exact relations in the theory of developed hydrodynamic turbulence exact relations of two types in the statistical theory of fully developed homogeneous isotropic turbulence in an incompressible fluid were found the relations of the first type connect two point and three point objects of the theory which are correlation functions and susceptibilities the second type of relations are the frequency sum rules which express some frequency integrals from fully dressed many point objects like vertices via corresponding bare values our approach is based on the navier stokes equation in quasi lagrangian variables and on the generating functional technique for correlation functions and susceptibilities the derivation of these relations uses no perturbation expansions and no additional assumptions this means that the relations are exact in the framework of the statistical theory of turbulence we showed that a many point scaling gives birth to the global scaling here many point scaling is the assumption that two point three point etc objects of the theory of turbulence are uniform functions in the inertial interval and may be characterized by some scaling exponents under this assumption the only global scale invariant model of fully developed turbulence suggested by kolmogorov dokl akad nauk ssr 32 19 1941 is consistent with the exact relations deduced

connes embedding problem and winning strategies for quantum xor games we consider quantum xor games defined in the work of regev and vidick acm trans comput theory 7 43 2015 from the perspective of unitary correlations defined in the work of harris and paulsen integr equations oper theory 89 125 2017 we show that the winning bias of a quantum xor game in the tensor product model respectively the commuting model is equal to the norm of its associated linear functional on the unitary correlation set from the appropriate model we show that connes embedding problem has a positive answer if and only if every quantum xor game has entanglement bias equal to the commuting bias in particular the embedding problem is equivalent to determining whether every quantum xor game g with a winning strategy in the commuting model also has a winning strategy in the approximate finite dimensional model

temporal multiscaling in hydrodynamic turbulence on the basis of the navier stokes equations we develop the high reynolds number statistical theory of different time many point spatial correlation functions of velocity differences we find that their time dependence is not scale invariant n order correlation functions exhibit infinitely many distinct decorrelation times that are characterized by anomalous dynamical scaling exponents we derive exact scaling relations that bridge all these dynamical exponents to the static anomalous exponents mathrm ensuremath zeta mathrm q of the standard structure functions we propose a representation of the time dependence using the legendre transform formalism of multifractals that automatically reproduces all the newly found bridge relationships

indeterminate sub critical bifurcations in parametric resonance in this paper we show that an indeterminate sub critical bifurcation with an unpredictable outcome arises in typical parametrically excited systems this occurs when the trivial solution is located on a fractal boundary as it loses its stability at a sub critical bifurcation here in the presence of even infinitesimal noise the outcome is indeterminate in the sense that we cannot predict on to which co existing attractor the system will settle

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

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

connes embedding problem and tsirelson s problem we show that tsirelson s problem concerning the set of quantum correlations and connes embedding problem on finite approximations in von neumann algebras known to be equivalent to kirchberg s qwep conjecture are essentially equivalent specifically tsirelson s problem asks whether the set of bipartite quantum correlations generated between tensor product separated systems is the same as the set of correlations between commuting c algebras connes embedding problem asks whether any separable ii1 factor is a subfactor of the ultrapower of the hyperfinite ii1 factor we show that an affirmative answer to connes question implies a positive answer to tsirelson s conversely a positive answer to a matrix valued version of tsirelson s problem implies a positive one to connes problem

passive motion in dynamical disorder as a model for stock market prices abstract a new model for stock price fluctuations is proposed based upon an analogy with the motion of tracers in gaussian random fields as used in turbulent dispersion models and in studies of transport in dynamically disordered media analytical and numerical results for this model in a special limiting case of a single scale field show characteristics similar to those found in empirical studies of stock market data specifically short term returns have a non gaussian distribution with super diffusive volatility assuming a power law decay of the time correlation of the disorder the returns correlation decays rapidly but the correlation function of the absolute returns exhibits a slow power law decay the returns distribution converges towards gaussian over long times some important characteristics of empirical data are not however reproduced by the model notably the scaling of tails of the cumulative distribution function of returns implied volatilities for options pricing are found by numerical simulation

