anomalous scaling of a passive scalar advected by the synthetic compressible flow abstract the field theoretic renormalization group and operator product expansion are applied to the problem of a passive scalar advected by a gaussian nonsolenoidal velocity field with finite correlation time in the presence of large scale anisotropy the energy spectrum of the velocity in the inertial range has the form e k k1 e and the correlation time at the wave number k scales as k 2 it is shown that depending on the values of the exponents e and the model exhibits various types of inertial range scaling regimes with nontrivial anomalous exponents explicit asymptotic expressions for the structure functions and other correlation functions are obtained they are represented by superpositions of power laws with nonuniversal amplitudes and universal independent of the anisotropy anomalous exponents calculated to the first order in e and in any space dimension these anomalous exponents are determined by the critical dimensions of tensor composite operators built of the scalar gradients and exhibit a kind of hierarchy related to the degree of anisotropy the less is the rank the less is the dimension and consequently the more important is the contribution to the inertial range behavior the leading terms of the even odd structure functions are given by the scalar vector operators the anomalous exponents depend explicitly on the degree of compressibility

phase diagram for turbulent transport sampling drift eddy diffusivity and variational principles we study the long time large scale transport in a three parameter family of isotropic incompressible velocity fields with power law spectra scaling law for transport is characterized by the scaling exponent q and the hurst exponent h as functions of the parameters the parameter space is divided into regimes of scaling laws of different em functional forms of the scaling exponent and the hurst exponent we present the full three dimensional phase diagram the limiting process is one of three kinds brownian motion h 1 2 persistent fractional brownian motions 1 2

critical behaviour of a fluid in a random shear flow renormalization group analysis of a simplified model critical behaviour of a fluid binary mixture or liquid crystal subjected to strongly anisotropic turbulent mixing is studied by means of the field theoretic renormalization group as a simplified model relaxational stochastic dynamics of a non conserved scalar order parameter coupled to a random velocity field with prescribed statistics is considered the velocity is taken gaussian white in time with a correlation function of the form t t k d where k is the component of the wave vector perpendicular to the distinguished direction direction of the flow the d dimensional generalization of the ensemble introduced by avellaneda and majda 1990 commun math phys 131 381 within the context of passive scalar advection it is shown that depending on the relation between the exponent and the space dimensionality d the system exhibits various types of large scale self similar behaviour associated with different infrared attractive fixed points of the renormalization group equations in addition to well known asymptotic regimes model a of equilibrium critical dynamics and a passively advected scalar with no self interaction the existence of a new non equilibrium and strongly anisotropic type of critical behaviour universality class is established and the corresponding critical dimensions are calculated to the second order of the double expansion in and e 4 d two loop approximation the most realistic values of the model parameters for example d 3 and the kolmogorov exponent 4 3 belong to this class the scaling behaviour appears anisotropic in the sense that the critical dimensions related to the directions parallel and perpendicular to the flow are essentially different the results are in qualitative agreement with the results obtained in experiments and simulations of fluid systems subjected to various kinds of regular and chaotic anisotropic flows

correlation functions in isotropic and anisotropic turbulence the role of the symmetry group the theory of fully developed turbulence is usually considered in an idealized homogeneous and isotropic state real turbulent flows exhibit the effects of anisotropic forcing the analysis of correlation functions and structure functions in isotropic and anisotropic situations is facilitated and made rational when performed in terms of the irreducible representations of the relevant symmetry group which is the group of all rotations so 3 in this paper we first consider the needed general theory and explain why we expect different universal scaling exponents in the different sectors of the symmetry group we exemplify the theory context of isotropic turbulence for third order tensorial structure functions and in weakly anisotropic turbulence for the second order structure function the utility of the resulting expressions for the analysis of experimental data is demonstrated in the context of high reynolds number measurements of turbulence in the atmosphere

logarithmic violation of scaling in strongly anisotropic turbulent transfer of a passive vector field inertial range asymptotic behavior of a vector e g magnetic field passively advected by a strongly anisotropic turbulent flow is studied by means of the field theoretic renormalization group and the operator product expansion the advecting velocity field is gaussian not correlated in time with the pair correlation function of the form t t k d 1 where k k and k is the component of the wave vector perpendicular to the distinguished direction direction of the flow the d dimensional generalization of the ensemble introduced by avellaneda and majda commun math phys 131 381 1990 the stochastic advection diffusion equation for the transverse divergence free vector field includes as special cases the kinematic dynamo model for magnetohydrodynamic turbulence and the linearized navier stokes equation in contrast to the well known isotropic kraichnan s model where various correlation functions exhibit anomalous scaling behavior with infinite sets of anomalous exponents here the dependence on the integral turbulence scale l has a logarithmic behavior instead of powerlike corrections to ordinary scaling determined by naive canonical dimensions the anomalies manifest themselves as polynomials of logarithms of l the key point is that the matrices of scaling dimensions of the relevant families of composite operators appear nilpotent and cannot be diagonalized the detailed proof of this fact is given for the correlation functions of arbitrary order

renormalization methods for higher order differential equations we adapt methodology of statistical mechanics and quantum field theory to approximate solutions to an arbitrary order ordinary differential equation boundary value problem by a second order equation in particular we study equations involving the derivative of a double well potential such as u u3 or u 2u3 using momentum fourier space variables we average over short length scales and demonstrate that the higher order derivatives can be neglected within the first cumulant approximation once length is properly renormalized yielding an approximation to solutions of the higher order equation from the second order the results are confirmed using numerical computations additional numerics confirm that the main role of the higher order derivatives is in rescaling the length

