Directed percolation
Encyclopedia
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
.
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
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.
that evolves in time. In the case of bond DP the time parameter
is discrete and all sites are updated in parallel. Activating a certain site (called initial seed) at time 
the resulting cluster can be constructed row by row. As shown in the figure, the corresponding number of active sites 
varies as time evolves.
. Above the so-called upper critical dimension 
they are given by their mean-field values while in
dimensions they have been estimated numerically. Current estimates are summarized in the following table:
) has the same mathematical underpinnings as the flow of electricity
through two-dimensional random networks of resistor
s. In chemistry, chromatography
can be understood with similar models.
The propagation of a tear or rip in a sheet of paper, in a sheet of metal, or even the formation of a crack in ceramic
bears broad mathematical resemblance to the flow of electricity through a random network of electrical fuses. 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. The study of percolation helps indicate how the flow of electricity will redistribute itself in the fuse network, thus modeling which fuses are most likely to pop next, and how fast they will pop, and what direction the crack may curve in.
Examples can be found not only in physical phenomena, but also in biological and ecological ones (evolution
), and also in economic and social ones (see diffusion of innovation).
Percolation can be considered to be a branch of the study of dynamical system
s or statistical mechanics
. In particular, percolation networks exhibit a phase change around a critical threshold.
Statistical physics
Statistical physics is the branch of physics that uses methods of probability theory and statistics, and particularly the mathematical tools for dealing with large populations and approximations, in solving physical problems. It can describe a wide variety of fields with an inherently stochastic...
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
Phase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
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
Thermal equilibrium
Thermal equilibrium is a theoretical physical concept, used especially in theoretical texts, that means that all temperatures of interest are unchanging in time and uniform in space...
.
Lattice models of Directed Percolation
One of the simplest realizations of DP is bond directed percolation. This model is a directed variant of ordinary (isotropic) percolationPercolation
In physics, chemistry and materials science, percolation concerns the movement and filtering of fluids through porous materials...
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
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
Percolation
In physics, chemistry and materials science, percolation concerns the movement and filtering of fluids through porous materials...
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.
Directed Percolation as a dynamical process
Interpreting the preferred direction as a temporal degree of freedom, directed percolation can be regarded as a stochastic processStochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
that evolves in time. In the case of bond DP the time parameter
is discrete and all sites are updated in parallel. Activating a certain site (called initial seed) at time the resulting cluster can be constructed row by row. As shown in the figure, the corresponding number of active sites varies as time evolves.Universal scaling behavior
The DP universality class is characterized by a certain set of critical exponents. These exponents depend on the spatial dimension. Above the so-called upper critical dimension they are given by their mean-field values while in dimensions they have been estimated numerically. Current estimates are summarized in the following table:| exponent |  |  |  |  | |
|---|---|---|---|---|---|
 |
 |
 |
 |
 |
|
 |
 |
 |
 |
 |
|
|  |
 |
 |
 |
dimensions (September 2006)Other examples
In two dimensions, the percolation of water through a thin tissue (such as toilet paperToilet paper
Toilet paper is a soft paper product used to maintain personal hygiene after human defecation or urination. However, it can also be used for other purposes such as blowing one's nose when one has a cold or absorbing common spills around the house, although paper towels are more used for the latter...
) has the same mathematical underpinnings as the flow of electricity
Electricity
Electricity is a general term encompassing a variety of phenomena resulting from the presence and flow of electric charge. These include many easily recognizable phenomena, such as lightning, static electricity, and the flow of electrical current in an electrical wire...
through two-dimensional random networks of resistor
Resistor
A linear resistor is a linear, passive two-terminal electrical component that implements electrical resistance as a circuit element.The current through a resistor is in direct proportion to the voltage across the resistor's terminals. Thus, the ratio of the voltage applied across a resistor's...
s. In chemistry, chromatography
Chromatography
Chromatography is the collective term for a set of laboratory techniques for the separation of mixtures....
can be understood with similar models.
The propagation of a tear or rip in a sheet of paper, in a sheet of metal, or even the formation of a crack in ceramic
Ceramic
A ceramic is an inorganic, nonmetallic solid prepared by the action of heat and subsequent cooling. Ceramic materials may have a crystalline or partly crystalline structure, or may be amorphous...
bears broad mathematical resemblance to the flow of electricity through a random network of electrical fuses. 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. The study of percolation helps indicate how the flow of electricity will redistribute itself in the fuse network, thus modeling which fuses are most likely to pop next, and how fast they will pop, and what direction the crack may curve in.
Examples can be found not only in physical phenomena, but also in biological and ecological ones (evolution
Evolution
Evolution is any change across successive generations in the heritable characteristics of biological populations. Evolutionary processes give rise to diversity at every level of biological organisation, including species, individual organisms and molecules such as DNA and proteins.Life on Earth...
), and also in economic and social ones (see diffusion of innovation).
Percolation can be considered to be a branch of the study of dynamical system
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
s or statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
. In particular, percolation networks exhibit a phase change around a critical threshold.
Experimental realizations
In spite of vast success in the theoretical and numerical studies of DP, obtaining convincing experimental evidence has proved challenging. However, 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 .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
- L. Canet: "Processus de réaction-diffusion : une approche par le groupe de renormalisation non perturbatif", Thèse. Thèse en ligne
- Muhammad Sahimi. Applications of Percolation Theory. Taylor & Francis, 1994. ISBN 0-7484-0075-3 (cloth), ISBN 0-7484-0076-1 (paper)
- Geoffrey Grimmett. Percolation (2. ed). Springer Verlag, 1999.
- K. A. Takeuchi, M. Kuroda, H. Chaté, and M. Sano: Directed Percolation Criticality in Turbulent Liquid Crystals, Phys. Rev. Lett. 99, 234503 (2007). cond-mat