fully developed isotropic turbulence symmetries and exact identities we consider the regime of fully developed isotropic and homogeneous turbulence of the navier stokes equation with a stochastic forcing we present two gauge symmetries of the corresponding navier stokes field theory and derive the associated general ward identities furthermore by introducing a local source bilinear in the velocity field we show that these symmetries entail an infinite set of exact and local relations between correlation functions they include in particular the k arm an howarth relation and another exact relation for a pressure velocity correlation function recently derived in g falkovich i fouxon and y oz j fluid mech 644 465 2010 that we further generalize

perceptual feedback in multigrid motion estimation using an improved dct quantization in this paper a multigrid motion compensation video coder based on the current human visual system hvs contrast discrimination models is proposed a novel procedure for the encoding of the prediction errors has been used this procedure restricts the maximum perceptual distortion in each transform coefficient this subjective redundancy removal procedure includes the amplitude nonlinearities and some temporal features of human perception a perceptually weighted control of the adaptive motion estimation algorithm has also been derived from this model perceptual feedback in motion estimation ensures a perceptual balance between the motion estimation effort and the redundancy removal process the results show that this feedback induces a scale dependent refinement strategy that gives rise to more robust and meaningful motion estimation which may facilitate higher level sequence interpretation perceptually meaningful distortion measures and the reconstructed frames show the subjective improvements of the proposed scheme versus an h 263 scheme with unweighted motion estimation and mpeg like quantization

nonperturbative renormalization group for the stationary kardar parisi zhang equation scaling functions and amplitude ratios in 1 1 2 1 and 3 1 dimensions we investigate the strong coupling regime of the stationary kardar parisi zhang equation for interfaces growing on a substrate of dimension d 1 2 and 3 using a nonperturbative renormalization group nprg approach we compute critical exponents correlation and response functions extract the related scaling functions and calculate universal amplitude ratios we work with a simplified implementation of the second order in the response field approximation proposed in a previous work phys rev e 84 061150 2011 and phys rev e 86 019904 e 2012 which greatly simplifies the frequency sector of the nprg flow equations while keeping a nontrivial frequency dependence for the two point functions the one dimensional scaling function obtained within this approach compares very accurately with the scaling function obtained from the full second order nprg equations and with the exact scaling function furthermore the approach is easily applicable to higher dimensions and we provide scaling functions and amplitude ratios in d 2 and d 3 we argue that our ansatz is reliable up to d symbol see text 3 5

renormalization group operator product expansion and anomalous scaling in models of turbulent advection recent progress on the anomalous scaling in models of turbulent heat and mass transport is reviewed with the emphasis on the approach based on the field theoretic renormalization group rg and operator product expansion ope in that approach the anomalous scaling is established as a consequence of the existence in the corresponding field theoretic models of an infinite number of dangerous composite fields operators with negative critical dimensions which are identified with the anomalous exponents this allows one to calculate the exponents in a systematic perturbation expansion similar to the e expansion in the theory of critical phenomena the rg and ope approach is presented in a self contained way for the example of a passive scalar field temperature concentration of an impurity etc advected by a self similar gaussian velocity ensemble with vanishing correlation time the so called kraichnan s rapid change model where the anomalous exponents are known up to order o e3 effects of anisotropy compressibility and the correlation time of the velocity field are discussed passive advection by non gaussian velocity field governed by the stochastic navier stokes equation and passively advected vector e g magnetic fields are considered

anomalous scaling in the n point functions of a passive scalar a recent analysis of the four point correlation function of the passive scalar advected by a time decorrelated random flow is extended to the n point case it is shown that all stationary state inertial range correlations are dominated by homogeneous zero modes of singular operators describing their evolution we compute analytically the zero modes governing the n point structure functions and the anomalous dimensions corresponding to them to the linear order in the scaling exponent of the two point function of the advecting velocity field the implications of these calculations for the dissipation correlations are discussed textcopyright 1996 the american physical society

exact scaling functions for one dimensional stationary kpz growth we determine the stationary two point correlation function of the one dimensional kpz equation through the scaling limit of a solvable microscopic model the polynuclear growth model the equivalence to a directed polymer problem with specific boundary conditions allows one to express the corresponding scaling function in terms of the solution to a riemann hilbert problem related to the painleve ii equation we solve these equations numerically with very high precision and compare our up to numerical rounding exact result with the prediction of colaiori and moore 1 obtained from the mode coupling approximation