mathematical models with exact renormalization for turbulent transport ii fractal interfaces non gaussian statistics and the sweeping effect this paper continues the study of a model for turbulent transport with an exact renormalization theory which has recently been proposed and developed by the authors three important topics are analyzed with complete mathematical rigor for this model 1 renormalized higher order statistics of a passively advected scalar such as the pair distance distribution and the fractal dimension of interfaces 2 the effect of non gaussian turbulent velocity statistics on renormalization theory 3 the sweeping effect of additional large scale mean velocities a special emphasis is placed on renormalization theory in the vicinity of the value of the analogue of the kolmogorov spectrum in the model in the authors earlier paper it was established that the kolmogorov value is at a phase transition boundary in the exact renormalization theory it is found here that the qualitative model despite its simplicity contains in the vicinity of the kolmogorov value a remarkable amount of the qualitative behavior of turbulent transport which has been uncovered in recent experiments and proposed in phenomenological theories in particular the richardson 4 3 law for pair dispersion and interfaces with fractal dimension defect of 2 3 occur in the model rigorously as limits when the kolmogorov spectrum is approached as a limit from one side of the phase transition boundary alternative corrections to the richardson law with the same form as those proposed heuristically in the recent literature and interfaces with fractal dimension defect 1 3 occur in the model when the kolmogorov spectrum is approached from the other side of the phase transition it is very interesting that fractal dimension defects of roughly the value either 1 3 or 2 3 for level sets and interfaces of passive scalars have been ubiquitous in recent turbulence experiments as regards non gaussian the asymptotic normality of normalized integrals b 56 corresponding to compactly supported blobs with mean zero the proof of this latter fact is done in the same way as step 2 proposition b 3 using the fact that the corresponding random processes tilde v delta s have finite domain of dependence this concludes the proof of proposition b 4

instanton for random advection a path integral over trajectories of 2n fluid particles is identified with a 2nth order correlation function of a passive scalar convected by d dimensional short correlated multiscale incompressible random velocity flow strong intermittency of the scalar is described by means of an instanton calculus saddle point plus fluctuations about it in the path integral at n ensuremath gg d the anomalous scaling exponent of the 2nth scalar s structural function is found analytically

passive advection of a vector field anisotropy finite correlation time exact solution and logarithmic corrections to ordinary scaling in this work we study the generalization of the problem considered in phys rev e 91 013002 2015 to the case of finite correlation time of the environment velocity field the model describes a vector e g magnetic field passively advected by a strongly anisotropic turbulent flow inertial range asymptotic behavior is studied by means of the field theoretic renormalization group and the operator product expansion the advecting velocity field is gaussian with finite correlation time and preassigned pair correlation function due to the presence of distinguished direction n all the multiloop diagrams in this model vanish so that the results obtained are exact the inertial range behavior of the model is described by two regimes the limits of vanishing or infinite correlation time that correspond to the two nontrivial fixed points of the rg equations their stability depends on the relation between the exponents in the energy spectrum e k 1 and the dispersion law k 2 in contrast to the well known isotropic kraichnan s model where various correlation functions exhibit anomalous scaling behavior with infinite sets of anomalous exponents here the corrections to ordinary scaling are polynomials of logarithms of the integral turbulence scale l

large order asymptotes of the quantum field expansions for the kraichnan model of passive scalar advection a simple model of a passive scalar quantity advected by a gaussian non solenoidal compressible velocity field is considered large order asymptotes of quantum field expansions are investigated by the instanton approach the existence of a finite convergence radius of the series is proved the type and the position of the singularity of the series in a regularization parameter e are determined anomalous exponents of the main contributions to the structural functions are resummed using new information about the series convergence and two known orders of the e expansion

bilinear realizability is equivalent to existence of a singular affine differential i o equation abstract for continuous time analytic input output maps existence of a singular differential equation relating derivatives of controls and outputs is shown to be equivalent to bilinear realizability a similar result holds for the problem of immersion into bilinear systems the proof is very analogous to that of the corresponding and previously known result for discrete time

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

modeling of nonlinear discrete time systems from input output data abstract an algorithm is presented that produces a polynomial state affine model for a discrete time nonlinear system the method uses only input output information and is based on a difference equation approximation to the input output map from the difference equation a behaviour matrix is constructed to be used in the state affine realization algorithm the difference equation approach requires the estimation of fewer parameters than the usual volterra series a state space model is then obtained directly from the difference equation as a byproduct a method for calculating the volterra series directly from the difference equation is obtained practical applications to nonlinear systems are included

scaling structure of the velocity statistics in atmospheric boundary layers the statistical objects characterizing turbulence in real turbulent flows differ from those of the ideal homogeneous isotropic model they contain contributions from various two and three dimensional aspects and from the superposition of inhomogeneous and anisotropic contributions we employ the recently introduced decomposition of statistical tensor objects into irreducible representations of the so 3 symmetry group characterized by j and m indices where j5 0 2 j m j to disentangle some of these contributions separating the universal and the asymptotic from the specific aspects of the flow the different j contributions transform differently under rotations and so form a complete basis in which to represent the tensor objects under study the experimental data are recorded with hot wire probes placed at various heights in the atmospheric surface layer time series data from single probes and from pairs of probes are analyzed to compute the amplitudes and exponents of different contributions to the second order statistical objects characterized by j 50 1 and 2 the analysis shows the need to make a careful distinction between long lived quasi twodimensional turbulent motions close to the ground and relatively short lived three dimensional motions we demonstrate that the leading scaling exponents in the three leading sectors j50 1 and 2 appear to be different but universal independent of the positions of the probe the tensorial component considered and the large scale properties the measured values of the scaling exponent are z 2 j50 50 6860 01 z 2 j51 51 0 60 15 and z 2 j52 51 3860 10 we present theoretical arguments for the values of these exponents using the clebsch representation of the euler equations neglecting anomalous corrections the values obtained are 2 3 1 and 4 3 respectively some enigmas and questions for the future are sketched

