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

2.4 MB·

by  N. V. Antonov

Directed percolation process in the presence of velocity fluctuations: Effect of
compressibility and finite correlation time
N. V. Antonov,1 M. Hnatiˇc,2,3,4 A. S. Kapustin,1 T. Luˇcivjansky´,3,5 and L. Miˇziˇsin3
1Department of Theoretical Physics, St. Petersburg University,
Ulyanovskaya 1, St. Petersburg, Petrodvorets, 198504 Russia
2Institute of Experimental Physics, SAS, 04001 Koˇsice, Slovakia
3Faculty of Sciences, P.J. Sˇafarik University, 04154 Koˇsice, Slovakia
4Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Moscow Region, Russia
5Fakult¨at fu¨r Physik, Universita¨t Duisburg-Essen, D-47048 Duisburg, Germany
6 (Dated: February 2, 2016)
1
The direct bond percolation process (Gribov process) is studied in the presence of random ve-
0
locityfluctuationsgeneratedbytheGaussianself-similarensemblewithfinitecorrelationtime. We
2
employ the renormalization group in order to analyze a combined effect of the compressibility and

n finite correlation time on the long-time behavior of the phase transition between an active and an
a absorbing state. The renormalization procedure is performed to the one-loop order. Stable fixed
J
points of the renormalization group and their regions of stability are calculated in the one-loop

4 approximation within the three-parameter (ε,y,η)-expansion. Different regimes corresponding to
1 the rapid-change limit and frozen velocity field are discussed, and their fixed points’ structure is
determined in numerical fashion.
]
h
c
I. INTRODUCTION quenched disorder [19]. In general, the novel behavior is
e
m observed with a possibility that critical behavior is lost.
For example, the presence of a quenched disorder in the
- Thenon-equilibriumphysicalsystemsconstituteanex-
t citing research topic to which a lot of effort has been latter case causes a shift of the critical fixed point to the
a
unphysical region. This leads to such interesting phe-
t made during last decades [1–3]. Absorbing phase transi-
s nomena as an activated dynamical scaling or Griffiths
. tions between active (fluctuating) and inactive (absorb-
t singularities [20–23].
a ing) states are of particular importance. In these tran-
m sitions large scale spatio-temporal fluctuations of an un- In this paper, we focus on the directed bond perco-
derlying order parameter take place and the resulting lation process in the presence of advective velocity fluc-
-
d collective behavior is similar to equilibrium phase tran- tuations. Velocity fluctuations are hardly avoidable in
n sitions. Such behavior could be observed in many natu- any of experiments. For example, a vast majority of
o
ral phenomena ranging from physics, chemistry, biology, chemical reactions occurs at finite temperature, which
c
[ economy or even sociology. isinevitablyencompassedwiththepresenceofathermal
A fundamental part of this type of systems belongs to noise. Furthermore, disease spreading and chemical re-
1 thedirectedpercolation(DP)universalityclass[2,4]. As actions could be affected by the turbulent advection to a
v
pointedbyJanssenandGrassberger[5,6],necessarycon- great extent. Fluid dynamics is in general described by
2
ditions are: i) a unique absorbing state, ii) short-ranged the Navier-Stokes equations [24]. A general solution of
4
6 interactions,iii)apositiveorderparameterandiv)noex- these equations remains an open question [25, 26]. How-
0 tra symmetry or additional slow variables. Among a few ever, to provide more insight we restrict ourselves to a
0 models described within this framework we name popu- moredecentproblem. Namely,weassumethattheveloc-
2. lation dynamics, reaction-diffusion problems [7], perco- ity field is given by the Gaussian velocity ensemble with
0 lation processes [8], hadron interactions [9], etc. These prescribed statistical properties [27, 28]. Although this
6 models are usually considered without an inclusion of assumption appears as oversimplified, compared to the
1 additional interactions within the mode-mode coupling realistic flows at the first sight, it nevertheless captures
v: approach [10]. However, in realistic situations impurities essential physical information about advection processes
i anddefects,whicharenottakenintoaccountintheorig- [27, 29, 30].
X
inal DP formulation, are expected to cause a change in Recently, there has been increased interest in differ-
r the universal properties of the model. This is believed ent advection problems in compressible turbulent flows
a
to be one of the reasons why there are not so many di- [31–34]. These studies show that compressibility plays a
rect experimental realizations [11, 12] of the percolation decisive role for population dynamics or chaotic mixing
process itself. A study of deviations from the ideal situ- ofcolloids. Ourmainaimistoinvestigateaninfluenceof
ation could proceed in different routes and this still con- compressibility [35, 36] on the critical properties of the
stitutes a topic of an ongoing debate [2]. A substantial directed bond percolation process [2]. To this end, the
efforthasbeenmadeinstudyingalong-rangeinteraction advectivefieldisdescribedbytheKraichnanmodelwith
using L´evy-flight jumps [13–15], effects of immunization finitecorrelationtime,inwhichnotonlyasolenoidal(in-
[8, 16], mutations [17], feedback of the environment on compressible) but also a potential (compressible) part of
thepercolatingdensity[18],orinthepresenceofspatially thevelocitystatisticsisinvolved. Notethatinourmodel
2
there is no backward influence of percolating field on the the large-scale behavior of the non-equilibrium phase
velocity fluctuations. In other words, our model corre- transition between the active (ψ > 0) and the absorb-
sponds to the passive advection of the reacting scalar ing state (ψ = 0). The Gaussian noise term ξ with zero
field. mean has to satisfy the absorbing state condition. Its
A powerful tool for analysis of the critical behavior correlation function can be chosen in the following form
is the renormalization group (RG) [37–39] method. It
constitutes a theoretical framework which allows one to (cid:104)ξ(t ,x )ξ(t ,x )(cid:105)=g D ψ(t ,x )δ(t −t )δ(d)(x −x )
1 1 2 2 0 0 1 1 1 2 1 2
compute universal quantities in a controllable manner (2)
and also to determine universality classes of the physical up to irrelevant contributions [3]. Here δ(d)(x) is the
system. Here this method is employed in order to de- d-dimensional generalization of the standard Dirac δ(x)-
termine the scaling behavior in the vicinity of the phase function.