anomalous scaling in homogeneous isotropic turbulence the anomalous scaling exponents n of the longitudinal structure functions sn for homogeneous isotropic turbulence are derived from the navier stokes equations by using field theoretic methods to develop a low energy approximation in which the kolmogorov theory is shown to act effectively as a mean field theory the corrections to the kolmogorov exponents are expressed in terms of the anomalous dimensions of the composite operators which occur in the definition of sn these are calculated from the anomalous scaling of the appropriate class of nonlinear green function using an uv fixed point of the renormalization group which thereby establishes the connection with the dynamics of the turbulence the main result is an algebraic expression for n which contains no adjustable constants it is valid at orders n below g 1 where g is the fixed point coupling constant this expression is used to calculate n for orders in the range n 2 10 and the results are shown to be in good agreement with experimental data key examples being 2 0 7 3 1 and 6 1 8

analytic calculation of the anomalous exponents in turbulence using the fusion rules to flush out a small parameter the main difficulty of statistical theories of fluid turbulence is the lack of an obvious small parameter in this paper we show that the formerly established fusion rules can be employed to develop a theory in which kolmogorov s statistics of 1941 k41 acts as the zero order or background statistics and the anomalous corrections to the k41 scaling exponents zeta n of the nth order structure functions can be computed analytically the crux of the method consists of renormalizing a four point interaction amplitude on the basis of the fusion rules this amplitude includes a small dimensionless parameter which is shown to be of the order of the anomaly of zeta 2 delta 2 zeta 2 2 3 approximately 0 03 higher order interaction amplitudes are shown to be even smaller the corrections to k41 to o delta 2 result from standard logarithmically divergent ladder diagrams in which the four point interaction acts as a rung the theory allows a calculation of the anomalous exponents zeta n in powers of the small parameter delta 2 the n dependence of the scaling exponents zeta n stems from pure combinatorics of the ladder diagrams in this paper we calculate the exponents zeta n up to o delta32 previously derived bridge relations allow a calculation of the anomalous exponents of correlations of the dissipation field and of dynamical correlations in terms of the same parameter delta 2 the actual evaluation of the small parameter delta 2 from first principles requires additional developments that are outside the scope of this paper

exact resummations in the theory of hydrodynamic turbulence iii scenarios for anomalous scaling and intermittency elements of the analytic structure of anomalous scaling and intermittency in fully developed hydrodynamic turbulence are described we focus here on the structure functions of velocity differences that satisfy inertial range scaling laws s n r sim r zeta n and the correlation of energy dissipation k epsilon epsilon r sim r mu the goal is to understand the exponents zeta n and mu from first principles in paper ii of this series it was shown that the existence of an ultraviolet scale the dissipation scale eta is associated with a spectrum of anomalous exponents that characterize the ultraviolet divergences of correlations of gradient fields the leading scaling exponent in this family was denoted delta the exact resummation of ladder diagrams resulted in the calculation of delta which satisfies the scaling relation delta 2 zeta 2 in this paper we continue our analysis and show that nonperturbative effects may introduce multiscaling i e zeta n not being linear in n with the renormalization scale being the infrared outer scale of turbulence l it is shown that deviations from k41 scaling of s n r zeta n neq n 3 must appear if the correlation of dissipation is mixing i e mu 0 we derive an exact scaling relation mu 2 zeta 2 zeta 4 we present analytic expressions for zeta n for all n and discuss their relation to experimental data one surprising prediction is that the time decay constant tau n r propto r z n of s n r scales independently of n the dynamic scaling exponent z n is the same for all n order quantities z n zeta 2

improved epsilon expansion for three dimensional turbulence two loop renormalization near two dimensions an improved epsilon expansion in the d dimensional d 2 stochastic theory of turbulence is constructed at two loop order which incorporates the effect of pole singularities at d 2 in coefficients of the epsilon expansion of universal quantities for a proper account of the effect of these singularities two different approaches to the renormalization of the powerlike correlation function of the random force are analyzed near two dimensions by direct calculation it is shown that the approach based on the mere renormalization of the nonlocal correlation function leads to contradictions at two loop order on the other hand a two loop calculation in the renormalization scheme with the addition to the force correlation function of a local term to be renormalized instead of the nonlocal one yields consistent results in accordance with the ultraviolet uv renormalization theory the latter renormalization prescription is used for the two loop renormalization group analysis amended with partial resummation of the pole singularities near two dimensions leading to a significant improvement of the agreement with experimental results for the kolmogorov constant