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

on two definitions of observation spaces this paper establishes the equality of the observation spaces defined by means of piecewise constant controls with those defined in terms of differentiable controls

structure of the three point correlation function of a passive scalar in the presence of a mean gradient the three point correlation function of a passive scalar advected by a random incompressible velocity field in the presence of a mean gradient is investigated by means of phenomenological hopf equations numerical solutions are provided in the case where the velocity is gaussian white in time and with a power law in space v stackrel ensuremath rightarrow r ensuremath v stackrel ensuremath rightarrow 0 2 ensuremath sim r 2 ensuremath ensuremath epsilon and in the model introduced by shraiman and siggia c r acad sci 321 279 1995 anomalous scaling exponents are found in both models the numerics agrees with all the available analytic perturbative results in addition the angular dependence of the correlation function is explicitly determined in the batchelor limit of random advection by a smooth velocity field the exponent is found to remain very close to 1 as found experimentally in this limit the three point correlation function is found to be very well represented away from collinearity by an explicit integral representation

algebraic differential equations and rational control systems an equivalence is shown between realizability of input output i o operators by rational control systems and high order algebraic differential equations for i o pairs this generalizes to nonlinear systems the equivalence between autoregressive representations and finite dimensional linear realizability

development of optimally disordered critical random excitation the problem of determining the critical power spectral density psd function of a partially specified stationary gaussian load process which maximizes the response of a linear system has been considered the partial specification of the load is given only in terms of its total average energy the critical input psd turns out to be highly narrow banded which fails to capture the erratic nature of the excitation consequently the trade off curve between the maximum linear system response and the disorder in the input process quantified in terms of its entropy rate has been generated the pareto optimization theory is used to tackle the conflicting objectives of simultaneous maximization of the system response and the input entropy rate consequently the non linear multi objective optimization has been carried out using a multi criteria genetic algorithm scheme an illustrative example of determining the critical input of an axially vibrating rod excited by a partially specified stationary gaussian load process has been considered

anomalous scaling nonlocality and anisotropy in a model of the passively advected vector field a model of the passive vector quantity advected by the gaussian velocity field with the covariance ensuremath propto ensuremath delta t ensuremath t ensuremath mathbf x ensuremath mathbf x ensuremath ensuremath varepsilon is studied the effects of pressure and large scale anisotropy are discussed the inertial range behavior of the pair correlation function is described by an infinite family of scaling exponents which satisfy exact transcendental equations derived explicitly in d dimensions by means of the functional techniques the exponents are organized in a hierarchical order according to their degree of anisotropy with the spectrum unbounded from above and the leading minimal exponent coming from the isotropic sector this picture extends to higher order correlation functions like in the scalar model the second order structure function appears nonanomalous and is described by the simple dimensional exponent s 2 ensuremath propto r 2 ensuremath ensuremath varepsilon for the higher order structure functions s 2n ensuremath propto r n 2 ensuremath ensuremath varepsilon ensuremath delta n the anomalous scaling behavior is established as a consequence of the existence in the corresponding operator product expansions of dangerous composite operators whose negative critical dimensions determine the anomalous exponents ensuremath delta n l0 a close formal resemblance of the model with the stirred navier stokes equation reveals itself in the mixing of relevant operators and is the main motivation of the paper using the renormalization group the anomalous exponents are calculated in the o ensuremath varepsilon approximation in large d dimensions for the even structure functions up to the twelfth order

two loop calculation of the turbulent prandtl number the turbulent prandtl number has been calculated in the two loop approximation of the epsilon expansion of the stochastic theory of turbulence the strikingly small value obtained for the two loop correction explains the good agreement of the earlier one loop result with the experiment this situation is drastically different from other available nontrivial two loop results which exhibit corrections of the magnitude of the one loop term the reason is traced to the mutual cancellation of additional divergences appearing in two dimensions which have had a major effect on the results of previous calculations of other quantities

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

calculation of the anomalous exponents in the rapid change model of passive scalar advection to order epsilon 3 the field theoretic renormalization group and operator product expansion are applied to the model of a passive scalar advected by the gaussian velocity field with zero mean and correlation function approximately equal to delta t t k d epsilon inertial range anomalous exponents identified with the critical dimensions of various scalar and tensor composite operators constructed of the scalar gradients are calculated within the epsilon expansion to order epsilon 3 three loop approximation including the exponents in anisotropic sectors the main goal of the paper is to give the complete derivation of this third order result and to present and explain in detail the corresponding calculational techniques the character and convergence properties of the epsilon expansion are discussed the improved inverse epsilon expansion is proposed and the comparison with the existing nonperturbative results is given

reliability based critical earthquake load models part 2 nonlinear structures the problem of determining critical stochastic earthquake excitation models for simple nonlinear systems under single point or multi point nonstationary seismic inputs is considered the earthquake acceleration components are obtained by multiplying known deterministic enveloping functions with zero mean gaussian stationary random processes with unknown auto power spectral density functions for single point excitations and power spectral density matrix for multi point excitations the definition of critical earthquake input is based on the notion of a performance function the system is considered to have failed if the maximum response over a given time interval exceeds specified limits the critical excitations are defined as those that minimize the hasofer lind reliability index associated with this performance function the computation of this index in turn is based on the use of response surface to model the limit surface near the check point here the quantity to be optimally determined is taken to be the unknown input power spectral density function for single point excitations or the input power spectral density matrix for multi point excitations the excitations are taken to satisfy constraints on total average energy zero crossing rate lower bounds on entropy rate and other positivity and bounding requirements that are of mathematical nature the resulting constrained nonlinear optimization problems are solved using the sequential quadratic programming method illustrative examples for computing random critical excitations for singly supported and multiply supported oscillators that have cubic force displacement relations are provided

anomalous scaling of passive scalar fields advected by the navier stokes velocity ensemble effects of strong compressibility and large scale anisotropy the field theoretic renormalization group and the operator product expansion are applied to two models of passive scalar quantities the density and the tracer fields advected by a random turbulent velocity field the latter is governed by the navier stokes equation for compressible fluid subject to external random force with the covariance t t k 4 d y where d is the dimension of space and y is an arbitrary exponent the original stochastic problems are reformulated as multiplicatively renormalizable field theoretic models the corresponding renormalization group equations possess infrared attractive fixed points it is shown that various correlation functions of the scalar field its powers and gradients demonstrate anomalous scaling behavior in the inertial convective range already for small values of y the corresponding anomalous exponents identified with scaling critical dimensions of certain composite fields operators in the quantum field terminology can be systematically calculated as series in y the practical calculation is performed in the leading one loop approximation including exponents in anisotropic contributions it should be emphasized that in contrast to gaussian ensembles with finite correlation time the model and the perturbation theory presented here are manifestly galilean covariant the validity of the one loop approximation and comparison with gaussian models are briefly discussed