transition between the active and absorbing state with A further step consists in incorporating of the velocity
an emphasis on a possible type of critical behavior. fluctuations into the model (1). The standard route [24]
The remainder of the paper proceeds as follows. In based on the replacement ∂ by the Lagrangian deriva-
t
Sec. II, we introduce a coarse-grained formulation of the tive∂ +(v·∇)isnotsufficientduetotheassumedcom-
t
problem, which we reformulate into the field-theoretic pressibility. As shown in [40], the following replacement
model. InSec.III,wedescribethemainstepsoftheper- is then adequate
turbative RG procedure. In Sec. IV, we present analysis
of possible regimes involved in the model. We analyze ∂ →∂ +(v·∇)+a (∇·v), (3)
t t 0
numericallyandtosomeextentanalyticallyfixedpoints’
structure. In Sec. V, we give a concluding summary. where a is an additional positive parameter, whose sig-
0
Technical details concerning calculation of RG constants nificance will be discussed later. Note that the last term
andfunctionsarepresentedinAppendixAandAppendix in (3) contains a divergence of the velocity field v and
B. The coordinates of all fixed points are given in Ap- thus ∇ operator does not act on what could possibly
pendix C. follow.
Following [36], we consider the velocity field to be a
randomGaussianvariablewithzeromeanandatransla-
II. THE MODEL tionally invariant correlator given as follows:
A continuum description of DP in terms of a density (cid:90) dω (cid:90) ddk
(cid:104)v (t,x)v (0,0)(cid:105)= D (ω,k)e−iωt+k·x,
ψ =ψ(t,x)ofinfectedindividualstypicallyarisesfroma i j 2π (2π)d v
coarse-grainingprocedureinwhichalargenumberoffast (4)
microscopic degrees of freedom are averaged out. A loss where the kernel function D (ω,k) takes the form
v
of the physical information is supplemented by a Gaus-
sian noise in a resulting Langevin equation. Obviously, g u D3k4−d−y−η
D (ω,k)=[Pk +αQk] 10 10 0 . (5)
a correct mathematical description has to be in confor- v ij ij ω2+u2 D2(k2−η)2
10 0
mity regarding the absorbing state condition: ψ = 0 is
always a stationary state and no microscopic fluctuation Here, Pk = δ − k k /k2 is a transverse and Qk =
ij ij i j ij
can change that. The coarse grained stochastic equation
k k /k2 a longitudinal projection operator, k = |k|, and
then reads [8] i j
d is the dimensionality of the x space. A positive pa-
rameter α > 0 can be interpreted as the simplest pos-
g D
∂tψ =D0(∇2−τ0)ψ− 02 0ψ2+ξ, (1) sible deviation [35] from the incompressibility condition
∇·v =0. The incompressible case, α=0, was analyzed
where ξ denotes the noise term, ∂ = ∂/∂t is the time in previous works [40–44]. The coupling constant g
t 10
derivative, ∇2 is the Laplace operator, D is the diffu- and the exponent y describe the equal-time velocity cor-
0
sionconstant,g isthecouplingconstantandτ measures relator or, equivalently, the energy spectrum [25, 28, 36]
0 0
a deviation from the threshold value for injected proba- of the velocity fluctuations. The constant u > 0 and
10
bility. It can be thought as an analog to the tempera- theexponentηarerelatedtothecharacteristicfrequency
ture variable in the standard ϕ4−theory [8, 38]. Due to ω (cid:39)u D k2−η of the mode with the wavelength k.
10 0
dimensional reasons, we have extracted the dimensional The momentum integral in (4) has an infrared (IR)
part from the interaction term (See later Sec. IIIA). cutoff at k = m, where m ∼ 1/L is the reciprocal of the
Hereandhenceforthwedistinguishbetweenunrenormal- integral scale L. A precise form of the cutoff [30, 45] is
ized(withthesubscript“0”)quantitiesandrenormalized actuallyunimportantanditsroleistoprovideuswithIR
terms (without the subscript “0”). The renormalized regularization. Further,dimensionalconsiderationsshow
fields will be later denoted by the subscript R. that the bare coupling constants g and u are related
10 10
Itcanberigorouslyproven[5]thattheLangevinequa- to the characteristic UV momentum scale Λ by
tion (1) captures the gross properties of the percolation
processandcontainsessentialphysicalinformationabout g (cid:39)Λy, u (cid:39)Λη. (6)
10 10
3
The choice y = 8/3 gives the famous Kolmogorov “five- D−1(t −t ,x −x )v (t ,x ), (11)
ij 1 2 1 2 j 2 2
thirds” law for the spatial velocity correlations, and η =
whereD−1 isthekerneloftheinverselinearoperationin
4/3 corresponds to the Kolmogorov frequency [25]. ij
The exponents y and η are analogous to the standard (4). The final interaction part can be written as
expansion parameter ε=4−d in the static critical phe- (cid:90) (cid:90) (cid:26)D λ u
nomena. It can be shown that the upper critical dimen- Sint[ϕ]= dt ddx 02 0[ψ−ψ˜]ψ˜ψ− 2D20 ψ˜ψv2
sion of the pure percolation problem [8] is also d = 4. 0
c (cid:27)
Therefore, we retain the standard notation for the expo- +ψ˜(v·∇)ψ+a ψ˜(∇·v)ψ . (12)
0
nent ε. According to the general rules [39] of the RG
approach, we formally assume that the exponents ε,y
All but the third term in (12) directly stem from the
and η are of the same order of magnitude and constitute
nonlinear terms in (1) and (3). The third term propor-
small expansion parameters of perturbation theory.