nonperturbative renormalization group for the kardar parisi zhang equation general framework and first applications we present an analytical method rooted in the nonperturbative renormalization group that allows one to calculate the critical exponents and the correlation and response functions of the kardar parisi zhang kpz growth equation in all its different regimes including the strong coupling one we analyze the symmetries of the kpz problem and derive an approximation scheme that satisfies the linearly realized ones we implement this scheme at the minimal order in the response field and show that it yields a complete qualitatively correct phase diagram in all dimensions with reasonable values for the critical exponents in physical dimensions we also compute in one dimension the full momentum and frequency dependent correlation function and the associated universal scaling function we find a very satisfactory quantitative agreement with the exact result from prahofer and spohn j stat phys 115 255 2004 in particular we obtain for the universal amplitude ratio g 0 1 149 18 to be compared with the exact value g 0 1 1504 the baik and rain j stat phys 100 523 2000 constant we emphasize that all these results which can be systematically improved are obtained with sole input the bare action and its symmetries without further assumptions on the existence of scaling or on the form of the scaling function

computing the scaling exponents in fluid turbulence from first principles the formal setup abstract we propose a scheme for the calculation from the navier stokes equations of the scaling exponents n of the nth order correlation functions in fully developed hydrodynamic turbulence the scheme is nonperturbative and constructed to respect the fundamental rescaling symmetry of the euler equation it constitutes an infinite hierarchy of coupled equations that are obeyed identically with respect to scaling for any set of scaling exponents n as a consequence the scaling exponents are determined by solvability conditions and not from power counting it is argued that in order to achieve such a formulation one must recognize that the many point space time correlation functions are not scale invariant in their time arguments the assumption of full scale invariance leads unavoidably to kolmogorov exponents it is argued that the determination of all the scaling exponents n requires equations for infinitely many renormalized objects one can however proceed in controlled successive approximations by successive truncations of the infinite hierarchy of equations clues as to how to truncate without reintroducing power counting can be obtained from renormalized perturbation theory to this aim we show that the fully resummed perturbation theory is equivalent in its contents to the exact hierarchy of equations obeyed by the nth order correlation functions and green s function in light of this important result we can safely use finite resummations to construct successive closures of the infinite hierarchy of equations this paper presents the conceptual and technical details of the scheme the analysis of the high order closure procedures which do not destroy the rescaling symmetry and the actual calculations for truncated models will be presented in a forthcoming paper in collaboration with v belinicher

nonperturbative renormalization group study of the stochastic navier stokes equation we study the renormalization group flow of the average action of the stochastic navier stokes equation with power law forcing using galilean invariance we introduce a nonperturbative approximation adapted to the zero frequency sector of the theory in the parametric range of the holder exponent 4 2e of the forcing where real space local interactions are relevant in any spatial dimension d we observe the convergence of the resulting renormalization group flow to a unique fixed point which yields a kinetic energy spectrum scaling in agreement with canonical dimension analysis kolmogorov s 5 3 law is thus recovered for e 2 as also predicted by perturbative renormalization at variance with the perturbative prediction the 5 3 law emerges in the presence of a saturation in the e dependence of the scaling dimension of the eddy diffusivity at e 3 2 when according to perturbative renormalization the velocity field becomes infrared relevant

solutions of renormalization group flow equations with full momentum dependence we demonstrate the power of a recently proposed approximation scheme for the nonperturbative renormalization group that gives access to correlation functions over their full momentum range we solve numerically the leading order flow equations obtained within this scheme and compute the two point functions of the o n theories at criticality in two and three dimensions excellent results are obtained for both universal and nonuniversal quantities at modest numerical cost