anomalous scaling of a passive scalar in the presence of strong anisotropy field theoretic renormalization group and the operator product expansion are applied to a model of a passive scalar quantity straight theta t x advected by the gaussian strongly anisotropic velocity field with the covariance infinity delta t t x x epsilon inertial range anomalous scaling behavior is established and explicit asymptotic expressions for the structure functions s n r identical with are obtained they are represented by superpositions of power laws the corresponding anomalous exponents which depend explicitly on the anisotropy parameters are calculated to the first order in epsilon in any space dimension d in the limit of vanishing anisotropy the exponents are associated with tensor composite operators built of the scalar gradients and exhibit a kind of hierarchy related to the degree of anisotropy the less is the rank the less is the dimension and consequently the more important is the contribution to the inertial range behavior the leading terms of the even odd structure functions are given by the scalar vector operators for the finite anisotropy the exponents cannot be associated with individual operators which are essentially mixed in renormalization but the aforementioned hierarchy survives for all the cases studied the second order structure function s2 is studied in more detail using the renormalization group and zero mode techniques the corresponding exponents and amplitudes are calculated within the perturbation theories in epsilon 1 d and in the anisotropy parameters if the anisotropy of the velocity is strong enough the skewness factor s 3 s 3 2 2 increases going down towards the depth of the inertial range the higher order odd ratios increase even if the anisotropy is weak

effects of mixing and stirring on the critical behaviour stochastic dynamics of a nonconserved scalar order parameter near its critical point subject to random stirring and mixing is studied using the field theoretic renormalization group the stirring and mixing are modelled by a random external gaussian noise with the correlation function t t k4 d y and the divergence free due to incompressibility velocity field governed by the stochastic navier stokes equation with a random gaussian force with the correlation function depending on the relations between the exponents y and y and the space dimensionality d the model reveals several types of scaling regimes some of them are well known model a of equilibrium critical dynamics and linear passive scalar field advected by a random turbulent flow but there are three new non equilibrium regimes universality classes associated with new nontrivial fixed points of the renormalization group equations the corresponding critical dimensions are calculated in the two loop approximation second order of the triple expansion in y y and e 4 d

nonperturbative spectrum of anomalous scaling exponents in the anisotropic sectors of passively advected magnetic fields we address the scaling behavior of the covariance of the magnetic field in the three dimensional kinematic dynamo problem when the boundary conditions and or the external forcing are not isotropic the velocity field is gaussian and delta correlated in time and its structure function scales with a positive exponent xi the covariance of the magnetic field is naturally computed as a sum of contributions proportional to the irreducible representations of the so 3 symmetry group the amplitudes are non universal determined by boundary conditions the scaling exponents are universal forming a discrete strictly increasing spectrum indexed by the sectors of the symmetry group when the initial mean magnetic field is zero no dynamo effect is found irrespective of the anisotropy of the forcing the rate of isotropization with decreasing scales is fully understood from these results

multicomponent turbulence the spherical limit and non kolmogorov spectra a set of models for homogeneous isotropic turbulence is considered in which the navier stokes equations for incompressible fluid flow are generalized to a set of n coupled equations in n velocity fields it is argued that in order to be useful these models must embody a new group of symmetries and a general formalism is laid out for their construction the work is motivated by similar techniques that have had extraordinary success in improving the theoretical understanding of equilibrium phase transitions in condensed matter systems we consider two classes of models a simpler class model i which does not contain an exact galilean symmetry and a more complicated extended class model ii which does the key result is that these models simplify when n is large the so called spherical limit n ensuremath rightarrow ensuremath infty can be solved exactly yielding closed sets of nonlinear integral equations for the response and correlation functions for model i these equations known as kraichnan s direct interaction approximation equations are solved fully in the scale invariant turbulent regime for model ii these equations are more complicated and their full solution is left for future work implications of these results for real turbulence n 1 are discussed in particular it is argued that previously applied renormalization group techniques based on an expansion in the exponent y that characterizes the driving spectrum are incorrect and that the kolmogorov exponent ensuremath zeta has a nontrivial dependence on n with ensuremath zeta n ensuremath rightarrow ensuremath infty 3 2 for both sets of models this value is close to the experimental result ensuremath zeta ensuremath simeq 5 3 which must therefore result from higher order corrections in powers of 1 n prospects for calculating these corrections are briefly discussed though daunting such calculations might provide a controlled perturbation expansion for the kolmogorov and other exponents our techniques may also be applied to other nonequilibrium dynamical problems such as the kardar parisi zhang equation for interface growth and perhaps to turbulence in nonlinear wave systems

anomalous scaling in the anisotropic sectors of the kraichnan model of passive scalar advection kraichnan s model of passive scalar advection in which the driving gaussian velocity field has fast temporal decorrelation is studied as a case model for understanding the anomalous scaling behavior in the anisotropic sectors of turbulent fields we show here that the solutions of the kraichnan equation for the n order correlation functions foliate into sectors that are classified by the irreducible representations of the so d symmetry group we find a discrete spectrum of universal anomalous exponents with a different exponent characterizing the scaling behavior in every sector generically the correlation functions and structure functions appear as sums over all these contributions with nonuniversal amplitudes that are determined by the anisotropic boundary conditions the isotropic sector is always characterized by the smallest exponent and therefore for sufficiently small scales local isotropy is always restored the calculation of the anomalous exponents is done in two complementary ways in the first they are obtained from the analysis of the correlation functions of gradient fields the theory of these functions involves the control of logarithmic divergences that translate into anomalous scaling with the ratio of the inner and the outer scales appearing in the final result in the second method we compute the exponents from the zero modes of the kraichnan equation for the correlation functions of the scalar field itself in this case the renormalization scale is the outer scale the two approaches lead to the same scaling exponents for the same statistical objects illuminating the relative role of the outer and inner scales as renormalization scales in addition we derive exact fusion rules which govern the small scale asymptotics of the correlation functions in all the sectors of the symmetry group and in all dimensions