perfect embezzlement of entanglement van dam and hayden introduced a concept commonly referred to as embezzlement where for any entangled quantum state there is an entangled catalyst state from which a high fidelity approximation of can be produced using only local operations we investigate a version of this where the embezzlement is perfect i e the fidelity is 1 we prove that perfect embezzlement is impossible in a tensor product framework even with infinite dimensional hilbert spaces and infinite entanglement entropy then we prove that perfect embezzlement is possible in a commuting operator framework we prove this using the theory of c algebras and we also provide an explicit construction next we apply our results to analyze perfect versions of a nonlocal game introduced by regev and vidick finally we analyze the structure of perfect embezzlement protocols in the commuting operator model showing that they require infinite dimensional hilbert spaces

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

variational methods for time dependent classical many particle systems abstract a variational method for the classical liouville equation is introduced that facilitates the development of theories for non equilibrium classical systems the method is based on the introduction of a complex valued auxiliary quantity that is related to the classical position momentum probability density via a functional of is developed whose extrema imply that satisfies the liouville equation multiscale methods are used to develop trial functions to be optimized by the variational principle the present variational principle with multiscale trial functions can capture both the microscopic and the coarse grained descriptions thereby yielding theories that account for the two way exchange of information across multiple scales in space and time equations of the smoluchowski form for the coarse grained state probability density are obtained constraints on the initial state of the n particle probability density for which the aforementioned equation is closed and conserves probability are presented the methodology has applicability to a wide range of systems including macromolecular assemblies ionic liquids and nanoparticles

self consistent formulations for stochastic nonlinear neuronal dynamics neural dynamics is often investigated with tools from bifurcation theory however many neuron models are stochastic mimicking fluctuations in the input from unknown parts of the brain or the spiking nature of signals noise changes the dynamics with respect to the deterministic model in particular classical bifurcation theory cannot be applied we formulate the stochastic neuron dynamics in the martin siggia rose de dominicis janssen msrdj formalism and present the fluctuation expansion of the effective action and the functional renormalization group frg as two systematic ways to incorporate corrections to the mean dynamics and time dependent statistics due to fluctuations in the presence of nonlinear neuronal gain to formulate self consistency equations we derive a fundamental link between the effective action in the onsager machlup om formalism which allows the study of phase transitions and the msrdj effective action which is computationally advantageous these results in particular allow the derivation of an om effective action for systems with non gaussian noise this approach naturally leads to effective deterministic equations for the first moment of the stochastic system they explain how nonlinearities and noise cooperate to produce memory effects moreover the msrdj formulation yields an effective linear system that has identical power spectra and linear response starting from the better known loopwise approximation we then discuss the use of the frg as a method to obtain self consistency beyond the mean we present a new efficient truncation scheme for the hierarchy of flow equations for the vertex functions by adapting the blaizot m endez and wschebor approximation from the derivative expansion to the vertex expansion the methods are presented by means of the simplest possible example of a stochastic differential equation that has generic features of neuronal dynamics

a path integral for turbulence in incompressible fluids in this paper a classical path integral is formulated for incompressible fluids that evolve according to the navier stokes equation the path integral propagates probability distributions deterministically on the space gvol of solenoidal velocity fields we construct a set of isp 2 charges associated with the geometry of gvol and its poisson structure and a pair of supersymmetry charges connected with the hamiltonian these charges generate exact symmetries of the classical path integral when the viscosity is set equal to zero when the effect of dissipation is included the charges associated with the poisson structure and the hamiltonian are no longer conserved charges that generate kolmogorov scaling and galilean transformations are also constructed the classical path integral is formulated in terms of vorticity as well