fully developed isotropic turbulence nonperturbative renormalization group formalism and fixed point solution we investigate the regime of fully developed homogeneous and isotropic turbulence of the navier stokes ns equation in the presence of a stochastic forcing using the nonperturbative functional renormalization group nprg within a simple approximation based on symmetries we obtain the fixed point solution of the nprg flow equations that corresponds to fully developed turbulence both in d 2 and 3 dimensions deviations to the dimensional scalings kolmogorov in d 3 or kraichnan batchelor in d 2 are found for the two point functions to further analyze these deviations we derive exact flow equations in the large wave number limit and show that the fixed point does not entail the usual scale invariance thereby identifying the mechanism for the emergence of intermittency within the nprg framework the purpose of this work is to provide a detailed basis for nprg studies of ns turbulence the determination of the ensuing intermittency exponents is left for future work

solving a specific thue mahler equation the diophantine equation x3 3xy2 y3 3no 17 n l9n2 is completely solved as follows first a large upper bound for the variables is obtained from the theory of linear forms in p adic and real logarithms of algebraic numbers then this bound is reduced to a manageable size by p adic and real computational diophantine approximation based on the l3 algorithm finally the complete list of solutions is found in a sieving process the method is in principle applicable to any thue mahler equation as the authors will show in a forthcoming paper

towards a nonperturbative theory of hydrodynamic turbulence fusion rules exact bridge relations and anomalous viscous scaling functions in this paper we address nonperturbative aspects of the analytic theory of hydrodynamic turbulence of paramount importance for this theory are the fusion rules that describe the asymptotic properties of n point correlation functions when some of the coordinates tend toward one other we first derive here on the basis of two fundamental assumptions a set of fusion rules for correlations of velocity differences when all the separations are in the inertial interval using this set of fusion rules we consider the standard hierarchy of equations relating the n mathrm th order correlations originating from the viscous term in the navier stokes equations to n 1 mathrm th order originating from the nonlinear term and demonstrate that for fully unfused correlations the viscous term is negligible consequently the hierarchic chain of equations is decoupled in the sense that the correlations of n 1 mathrm th order satisfy a homogeneous equation that may exhibit anomalous scaling solutions using the same hierarchy of equations when some separations go to zero we derive on the basis of the navier stokes equations a second set of fusion rules for correlations with differences in the viscous range the latter includes gradient fields we demonstrate that every n mathrm th order correlation function of velocity differences mathcal f n r 1 r 2 dots exhibits its own crossover length ensuremath eta n to dissipative behavior as a function of say r 1 this length depends on n and on the remaining separations r 2 r 3 dots when all these separations are of the same order r this length scales as ensuremath eta n r ensuremath sim ensuremath eta frac r l x n with x n frac ensuremath zeta n ensuremath ensuremath zeta n 1 ensuremath zeta 3 ensuremath ensuremath zeta 2 2 ensuremath ensuremath zeta 2 with ensuremath zeta n being the scaling exponent of the n mathrm th order structure function we derive a class of exact scaling relations bridging the exponents of correlations of gradient fields to the exponents ensuremath zeta n of the n mathrm th order structure functions one of these relations is the well known bridge relation for the scaling exponent of dissipation fluctuations ensuremath mu 2 ensuremath ensuremath zeta 6

passive scalar turbulence in high dimensions exploiting a lagrangian strategy we present a numerical study for both perturbative and nonperturbative regions of the kraichnan advection model the major result is the numerical assessment of the first order 1 d expansion by chertkov falkovich kolokolov and lebedev phys rev e 52 4924 1995 for the fourth order scalar structure function in the limit of high dimension d s in addition to the perturbative results the behavior of the anomaly for the sixth order structure functions versus the velocity scaling exponent xi is investigated and the resulting behavior is discussed

functional renormalization group approach to the kraichnan model we study the anomalous scaling of the structure functions of a scalar field advected by a random gaussian velocity field the kraichnan model by means of functional renormalization group techniques we analyze the symmetries of the model and derive the leading correction to the structure functions considering the renormalization of composite operators and applying the operator product expansion

