applications of field theoretic renormalization group methods to reaction diffusion problems we review the application of field theoretic renormalization group rg methods to the study of fluctuations in reaction diffusion problems we first investigate the physical origin of universality in these systems before comparing rg methods to other available analytic techniques including exact solutions and smoluchowski type approximations starting from the microscopic reaction diffusion master equation we then pedagogically detail the mapping to a field theory for the single species reaction ka a k we employ this particularly simple but non trivial system to introduce the field theoretic rg tools including the diagrammatic perturbation expansion renormalization and callan symanzik rg flow equation we demonstrate how these techniques permit the calculation of universal quantities such as density decay exponents and amplitudes via perturbative dc d expansions with respect to the upper critical dimension dc with these basics established we then provide an overview of more sophisticated applications to multiple species reactions disorder effects l evy flights persistence problems and the influence of spatial boundaries we also analyse field theoretic approaches to non equilibrium phase transitions separating active from absorbing states we focus particularly on the generic directed percolation universality class as well as on the most prominent exception to this class even offspring branching and annihilating random walks finally we summarize the state of the field and present our perspective on outstanding problems for the future

the mean field scaling function of the universality class of absorbing phase transitions with a conserved field we consider two mean field like models which belong to the universality class of absorbing phase transitions with a conserved field in both cases we analytically derive the order parameter as a function of the control parameter and of an external field conjugated to the order parameter this allows us to calculate the universal scaling function of the mean field behaviour the obtained universal function is in perfect agreement with recently obtained numerical data of the corresponding five and six dimensional models showing that four is the upper critical dimension of this particular universality class

spectral analysis of fractional kinetic equations with random data we present a spectral representation of the mean square solution of the fractional kinetic equation also known as fractional diffusion equation with random initial condition gaussian and non gaussian limiting distributions of the renormalized solution of the fractional in time and in space kinetic equation are described in terms of multiple stochastic integral representations

statistical description of interacting brownian particles abstract an alternative approach to the statistical description of interacting brownian particles based on the microscopic langevin equation is presented applying the formalism of microscopic phase space densities a stochastic form of the well known bbgky hierarchy for reduced distribution functions in phase space is produced special attention is paid to both the particle interaction and the hydrodynamic interaction between the brownons considered on the two particle level assuming fast momentum relaxation we find a hierarchy of diffusion like equations in configuration space on the smoluchowski time scale for reduced particle densities and corresponding particle flows these equations are an extension of the ebeling falkenhagen diffusion equations with respect to hydrodynamics finally we formulate the direct correlation force formalism for inhomogeneous non equilibrium systems in order to break off the hierarchy of coupled equations examples for closure relations are given

renormalization of stochastic differential equations with multiplicative noise using effective potential methods we present a method to renormalize stochastic differential equations subjected to multiplicative noise the method is based on the widely used concept of effective potential in high energy physics and has already been successfully applied to the renormalization of stochastic differential equations subjected to additive noise we derive a general formula for the one loop effective potential of a single ordinary stochastic differential equation with arbitrary interaction terms subjected to multiplicative gaussian noise provided the noise satisfies a certain normalization condition to illustrate the usefulness and limitations of the method we use the effective potential to renormalize a toy chemical model based on a simplified gray scott reaction in particular we use it to compute the scale dependence of the toy model s parameters in perturbation theory when subjected to a gaussian power law noise with short time correlations

analysis of thermoelastic stress concentration around oblate cavities in three dimensional generally anisotropic bodies by the boundary element method abstract in this article stress concentrations around cavities in 3d generally anisotropic bodies are investigated using the boundary element method bem where the associated volume integral is analytically transformed to the boundary all derivations are based upon the fourier series representations of the fundamental solutions including green s function of displacements and its derivatives this approach of analytical transformation has fully restored the bem s distinctive notion that only the boundary needs to be discretized the goal of the present work is to investigate the thermoelastic stress concentration around cavities in 3d anisotropic bodies by use of the analytically transformed boundary integral equation bie the work has fully recovered the bem s nature of boundary discretization for treating 3d generally anisotropic thermoelasticity this is the first implemented work that successfully treats thermoelastic problems for 3d generally anisotropic solids by the analytically transformed bie in the paper interesting phenomena are observed from the analyses of stress concentrations around oblate cavities and some discussions are made