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

generating series for bilinear hybrid systems in this paper we introduce the novel concept of a hybrid generating series and show that continuous state and output trajectories of bilinear hybrid systems can be described in terms of these series the results represent an extension of the fliess series expansion for bilinear systems to hybrid systems

inertial range scaling of laminar shear flow as a model of turbulent transport asymptotic scaling behavior characteristic of the inertial range is obtained for a fractal stochastic system proposed as a model for turbulent transport

anomalous scaling in two models of passive scalar advection effects of anisotropy and compressibility the problem of the effects of compressibility and large scale anisotropy on anomalous scaling behavior is considered for two models describing passive advection of scalar density and tracer fields the advecting velocity field is gaussian delta correlated in time and scales with a positive exponent epsilon explicit inertial range expressions for the scalar correlation functions are obtained they are represented by superpositions of power laws with nonuniversal amplitudes and universal anomalous exponents dependent only on epsilon and alpha the compressibility parameter the complete set of anomalous exponents for the pair correlation functions is found nonperturbatively in any space dimension d using the zero mode technique for higher order correlation functions the anomalous exponents are calculated to o epsilon 2 using the renormalization group as in the incompressible case the exponents exhibit a hierarchy related to the degree of anisotropy the leading contributions to the even correlation functions are given by the exponents from the isotropic shell in agreement with the idea of restored small scale isotropy as the degree of compressibility increases the corrections become closer to the leading terms the small scale anisotropy reveals itself in the odd ratios of correlation functions the skewness factor slowly decreases going down to small scales for the incompressible case but starts to increase if alpha is large enough the higher odd dimensionless ratios hyperskewness etc increase thus signaling persistent small scale anisotropy this effect becomes more pronounced for larger values of alpha

on local isotropy of passive scalars in turbulent shear flows an assessment of local isotropy and universality in high reynolds number turbulent flows is presented the emphasis is on the behaviour of passive scalar fields advected by turbulence but a brief review of the relevant facts is given for the turbulent motion itself experiments suggest that local isotropy is not a natural concept for scalars in shear flows except perhaps at such extreme reynolds numbers that are of no practical relevance on earth yet some type of scaling exists even at moderate reynolds numbers the relation between these two observations is a theme of this paper

instanton for the kraichnan passive scalar problem we consider high order correlation functions of the passive scalar in the kraichnan model using the instanton formalism we find the scaling exponents ensuremath zeta n of the structure functions s n for n ensuremath gg 1 under the additional condition d ensuremath zeta 2 ensuremath gg 1 where d is the dimensionality of space at nl n c where n c d ensuremath zeta 2 2 2 ensuremath ensuremath zeta 2 the exponents are ensuremath zeta n ensuremath zeta 2 4 2n ensuremath n 2 n c while at ng n c they are n independent ensuremath zeta n ensuremath zeta 2 n c 4 we also estimate n dependent factors in s n particularly their behavior at n close to n c

a remark on bilinear systems and moduli spaces of instantons abstract explicit equations are given for the moduli space of framed instantons as a quasi affine variety based on the representation theory of noncommutative power series or equivalently the minimal realization theory of bilinear systems

inverse cascade and intermittency of passive scalar in one dimensional smooth flow random advection of a lagrangian tracer scalar field ensuremath theta t x by a one dimensional spatially smooth and short correlated in time velocity field is considered scalar fluctuations are maintained by a source concentrated at the integral scale l the statistical properties of both scalar differences and the dissipation field are analytically determined exploiting the dynamical formulation of the model the gaussian statistics known to be present at small scales for incompressible velocity fields emerges here at large scales x ensuremath gg l these scales are shown to be excited by an inverse cascade of ensuremath theta 2 and the probability distribution function pdf of the corresponding scalar differences to approach the gaussian form as larger and larger scales are considered small scale x ensuremath ll l statistics is shown to be strongly non gaussian a collapse of scaling exponents for scalar structure functions takes place moments of order pg 1 all scale linearly independently of the order p smooth scaling x p is found for ensuremath 1lpl1 tails of the scalar difference pdf are exponential while at the center a cusped shape tends to develop when smaller and smaller ratios x l are considered the same tendency is present for the scalar gradient pdf with respect to the inverse of the p eclet number the pumping to diffusion scale ratio the tails of the latter pdf are however much more extended decaying as a stretched exponential of exponent 2 3 smaller than unity this slower decay is physically associated with the strong fluctuations of the dynamical dissipative scale

anomalous scaling of a passive scalar advected by the turbulent velocity field with finite correlation time two loop approximation the renormalization group and operator product expansion are applied to the model of a passive scalar quantity advected by the gaussian self similar velocity field with finite and not small correlation time the inertial range energy spectrum of the velocity is chosen in the form e k ensuremath propto k 1 ensuremath 2 ensuremath varepsilon and the correlation time at the wave number k scales as k ensuremath 2 ensuremath eta inertial range anomalous scaling for the structure functions and other correlation functions emerges as a consequence of the existence in the model of composite operators with negative scaling dimensions identified with anomalous exponents for ensuremath eta g ensuremath varepsilon these exponents are the same as in the rapid change limit of the model for ensuremath eta l ensuremath varepsilon they are the same as in the limit of a time independent quenched velocity field for ensuremath varepsilon ensuremath eta local turnover exponent the anomalous exponents are nonuniversal through the dependence on a dimensionless parameter the ratio of the velocity correlation time and the scalar turnover time the nonuniversality reveals itself however only in the second order of the ensuremath varepsilon expansion and the exponents are derived to order ensuremath varepsilon 2 including anisotropic contributions it is shown that for moderate order of the structure function n and the space dimensionality d finite correlation time enhances the intermittency in comparison with both the limits the rapid change and quenched ones the situation changes when n and or d become large enough the correction to the rapid change limit due to the finite correlation time is positive that is the anomalous scaling is suppressed it is maximal for the quenched limit and monotonically decreases as the correlation time tends to zero

turbulence without pressure we develop exact field theoretic methods to treat turbulence when the effect of pressure is negligible we find explicit forms of certain probability distributions demonstrate that the breakdown of galilean invariance is responsible for intermittency and establish the operator product expansion we also indicate how the effects of pressure can be turned on perturbatively

