Directed percolation process in the presence of velocity fluctuations: Effect of compressibility and finite correlation time

The direct bond percolation process (Gribov process) is studied in the presence of random velocity fluctuations generated by the Gaussian self-similar ensemble with finite correlation time. We employ the renormalization group in order to analyze a combined effect of the compressibility and finite correlation time on the long-time behavior of the phase transition between an active and an absorbing state. The renormalization procedure is performed to the one-loop order. Stable fixed points of the renormalization group and their regions of stability are calculated in the one-loop approximation within the three-parameter $(\varepsilon ,y,\eta )$ expansion. Different regimes corresponding to the rapid-change limit and frozen velocity field are discussed, and their fixed points' structure is determined in numerical fashion.

Figure 1

Diagrammatic representation of the bare propagators. The time flows from right to left.

Figure 2

Diagrammatic representation of the interaction vertices describing an ideal directed bond percolation process.

Figure 3

Interaction vertices describing the influence of the advecting velocity field with the order parameter fluctuations.

Figure 4

A qualitative sketch of the regions of stability for the fixed points in the limit of the rapid-change model. The borders between the regions are depicted with the bold lines.

Figure 5

Fixed points' structure for the thermal noise situation (53). From above to bottom the compressibility parameter $\alpha$ attains consecutively the values (a) $\alpha =0$, (b) $\alpha =5$, and (c) $\alpha =100$.

Figure 6

Regions of stability for the fixed points in the limit of the frozen velocity field. The borders between the regions are depicted with the bold lines.

Figure 7

Fixed points' structure for frozen velocity case with $\eta =0$. From top to bottom the compressibility parameter $\alpha$ attains consecutively the values (a) $\alpha =0$, (b) $\alpha =3.5$, and (c) $\alpha =8$.

Figure 8

Fixed points' structure for rapid change model with $\eta =4/3$. From top to bottom the compressibility parameter $\alpha$ attains consecutively the values (a) $\alpha =0$, (b) $\alpha =5$, and (c) $\alpha =\infty$. The dot denotes the coordinates of the three-dimensional Kolmogorov regime.

Figure 9

Fixed points' structure for the frozen velocity case with $\eta =4/3$. From top to bottom the compressibility parameter $\alpha$ attains consecutively the values (a) $\alpha =0$, (b) $\alpha =8$, and (c) $\alpha =\infty$. The dot denotes the coordinates of the three-dimensional Kolmogorov regime.

Figure 10

Demonstration of the RG trajectories flows' in (a) the plane $(g_1,u_1)$ and (b) the plane $(g_2,u_1)$ for three-dimensional ($\varepsilon =1$) turbulent advection with $\alpha =110$. The square $\blacksquare$ denotes frozen velocity regime ${\text{FP}}_7^{\mathrm{II}}$, triangle $◂$ corresponds to the unstable regime ${\text{FP}}_2^{\mathrm{III}}$, and circle $•$ to the nontrivial regime ${\text{FP}}_1^{\mathrm{III}}$ for which the time correlations are relevant. Dashed lines corresponds to the chosen flows to the point ${\text{FP}}_7^{\mathrm{II}}$, whereas the full lines to the flows to the other stable point ${\text{FP}}_1^{\mathrm{III}}$.