Influence of turbulent mixing on critical behavior of directed percolation process: Effect of compressibility

Universal behavior is a typical emergent feature of critical systems. A paramount model of the nonequilibrium critical behavior is the directed bond percolation process that exhibits an active-to-absorbing state phase transition in the vicinity of a percolation threshold. Fluctuations of the ambient environment might affect or destroy the universality properties completely. In this work, we assume that the random environment can be described by means of compressible velocity fluctuations. Using field-theoretic models and renormalization group methods, we investigate large-scale and long-time behavior. Altogether, 11 universality classes are found, out of which 4 are stable in the infrared limit and thus macroscopically accessible. In contrast to the model without velocity fluctuations, a possible candidate for a realistic three-dimensional case, a regime with relevant short-range noise, is identified. Depending on the dimensionality of space and the structure of the turbulent flow, we calculate critical exponents of the directed percolation process. In the limit of the purely transversal random force, critical exponents comply with the incompressible results obtained by previous authors. We have found intriguing nonuniversal behavior related to the mutual effect of compressibility and advection.

Figure 1

Schematic visualization of the DP process in the presence of the fully developed turbulent flow. The black path represents the DP process (starting from the left and going rightwards) and the fully developed turbulence is displayed in the background.

Figure 2

Perturbation elements of the velocity part of model (31).

Figure 3

Perturbation elements related to DP process and advecting interaction of model (31).

Figure 4

Relation between $w^{*}$ and $a^{*}$ for FPII (upper figure) and FPIV (lower figure), respectively. For FPII, the parameter $w^{*}\in \langle 0,1.0518\left(8\right)\rangle$ for any $a^{*}\in \langle 0,1\rangle$. The same situation occurs in the case of FPIV, where the corresponding maximum depends on $\mathrm{\Delta }$. The lower plot is made for the physically relevant choice $(y,\varepsilon )=(4,1)$ and three different values of $\alpha$. Note that all plots are symmetric around the point $a^{*}=\frac{1}{2}.$

Figure 5

Numerical solution for the fixed point's coordinate of the charge $g_3^{*}$ for three different values of parameter $\alpha \in \{0,1,\infty \}$ (depicted in a given order from top to bottom). Distinct regions of stability are separated by the solid black line, the line $\varepsilon =1(d=3)$ is represented by the gray dashed line, and black dot represents the case $(y,\varepsilon )=(4,1)$. Technical details concerning boundary between regions of stability for FPIV and FPV can be found in Appendix pp3-s5.

Figure 6

RG flow in the $(a,w,g_3)$ plane for the physical values $(y,\varepsilon )=(4,1)$ and for $\alpha \in \{0,1,\infty \}$ (from top to bottom). By increasing the value of $\alpha$, the fixed (dashed) line shifts from $w=1$.

Figure 7

Phase portraits in the $(d,y)$ plane for the values $\alpha \in \{0,1,\infty \}$ (from top to bottom). For $\alpha >0$, four regimes are present and in the limit $\alpha \to 0$ advected DP regime vanishes. The case $d=3(\varepsilon =1)$ is denoted with the dashed line, so that the crossover with growing $\alpha$ between distinct regimes is clearly visible. The most relevant point $(d,y)=(3,4)$ (denoted by the black dot) belongs to the universality class of passive advection.

Figure 8

Critical exponents for FPII for $\varepsilon =1$ as a function of $a^{*}$.

Figure 9

Numerical solution for critical exponents $\mathrm{\Theta },\delta$ and the difference $\mathrm{\Delta }\delta =|\delta -{\delta }^{\prime }|$ (from top to the bottom). Result for the exponent ${\delta }^{\prime }$ is horizontally symmetric to $\delta$ with respect to the plane $a=\frac{1}{2}$ (not displayed). For $\alpha \to 0$ exponents tend to the incompressible limit and the difference $\mathrm{\Delta }\delta$ vanishes.

Figure 10

The parameter $\xi$ as a function of $\alpha$. The results are consistent with the numerical solution shown in Fig. 5.