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 exotic conformal galilei algebra and nonlinear partial differential equations abstract the conformal galilei algebra cga and the exotic conformal galilei algebra ecga are applied to construct partial differential equations pdes and systems of pdes which admit these algebras we show that there are no single second order pdes invariant under the cga but systems of pdes can admit this algebra moreover a wide class of nonlinear pdes exists which are conditionally invariant under cga it is further shown that there are systems of nonlinear pdes admitting ecga with the realisation obtained very recently in d martelli y tachikawa comments on galilei conformal field theories and their geometric realisation preprint arxiv 0903 5184v2 hep th 2009 moreover wide classes of nonlinear systems invariant under two different 10 dimensional subalgebras of ecga are explicitly constructed and an example with possible physical interpretation is presented

direct evaluation of large deviation functions we introduce a numerical procedure to evaluate directly the probabilities of large deviations of physical quantities such as current or density that are local in time the large deviation functions are given in terms of the typical properties of a modified dynamics and since they no longer involve rare events can be evaluated efficiently and over a wider ranges of values we illustrate the method with the current fluctuations of the totally asymmetric exclusion process and with the work distribution of a driven lorentz gas

first integrals integrating factors and λ symmetries of second order differential equations for a given second order ordinary differential equation ode several relationships among first integrals integrating factors and symmetries are studied the knowledge of a symmetry of the equation permits the determination of an integrating factor or a first integral by means of coupled first order linear systems of partial differential equations if two nonequivalent symmetries of the equation are known then an algorithm to find two functionally independent first integrals is provided these methods include and complete other methods to find integrating factors or first integrals that are based on variational derivatives or in the prelle singer method these results are applied to several odes that appear in the study of relevant equations of mathematical physics

λ symmetries for discrete equations following the usual definition of symmetries of differential equations we introduce the analogous concept for difference equations and apply it to some examples

on the identification of quasiprimary scaling operators in local scale invariance the relationship between physical observables defined in lattice models and the associated quasi primary scaling operators of the underlying field theory is revisited in the context of local scale invariance we argue that this relationship is only defined up to a time dependent amplitude and derive the corresponding generalizations of predictions for two time response and correlation functions applications to non equilibrium critical dynamics of several systems with a fully disordered initial state and vanishing initial magnetization including the glauber ising model the frederikson andersen model and the ising spin glass are discussed the critical contact process and the parity conserving non equilibrium kinetic ising model are also considered

periodic solutions for nonlinear second order difference equations abstract this paper is devoted to introduce a new approach to investigate the existence of periodic solutions for nonlinear difference equations transforming a second order nonlinear difference equation on a finite discrete segment with periodic boundary conditions into a continuous system we obtain the existence conditions of periodic solutions of the equation the proof relies on coincidence degree theory and matrix theory without using green s function

on the dimension and the index of the solution set of nonlinear equations we study the covering dimension and the index of the solution set to multiparameter nonlinear and semilinear operator equations involving fredholm maps of positive index the classes of maps under consideration are pseudo a proper and either approximation essential or equivariant approximation essential applications are given to semilinear elliptic bvp s

periodic and subharmonic solutions for a class of superquadratic hamiltonian systems abstract some existence theorems are obtained for periodic and subharmonic solutions of a class of superquadratic hamiltonian systems by the minimax methods in critical point theory specially a local link theorem due to li and willem and the generalized mountain pass theorem respectively

the galilean relativistic principle and nonlinear partial differential equations the second order partial differential equations invariant under transformations of galilei rotation scale and projection are described

instantons in the burgers equation the instanton solution for the forced burgers equation is found this solution describes the exponential tail of the probability distribution function of velocity differences in the region where shock waves are absent that is for large positive velocity differences the results agree with the one found recently by polyakov who used the operator product conjecture if this conjecture is true then our wkb asymptotics of the wyld functional integral should be exact to all orders of perturbation expansion around the instanton solution we also generalized our solution for the arbitrary dimension of the burgers kpz equation as a result we found the asymptotics of the angular dependence of the velocity difference probability distribution function textcopyright 1996 the american physical society

