Classification of phase transitions in reaction-diffusion models

Equilibrium phase transitions are associated with rearrangements of minima of a (Lagrangian) potential. Treatment of nonequilibrium systems requires doubling of degrees of freedom, which may be often interpreted as a transition from the “coordinate”- to the “phase”-space representation. As a result, one has to deal with the Hamiltonian formulation of the field theory instead of the Lagrangian one. We suggest a classification scheme of phase transitions in reaction-diffusion models based on the topology of the phase portraits of corresponding Hamiltonians. In models with an absorbing state such a topology is fully determined by intersecting curves of zero “energy.” We identify four families of topologically distinct classes of phase portraits stable upon renormalization group transformations.

Figure 1

(Color online) Phase portrait of the binary annihilation system, $2A\stackrel{\lambda }{\to }0$. The corresponding classical Hamiltonian is given by $H_R(p,q)=\frac{\lambda }{2}(1-p^2)q^2$. Solid black lines show zero-energy trajectories: generic lines $p=1$ and double degenerate $q=0$ and the “accidental” line $p=-1$. Dashed colored curves indicate trajectories with nonzero energy. The arrows show the evolution direction.

Figure 2

(Color online) Phase portraits of the system exhibiting the first-order phase transition. Thick solid lines represent the zero-energy trajectories. (a)–(c) The set, Eq. (21): (a) extinction phase, (b) transition point, and (c) active phase. (d) Reaction of the type of Eq. (23) with $k=2$, $n=3$, and $i=4$ in the active phase.

Figure 3

(Color online) Phase portraits of the system, Eq. (25), exhibiting the first-order phase transition. Thick solid lines represent the zero-energy trajectories. (a)–(c) Reaction set with $k=l=m=1$: (a) extinction phase, (b) transition point, and (c) unlimited proliferation phase. (d) Reaction set with $k=l=2$ and $m=1$ in the extinction phase.

Figure 4

(Color online) Phase portrait of the DP system in the active phase. Thick solid lines represent zero-energy trajectories which divide the phase space into a number of disconnected regions. Point $B=\mathbf{(}1,2(\mu -\lambda )∕\sigma \mathbf{)}$ represents the active mean-field point. The system is brought to the phase transitions if points $A$, $B$, and $C$ coalesce.

Figure 5

(Color online) Generic phase portrait of DP models in the vicinity of the phase transition [after the shift, Eq. (30)]. (a) Active phase, $m>0$; (b) transition point, $m=0$; (c) extinction phase, $m<0$. The plus and minus signs show the sign of the Hamiltonian in each sector. (d) The one-loop diagram renormalizing $u$ vertex (vertexes $m$ and $v$ are renormalized in a similar way).

Figure 6

(Color online) Phase portrait of the 2CPD system in the active phase (cf. DP, Fig. 4). The zero-energy line $q=0$ is doubly degenerate and is depicted by the double line. At the transition points $A$, $B$, and $C$ coalesce.

Figure 7

(Color online) Generic phase portraits of (a) 2CPD models and (b) 3CPD models in the active phase.

Figure 8

Schematic phase boundary in the parameter space $(D,\mu )$ of the $\lambda =0$ reaction set (49). (a) $d\le d_c^{\prime }$, the transition along the entire line is of kCPD class. (b) $d>d_c^{\prime }$, to the left of the tricritical point $T$ the transition is of kCPD class, while to the right it is $\mu =0$ mean-field transition.

Figure 9

(Color online) Phase portrait evolution upon $\mu =0$ mean-field transitions: (a) $k=1$ and $\mu >0$, decreasing the branching rate $\mu$ is schematically shown by the dashed arrows. (b) $k=1$ and $\mu =0$, the “accidental” line becomes horizontal and the local topology of the phase portrait is that of the binary annihilation reaction; cf. Fig. 1. (c) $k=2$ and $\mu >0$. (d) $k=2$ and $\mu =0$, the topology is that of the $3$-plet annihilation reaction, $3A\to 0$.

Figure 10

(Color online) Phase portrait of the parity conserving model, Eq. (52). Notice the reflection symmetry around the origin.

Figure 11

(Color online) Phase diagrams of kPC models. (a) 2PC model, the inset shows the diagram leading to the logarithmic corrections in $d=1$. (b) 3PC model. (c) 2PC model with all reactions including only an even number of particles. This topology is not stable and evolves towards (a). (d) 2PC with $2A\to 0$, the corresponding topology is essentially equivalent to the 2CPD model, Fig. 6.

Figure 12

(Color online) (a)–(c) Phase portraits of reversible reactions. (a) $A\leftrightarrow 2A$ and (b) $2A\leftrightarrow 3A$. (c) Parity conserving $A\leftrightarrow 3A$. (d) A topology with $q\to -q$ symmetry unstable against RG transformations; see Sec. 7.