tional to ∝ ψ˜ψv2 deserves a special consideration. The
The kernel function in (5) is chosen in a quite general
presenceofthistermisprohibitedintheoriginalKraich-
form and as such it contains various special limits. They
nan model due to the underlying Galilean invariance.
simplifynumericalanalysisoftheresultingequationsand
However,inourcasethegeneralformofthevelocityker-
allowsustogainadeeperphysicalinsightintothemodel.
nel function does not lead to such restriction. Moreover,
Possible limiting cases are
by direct inspection of the perturbative expansion, one
i) The rapid-change model, which corresponds to the canshowthatthiskindoftermisindeedgeneratedunder
limit u →∞,g(cid:48) ≡g /u =const. Then for the RG transformation (consider second Feynman graph in
10 10 10 10
kernel function we have the expression (A5)). This term was considered for the
first time in our previous work [44], where the incom-
D (ω,k)∝g(cid:48) D k−d−y+η (7)
v 10 0 pressible case is analyzed.
andobviouslythevelocitycorrelatorisδ−correlated Let us also note that for the linear advection-diffusion
in a time variable. equation [24, 36], the choice a0 = 1 corresponds to
the conserved quantity ψ (advection of a density field),
ii) The frozen velocity field, which arises in the limit
whereas for the choice a = 0 the conserved quantity is
0
u10 →0 and the kernel function corresponds to ψ˜ (advection of a tracer field). From the point of view
Dv(ω,k)∝g0D02πδ(ω)k2−d−y. (8) of the renormalization group, the introduction of a0 is
necessary, because it ensures multiplicative renormaliz-
iii) Thepurelypotentialvelocityfield,whichisobtained ability of the model [40].
forα→∞withαg =constant. Thislimitissimilar Inprinciple,basicingredientsofanystochastictheory,
10
to the model of random walks in a random environ- correlation and response functions of the concentration
ment with long-range correlations [46, 47]. fieldψ(t,x),canbecomputedasfunctionalaverageswith
respecttotheweightfunctionalexp(−S)withaction(9).
iv) The turbulent advection, for which the y = 2η =
Further, the field-theoretic formulation summarized in
8/3. This choice mimics properties of the genuine
(10)-(12) has an additional advantage to be amenable to
turbulence and leads to the celebrated Kolmogorov
the full machinery of (quantum) field theory [38, 39]. In
scaling [25].
the subsequent section, we apply the RG perturbative
For an effective use of the RG method it is advanta- technique [39] that allows us to study the model in the
geous to rewrite the stochastic problem (1-5) into the vicinity of its upper critical dimension dc =4.
field-theoretic formulation. This could be achieved in
the standard fashion [48–50] and the resulting dynamic
functional is III. RENORMALIZATION GROUP ANALYSIS
S[ϕ]=S [ϕ]+S [ϕ]+S [ϕ], (9)
diff vel int An important goal of statistical theories is the deter-
where ϕ={ψ˜,ψ,v} stands for the complete set of fields mination of correlation and response functions (usually
andψ˜istheauxiliary(Martin-Siggia-Rose)responsefield called Green functions) of the dynamical fields as func-
tions of the space-time coordinates. Traditionally, these
[51]. Thefirsttermrepresentsafreepartoftheequation
functions are represented in the form of sums over the
(1) and is given by the following expression:
Feynman diagrams [38, 39]. The functional formulation
(cid:90) (cid:90) (cid:26) (cid:27)
provides a convenient theoretical framework suitable for
S [ϕ]= dt ddx ψ˜[∂ −D ∇2+D τ ]ψ . (10)
diff t 0 0 0 applying methods of quantum field theory. Using the
RG method [38, 52] it is possible to determine the in-
SincethevelocityfluctuationsaregovernedbytheGaus-
frared (IR) asymptotic (large spatial and time scales)
sian statistics, the corresponding averaging procedure is
behavior of the correlation functions. A proper renor-
performed with the quadratic functional
malization procedure is needed for the elimination of ul-
1(cid:90) (cid:90) (cid:90) (cid:90) traviolet (UV) divergences. There are various renormal-
S [v]= dt dt ddx ddx v (t ,x )
vel 2 1 2 1 2 i 1 1 ization prescriptions applicable for such problem, each
4
with its own advantages [38]. In this work, we employ Q ψ,ψ˜ v D τ g λ u u ,a ,α
0 0 10 0 10 20 0
the minimal subtraction (MS) scheme. UV divergences
manifest themselves in the form of poles in the small ex- dk d/2 −1 −2 2 y ε/2 η 0
Q
pansionparameters,andtheminimalsubtractionscheme
dω 0 1 1 0 0 0 0 0
ischaracterizedbydiscardingallfinitepartsoftheFeyn- Q
mangraphsinthecalculationoftherenormalizationcon-
d d/2 1 0 2 y ε/2 η 0
Q
stants. Inthevicinityofcriticalpointslargefluctuations
on all spatio-temporal scales dominate the behavior of
TableI.Canonicaldimensionsofthebarefieldsandbarepa-
the system, which in turn results in the divergences in
rameters for the model (10)-(12).