symmetry based determination of space time functions in nonequilibrium growth processes we study the space time correlation and response functions in nonequilibrium growth processes described by linear stochastic langevin equations exploiting exclusively the existence of space and time dependent symmetries of the noiseless part of these equations we derive expressions for the universal scaling functions of two time quantities which are found to agree with the exact expressions obtained from the stochastic equations of motion the usefulness of the space time functions is illustrated through the investigation of two atomistic growth models the family model and the restricted family model which are shown to belong to a unique universality class in 1 1 and 2 1 space dimensions this corrects earlier studies which claimed that in 2 1 dimensions the two models belong to different universality classes

remark on higher derivative mechanics with l conformal galilei symmetry based on the results in a galajinsky and i masterov nucl phys b 866 212 2013 we consider a way to construct a higher derivative mechanical model which possesses the l conformal galilei symmetry the dynamical system describes generalized pais uhlenbeck oscillator in the presence of an effective external field

variational c symmetries and euler lagrange equations abstract a generalization of the concept of variational symmetry based on prolongations allows us to construct new methods of reduction for euler lagrange equations an adapted formulation of the noether s theorem for the new class of symmetries is presented some examples illustrate how the method works in practice

subharmonics for convex nonautonomous hamiltonian systems on etudie des solutions periodiques du systeme hamiltonien jx h t x ou j est la matrice symplectique standard et h c 2 r r 2n r est t periodique en la premiere variable pour tout t x h t x note le gradient partiel par rapport a la seconde variable

on the relation between standard and μ symmetries for pdes we give a geometrical interpretation of the notion of prolongations of vector fields and of the related concept of symmetry for partial differential equations extending to pdes the notion of symmetry for odes we give in particular a result concerning the relationship between symmetries and standard exact symmetries the notion is also extended to the case of conditional and partial symmetries and we analyse the relation between local symmetries and nonlocal standard symmetries

large deviations of the current for driven periodic diffusions we study the large deviations of the time integrated current for a driven diffusion on the circle often used as a model of nonequilibrium systems we obtain the large deviation functions describing the current fluctuations using a fourier bloch decomposition of the so called tilted generator and we also construct from this decomposition the effective biased auxiliary or driven markov process describing the diffusion as current fluctuations are observed in time this effective process provides a clear physical explanation of the various fluctuation regimes observed it is used here to obtain an upper bound on the current large deviation function which we compare to a recently derived entropic bound and to study the low noise limit of large deviations

non local representations of the ageing algebra in higher dimensions the ageing lie algebra and especially its local representations for a dynamical exponent z 2 have played an important role in the description of systems undergoing simple ageing after a quench from a disordered state to the low temperature phase here the construction of representations of for generic values of z is described for any space dimension d 1 generalizing upon earlier results for d 1 the mechanism for the closure of the lie algebra is explained the lie algebra generators contain higher order differential operators or the riesz fractional derivative co variant two time response functions are derived some simple applications to the exactly solvable models of phase separation or interface growth with conserved dynamics are discussed

nonrelativistic conformal groups ii further developments and physical applications the finite dimensional conformal groups associated with the galilei and oscillating or expanding newton hooke space time manifolds was characterized by the present authors in a recent work three isomorphic group families one for each nonrelativistic kinematics were obtained whose members are labeled by a half integer number l since the action of these groups on their corresponding space time manifolds is only local a linearization is introduced here such that the corresponding action is well defined everywhere in particular the l 1 conformal cases that can be obtained by contraction from the well known minkowskian conformal group are treated in more detail as an application of physical interest the conformal invariance of the galilean electromagnetism is studied in order to achieve it the pertinent local representations of the galilean conformal algebras are derived