on the elimination of the sweeping interactions from theories of hydrodynamic turbulence in this paper we revisit the claim that the eulerian and quasi lagrangian same time correlation tensors are equal this statement allows us to transform the results of an msr quasi lagrangian statistical theory of hydrodynamic turbulence back to the eulerian representation we define a hierarchy of homogeneity symmetries between incremental homogeneity and global homogeneity it is shown that both the elimination of the sweeping interactions and the derivation of the 4 5 law require a homogeneity assumption stronger than incremental homogeneity but weaker than global homogeneity the quasi lagrangian transformation on the other hand requires an even stronger homogeneity assumption which is many time rather than one time but still weaker than many time global homogeneity we argue that it is possible to relax this stronger assumption and still preserve the conclusions derived from theoretical work based on the quasi lagrangian transformation

bifurcational precedences in the braids of periodic orbits of spiral 3 shoes in driven oscillators we conjecture the existence of a 3 striped spiral horseshoe under one iterate of the poincare map that arises in the analysis of a class of driven oscillators these systems represent escape from a smooth potential well under periodic forcing we assume topological conjugacy between the flow of the differential equation and an idealized suspension of a spiral 3 shoe and deduce bifurcational precedences by consideration of intertwining in the braid of periodic orbits we argue that many significant subharmonic bifurcations observed in such systems can then be understood in terms of the creation of this 3 shoe

indeterminate trans critical bifurcations in parametrically excited systems this paper may be regarded as a companion paper to a recent article m s soliman j m t thompson proc r soc lond a 438 511 518 1992 that addressed the phenomena of indeterminate sub critical bifurcations in typical parametrically excited systems this earlier work focused on the principal subharmonic resonance which occurs when the excitation frequency is almost twice the linear natural frequency in contrast we here focus on the fundamental harmonic resonance that arises when the two frequencies are almost the same we show that under this condition indeterminate trans critical bifurcations with unpredictable outcome can arise when the trivial solution is located on a fractal boundary as it loses its stability at the bifurcation the outcome is then unpredictable even under infinitesimal perturbations from the trivial solution

multiple point and multiple time correlation functions in a hard sphere fluid a recent mode coupling theory of higher order correlation functions is tested on a simple hard sphere fluid system at intermediate densities multiple point and multiple time correlation functions of the densities of conserved variables are calculated in the hydrodynamic limit and compared to results obtained from event based molecular dynamics simulations it is demonstrated that the mode coupling theory results are in excellent agreement with the simulation results provided that dissipative couplings are included in the vertices appearing in the theory in contrast simplified mode coupling theories in which the densities obey gaussian statistics neglect important contributions to both the multiple point and multiple time correlation functions on all time scales

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

a note on third order structure functions in turbulence starting from the navier stokes equation we rigorously prove that a modified third order structure function stilde3 r asymptotically equals 4 over 3 r is the dissipation rate in an inertial regime from this result we rigorously confirm the kolmogorov four fifths law without the kolmogorov assumption on isotropy our definition of the structure function involves a solid angle averaging over all possible orientations of the displacement vector y besides space time averaging direct numerical simulation for a highly symmetric flow for a taylor reynolds number of up to 155 shows that the flow remains significantly anisotropic and that without solid angle averaging the resulting structure functions approximately satisfy these scaling relations over some range of r l y l for some orientation of y but not for another

inverse energy cascade in two dimensional turbulence deviations from gaussian behavior high resolution numerical simulations of stationary inverse energy cascade in two dimensional turbulence are presented deviations from gaussian behavior of velocity differences statistics are quantitatively investigated the level of statistical convergence is pushed enough to permit reliable measurement of the asymmetries in the probability distribution functions of longitudinal increments and odd order moments which bring the signature of the inverse energy flux no measurable intermittency corrections could be found in their scaling laws the seventh order skewness increases by almost two orders of magnitude with respect to the third thus becoming of order unity

