Directed percolation process advected by the compressible flow
T. Lučivjanský, N. V. Antonov, M. Hnatič, A. S. Kapustin, L. Mižišin
aa r X i v : . [ n li n . C D ] D ec Directed percolation process advected by the compressible flow
T. Luˇcivjansk´y ∗ , N. V. Antonov , M. Hnatiˇc , A. S. Kapustin , L. Miˇziˇsin , Faculty of Sciences, ˇSaf´arik University, Koˇsice, Slovakia Fakult¨at f¨ur Physik, Universit¨at Duisburg-Essen, D-47048 Duisburg, Germany Department of Theoretical Physics, St. Petersburg University, Ulyanovskaya 1, St.Petersburg, Petrodvorets, 198504 Russia BLTP, Joint Institute for Nuclear Research, 141 980 Dubna, Russia
ANNOTATION
It will be shown how the directed percolation process in the presence of compressiblevelocity fluctuations could be formulated within the means of field-theoretic formalism,which is suitable for the renormalization group treatment.
INTRODUCTION
The directed percolation (DP) process [1] is one of the most important model, that de-scribes formation of the fractal structures. The distinctive property of DP is the exhibitionof non-equilibrium second order phase transition [2] between absorbing (inactive) and ac-tive state. Similar to the equilibrium critical behavior, emerging scale invariant behavior,can be analyzed with the help of renormalization group (RG) technique. The deviationsfrom the ideal models are known to have a profound effect. The main aim of this study isto describe how the directed percolation process in the presence of compressible velocityfluctuations can be analyzed in the framework of field-theoretic formulation.
THE MODEL
The continuum description of DP in terms of a density field ψ = ψ ( t, x ) arises from acoarse-graining procedure in which a large number of microscopic degrees of freedom wereaveraged out. The mathematical model has to respect the absorbing state condition, thatis ψ = 0 is always a stationary state. The coarse grained stochastic equation then reads [3] ∂ t ψ = D ( ∇ − τ ) ψ − g D ψ + η, (1)where η denotes the noise term, ∂ t = ∂/∂t is the time derivative, ∇ is the Laplaceoperator, D is the diffusion constant, g is the coupling constant and τ measures deviation ∗ [email protected] rom the criticality. The Gaussian noise term η with zero mean stands for the neglectedfast microscopic degrees of freedom. Its correlation function must respect absorbing statecondition and it can be chosen in the following form h η ( t , x ) η ( t , x ) i = g D ψ ( t , x ) δ ( t − t ) δ ( d ) ( x − x ) . (2)The next step consists in the incorporation of the velocity fluctuations into the equation(1). The standard route based on the the replacement ∂ t by the Lagrangian derivative ∂ t + ( v · ∇ ) is not sufficient due to the assumed compressibility. As was shown in [4] theadditional parameter a has to be introduced via following replacement ∂ t → ∂ t + ( v · ∇ ) + a ( ∇ · v ) . (3)The choice a = 1 corresponds to the conserved quantity ψ , whereas for the choice a = 0the conserved quantity is ˜ ψ . The full description of the model requires specification ofthe velocity field. Following the work [5] the velocity field is considered to be a randomGaussian variable with zero mean and correlator h v i ( t, x ) v j (0 , ) i = Z d ω π Z d d k (2 π ) d D v ( ω, k )e − iωt + k · x , (4)where d is dimension of the space and the kernel function D v ( ω, k ) is chosen in the form D v ( ω, k ) = [ P kij + αQ kij ] g u D k − d − y − η ω + u D ( k − η ) . (5)Here P kij = δ ij − k i k j /k is transverse and Q kij longitudinal projection operator, k = | k | ,positive parameter α > ∇ · v = 0. The coupling constant g and exponent y describe the equal-timevelocity correlator or equivalently, the energy spectrum of the velocity fluctuations. On theother hand parameter u > η describe dispersion behavior of the mode k .The exponents y and η are analogous to the standard expansion parameter ε = 4 − d inthe static critical phenomena [6]. According to the general rules of the RG approach weformally assume that the exponents ε, y and η are of the same order of magnitude and inprinciple they constitute small expansion parameters in a perturbation sense.For the effective use of RG method it is advantageous to reformulate the stochasticproblem (1-5) into the field-theoretic language. This can be achieved in the standard ˜ ψ ˜ ψ ψ v i v j ˜ ψ ˜ ψψ ψ v j ˜ ψ ψψ ˜ ψ v i v j ψ ˜ ψ Figure 1. Elements of the perturbation theory in the graphical representation. fashion [7, 8] and the resulting dynamic functional can be written as a sum J [ ϕ ] = J diff [ ϕ ] + J vel [ ϕ ] + J int [ ϕ ] , (6)where ϕ = { ˜ ψ, ψ, v } stands for the complete set of fields and ψ † is the response field. Thecorresponding terms have the following form J diff [ ϕ ] = Z d t Z d d x (cid:26) ˜ ψ [ ∂ t − D ∇ + D τ ] ψ (cid:27) , (7) J vel [ v ] = − Z d t Z d t Z d d x Z d d x v i ( t , x ) D − ij ( t − t , x − x ) v j ( t , x ) , (8) J int [ ϕ ] = Z d t Z d d x ˜ ψ (cid:26) D λ ψ − ˜ ψ ] − u D v + ( v · ∇ ) + a ( ∇ · v ) (cid:27) ψ. (9)All but third term in (9) stems directly stems from the nonlinear terms in (1) and (3).The third term proportional to ∝ ˜ ψψ v deserves a special consideration. Presence ofsuch term is prohibited in the original Kraichnan model due to the underlying Galileaninvariance. However in our case the finite time correlations of the velocity fluctuationsdoes not impose such restriction. In the language of Feynman graphs, one can show thatsuch term will indeed be generated as can be readily seen considering first three graphs inthe following expansionΓ ˜ ψψ vv = u D δ ij Z + + + 12 ++ + + + . (10)We conclude that compressibility and non-Galilean nature of the velocity correlator leadto the quite involved situation, which requires analysis. Note, that in the incompressiblecase [9] presence of a given term does not lead to the significant effects. ENORMALIZATION GROUP ANALYSIS
The field-theoretic formulation summarized in (8)-(9) has an advantage to be amenableto the machinery of field theory [6]. Near criticality τ = 0 large fluctuations on all scalesdominate the behavior of the system, which results into the divergences in Feynman graphs.The RG technique allows us to deal with them and as a result it allows for pertubativecomputation of critical exponent in a formal expansion around upper critical dimension.Thus provides us with information about the scaling behavior of Green functions. Therenormalization of the model can be achieved through the relations D = DZ D , τ = τ Z τ + τ C , a = aZ a , g = g µ y + η Z g ,u = u µ η Z u , λ = λµ ε Z λ , u = u Z u , ˜ ψ = Z ˜ ψ ˜ ψ R , ψ = Z ψ ψ R , v = Z v v R . (11)where µ is the reference mass scale in the MS scheme [6]. CONCLUSIONS
In this brief article we have summarized main points of the field-theoretic study of di-rected percolation process in the presence of compressible velocity field. The detailedresults for the renormalization constants and analysis of the scaling behavior will be pub-lished elsewhere [10].