universal large deviation function of the kardar parisi zhang equation in one dimension using the bethe ansatz we calculate the whole large deviation function of the displacement of particles in the asymmetric simple exclusion process asep on a ring when the size of the ring is large the central part of this large deviation function takes a scaling form independent of the density of particles we suggest that this scaling function found for the asep is universal and should be characteristic of all the systems described by the kardar parisi zhang equation in 1 1 dimension simulations done on two simple growth models are in reasonable agreement with this conjecture

schrodinger invariance and strongly anisotropic critical systems the extension of strongly anisotropic or dynamical scaling to local scale invariance is investigated for the special case of an anisotropy or dynamical exponent z 2 the group of local scale transformation considered is the schrodinger group which can be obtained as the nonrelativistic limit of the conformal group the requirement of schrodinger invariance determines the two point function in the bulk and reduces the three point function to a scaling form of a single variable scaling forms are also derived for the two point function close to a free surface which can be either spacelike or timelike these results are reproduced in several exactly solvable statistical systems namely the kinetic ising model with glauber dynamics lattice diffusion lifshitz points in the spherical model and critical dynamics of the spherical model with a nonconserved order parameter for generic values of evidence from higher order lifshitz points in the spherical model and from directed percolation suggests a simple scaling form of the two point function

applications of a zp index theory to periodic solutions for a class of functional differential equations abstract by means of a new geometrical index with z p group actions multiplicity results for a certain class of nonautonomous time periodic functional differential systems are obtained

deformation of lie derivative and μ symmetries we introduce in the spirit of witten s gauging of exterior differential a deformed lie derivative that allows a geometrical interpretation of and symmetries in complete analogy with standard symmetries the case of variational symmetries both for odes and for pdes is also considered in this approach leading to and conservation laws

symmetries of the stochastic burgers equation all lie symmetries of the burgers equation driven by an external random force are found besides the generalized galilean transformations this equation is also invariant under the time reparametrizations it is shown that the gaussian distribution of a pumping force is not invariant under the symmetries and breaks them down leading to the non trivial vacuum instanton integration over the volume of the symmetry groups provides the description of fluctuations around the instanton and leads to an exactly solvable quantum mechanical problem

periodic and subharmonic solutions of second order hamiltonian systems abstract some solvability conditions of periodic solutions and subharmonic solutions are obtained for a class of new superquadratic non autonomous second order hamiltonian systems by the minimax methods in critical point theory

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

large deviations of the dynamical activity in the east model analysing structure in biased trajectories we consider large deviations of the dynamical activity in the east model we bias this system to larger than average activity and investigate the structure that emerges to best characterize this structure we exploit the fact that there are effective interactions that would reproduce the same behaviour in an equilibrium system we combine numerical results with linear response theory and variational estimates of these effective interactions giving the first insights into such interactions in a many body system across a wide range of biases the system exhibits a hierarchy of responses to the bias remaining quasi equilibrated on short length scales but deviating far from equilibrium on large length scales we discuss the connection between this hierarchy and the hierarchical ageing behaviour of the system

nonrelativistic conformal groups in this work a systematic study of finite dimensional nonrelativistic conformal groups is carried out under two complementary points of view first the conformal killing equation is solved to obtain a whole family of finite dimensional conformal algebras corresponding to each of the galilei and newton hooke kinematical groups some of their algebraic and geometrical properties are studied in a second step among the groups included in these families one can identify for example the contraction of the minkowski conformal group the analog for a nonrelativistic de sitter space or the nonextended schrodinger group

subharmonic solutions with prescribed minimal period for nonautonomous hamiltonian systems on etablit un resultat de multiplicite pour une certaine classe de systemes hamiltoniens periodiques en temps

nonlinear galilei invariant pdes with infinite dimensional lie symmetry a class of nonlinear galilei invariant generalizations of the heat equation admitting infinite dimensional lie symmetry is presented lie symmetries and examples of exact solutions are found for some nonlinear equations generalizing the heat equation full sets of the absolute differential invariants for two non equivalent representations of the galilei algebra are constructed