kardar parisi zhang equation with spatially correlated noise a unified picture from nonperturbative renormalization group we investigate the scaling regimes of the kardar parisi zhang kpz equation in the presence of spatially correlated noise with power law decay d p ensuremath sim p ensuremath 2 ensuremath rho in fourier space using a nonperturbative renormalization group approach we determine the full phase diagram of the system as a function of ensuremath rho and the dimension d in addition to the weak coupling part of the diagram which agrees with the results from europhys lett 47 14 1999 and eur phys j b 9 491 1999 we find the two fixed points describing the short range sr and long range lr dominated strong coupling phases in contrast with a suggestion in the references cited above we show that for all values of ensuremath rho there exists a unique strong coupling sr fixed point that can be continuously followed as a function of d we show in particular that the existence and the behavior of the lr fixed point do not provide any hint for 4 being the upper critical dimension of the kpz equation with sr noise

a new approach to computing the scaling exponents in fluid turbulence from first principles in this short paper we describe the essential ideas behind a new consistent closure procedure for the calculation of the scaling exponents n of the nth order correlation functions in fully developed hydrodynamic turbulence starting from first principles the closure procedure is constructed to respect the fundamental rescaling symmetry of the euler equation the starting point of the procedure is an infinite hierarchy of coupled equations that are obeyed identically with respect to scaling for any set of scaling exponents n this hierarchy was discussed in detail in a recent publication v s l vov and i procaccia physica a 1998 in press chao dyn 9707015 the scaling exponents in this set of equations cannot be found from power counting in this short paper we discuss in detail low order non trivial closures of this infinite set of equations and prove that these closures lead to the determination of the scaling exponents from solvability conditions the equations under consideration after this closure are nonlinear integro differential equations reflecting the nonlinearity of the original navier stokes equations nevertheless they have a very special structure such that the determination of the scaling exponents requires a procedure that is very similar to the solution of linear homogeneous equations in which amplitudes are determined by fitting to the boundary conditions in the space of scales the renormalization scale that is necessary for any anomalous scaling appears at this point the holder inequalities on the scaling exponents select the renormalization scale as the outer scale of turbulence l

on the perturbation expansion of the kpz equation we present a simple argument to show that the function of the d dimensional kpz equation d 2 is to all orders in perturbation theory given by beta g r d 2 g r 2 8 pi d 2 text gamma 2 d 2 g r 2 neither the dynamical exponent z nor the roughness exponent have any correction in any order of perturbation theory this shows that standard perturbation theory cannot attain the strong coupling regime and in addition breaks down at d 4 we also calculate a class of correlation functions exactly

computing the scaling exponents in fluid turbulence from first principles demonstration of multiscaling we develop a consistent closure procedure for the calculation of the scaling exponents n of the nth order correlation functions in fully developed hydro dynamic turbulence starting from first principles the closure procedure is constructed to respect the fundamental rescaling symmetry of the euler equation the starting point of the procedure is an infinite hierarchy of coupled equations that are obeyed identically with respect to scaling for any set of scaling exponents n this hierarchy was discussed in detail in a recent publication by v s l vov and i procaccia the scaling exponents in this set of equations cannot be found from power counting in this paper we present in detail the lowest non trivial closure of this infinite set of equations and prove that this closure leads to the determination of the scaling exponents from solvability conditions the equations under consideration after this closure are nonlinear integro differential equations reflecting the nonlinearity of the original navier stokes equations nevertheless they have a very special structure such that the determination of the scaling exponents requires a procedure that is very similar to the solution of linear homogeneous equations in which amplitudes are determined by fitting to the boundary conditions in the space of scales the renormalization scale that is necessary for any anomalous scaling appears at this point the holder inequalities on the scaling exponents select the renormalization scale as the outer scale of turbulence l we demonstrate that the solvability condition of our equations leads to non kolmogorov values of the scaling exponents n finally we show that this solutions is a first approximation in a systematic series of improving approximations for the calculation of the anomalous exponents in turbulence

normal and anomalous scaling of the fourth order correlation function of a randomly advected passive scalar for a delta correlated velocity field simultaneous correlation functions of a passive scalar satisfy closed equations we analyze the equation for the four point function to describe a solution completely one has to solve the matching problems at the scale of the source and at the diffusion scale we solve both the matching problems and thus find the dependence of the four point correlation function on the diffusion and pumping scale for large space dimensionality d it is shown that anomalous scaling appears in the first order of 1 d perturbation theory anomalous dimensions are found analytically both for the scalar field and for it s derivatives in particular for the dissipation field