the Feynman graphs. The resulting RG functions satisfy
certain differential equations and their analysis provides
us with an efficient computational technique for estima- Γ1−ir Γψ˜ψ Γψ˜ψv Γψ˜2ψ Γψ˜ψ2 Γψ˜ψv2
tion of universal quantities.
d 2 1 ε/2 ε/2 0
Γ
δ 2 1 0 0 0
Γ
A. Canonical dimensions
TableII.Canonicaldimensionsforthe(1PI)divergentGreen
In order to apply the dimensional regularization for functions of the model.
evaluation of renormalization constants, an analysis of
possiblesuperficialdivergenceshastobeperformed. For
Dimensional analysis should be augmented by certain
translationally invariant systems, it is sufficient to ana-
additional considerations. In dynamical models of the
lyze only 1-particle irreducible (1PI) graphs [37, 38]. In
type(10)-(12),allthe1-irreduciblediagramswithoutthe
contrasttostaticmodels,dynamicmodels[3,39]contain
two independent scales: a frequency scale dω and a mo- fields ψ˜ vanish, and it is sufficient to consider the func-
Q
tions with N ≥ 1. As was shown in [40], the rapidity
mentumscaledk foreachquantityQ. Thecorresponding ψ˜
Q symmetry(24)requiresalsoN ≥1tohold. Usingthese
dimensions are found using the standard normalization ψ
considerations together with relation (15), possible UV
conditions
divergent structures are expected only for the 1PI Green
dk =−dk =1, dk =dk =0, functions listed in Table II.
k x ω t
dω =dω =0, dω =−dω =1 (13)
k x ω t
B. Computation of the RG constants
together with a condition for field-theoretic action to be
a dimensionless quantity. Using the quantities dω and
Q
dk, the total canonical dimension d , In this section, the main steps of the perturbative RG
Q Q
approach are summarized, deferring the explicit results
of the RG constants and RG functions (anomalous di-
d =dk +2dω (14)
Q Q Q mensions and beta functions) to Appendices A and B.
A starting point of the perturbation theory is a free
can be introduced, whose precise form is obtained from
a comparison of the IR most relevant terms (∂ ∝ ∇2) part of the action given by expressions (10) and (11).
t
By graphical means, they are represented as lines in the
in the action (10). The total dimension d for the dy-
Q
Feynmandiagrams, whereasthenon-lineartermsin(12)
namical models plays the same role as the conventional
correspond to vertices connected by the lines.
(momentum) dimensiondoesinstatic problems. The di-
For the calculation of the RG constants we have em-
mensions of all quantities for the model are summarized
ployed dimensional regularization in the combination
inTableI.Itfollowsthatthemodelislogarithmic(when
with the MS scheme [38]. Since the finite correlated
coupling constants are dimensionless) at ε = y = η = 0,
case involves two different dispersion laws: ω ∝ k2 for
and the UV divergences are in principle realized as poles
the scalar and ω ∝ k2−η for the velocity fields, the
intheseparameters. Thetotalcanonicaldimensionofan
calculations for the renormalization constants become
arbitrary 1− irreducible Green function is given by the
rather cumbersome already in the one-loop approxima-
relation
tion[28,36]. However,itwasshown[53]thattothetwo-
d =dk+2dω =d+2−(cid:88)N d , ϕ∈{ψ˜,ψ,v}. (15) loop order it is sufficient to consider the choice η = 0.
Γ Γ Γ ϕ ϕ
This significantly simplifies practical calculations and as
ϕ
can be seen in (A6), the only poles to the one-loop or-
The total dimension d in the logarithmic theory is the der are of two types: either 1/ε or 1/y. This simple
Γ
formal degree of the UV divergence δ = d | . picture pertains only to the lowest orders in a perturba-
Γ Γ ε=y=η=0
SuperficialUVdivergences,whoseremovalrequirescoun- tion scheme. In higher order terms, poles in the form of
terterms, could be present only in those functions Γ for general linear combinations in ε,η and y are expected to
which δ is a non-negative integer [39]. arise.
Γ
5
The perturbation theory of the model (9) is amenable
to the standard Feynman diagrammatic expansion [37– ψ ψ˜
39]. The inverse matrix of the quadratic part in the ac-
tions determines a form of the bare propagators. The
propagatorsarepresentedinthewave-number-frequency
representation, which is for the translationally invariant ψ˜ ψ
systems the most convenient way for doing explicit cal-
culations. The bare propagators are easily read off from
the Gaussian part of the model given by (10) and (11),
v v
respectively. Their graphical representation is depicted i j
in Fig. 1. The corresponding algebraic expressions can
beeasilyreadoffandinthefrequency-momentumrepre- Figure 1. Diagrammatic representation of the bare propaga-
sentation are given by tors. The time flows from right to left.
1
(cid:104)ψψ˜(cid:105) =(cid:104)ψ˜ψ(cid:105)∗ = , (16)
0 0 −iω+D (k2+τ ) ψ˜ ψ
0 0
g u D3k4−d−y−η
(cid:104)vv(cid:105) =[Pk +αQk] 10 10 0 (17) ,
0 ij ij ω2+u210D02(k2−η)2 ψ ψ˜
ψ˜ ψ
or in the time-momentum representation as
(cid:104)ψψ˜(cid:105) =θ(t)exp(−D [k2+τ ]t), (18)
0 0 0
Figure2. Diagrammaticrepresentationoftheinteractionver-
(cid:104)ψ˜ψ(cid:105)0 =θ(−t)exp(D0[k2+τ0]t), (19) tices describing an ideal directed bond percolation process.
g D2
(cid:104)vv(cid:105) =[Pk +αQk] 10 0 e−u10D0k2−η|t|, (20)
0 ij ij kd+y−2
Therefore,weintroduceanewchargeg viatherelation
20
where θ(t) is the Heaviside step function.
g =λ2 (25)
The interaction vertices from the nonlinear part (12) 20 0
describethefluctuationeffectsconnectedwiththeperco-
and express the perturbation calculation in terms of this
lation process itself, advection of the concentration field
parameter.
and the interactions between the velocity components.
