This review addresses recent developments in non-equilibrium statistical physics. Focusing on phase transitions from fluctuating phases into absorbing states, the universality class of directed percolation is investigated in detail. The survey gives a general introduction to various lattice models of directed percolation and studies their scaling properties, field-theoretic aspects, numerical techniques, as well as possible experimental realizations. In addition, several examples of absorbing-state transitions which do not belong to the directed percolation universality class will be discussed. As a closely related technique, we investigate the concept of damage spreading. It is shown that this technique is ambiguous to some extent, making it impossible to define chaotic and regular phases in stochastic non-equilibrium systems. Finally, we discuss various classes of depinning transitions in models for interface growth which are related to phase transitions into absorbing states.

Non-equilibrium critical phenomena and phase transitions into absorbing states

A simplified prisoner's game is studied on a square lattice when the players interacting with their neighbors can follow two strategies: to cooperate $(C)$ or to defect $(D)$ unconditionally. The players updated in random sequence have a chance to adopt one of the neighboring strategies with a probability depending on the payoff difference. Using Monte Carlo simulations and dynamical cluster techniques, we study the density $c$ of cooperators in the stationary state. This system exhibits a continuous transition between the two absorbing states when varying the value of temptation to defect. In the limits $\stackrel{\ensuremath{\rightarrow}}{c}0$ and 1 we have observed critical transitions belonging to the universality class of directed percolation.

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Evolutionary prisoner's dilemma game on a square lattice

This review addresses recent developments in nonequilibrium statistical physics. Focusing on phase transitions from fluctuating phases into absorbing states, the universality class of directed percolation is investigated in detail. The survey gives a general introduction to various lattice models of directed percolation and studies their scaling properties, field-theoretic aspects, numerical techniques, as well as possible experimental realizations. In addition, several examples of absorbing-state transitions which do not belong to the directed percolation universality class will be discussed. As a closely related technique, we investigate the concept of damage spreading. It is shown that this technique is ambiguous to some extent, making it impossible to define chaotic and regular phases in stochastic nonequilibrium systems. Finally, we discuss various classes of depinning transitions in models for interface growth which are related to phase transitions into absorbing states.

Nonequilibrium Critical Phenomena and Phase Transitions into Absorbing States

Non-equilibrium critical phenomena have attracted a lot of research interest in the recent decades.
Similar to equilibrium critical phenomena, the concept of universality remains the major tool to order the great variety of non-equilibrium phase transitions systematically.
All systems belonging to a given universality class share the same set of critical exponents, and certain scaling functions become identical near the critical point.
 It is known that the scaling functions vary
more widely between different universality classes
than the exponents. Thus, universal scaling functions offer a sensitive and accurate test for a system's universality class.
 On the other hand, universal scaling functions 
demonstrate the robustness of a given universality class impressively. Unfortunately, most studies focus on the determination of the critical exponents, neglecting the universal scaling functions.
 
In this work a particular class of non-equilibrium
critical phenomena is considered, the so-called 
absorbing phase transitions. Absorbing phase transitions are expected to occur in physical, chemical as well as biological systems, and a detailed introduction is presented. The universal scaling behavior of two different universality 
classes is analyzed in detail, namely the directed percolation and the Manna universality class.
Especially, directed percolation is the most common universality class of absorbing phase transitions. The presented picture gallery of universal scaling functions includes steady state, dynamical as well as finite size scaling functions.
 In particular, the effect of an external field conjugated to the order parameter is investigated.
Incorporating the conjugated field, it is possible
to determine the equation of state, the susceptibility, and to perform a modified
finite-size scaling analysis appropriate for absorbing phase transitions. Focusing on these equations, the obtained results can be applied to other non-equilibrium continuous phase transitions observed in numerical simulations or experiments.
Thus, we think that the presented picture gallery of universal scaling functions is valuable for future work.
 Additionally to the manifestation of universality classes, universal scaling functions are useful in order to check renormalization group results quantitatively. Since the renormalization group theory is the basis of our understanding of critical phenomena, it is of fundamental interest to examine the accuracy of the obtained results.
Due to the continuing improvement of computer hardware, accurate numerical data have become available, resulting in a fruitful interplay
between numerical investigations and renormalization group analyzes.

Universal scaling behavior of non-equilibrium phase transitions

We present numerical evidence for the fact that the damage spreading transition in the Domany-Kinzel automaton found by Martinset al. is in the same universality class as directed percolation. We conjecture that also other damage spreading transitions should be in this universality class, unless they coincide with other transitions (as in the Ising model with Glauber dynamics) and provided the probability for a locally damaged state to become healed is not zero.

Are Damage Spreading Transitions Generically in the Universality Class of Directed Percolation

We present numerical and analytical results for a special kind of one-dimensional probabilistic cellular automaton, the so-called Domany-Kinzel automaton. It is shown that the phase boundary separating the active and the recently found chaotic phase exhibits re-entrant behaviour. Furthermore exact results for the p2=0 line are discussed.

Re-entrant behaviour in the Domany-Kinzel cellular automaton

We present numerical and analytical results for a special kind of one-dimensional probabilistic cellular automaton, the so called Domany-Kinzel automaton. It is shown that the phase boundary separating the active and the recently found chaotic phase exhibits reentrant behavior. Furthermore exact results for the $p_2$=0-line are discussed.

H. Rieger

Papers

Re-entrant behaviour in the Domany-Kinzel cellular automaton

Reentrant Behavior in the Domany-Kinzel Cellular Automaton