clustering and coalescence from multiplicative noise the kraichnan ensemble we study the dynamics of the two point statistics of the kraichnan ensemble which describes the transport of a passive pollutant by a stochastic turbulent flow characterized by scale invariant structure functions the fundamental equation of this problem consists of the fokker planck equation for the two point correlation function of the density of particles performing spatially correlated brownian motions with scale invariant correlations this problem is equivalent to the stochastic motion of an effective particle driven by a generic multiplicative noise in this paper we propose an alternative and more intuitive approach to the problem than the original one gawedzki and vergassola 2000 physica d 138 63 leading to the same conclusions the general features of this new approach make possible to fit it to other more complex contexts

unique continuation property for multi terms time fractional diffusion equations this paper concens about a carleman estimate which can give the unique continuation property of solutions for a multi terms time fractional diffusion equation up to order alpha 0 alpha 2 with general time dependent second order strongly elliptic operator for the diffusion by using a special holmgren type transformation which is linear with respect to time the estimate giving a local unique continuation of solutions is derived via some subelliptic estimate for an operator associated to this transformed equation using calculus of pseudo differential operators after that we have given a new argument to derive the global unique continuation of solutions here the global unique continuation means as follows if u is a solution of the multi terms time fractional diffusion equation in a domain over the time interval 0 t and it is supported on t ge 0 then a zero set of solution over a subdomain of omega can be continued to 0 t times omega

evolution of the system with multiplicative noise the governed equations for the order parameter one and two time correlators are obtained for systems with white multiplicative noise we consider the noise whose amplitude depends on stochastic variable as xa where 0 a 1 it turns out that the equation for autocorrelator includes an anomalous average of the power law function with the fractional exponent 2a determination of this average for the stochastic system with a self similar phase space is performed it is shown that at when the system is disordered the correlator behaves in the course of time non monotonically whereas the autocorrelator increases monotonically at the phase portrait of the system divides into two domains at small initial values of the order parameter the system evolves to a disordered state as above within the ordered domain it is attracted to the point with finite values of the autocorrelator and order parameter the long time asymptotes are defined to show that within the disordered domain the autocorrelator decays hyperbolically and the order parameter behaves as a power law function with fractional exponent 2 1 a correspondingly within the ordered domain the behaviour of both dependencies is exponential with an index proportional to

existence of almost everywhere solution for nonlinear hyperbolic parabolic system in this paper the existence theorem for an almost everywhere solution of a mixed problem for more complicated nonlinear hyperbolic parabolic nonlinear thermoelasticity system was proved using galerkin s method and suitable a priori estimates were obtained

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

cauchy problem for fractional diffusion equations abstract we consider an evolution equation with the regularized fractional derivative of an order 0 1 with respect to the time variable and a uniformly elliptic operator with variable coefficients acting in the spatial variables such equations describe diffusion on inhomogeneous fractals a fundamental solution of the cauchy problem is constructed and investigated

nature of different types of absorbing states we present a comparison of three different types of langevin equations exhibiting absorbing states the langevin equation defining the reggeon field theory one with multiplicative noise and a third type in which the noise is complex each one is found to describe a different underlying physical mechanism in particular the nature of the different absorbing states depends on the type of noise considered by studying the stationary single site effective potential we analyze the impossibility of finding a reaction diffusion model in the multiplicative noise universality class we also discuss some theoretical questions related to the nature of complex noise as for example whether it is necessary or not to consider a complex equation in order to describe processes as the annihilation reaction a stackrel ensuremath rightarrow a 0