Inthepresenceofcompressiblevelocityfieldthetrans-
With every such vertex the following algebraic factor
formation (24) has to be augmented by the transforma-
δNS [ϕ] tion
V (x ,...,x ;ϕ)= int , ϕ∈{ψ˜,ψ,v}
N 1 N δϕ(x )...δϕ(x )
1 N a →1−a , (26)
0 0
is associated [39]. In our model there are four differ-
as can be easily seen by inserting (24) in (12) and per-
ent interaction vertices, which are graphically depicted
forming integration by parts.
in Fig. 2 and Fig. 3, respectively. The corresponding
With the help of Table I the renormalized parameters
vertex factors are
can be introduced in the following manner:
V =−V =D λ , (21)
ψ˜ψψ ψ˜ψ˜ψ 0 0 D0 =DZD, τ0 =τZτ +τc, a0 =aZa,
V =ik +ia q , (22)
ψ˜ψv j 0 j g =g µyZ , u =u µηZ , λ =λµε/2Z ,
u 10 1 g1 10 1 u1 0 λ
V =− 20δ . (23) g =g µεZ , u =u Z , (27)
ψ˜ψvv D ij 20 2 g2 20 2 u2
0
whereµisthereferencemassscaleintheMSscheme[38].
IntheexpressionforV ,wehaveadoptedthefollowing
ψ˜ψv Notethatthetermτ isanon-perturbativeeffect[54,55],
convention: k is the momentum of the field ψ and q is c
j j
the momentum of the velocity field v. The presence of
the interaction vertex V leads to the proliferation of
ψ˜ψvv
the new Feynman graphs (see Appendix A), which were vj vj ψ˜
absent in the previous studies [40, 41, 44].
BydirectinspectionoftheFeynmandiagramsonecan ,
observe that the real expansion parameter is rather λ2 ψ˜
0 ψ v ψ
thanλ . Thisisadirectconsequenceofthedualitysym- i
0
metry [8] of the action for the pure percolation problem
with respect to time inversion
Figure 3. Interaction vertices describing the influence of the
ψ(t,x)→−ψ˜(−t,x), ψ˜(t,x)→−ψ(−t,x). (24) advectingvelocityfieldwiththeorderparameterfluctuations.
6
which is not captured by the dimensional regularization. Allfixedpointscanbefoundfromarequirementthatall
The renormalization prescription (27) together with the beta-functions of the model simultaneously vanish
renormalization of fields
β (g∗)=β (g∗)=β (g∗)=β (g∗)=β (g∗)=0,
g1 g2 u1 u2 a
ψ˜=Zψ˜ψ˜R, ψ =ZψψR, v =ZvvR (28) (32)
is sufficient for obtaining a fully renormalized theory. where g∗ stands for an entire set of charges
Thus, the total renormalized action for the renormalized {g∗,g∗,u∗,u∗,a∗}. In what follows, the asterisk will al-
1 2 1 2
fields ϕ ≡ {ψ˜ ,ψ ,v } can be written in a compact ways refer to coordinates of some fixed point. Whether
R R R R
form the given FP could be realized in physical systems (IR
stable)ornot(IRunstable)isdeterminedbyeigenvalues
(cid:90) (cid:90) (cid:26) (cid:20)
SR[ϕR]= dt ddx ψ˜R Z1∂t−Z2D∇2+Z3Dτ of the matrix Ω={Ωij} with the elements
(cid:21) ∂β
+Z4(vR·∇)+aZ5(∇·vR) ψR− D2λ[Z6ψ˜R Ωij = ∂gji, (33)
(cid:27)
−Z ψ ]ψ˜ ψ −Z u2 ψ˜ ψ v2 + where βi is a full set of beta-functions and gj is the full
7 R R R 82D R R R setofcharges{g1,g2,u1,u2,a}. FortheIRstableFPthe
1(cid:90) (cid:90) (cid:90) (cid:90) real parts of the eigenvalues of the matrix Ω have to be
2 dt1 dt2 ddx1 ddx2 vRi(t1,x1) strictly positive. In general, these conditions determine
a region of stability for the given FP in terms of ε,η and
DR−i1j(t1−t2,x1−x2)vRj(t2,x2). y.
(29) Furthermore, to obtain the RG equation, one can ex-
ploitafactthatthebareGreenfunctionsareindependent
The latter term is a renormalized version of (11). The of the momentum scale µ [37]. Applying the differential
relations between the renormalization constants follow operator µ∂ at the fixed bare quantities leads to the
µ
directly from the action (29) following equation for the renormalized Green function
G
R
Z =Z Z , Z =Z Z Z ,
1 ψ ψ˜ 2 ψ ψ˜ D
{D +N γ +N γ +N γ }G (e,µ,...)=0, (34)
Z =Z Z Z Z , Z =Z Z Z , RG ψ ψ ψ˜ ψ˜ v v R
3 ψ ψ˜ D τ 4 ψ ψ˜ v
Z =Z Z Z Z , Z =Z Z2Z Z , where GR is a function of the full set e of renor-
5 ψ ψ˜ v a 6 ψ ψ˜ D λ malized counterparts to the bare parameters e =
0
Z7 =Zψ2Zψ˜ZDZλ, Z8 =ZψZψ˜Zv2Zu2ZD−1. (30) {D0,τ0,u10,u20,g10,g20,a0}, the reference mass scale µ
and other parameters, e.g. spatial and time variables.