brownian motion in dynamically disordered media the motion of brownian test particles in a model random potential with time dependent correlations is investigated using four methods renormalized perturbation perturbation using martin siggia and rose functional formalism msr the edwards variational method on the msr functional and renormalization group with the msr function the disorder averaged one particle propagators determined by the renormalized perturbation expansion and msr perturbation expansion are identical to the second and possibly higher order and the two particle propagators determined by these perturbation methods are identical at the first and possibly higher order the one particle propagator determined by the edwards method is identical to the perturbation expansions at the first order but the second order analogue of the edwards method has a more complex expression which reduces to the second order perturbation expression with additional higher order terms the diffusion constant and two particle correlations are calculated from these propagators and are used to determine the effects of the random potential on the brownian particles generally the diffusion rate decreases with the disorder strength and increases with the temporal decay rate the two competing mechanisms result in an enhancement of the diffusion constant for weak potentials with fast temporal fluctuations the system exhibits two particle correlations that are inherently non gaussian and indicate clustering behavior the diffusion constant is also determined from a simple one loop renormalization group calculation in the static limit the diffusion constant calculated by the renormalization group recovers the results of deem and chandler m w deem and d chandler j stat phys 76 911 1994

lobe dynamics and the escape from a potential well periodically forced nonlinear oscillators that permit escape from a potential well frequently possess unstable periodic orbits whose invariant manifolds are homoclinically tangled the trellises formed by such overlapping manifolds separate the poincare plane into regions called lobes whose fates at successive time periods are amenable to analysis the lobe configurations observed in simple escape systems are discussed together with their interrelation with the location of periodic orbits and other invariant sets three applications of this procedure are illustrated i a numerical technique for localizing the non wandering set ii a demonstration that the existence of a saddle possessing a bounded homoclinically tangled unstable manifold branch implies the existence of safe open regions in the global basin of attraction iii the proposal of definitions of global integrity and escape times which may be used to relate safe basin erosion to the evolution of a homoclinic tangle

chaotic phenomena triggering the escape from a potential well this paper explores the manner in which a driven mechanical oscillator escapes from the cubic potential well typical of a metastable system close to a fold the aim is to show how the well known atoms of dissipative dynamics saddle node folds period doubling flips cascades to chaos boundary crises etc assemble to form molecules of overall response hierarchies of cusps incomplete feigenbaum trees etc particular attention is given to the basin of attraction and the loss of engineering integrity that is triggered by a homoclinic tangle the latter being accurately predicted by a melnikov analysis after escape chaotic transients are shown to conform to recent scaling laws analytical constraints on the mapping eigenvalues are used to demonstrate that sequences of flips and folds commonly predicted by harmonic balance analysis are in fact physically inadmissible

eulerian field theoretic closure formalisms for fluid turbulence the formalisms of wyld ann phys 14 143 1961 and martin siggia and rose msr phys rev a 8 423 1973 address the closure problem of a statistical treatment of homogeneous isotropic turbulence hit based on techniques primarily developed for quantum field theory in the wyld formalism there is a well known double counting problem for which an ad hoc solution was suggested by lee ann phys 32 292 1965 we show how to implement this correction in a more natural way from the basic equations of the formalism this leads to what we call the improved wyld lee renormalized perturbation theory msr had noted that their formalism had more vertex functions than wyld s formalism and based on this felt wyld s formalism was incorrect however a careful comparison of both formalisms here shows that the wyld formalism follows a different procedure to that of the msr formalism and so the treatment of vertex corrections appears in different ways in the two formalisms taking that into account along with clarifications made to both formalisms we find that they are equivalent and we demonstrate this up to fourth order

invariant sets of planar diffeomorphisms in nonlinear vibrations using the standard technique of poincare sampling many simple periodically forced nonlinear oscillators can be represented by area contracting planar diffeomorphisms a class of such diffeomorphisms referred to as single sided systems are considered these permit escape of orbits to arbitrarily large displacement six closure relations which relate the invariant manifolds of the hilltop saddle cycle to other invariant sets are derived these relations embody much of the underlying organization that is observed in the dynamics of these systems

classical diffusion in strong random media we study classical diffusion of particles in random media although many of our results are general we focus on the case of an ion in a three dimensional medium with random quenched charge centers obeying bulk charge neutrality within a functional integral framework we calculate the effective diffusion coefficients by first order and second order self consistent perturbation theory with a gaussian reference in both cases we also carry out a one loop order momentum space renormalization group calculation the self consistent methods are complicated numerically and fail beyond intermediate disorder strengths in contrast the renormalization group calculation gives an analytical result that appears valid even to high disorder strengths the methodology generally applicable to a quantitative calculation of effective diffusion coefficients in disordered media resolves deficiencies in self consistent perturbation theory approaches to this class of problems

