Directed percolation

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In statistical physics Directed Percolation (DP) refers to a class of models that mimic filtering of fluids through porous materials along a given direction. 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.

1 Lattice models of Directed Percolation
2 Directed Percolation as a dynamical process
3 Universal scaling behavior
4 Open problems
5 Literature

[edit] Lattice models of Directed Percolation

Realization of permeable (solid lines) and closed bonds (dashed lines) on a tilted square lattice.

One of the simplest realizations of DP is bond directed percolation. This model is a directed variant of ordinary (isotropic) percolation and can be introduced as follows. The figure shows a tilted square lattice with bonds connecting neighboring sites. The bonds are permeable (open) with probability $p\,$ and impermeable (closed) otherwise. The sites and bonds may be interpreted as holes and randomly distributed channels of a porous medium.

The difference between ordinary and directed percolation is illustrated below. In isotropic percolation a spreading agent (e.g. water) introduced at a particular site marked by a green circle percolates along open bonds, generating a certain cluster of wet sites. Contrarily, in directed percolation the spreading agent can pass open bonds only along a preferred direction in space, as indicated by the arrow. The resulting cluster is directed in space.

[edit] Directed Percolation as a dynamical process

Interpreting the preferred direction as a temporal degree of freedom, directed percolation can be regarded as a stochastic process that evolves in time. In the case of bond DP the time parameter $t$ is discrete and all sites are updated in parallel. Activating a certain site (called initial seed) at time $t = 0$ the resulting cluster can be constructed row by row. As shown in the figure, the corresponding number of active sites $N (t)$ varies as time evolves.

[edit] Universal scaling behavior

The DP universality class is characterized by a certain set of critical exponents. These exponents depend on the spatial dimension $d\,$ . Above the so-called upper critical dimension $d\geq d_c=4\,$ they are given by their mean-field values while in $d<4\,$ dimensions they have been estimated numerically. Current estimates are summarized in the following table:

Critical exponents of directed percolation in $d$ dimensions (September 2006)
exponent	$d=1\,$	$d=2\,$	$d=3\,$	$d\geq 4\,$
$\beta\,$	$0.276486 \pm 0.000008$	$0.583 \pm 0.003$	$0.813 \pm 0.009$	$1\,$
$\nu_\perp$	$1.096854 \pm 0.000004$	$0.733 \pm 0.008$	$0.584 \pm 0.005$	$1/2\,$
$\nu_{\|\|}\,$	$1.733847 \pm 0.000006$	$1.295 \pm 0.006$	$1.11 \pm 0.01$	$1\,$

[edit] Open problems

So far no experiment is known that reproduces the critical behavior of DP quantitatively, in particular the values of the critical exponents. Designing and performing such an experiment remains a challenging open problem.

[edit] Literature

H. Hinrichsen: Nonequilibrium critical phenomena and phase-transitions into absorbing states, Adv. Phys. 49, 815 (2000). cond-mat
G. Ódor: Universality classes in nonequilibrium lattice systems, Rev. Mod. Phys. 76, 663 (2004). cond-mat
S. Lübeck: Universal scaling behaviour of non-equilibrium phase-transitions, Int. J. Mod. Phys. B 18, 3977 (2004). cond-mat