The theory is made UV finite through the appropriate The RG operator D is given by
RG
choice of the RG constants Z ,...,Z . Afterwards, rela-
1 8
(cid:88)
tions (30) yield the corresponding RG constants for the D ≡µ∂ | =µ∂ + β ∂ −γ D −γ D , (35)
RG µ 0 µ g g D D τ τ
fields and parameters appearing in relations (27). The
g
explicit results for the RG constants are given in Ap-
pendix A. where g ∈ {g1,g2,u1,u2,a}, Dx = x∂x for any variable
According to the general rules of the RG method [39], x, ...|0 stands for fixed bare parameters and γx are the
thenonlocalterminaction(29)shouldnotberenormal- so-calledanomalousdimensionsofthequantityxdefined
ized. From the inspection of the kernel function (5) two as
additional relations
γ ≡µ∂ lnZ | . (36)
x µ x 0
1=Z Z , 1=Z Z Z3Z−2 (31)
u1 D u1 g1 D v The beta-functions, which express the flows of param-
eters under the RG transformation [37], are defined
follow, which have to be satisfied to all orders in the
through
perturbation scheme.
β =µ∂ g| . (37)
g µ 0
IV. FIXED POINTS AND SCALING REGIMES Applying this definition to relations (27) yields
β =g (−y+2γ −2γ ), β =g (−ε−γ ),
Oncetherenormalizationproceduretoagivenorderof g1 1 D v g2 2 g2
perturbationschemeisperformed,wecanfindthescaling β =u (−η+γ ), β =−u γ ,
u1 1 D u2 2 u2
behavior in the infrared IR limit by studying the flow β =−aγ . (38)
a a
as µ → 0. According to the general statement of the
RG theory [37, 39], a possible IR asymptotic behavior is Thelastequationsuggeststhatforthefixedpoints’equa-
governed by the fixed point (FP) of the beta-functions. tion β (g∗)=0 either a=0 or a(cid:54)=0 has to be satisfied.
a
7
However, as the explicit results (B4) show, this is not N(t)∼t−(γψ+γψ˜)/∆ω, (46)
true (parameter a appears also in the denominator of
P(t)∼t−(d+γψ+γψ˜)/2∆ω. (47)
γ )andtheright-handsideofβ hastobeconsideredas
a a
a whole expression. A similar reasoning also applies for
From the structure of anomalous dimensions (B4-B5)
the function β .
u2 itisclearthattheresultingsystemofequationsforFPsis
Itturnsoutthatforsomefixedpointsthecomputation
quitecomplicated. Althoughtosomeextentitispossible
of the eigenvalues of the matrix (33) is cumbersome and
to obtain coordinates of the fixed points, the eigenvalues
rather unpractical. In those cases it is possible to obtain
of the matrix (33) pose a more severe technical problem.
information about the stability from analyzing RG flow
Hence, in order to gain some physical insight into the
equations[39]. Itsessentialideaistostudyasetofinvari-
structure of the model, we divide overall analysis into
ant charges g = g(s,g) with the initial data g| = g.
s=1 special cases and analyze them separately.
The parameter s stands for a scaling parameter and one
isinterestedinthebehaviorofchargesinthelimits→0.
The evolution of invariant charges is given by the equa- A. Rapid change
tion
First,weperformananalysisoftherapid-changelimit
D g =β(g). (39)
s
of the model. It is convenient [28, 36] to introduce the
The very existence of IR stable solutions of the RG new variables g(cid:48) and w given by
1
equations leads to the existence of the scaling behavior
g 1
ofGreenfunctions. Indynamicalmodels, criticaldimen- g(cid:48) = 1, w = . (48)
sions of the quantity Q is given by the relations 1 u1 u1
∆ =dk +∆ dω +γ∗, ∆ =2−γ∗. (40) The rapid change limit then corresponds to fixed points
Q Q ω Q Q ω D
with a coordinate w∗ = 0. Using the definition (37)
The dQ and dQ are canonical dimensions of the quantity together with expressions (38) the beta-functions for the
k ω
Q calculated with the help of Tab. II, γ∗ is the value of charges (48) are easily obtained
Q
its anomalous dimension. Using Eqs. (40) we obtain the
β =g(cid:48)(η−y+γ −2γ ), β =w(η−γ ). (49)
following relations g(cid:48) 1 D v w D
1
d d Analyzing the resulting system of equations seven possi-
∆ = +γ , ∆ = +γ , ∆ =2+γ∗. (41)
ψ˜ 2 ψ˜ ψ 2 ψ τ τ ble regimes can be found. Their coordinates are listed in
Tab. III in Appendix C. Due to the cumbersome form
Important information about the physical system can
of the matrix (33), we were not able to determine all the
be read out from the behavior of correlation functions,
corresponding eigenvalues in an explicit form. In par-
which can be expressed in terms of the cumulant Green
ticular, for nontrivial fixed points (with non-zero coor-
functions. In the percolation problems one is typically dinates of g(cid:48),g and u ) the resulting expressions are of
interested[2,8]inthebehaviorofthefollowingfunctions 1 2 2
a quite unpleasant form. Nevertheless, using numerical
a) The number N(t,τ) of active particles generated by a software [56] it is possible to obtain all the necessary in-
seed at the origin formation about the fixed points’ structure and in this
way the boundaries between the corresponding regimes
(cid:90)
N(t)= ddx G (t,x). (42) havebeenobtained. Intheanalysisitisadvantageousto
ψψ˜ exploitadditionalconstraintsfollowingfromthephysical
interpretation of the charges. For example, g(cid:48) describes
1
b) ThemeansquareradiusR2(t)ofpercolatingparticles, the density of kinetic energy of the velocity fluctuations,
which started from the origin at time t=0 g is equal to λ2 and a(cid:48) will be later on introduced (see
2
(cid:82) ddx x2G (t,x) Appendix C) as (1−2a)2. Hence, it is clear that these
R2(t)= (cid:82) ψψ˜ . (43) parameters have to be non-negative real numbers. Fixed
2d ddx Gψψ˜(t,x) pointsthatviolatethisconditioncanbeimmediatelydis-
carded as non-physical.