mode coupling theory for multiple point and multiple time correlation functions we present a theoretical framework for higher order correlation functions involving multiple times and multiple points in a classical many body system such higher order correlation functions have attracted much interest recently in the context of various forms of multidimensional spectroscopy and have found an intriguing application as proposed measures of dynamical heterogeneities in structural glasses the theoretical formalism is based upon projection operator techniques that are used to isolate the slow time evolution of dynamical variables by expanding the slowly evolving component of arbitrary variables in an infinite multilinear basis composed of the products of slow variables of the system using the formalism a formally exact mode coupling theory is derived for multiple point and multiple time correlation functions the resulting expressions for higher order correlation functions are made tractable by applying a rigorous perturbation scheme called the n ordering method which is exact for systems with finite correlation lengths in the thermodynamic limit the theory is contrasted with standard mode coupling theories in which the noise or fluctuating force appearing in the generalized langevin equation is assumed to be gaussian and it is demonstrated that the non gaussian nature of the fluctuating forces leads to important contributions to higher order correlation functions finally the higher order correlation functions are evaluated analytically for an ideal gas system for which it is shown that the mode coupling theory is exact

from global scaling a la kolmogorov to local multifractal scaling in fully developed turbulence in the first part of the paper a modern presentation of scaling ideas is made it includes a reformulation of kolmogorov s 1941 theory bypassing the universality problem pointed out by landau and a presentation of the multifractal theory with emphasis on scaling rather than on cascades in the second part various historical aspects are discussed the importance of kolmogorov s rigorous derivation of the 4 5 l law for the third order structure function in his last 1941 turbulence paper is stressed this paper also contains evidence that he was aware of universality not being essential to the 1941 theory an inequality is established relating the exponents 2 p of the structure functions of order 2 p and the maximum velocity excursion it follows that models such as the obukhov kolmogorov 1962 log normal model in which 2 p does not increase monotonically are inconsistent with the basic physics of incompressible flow this result is independent of novikov s 1971 inequality in particular the proof presented here does not rely on the questionable relation proposed by obukhov and kolmogorov between instantaneous velocity increments and local averages of the dissipation

perturbation schemes for flow in random media we utilize a many body style formalism to compute the long range effective diffusivity kappa e for dispersal of a passive scalar field in a random irrotational flow with molecular diffusivity by comparing the theoretical results with the outcome of computer simulations we show that a renormalization group approach that includes an appropriate vertex renormalization is superior to low order perturbation theory and to the hartree fock approximation and gives a rather accurate method of computing kappa e for a range of parameter values

fock space for fermion like lattices and the linear glauber model abstract the concept of fock space representation is developed to deal with stochastic spin lattices written in terms of fermion operators a density operator is introduced in order to follow in parallel the developments of the case of bosons in the literature some general conceptual quantities for spin lattices are then derived including the notion of generating function and path integral via grassmann variables the formalism is used to derive the liouvillian of the d dimensional linear glauber dynamics in the fock space representation then the time evolution equations for the magnetization and the two point correlation function are derived in terms of the number operator

recovering isotropic statistics in turbulence simulations the kolmogorov 4 5th law one of the main benchmarks in direct numerical simulations of three dimensional turbulence is the kolmogorov prediction for third order structure functions with homogeneous and isotropic statistics in the infinite reynolds number limit previous direct numerical simulations dns techniques to obtain isotropic statistics have relied on time averaging structure functions in a few directions over many eddy turnover times using forcing schemes carefully constructed to generate isotropic data motivated by recent theoretical work which removes isotropy requirements by spherically averaging the structure functions over all directions we will present results which supplement long time averaging by angle averaging over up to 73 directions from a single flow snapshot the directions are among those natural to a square computational grid and are weighted to approximate the spherical average we use this angle averaging procedure to compare the statistically steady flows generated by two different forcing schemes in a periodic box our results show that despite the apparent differences in the two flows their isotropic components as measured by the kolmogorov laws are essentially identical this procedure may be used to investigate the isotropic part of the small scale statistics of any quantity of interest the averaging process is inexpensive and for the kolmogorov 4 5 law reasonable results can be obtained from a single snapshot of data this implies consistency with the recently derived local versions of the kolmogorov laws which do not require long time averages

