Directed percolation

Varying the microscopic connectivity of the pores, these models display a phase transition from a macroscopically permeable (percolating) to an impermeable (non-percolating) state.

Directed percolation is also used as a simple model for epidemic spreading with a transition between survival and extinction of the disease depending on the infection rate.

More generally, the term directed percolation stands for a universality class of continuous phase transitions which are characterized by the same type of collective behavior on large scales.

Directed percolation is probably the simplest universality class of transitions out of thermal equilibrium.

The figure shows a tilted square lattice with bonds connecting neighboring sites.

Current estimates are summarized in the following table: In two dimensions, the percolation of water through a thin tissue (such as toilet paper) has the same mathematical underpinnings as the flow of electricity through two-dimensional random networks of resistors.

Above a certain critical point, the electrical flow will cause a fuse to pop, possibly leading to a cascade of failures, resembling the propagation of a crack or tear.

Examples can be found not only in physical phenomena, but also in biology, neuroscience, ecology (e.g. evolution), and economics (e.g. diffusion of innovation).

In spite of vast success in the theoretical and numerical studies of DP, obtaining convincing experimental evidence has proved challenging.

In 1999 an experiment on flowing sand on an inclined plane was identified as a physical realization of DP.

[3] In 2007, critical behavior of DP was finally found in the electrohydrodynamic convection of liquid crystal, where a complete set of static and dynamic critical exponents and universal scaling functions of DP were measured in the transition to spatiotemporal intermittency between two turbulent states.