quasi stationary distributions for stochastic processes with an absorbing state we study the long time behaviour of stochastic models with an absorbing state conditioned on survival for a large class of processes in which saturation prevents unlimited growth statistical properties of the surviving sample attain time independent limiting values we may then define a quasi stationary probability distribution as one in which the ratios pn t pm t for any pair of non absorbing states n and m are time independent this is not a true stationary distribution since the overall normalization decays as probability flows irreversibly to the absorbing state we construct quasi stationary solutions for the contact process on a complete graph the malthus verhulst process

langevin equation for the density of a system of interacting langevin processes we present a simple derivation of the stochastic equation obeyed by the density function for a system of langevin processes interacting via a pairwise potential the resulting equation is considerably different from the phenomenological equations usually used to describe the dynamics of non conserved model a and conserved model b particle systems the major feature is that the spatial white noise for this system appears not additively but multiplicatively this simply expresses the fact that the density cannot fluctuate in regions devoid of particles the steady state for the density function may however still be recovered formally as a functional integral over the coursed grained free energy of the system as in models a and b

mixed boundary value problems of two dimensional anisotropic thermoelasticity with elliptic boundaries abstract the two dimensional mixed boundary value problems for an anisotropic thermoelastic body containing an elliptic hole boundary are considered in this paper by using the formalism of stroh phil mag 7 625 646 the approach of analytic function continuation and the technique of conformal mapping a unified analytical solution for elliptic hole boundaries and for general anisotropic thermoelastic media is provided as an application two typical examples associated with mixed boundary value problems are solved completely one is an indentation problem over an elliptic hole boundary the other is a partially reinforced elliptic hole under a remote uniform heat flow both the contact stress under the rigid stamp and the bonded stress along the reinforced segment are studied in detail and shown in graphic form

wilson renormalization of a reaction diffusion process abstract healthy and sick individuals a and b particles diffuse independently with diffusion constants da and db sick individuals upon encounter infect healthy ones at rate k but may also spontaneously recover at rate 1 the propagation of the epidemic therefore couples to the fluctuations in the total population density global extinction occurs below a critical value c of the spatially averaged total density the epidemic evolves as the diffusion reaction decay process a b 2b b a for which we write down the field theory the stationary state properties of this theory when da db were obtained by kree et al the critical behavior for da db remains unsolved

cusps self organization and absorbing states elastic interfaces embedded in quenched random media exhibit metastability and stick slip dynamics these nontrivial dynamical features have been shown to be associated with cusp singularities of the coarse grained disorder correlator here we show that annealed systems with many absorbing states and a conservation law but no quenched disorder exhibit identical cusps on the other hand similar nonconserved systems in the directed percolation class are also shown to exhibit cusps but of a different type these results are obtained both by a recent method to explicitly measure disorder correlators and by defining an alternative new protocol inspired by self organized criticality which opens the door to easily accessible experimental realizations

critical density of the abelian manna model via a multitype branching process a multitype branching process is introduced to mimic the evolution of the avalanche activity and determine the critical density of the abelian manna model this branching process incorporates partially the spatiotemporal correlations of the activity which are essential for the dynamics in particular in low dimensions an analytical expression for the critical density in arbitrary dimensions is derived which significantly improves the results over mean field theories as confirmed by comparison to the literature on numerical estimates from simulations the method can easily be extended to lattices and dynamics other than those studied in the present work

relationship between directed percolation and the synchronization transition in spatially extended systems we study the nature of the synchronization transition in spatially extended systems by discussing a simple stochastic model an analytic argument is put forward showing that in the limit of discontinuous processes the transition belongs to the directed percolation dp universality class the analysis is complemented by a detailed investigation of the dependence of the first passage time for the amplitude of the difference field on the adopted threshold we find the existence of a critical threshold separating the regime controlled by linear mechanisms from that controlled by collective phenomena as a result of this analysis we conclude that the synchronization transition belongs to the dp class also in continuous models the conclusions are supported by numerical checks on coupled map lattices too