c) SurvivalprobabilityP(t)ofanactiveclusteroriginat- Out of seven possible fixed points, only four are IR
ing from a seed at the origin stable: FPI, FPI, FPI and FPI. Thus, only regimes
1 2 5 6
which correspond to those points could be in principle
P(t)=− lim (cid:104)ψ˜(−t,0)e−k(cid:82)ddx ψ(0,x)(cid:105). (44) realized in real physical systems. As expected [36], the
k→∞
coordinates of these fixed points (see Tab. III) and the
scaling behavior of the Green functions (see Tab. VII)
By straightforward analysis [8] it can be shown that
dependonlyontheparameterξ =y−η. Inwhatfollows,
the scaling behavior of these functions is given by the
we restrict our discussion only to them.
asymptotic relations
The FPI represents the free (Gaussian) FP for which
1
R2(t)∼t2/∆ω, (45) all interactions are irrelevant and ordinary perturbation
8
6
HaL
4 FPI
6
2
y FPI FPI
1 2 FPI
0 5
-2
-4
Figure 4. A qualitative sketch of the regions of stability for -2 0 2 4 6 8 10
the fixed points in the limit of the rapid-change model. The Ε
bordersbetweentheregionsaredepictedwiththeboldlines.
6
HbL
theoryisapplicable. Asexpected,thisregimeisIRstable 4 FPI
6
in the region
y <η, η >0, ε<0. (50) 2
y FPI FPI
1 2 FPI
Thelatterconditionensuresthatweareabovetheupper 0 5
critical dimension d = 4. For FPI the correlator of
c 2
the velocity field is irrelevant and this point describes
standard the DP universality class [8] and is IR stable in -2
the region
-4
ε>0, ε/12+η >y, ε<12η. (51) -2 0 2 4 6 8 10
Ε
The remaining two fixed points constitute nontrivial
6
regimes for which velocity fluctuations as well as perco- HcL
lation interaction become relevant. The FPI is IR stable
5
in the region given by 4 FPI
6
(α+3)ε>3(2α+7)(y−η), 12(y−η)>ε, 2η >y.
2
(52)
TheboundariesforFPI6 canbeonlycomputedbynumer- y FP1I FP2I FPI
ical calculations. 0 5
Using the information about the phase boundaries,
a qualitative picture of the phase diagram can be con-
-2
structed. In Fig. 4 the situation in the plane (ε,y) is de-
picted. We observe that compressibility affects only the
outer boundary of FPI. The larger value of α the larger -4
5
areaofstability. Wealsoobservethattherealizabilityof -2 0 2 4 6 8 10
the regime FPI crucially depends on the nonzero value Ε
5
of η.
The important subclass of the rapid-change limit con-
Figure5. Fixedpoints’structureforthethermalnoisesitua-
stitutes thermal velocity fluctuations, which are charac- tion (53). From above to bottom the compressibility param-
terized by the quadratic dispersion law [57]. In our for- eter α attains consecutively the values: (a) α=0, (b) α=5
mulationthisisachievedbyconsideringthefollowingre- and (c) α=100.
lation:
η =6+y−ε (53)
is that of pure DP. The nontrivial regimes FPI and FPI
5 6
which follows directly from expression (7). The situa- are realized only in the nonphysical region for large val-
tion for increasing values of the parameter α is depicted ues of ε. This numerical result confirms our previous
in Fig. 5. We see that for physical space dimensions expectations [40, 41]. It was pointed out [58, 59] that
d = 3 (ε = 1) and d = 2 (ε = 2) the only stable regime genuinethermalfluctuationscouldchangeIRstabilityof
9
thegivenuniversalityclass. However, thisisnotrealized 4
for the percolation process. HaL
3
B. Regime of frozen velocity field FPII
2 7
Accordingtoequation(8),theregimeofthefrozenve- y
locity field corresponds to the constraint u∗ = 0. Using 1
1
0 FPII
2
FPII
1
-1
-2 0 2 4 6 8 10 12
Ε
4
HbL
3 FPII
7
2
Figure6. Regionsofstabilityforthefixedpointsinthelimit
of the frozen velocity field. The borders between the regions y
are depicted with the bold lines.
