, the outward energy flow through the mantle is injected into it and drives plate tectonics. The plates relative movement induces fractures in which earthquakes occur along interacting quasi-planar faults distributed through a wide range of scales from nanometric crystal-structures to kilometric subduction-zones. Seismicity is basically brittle deformation and a manifestation of the lithosphere energy-dissipation process that is continually in a critical state. It can be seen as a delicate balance between two main factors: fault-interactions organizing displacements and tectonic-driven forces acting over single elements. From this local rules complex-patterns emerge in the form of global interactions in space and time.

Earthquake complex behavior and the observation of their scaling laws is typical of systems showing phase transitions, like nucleation and critical phenomena observed in thermal and magnetic systems. The most simple model with a phase transition is the 2-dimensional Ising model: a regular lattice of sites with two possible values representing spin directions. The spins interact between near-neighbors and can have the influence of an external force. As the system reaches its critical point a very-ordered low-energy phase appears where multi-scale clusters of aligned sites dominate. This critical state is characterized by bifurcations, non-differentiability and power-law scaling of thermodynamic state-variables.

We propose a 2D spin-lattice fault-plane model. Following the classic description of earthquake kinematics, a spin-site in the lattice represents a double-couple point-source. A Hamiltonian representing interactions and tectonics forces is set allowing us to sample the partition function through state-of-the-art Montecarlo methods. The critical state-variables are then compared with earthquake data using Renormalization Group methods. Preliminary results and insights into the energy dissipation rate of the lithosphere are presented.