singularities interfaces and cracks in dissimilar anisotropic media for a non pathological bimaterial in which an interface crack displays no oscillatory behaviour it is observed that apart possibly from the stress intensity factors the structure of the near tip field in each of the two blocks is independent of the elastic moduli of the other block collinear interface cracks are analysed under this non oscillatory condition and a simple rule is formulated that allows one to construct the complete solutions from mode iii solutions in an isotropic homogeneous medium the general interfacial crack tip field is found to consist of a two dimensional oscillatory singularity and a one dimensional square root singularity a complex and a real stress intensity factors are proposed to scale the two singularities respectively owing to anisotropy a peculiar fact is that the complex stress intensity factor scaling the oscillatory fields however defined does not recover the classical stress intensity factors as the bimaterial degenerates to be non pathological collinear crack problems are also formulated in this context and a strikingly simple mathematical structure is identified interactive solutions for singularity interface and singularity interface crack are obtained the general results are specialized to decoupled antiplane and in plane deformations for this important case it is found that if a material pair is non pathological for one set of relative orientations of the interface and the two solids it is non pathological for any set of orientations for bonded orthotropic materials an intuitive choice of the principal measures of elastic anisotropy and dissimilarity is rationalized a complex variable representation is presented for a class of degenerate orthotropic materials throughout the paper the equivalence of the lekhnitskii and stroh formalisms is emphasized the article concludes with a formal statement of interfacial fracture mechanics for anisotropic solids

coherent state path integral versus coarse grained effective stochastic equation of motion from reaction diffusion to stochastic sandpiles we derive and study two different formalisms used for non equilibrium processes the coherent state path integral and an effective coarse grained stochastic equation of motion we first study the coherent state path integral and the corresponding field theory using the annihilation process a a to a as an example the field theory contains counter intuitive quartic vertices we show how they can be interpreted in terms of a first passage problem reformulating the coherent state path integral as a stochastic equation of motion the noise generically becomes imaginary this renders it not only difficult to interpret but leads to convergence problems at finite times we then show how alternatively an effective coarse grained stochastic equation of motion with real noise can be constructed the procedure is similar in spirit to the derivation of the mean field approximation for the ising model and the ensuing construction of its effective field theory we finally apply our findings to stochastic manna sandpiles we show that the coherent state path integral is inappropriate or at least inconvenient as an alternative we derive and solve its mean field approximation which we then use to construct a coarse grained stochastic equation of motion with real noise

path integral solution of the system with coloured multiplicative noise the behaviour of most probable values of the order parameter x and the amplitude of the conjugate force fluctuations is studied for a stochastic system with a coloured multiplicative noise and absorbing state the phase diagrams introduced as dependencies of the noise self correlation time vs temperature and noise growth velocity are defined it is shown that phase plane x is divided into isolated domains of large intermediate and small values of x the behaviour of the system in these domains is studied in terms of the path integral in the region x 1 the trajectories converge to the point x 0 at 0 a 12 and to x 0 at 12 a 1 in the former case the probability of realization of trajectories is finite while in the latter case it is vanishing small and an absorbing state can be formed

unique continuation property for the anomalous diffusion and its application abstract the unique continuation for anomalous diffusion operators with fractional time derivative of order 1 2 is proved by deriving carleman estimates for the operators by applying operators which have a negative sign in front of the diffusion terms of anomalous diffusion operators the anomalous diffusion operators are transformed to parabolic operators of order 4 in the space variables the carleman estimates are derived for these transformed operators considered as semi elliptic operators the usual calderon uniqueness argument for semi elliptic operators does not work due to the singularities in factors of the factorization of their principal parts in order to have smooth factorizations of the operators we took into account the lower order terms of the operators

on an imaginary exponential functional of brownian motion we investigate a random integral which provides a natural example of an imaginary exponential functional of brownian motion this functional shows up in the study of the binary annihilation process within the doi peliti formalism for reaction diffusion systems the main emphasis is put on the complementarity between the usual langevin approach and another approach based on the similarity with kesten variables and other one dimensional disordered systems even though neither of these routes leads to the full solution of the problem we have obtained a collection of results describing various regimes of interest