multilocality and fusion rules on the generalized structure functions in two dimensional and three dimensional navier stokes turbulence using the fusion rules hypothesis for three dimensional and two dimensional navier stokes turbulence we generalize a previous nonperturbative locality proof to multiple applications of the nonlinear interactions operator on generalized structure functions of velocity differences we call this generalization of nonperturbative locality to multiple applications of the nonlinear interactions operator multilocality the resulting cross terms pose a new challenge requiring a new argument and the introduction of a new fusion rule that takes advantage of rotational symmetry our main result is that the fusion rules hypothesis implies both locality and multilocality in both the ir and uv limits for the downscale energy cascade of three dimensional navier stokes turbulence and the downscale enstrophy cascade and inverse energy cascade of two dimensional navier stokes turbulence we stress that these claims relate to nonperturbative locality of generalized structure functions on all orders and not the term by term perturbative locality of diagrammatic theories or closure models that involve only two point correlation and response functions

anomalous scaling regimes of a passive scalar advected by the synthetic velocity field the field theoretic renormalization group rg is applied to the problem of a passive scalar advected by the gaussian self similar velocity field with finite correlation time and in the presence of an imposed linear mean gradient the energy spectrum in the inertial range has the form e k ensuremath propto k 1 ensuremath ensuremath varepsilon and the correlation time at the wave number k scales as k ensuremath 2 ensuremath eta it is shown that depending on the values of the exponents ensuremath varepsilon and ensuremath eta the model in the inertial convective range exhibits various types of scaling regimes associated with the infrared stable fixed points of the rg equations diffusive type regimes for which the advection can be treated within ordinary perturbation theory and three nontrivial convection type regimes for which the correlation functions exhibit anomalous scaling behavior the explicit asymptotic expressions for the structure functions and other correlation functions are obtained the anomalous exponents determined by the scaling dimensions of the scalar gradients are calculated to the first order in ensuremath varepsilon and ensuremath eta in any space dimension for the first nontrivial regime the anomalous exponents are the same as in the rapid change version of the model for the second they are the same as in the model with time independent frozen velocity field in these regimes the anomalous exponents are universal in the sense that they depend only on the exponents entering into the velocity correlator for the last regime the exponents are nonuniversal they can depend also on the amplitudes however the nonuniversality can reveal itself only in the second order of the rg expansion a brief discussion of the passive advection in the non gaussian velocity field governed by the nonlinear stochastic navier stokes equation is also given

spectrum of anisotropic exponents in hydrodynamic systems with pressure we discuss the scaling exponents characterizing the power law behavior of the anisotropic components of correlation functions in turbulent systems with pressure the anisotropic components are conveniently labeled by the angular momentum index mathcal l of the irreducible representation of the so 3 symmetry group such exponents govern the rate of decay of anisotropy with decreasing scales it is a fundamental question whether they ever increase as mathcal l increases or they are bounded from above the equations of motion in systems with pressure contain nonlocal integrals over all space one could argue that the requirement of convergence of these integrals bounds the exponents from above it is shown here on the basis of a solvable model the linear pressure model that this is not necessarily the case the model introduced here is of a passive vector advection by a rapidly varying velocity field the advected vector field is divergent free and the equation contains a pressure term that maintains this condition the zero modes of the second order correlation function are found in all the sectors of the symmetry group we show that the spectrum of scaling exponents can increase with mathcal l without bounds while preserving finite integrals the conclusion is that contributions from higher and higher anisotropic sectors can disappear faster and faster upon decreasing the scales also in systems with pressure

anisotropic nonperturbative zero modes for passively advected magnetic fields an analytic assessment of the role of anisotropic corrections to the isotropic anomalous scaling exponents is given for the d dimensional kinematic magnetohydrodynamics problem in the presence of a mean magnetic field the velocity advecting the magnetic field changes very rapidly in time and scales with a positive exponent xi inertial range anisotropic contributions to the scaling exponents zeta j of second order magnetic correlations are associated with zero modes and have been calculated nonperturbatively for d 3 the limit xi 0 yields zeta j j 2 xi 2j 3 j 2 5j 4 2 4j 2 1 where j j or 2 is the order in the legendre polynomial decomposition applied to correlation functions conjectures on the fact that anisotropic components cannot change the isotropic threshold to the dynamo effect are also made

anomalous scaling of a passive vector advected by the navier stokes velocity field using the field theoretic renormalization group and the operator product expansion the model of a passive vector field a weak magnetic field in the framework of the kinematic mhd advected by the velocity field which is governed by the stochastic navier stokes equation with the gaussian random stirring force correlated in time and with the correlator proportional to k4 d 2e is investigated to the first order in e one loop approximation it is shown that the single time correlation functions of the advected vector field have anomalous scaling behavior and the corresponding exponents are calculated in the isotropic case as well as in the case with the presence of large scale anisotropy the hierarchy of the anisotropic critical dimensions is briefly discussed and the persistence of the anisotropy inside the inertial range is demonstrated on the behavior of the skewness and hyperskewness dimensionless ratios of correlation functions as functions of the reynolds number re it is shown that even though the present model of a passive vector field advected by the realistic velocity field is mathematically more complicated than on one hand the corresponding models of a passive vector field advected by synthetic gaussian velocity fields and on the other hand than the corresponding model of a passive scalar quantity advected by the velocity field driven by the stochastic navier stokes equation the final one loop approximate asymptotic scaling behavior of the single time correlation or structure functions of the advected fields of all models are defined by the same anomalous dimensions up to normalization

controllability of switched bilinear systems the controllability of switched bilinear systems sblss is considered three kinds of controllabilities including weak controllability approximate controllability and global controllability are investigated one by one sets of easily verifiable sufficient conditions are obtained for each case which are applicable to a large class of switched bilinear systems

comment on the paper on the scaling of three dimentsional homogeneous and isotropic turbulence by benzi et al abstract benzi et al presented experimental data and an analysis to the effect that the well known 2 3 kolmogorov obukhov exponent in the inertial range of local structure in turbulence should be corrected by a small but definitely nonzero amount we reexamine the very same data and show that this conclusion is unjustified the data are in fact consistent with incomplete similarity in the inertial range and with an exponent that depends on the reynolds number and tends to 2 3 in the limit of vanishing viscosity

