Tricritical Directed Percolation

Abstract

We consider a modification of the contact process incorporating higher-order reaction terms. The original contact process exhibits a non-equilibrium phase transition belonging to the universality class of directed percolation. The incorporated higher-order reaction terms lead to a non-trivial phase diagram. In particular, a line of continuous phase transitions is separated by a tricritical point from a line of discontinuous phase transitions. The corresponding tricritical scaling behavior is analyzed in detail, i.e., we determine the critical exponents, various universal scaling functions as well as universal amplitude combinations.

Cluster geometry and survival probability in systems driven by reaction–diffusion dynamics

“…From these simulations we find that for 1 + 1 dimensions, continuous phase transitions are observed for all k. In order to examine tricritical behaviour we therefore proceed in 2 + 1 dimensions. Since the focus in this paper is on the geometrical structure of the population at criticality, we give here only a brief summary of how we obtained the critical points and refer the reader to [7,8] for a more detailed overview of the techniques involved.…”

“…In fact, the robustness of DP led Janssen [5] and Grassberger [6] to the conjecture that all models involving short-range processes, with a single, scalar order parameter that exhibit a continuous phase transition from an active to a single absorbing state belong to the class. A universality class closely associated with DP is tricritical directed percolation (TDP), which has recently been studied by both steady-state [7] and dynamical [8] simulations.…”

“…Lübeck [7] examined tricritical behaviour by adding the pair reaction 2A → 3A to the contact process [10], whereas Grassberger [8] used a generalization of the Domany-Kinzel model [11] in 2 + 1 dimensions.…”

“…Although, from the mean field, the first-order phase transition was expected in all dimensions, the larger fluctuations in the (1 + 1)-dimensional case are likely to destabilize the ordered phase [14], resulting in the observed DP transition. Following on from Lübeck [7], we introduce a lower order proliferation reaction to our original model. By allowing both asexual and sexual reproduction, we have a phase space exhibiting both first-order and continuous phase transitions.…”

“…By allowing both asexual and sexual reproduction, we have a phase space exhibiting both first-order and continuous phase transitions. While the critical behaviour of such a system has been previously studied [7,8], the geometrical structure of the population at the different types of critical behaviour has so far been ignored. Sample snapshots of the population for sexual reproduction only and both asexual and sexual reproduction are shown in figure 1, highlighting the range of structures that are observed.…”