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

functional and graphical methods for classical statistical dynamics i a formulation of the martin siggia rose method a formulation of the martin siggia rose msr method for describing the statistical dynamics of classical systems is presented the present formulation is very similar in structure to the original msr operator formalism and very different from the alternative functional integral formalism of janssen de dominicis peliti and others the need for imposing certain boundary conditions in the msr formalism as pointed out by deker is clarified the basic results of this paper include a construction of the msr formalism in a way that demonstrates its internal consistency a definition of a functional whose functional derivatives give all the correlation functions and response functions of an ensemble of mechanical systems a graphical expression for the correlation functions and response functions a graphical expression for the legendre transform of the functional and of the resulting vertex functions and a graphical derivation of the appropriate dyson equation the present formulation is applicable to systems with highly non gaussian statistics including systems of classical particles described in terms of the particle density in single particle phase space in this paper we consider only the case of ensembles of systems whose coordinates are continuous and whose time evolution is described by deterministic first order differential equations that are local in time the method is easily extended to systems whose dynamics is governed by stochastic differential equations and to spin systems

effective diffusivity in non isotropic gradient flows we study the effective diffusivity tensor for particles in a random gradient flow that statistically lacks rotational symmetry the effective diffusivity tensor is computed up to two loop order in perturbation theory and the ward identity that relates this tensor to the effective coupling is verified to the same order we re examine the renormalization group calculation that produced a very accurate numerical value for the effective diffusivity in the rotationally symmetric case and formulate two versions based on distinct divisions of the random potential field that gives rise to the flow both types of renormalization group calculation give good results when compared with numerical simulations however at two loop order in perturbation theory the two methods differ in detail from each other and from the exact perturbation calculation this is in contrast to the corresponding results in the isotropic case

replica field theory for composite media abstract in this paper we use the replica trick together with a variational method in order to compute the effective permittivity of a medium where the local permittivity is a random binary variable we obtain an effective medium formula previously derived by a cumulant expansion this formula is satisfactory from many points of view hashin and shtrikman bounds low density expansion etc showing that the replica method does not violate some basic principles

single molecule tracking of heterogeneous diffusion the mean square displacement of heterogeneous diffusion obeys the einstein relation thereby showing no sign of heterogeneities in the ensemble measurement of the diffusion constant the signature of spatial heterogeneities appears in the time evolution of the non gaussian distribution and in the cross correlation between the square displacements at different times both available from single molecule diffusional trajectories as a quantitative measure the non gaussian indicator g t decays asymptotically to zero according to 1 t for finite time correlation but saturates at a plateau value for power law correlation in addition the joint moment correlation function f t tau provides a direct probe of the memory effect of the fluctuating rate constant a two state diffusion model and a stochastic gaussian model are constructed to evaluate these quantities and are shown to yield the same result within the second cumulant expansion

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

long range correlation effects generalized brownian motion and anomalous diffusion we investigate the diffusion motion of a brownian particle which is acted upon by both a friction force with memory effect and a noise with long range correlation effects the noise is expressed as f x t approximately x sigma f t sigma 0 where x and t are the displacement and time respectively and f t has the long time correlation effect approximately t beta 0 beta 1 beta 1 1 beta 2 the generalized langevin equation corresponding fokker planck equation and its solution at large time are established a variety of anomalous diffusion patterns are derived due to the long range correlation effects the effective diffusion coefficient is dependent on both the displacement and time and the probability density for finding the brownian particle at displacement x and time t is non gaussian distribution when this model is applied to diffusion on fractals o shaughnessy and procaccia s results 1985 can be naturally derived