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

improved epsilon expansion for three dimensional turbulence summation of nearest dimensional singularities an improved epsilon expansion in the d dimensional d 2 stochastic theory of turbulence is constructed by taking into account pole singularities at d 2 in coefficients of the epsilon expansion of universal quantities effectiveness of the method is illustrated by a two loop calculation of the kolmogorov constant in three dimensions

turbulent compressible fluid renormalization group analysis scaling regimes and anomalous scaling of advected scalar fields we study a model of fully developed turbulence of a compressible fluid based on the stochastic navier stokes equation by means of the field theoretic renormalization group in this approach scaling properties are related to the fixed points of the renormalization group equations previous analysis of this model near the real world space dimension 3 identified a scaling regime n v antonov et al theor math phys 110 305 1997 tmphah0040 577910 1007 bf02630456 the aim of the present paper is to explore the existence of additional regimes which could not be found using the direct perturbative approach of the previous work and to analyze the crossover between different regimes it seems possible to determine them near the special value of space dimension 4 in the framework of double y and expansion where y is the exponent associated with the random force and 4 d is the deviation from the space dimension 4 our calculations show that there exists an additional fixed point that governs scaling behavior turbulent advection of a passive scalar density field by this velocity ensemble is considered as well we demonstrate that various correlation functions of the scalar field exhibit anomalous scaling behavior in the inertial convective range the corresponding anomalous exponents identified as scaling dimensions of certain composite fields can be systematically calculated as a series in y and all calculations are performed in the leading one loop approximation

span reachability and observability of bilinear hybrid systems the paper presents an algebraic characterization of observability and span reachability of bilinear hybrid systems without guards i e hybrid systems whose continuous dynamics are determined by bilinear control systems and whose discrete transitions are triggered externally the proposed characterization provides necessary and sufficient conditions for observability and span reachability in terms of rank conditions for generalized observability and controllability matrices this characterization is useful for the system theory of bilinear hybrid systems in particular it is used in the realization theory of bilinear hybrid systems petreczky 2006

some specific features of the e expansion in the theory of turbulence and the possibility of its improvement specific problems arising in the use of the method of renormalization group and e expansion in the theory of developed turbulence are discussed the necessity to take into account the dependence of the model parameters on the expansion parameter e and the large magnitude of the physical value of the expansion parameter it is shown that quantities independent of the amplitude of the correlation function of the random force possess a uniquely defined e expansion on the example of the two loop calculation of the kolmogorov constant and the turbulent prandtl number effectiveness of the improved expansion is demonstrated which uses an approximate summation of the high order terms of the e expansion

mathematical models with exact renormalization for turbulent transport the advection diffusion of a passive scalar by incompressible velocity fields which admit a statistical description and involve a continuous range of excited spatial and or temporal scales is very important in applications ranging from fully developed turbulence to the diffusion of tracers in heterogeneous porous media a variety of renormalization theories which typically utilize partial resummation of divergent perturbation series according to various recipes have been applied to this problem in various contexts in this paper a simple model problem for the advection diffusion of a passive scalar is introduced and the complete renormalization theory is developed with full mathematical rigor explicit formulas for the anomalous time scaling in various regimes as well as the green s function for the large scale long time ensemble average are developed here formulas for the renormalized higher order statistics are also developed the simple form of the model problem is deceptive the renormalization theory for this problem exhibits a remarkable range of different renormalization phenomena as parameters in the velocity statistics are varied these phenomena include the existence of several distinct anomalous scaling regimes as the spectral parameter tilde varepsilon is varied as well as explicit regimes in tilde varepsilon where the effective equation for the ensemble average is not a simple diffusion equation but instead involves an explicit random nonlocal eddy diffusivity we use fourier analysis and the feynman kac formula as main tools in the explicit exact renormalization theory developed here

anomalous scaling of a passive scalar advected by the navier stokes velocity field two loop approximation the field theoretic renormalization group and operator product expansion are applied to the model of a passive scalar quantity advected by a non gaussian velocity field with finite correlation time the velocity is governed by the navier stokes equation subject to an external random stirring force with the correlation function proportional to ensuremath delta t ensuremath t ensuremath k 4 ensuremath d ensuremath 2 ensuremath epsilon it is shown that the scalar field is intermittent already for small ensuremath epsilon its structure functions display anomalous scaling behavior and the corresponding exponents can be systematically calculated as series in ensuremath epsilon the practical calculation is accomplished to order ensuremath epsilon 2 two loop approximation including anisotropic sectors as for the well known kraichnan rapid change model the anomalous scaling results from the existence in the model of composite fields operators with negative scaling dimensions identified with the anomalous exponents thus the mechanism of the origin of anomalous scaling appears similar for the gaussian model with zero correlation time and the non gaussian model with finite correlation time it should be emphasized that in contrast to gaussian velocity ensembles with finite correlation time the model and the perturbation theory discussed here are manifestly galilean covariant the relevance of these results for real passive advection and comparison with the gaussian models and experiments are briefly discussed

anomalous scaling for passively advected magnetic fields the kraichnan model 1 for passive scalar advection is by now considered a paradigm for intermittency problems a first systematic understanding of the scalar field scaling properties has been attained the crucial property of the model is that equal time correlation functions obey closed equations of motion this stems from the small correlation time of the velocity field advecting the scalar it has been recognized that a general mechanism for intermittency and anomalous scaling is the presence of nontrivial zero modes of the closed equations satisfied by the correlation functions 2 3 4 such mechanism can be illustrated in its simplest form by considering the following linear differential equation frac d 2 y r d r 2 frac a r frac dy r dr b frac y r r 2 f frac r l 1 where r varies on the positive axis y r is the unknown function a and b are two constants and f r l varies over distances of order l and is almost constant for r l in the physical situations l is the integral scale and the mechanisms maintaining the turbulence are supposed to act at this scale as for intermittency in navier stokes turbulence we are interested in the scaling behavior in the inertial range i e for r l note that the l h s of 1 is invariant under scale transformations reflecting the scale invariance of the general solution of 1 is given by a non homogeneous and a linear combination of two homogeneous solutions the combination is fixed by the conditions that y r being a correlation function should be regular and vanish at infinity

