Simplest nonequilibrium phase transition into an absorbing state

We study in further detail particle models displaying a boundary-induced absorbing state phase transition [Deloubrière and van Wijland Phys. Rev. E 65, 046104 (2002) and Barato and Hinrichsen, Phys. Rev. Lett. 100, 165701 (2008)]. These are one-dimensional systems consisting of a single site (the boundary) where creation and annihilation of particles occur, and a bulk where particles move diffusively. We study different versions of these models and confirm that, except for one exactly solvable bosonic variant exhibiting a discontinuous transition and trivial exponents, all the others display nontrivial behavior, with critical exponents differing from their mean-field values, representing a universality class. Finally, the relation of these systems with a $\left(0+1\right)$-dimensional non-Markovian process is discussed.

Figure 1

(Color online) Model on a semi-infinite lattice with symmetric diffusion in the bulk and special dynamical rules at the left boundary (see main text).

Figure 2

(Color online) Density of particles at the leftmost site ${\rho }_0$ (left) and the average total number of particles $N$ (right) as functions of time for different values of △, below, above, and at criticality. At the critical point, ${\rho }_0\left(t\right)$ decays as $t^{-1/2}$ while and $⟨N\left(t\right)⟩$ is essentially constant.

Figure 3

(Color online) Left panel: Density of particles at the leftmost site, at the time when it reaches its minimum value, as a function of the distance from criticality △. This gives the exponent $\beta =0.68\left(5\right)$. Right panel: The same quantity, at criticality and in the stationary state, as a function of the external field $h$, giving ${\delta }_h^{-1}=0.29\left(5\right)$, compatible with the conjectured value 1/3.

Figure 4

(Color online) Survival probability $P_s\left(t\right)$ as function of time below, above, and at criticality. At the critical point, it decays with the exponent $\delta =0.15\left(2\right)$, different from $\beta /{\nu }_{\parallel }$, and in agreement with the conjectured value 1/6.

Figure 5

(Color online) Left: Data collapse of the rescaled profiles of the particle density at criticality for $t_0=64,128,\dots ,8192$ (blue) compared to a Gaussian distribution (red). Inset: The same data collapse in a double-logarithmic representation. Right: Density of particles (blue) and pairs (green) at $t_0=1\times {10}^6$, showing the presence of correlations which decay in space as $x^{-1/2}$, indicating that $\beta /{\nu }_{\perp }=1/2$.

Figure 6

(Color online) Temporal behavior of ${\rho }_0$ for the bosonic pair-annihilating model, starting from a homogeneous initial condition for different system sizes (from $L=64$ to $L=2048$). The exponent $\beta /{\nu }_{\perp }$ can be measured from the scaling of the different saturation values as a function of system size (see inset; yellow line). Also, in the inset (dashed green line), we show the scaling of saturation values for the fermionic version of the same model, showing the same type of scaling.

Figure 7

(Color online) Off-critical data collapse with the one-site model: $⟨s\left(t\right)⟩{\Delta }^{-2\beta }$ as a function of $t{\Delta }^{4\beta }$ for different values of △, with $\beta =0.71\left(2\right)$.