1
a general form of anomalous dimensions (B4) with the
given constraint on u1 eight possible fixed points are ob- 0 FP2II
tained. Their coordinates are listed in Tab. IV in Ap- FPII
1
pendix C. However, only three of them (FPI1I, FPI2I and -1
FPII) could be physically realized (IR stable). -2 0 2 4 6 8 10 12
7
ThefixedpointFPII describesthefree(Gaussian)the- Ε
1
ory. It is stable in the region
4
HcL
y <0, ε<0, η <0. (54)
3
For FPI2I the velocity field is asymptotically irrelevant FPII
andtheonlyrelevantinteractionisduetothepercolation 7
2
process itself. This regime is stable in the region
y
ε>6y, ε>0, ε>12η. (55)
1
On the other hand, FPII represents a truly nontrivial
regime for which both vel7ocity and percolation are rele- 0 FPII
2
vant. The regions of stability for the FPII and FPII are FPII
1 2 1
depictedinFig. 6. Sinceforthesetwopointsthevelocity
-1
fieldcouldbeeffectivelyneglected,thetrivialobservation -2 0 2 4 6 8 10 12
is that these boundaries do not depend on the value of Ε
the parameter α. The stability region of FPII can be
7
computed only numerically.
Figure7. Fixedpoints’structureforfrozenvelocitycasewith
In order to study the influence of compressibility on
η =0. From above to bottom the compressibility parameter
thestabilityinthenontrivialregimeFPII, wehavestud-
7 α attains consecutively the values: (a) α = 0, —(b) α = 3.5
iedsituationforη =0. Forothervaluesofηthesituation
and (c) α=8.
remainsqualitativelythesame. Thesituationforincreas-
ingvaluesofα isdepictedinFig. 7. Weobservethatfor
α=0thereisaregionofstabilityforFPII,whichshrinks
7
for the immediate value α = 3.5 to a smaller area. Nu- the (y,ε) plane. The compressibility thus changes pro-
merical analysis shows that this shrinking continues well foundlyasimplepictureexpectedfromanincompressible
down to the value α = 6. A further increase of α leads case. Altogether the advection process becomes more ef-
to a substantially larger region of stability for the given ficientduetothecombinedeffectsofcompressibilityand
FP. Already for α = 8 this region covers all the rest of the nonlinear terms.
10
6
C. Turbulent advection
HaL
5
In the last part we focus on a special case of the tur-
bulentadvection. Ourmainaimistodeterminewhether 4 FPI
6
Kolmogorovregime[25],whichcorrespondstothechoice
FPI
y = 2η = 8/3, could lead to a new nontrivial regime for 3 5
the percolation process. In this section, the parameter η y
2
isalwaysconsideredtoattainitsKolmogorovvalue,4/3.
Forabettervisualizationwepresenttwo-dimensionalre-
1
gionsofstabilityintheplane(ε,y)fordifferentvaluesof
the parameter α. 0 FPI FPI
1 2
First of all, we reanalyze the situation for the rapid-
change model. The result is depicted in Fig. 8. It is -1
clearly visible that for this case a realistic turbulent sce- 0 5 10 15
nario (ε = 1 or ε = 2) falls out of the possible stable Ε
regions. Thisresultisexpectedbecausetherapid-change 6
model with vanishing time-correlations could not prop- HbL
erly describe well-known turbulent properties [25, 26]. 5
We also observe that compressibility mainly affects the
boundaries between the regions FPI5 and FPI6. However, 4 FP6I
this happens mainly in the nonphysical region. FPI
3 5
Next, we turn our attention to a similar analysis for
y
the frozen velocity field. The corresponding stability re- 2
gions are depicted in Fig. 9. Here we see that the sit-
uation is more complex. The regime FPII is situated in 1
2
the non-physical region and could not be realized. For
small values of the parameter α the Kolmogorov regime 0 FP1I FP2I
(depicted by a point) does not belong to the frozen ve-
-1
locity limit. However, from a special value α = 6 up to
0 5 10 15
α→∞theKolmogorovregimebelongstothefrozenve-
Ε
locity limit. Note that the bottom line for the region of
stability of FPI7I is exactly given by y =4/3. We observe 6 HcL
that compressibility affects mainly the boundary of the
5
nontrivialregion. Weconcludethatthepresenceofcom-
pressibility has a stabilizing effect on the regimes where 4 FPI
nonlinearities are relevant. 6
FPI
Finally, we look carefully at the nontrivial regime, 3 5
whichmeansthatnospecialrequirementswerelaidupon y
the parameter u . As obtaining of analytical results 2
1
proves to be too difficult, we have analyzed numerically
1
thedifferentialequationsfortheRGflow(39). Wefound
thatthebehavioroftheRGflowsisasfollows. Thereex- 0 FPI FPI
1 2
istsaborderlineintheplane(ε,α )givenapproximately
c
by the expression -1
0 5 10 15
α =−12.131ε+117.165. (56)
c Ε
Below α , only the frozen velocity regime correspond-
c
ing to FPII is stable. Above α , three fixed points FPII, Figure 8. Fixed points’ structure for rapid change model
7 c 7
FPIIIandFPIIIareobserved. Whereastwoofthem(FPII with η = 4/3. From above to bottom the compressibility
1 2 7
and FPIII) are IR stable, the remaining one FPIII is un- parameter α attains consecutively the values: (a) α=0, (b)
1 2
α=5and(c)α=∞. Thedotdenotesthecoordinatesofthe
stable in the IR regime. Again one of the stable FPs
corresponds to FPII, but the new FP is a regime with three-dimensional Kolmogorov regime.
7
finite correlation time. For the reference the coordinates
of these two points for the value α = 110 are given in
Tabs. VandVIinAppendixC.Sinceallfreeparameters stochastic magnetohydrodynamic turbulence [60], where
(ε,η,y,α)arethesameforbothpoints, whichofthetwo the crucial role is played by a forcing decay-parameter.
pointswillberealizeddependsontheinitialvaluesofthe ForillustrationpurposestheprojectionsoftheRGflow
bare parameters. A similar situation is observed for the onto the planes (g ,u ) and (g ,u ) are depicted in Fig.
1 1 2 1