renormalization group operator product expansion and anomalous scaling in a model of advected passive scalar field theoretical renormalization group methods are applied to the obukhov kraichnan model of a passive scalar advected by the gaussian velocity field with the covariance mathbf v t mathbf x mathbf v t ensuremath mathbf x ensuremath mathbf v t mathbf x mathbf v t ensuremath mathbf x ensuremath ensuremath propto ensuremath delta t ensuremath t ensuremath mathbf x ensuremath mathbf x ensuremath ensuremath varepsilon inertial range anomalous scaling for the structure functions and various pair correlators is established as a consequence of the existence in the corresponding operator product expansions of certain essential or dangerous composite operators powers of the local dissipation rate whose negative critical dimensions determine anomalous exponents the main technical result is the calculation of the anomalous exponents in the order ensuremath varepsilon 2 of the ensuremath varepsilon expansion generalization of the results obtained to the case of a slow velocity field is also presented

turbulence with pressure anomalous scaling of a passive vector field the field theoretic renormalization group rg and the operator product expansion are applied to the model of a transverse divergence free vector quantity passively advected by the synthetic turbulent flow with a finite and not small correlation time the vector field is described by the stochastic advection diffusion equation with the most general form of the inertial nonlinearity it contains as special cases the kinematic dynamo model linearized navier stokes ns equation the special model without the stretching term that possesses additional symmetries and has a close formal resemblance with the stochastic ns equation the statistics of the advecting velocity field is gaussian with the energy spectrum e k ensuremath propto k 1 ensuremath ensuremath varepsilon and the dispersion law ensuremath omega ensuremath propto k ensuremath 2 ensuremath eta k being the momentum wave number the inertial range behavior of the model is described by seven regimes or universality classes that correspond to nontrivial fixed points of the rg equations and exhibit anomalous scaling the corresponding anomalous exponents are associated with the critical dimensions of tensor composite operators built solely of the passive vector field which allows one to construct a regular perturbation expansion in ensuremath varepsilon and ensuremath eta the actual calculation is performed to the first order one loop approximation including the anisotropic sectors universality of the exponents their in dependence on the forcing effects of the large scale anisotropy compressibility and pressure are discussed in particular for all the scaling regimes the exponents obey a hierarchy related to the degree of anisotropy the more anisotropic is the contribution of a composite operator to a correlation function the faster it decays in the inertial range the relevance of these results for the real developed turbulence described by the stochastic ns equation is discussed

intermittency and anomalous scaling of passive scalars in any space dimension we establish exact inequalities for the structure function scaling exponents of a passively advected scalar in both the inertial convective and viscous convective ranges these inequalities involve the scaling exponents of the velocity structure functions and in a refined form an intermittency exponent of the convective range scalar flux they are valid for three dimensional navier stokes turbulence and satisfied within errors by present experimental data the inequalities also hold for any synthetic turbulent velocity statistics with a finite correlation in time we show that for time correlation exponents of the velocity smaller than the local turnover exponent the scalar spectral exponent is strictly less than that in kraichnan s phys rev lett 72 1016 1994 phys fluids 11 945 1968 j fluid mech 64 737 1974 62 305 1974 soluble rapid change model with velocity ensuremath delta correlated in time our results include as a special case an exponent inequality derived previously by constantin and procaccia nonlinearity 7 1045 1994 but with a more direct proof the inequalities in their simplest form follow from a kolmogorov type relation for the turbulent passive scalar valid in each space dimension d our improved inequalities are based upon a rigorous version of the refined similarity hypothesis for passive scalars these are compared with the relations implied by fusion rules hypothesized for scalar gradients textcopyright 1996 the american physical society

ageing properties of critical systems in the past few years systems with slow dynamics have attracted considerable theoretical and experimental interest ageing phenomena are observed during this everlasting non equilibrium evolution a simple instance of such a behaviour is provided by the dynamics that takes place when a system is quenched from its high temperature phase to the critical point the aim of this review is to summarize the various numerical and analytical results that have been recently obtained for this case particular emphasis is put on the field theoretical methods that can be used to provide analytical predictions for the relevant dynamical quantities fluctuation dissipation relations are discussed and in particular the concept of fluctuation dissipation ratio fdr is reviewed emphasizing its connection with the definition of a possible effective temperature the renormalization group approach to critical dynamics is summarized and the scaling forms of the time dependent non equilibrium correlation and response functions of a generic observable are discussed from them the universality of the associated fdr follows as an amplitude ratio it is then possible to provide predictions for ageing quantities in a variety of different models in particular the results for models a b and c dynamics of the o n ginzburg landau hamiltonian and model a dynamics of the weakly dilute ising magnet and of the 3 theory are reviewed and compared with the available numerical results and exact solutions the effect of a planar surface on the ageing behaviour of model a dynamics is also addressed within the mean field approximation

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

renormalization group and anomalous scaling in a simple model of passive scalar advection in compressible flow field theoretical renormalization group methods are applied to a simple model of a passive scalar quantity advected by the gaussian non solenoidal compressible velocity field with the covariance propto delta t t x x epsilon convective range anomalous scaling for the structure functions and various pair correlators is established and the corresponding anomalous exponents are calculated to the order epsilon 2 of the epsilon expansion these exponents are non universal as a result of the degeneracy of the rg fixed point in contrast to the case of a purely solenoidal velocity field obukhov kraichnan model the correlation functions in the case at hand exhibit nontrivial dependence on both the ir and uv characteristic scales and the anomalous scaling appears already at the level of the pair correlator the powers of the scalar field without derivatives whose critical dimensions determine the anomalous exponents exhibit multifractal behaviour the exact solution for the pair correlator is obtained it is in agreement with the result obtained within the epsilon expansion the anomalous exponents for passively advected magnetic fields are also presented in the first order of the epsilon expansion