aspects of infinite dimensional l super galilean conformal algebra in this work we construct an infinite dimensional l super galilean conformal algebra which is a generalization of the l 1 algebra found in the literature we give a classification of central extensions the vector field representation the coadjoint representation and the operator product expansion of the infinite dimensional l super galilean conformal algebra keeping possible applications in physics and mathematics in mind

the totally asymmetric exclusion process on a ring exact relaxation dynamics and associated model of clustering transition abstract the totally asymmetric simple exclusion process in discrete time is considered on finite rings with fixed number of particles a translation invariant version of the backward ordered sequential update is defined for periodic boundary conditions we prove that the so defined update leads to a stationary state in which all possible particle configurations have equal probabilities using the exact analytical expression for the propagator we find the generating function for the conditional probabilities average velocity and diffusion constant at all stages of evolution an exact and explicit expression for the stationary velocity of tasep on rings of arbitrary size and particle filling is derived the evolution of small systems towards a steady state is clearly demonstrated considering the generating function as a partition function of a thermodynamic system we study its zeros in planes of complex fugacities at long enough times the patterns of zeroes for rings with increasing size provide evidence for a transition of the associated two dimensional lattice paths model into a clustered phase at low fugacities

a λ symmetry based method for the linearization and determination of first integrals of a family of second order ordinary differential equations we present a characterization of the second order ordinary differential equations odes that can be linearized by means of certain nonlocal transformations this characterization is given in terms of the coefficients of the equation and also determines the second order odes that admit symmetries and first integrals of some specific forms systematic methods to find these first integrals symmetries and the linearizing transformations are also derived

response functions in phase ordering kinetics we discuss the behavior of response functions in phase ordering kinetics within the perturbation theory approach developed earlier at zeroth order the results agree with previous gaussian theory calculations at second order the nonequilibrium exponents lambda and lambda r are changed but remain equal

transition probabilities and dynamic structure function in the asep conditioned on strong flux we consider the asymmetric simple exclusion processes asep on a ring constrained to produce an atypically large flux or an extreme activity using quantum free fermion techniques we find the time dependent conditional transition probabilities and the exact dynamical structure function under such conditioned dynamics in the thermodynamic limit we obtain the explicit scaling form this gives a direct proof that the dynamical exponent in the extreme current regime is z 1 rather than the kpz exponent z 3 2 which characterizes the asep in the regime of typical currents some of our results extend to the activity in the partially asymmetric simple exclusion process including the symmetric case

periodic and subharmonic solutions of a class of superquadratic hamiltonian systems abstract periodic solutions and infinitely distinct subharmonic solutions are obtained for a class of nonconvex and nonautonomous superquadratic hamiltonian systems z j h z z t by using the minimax methods in critical point theory

on the reduction methods for ordinary differential equations as tandard method based on the use of differential invariants of a lie group g e nables us to reduce any ordinary differential equation invariant under the action of g w e show that this method is applicable to vector fields more general than those associated with lie symmetries we characterize all such vector fields and study their relationship with nonlocal symmetries and symmetries govinder k s and leach p g l 1995 j phys a math gen 28 5349 59 muriel c and romero l 2001 ima j appl math 66 111 25

antishocks in the asep with open boundaries conditioned on low current we study the time evolution of the asep on a finite lattice with l sites and open boundaries conditioned on an atypically low current up to a finite time t by an exact computation we show that for a one parameter family of boundary densities and a special value of the conditioned current an initial product measure with an antishock at site k evolves into a convex combination of such antishocks at sites k the weights p k t k 0 are shown to be the transition probabilities of simple biased random walk with reflecting boundaries we compute explicitly these transition rates our result implies that the antishock remains microscopically stable under the locally conditioned dynamics

a generalization of λ symmetry reduction for systems of odes σ symmetries we consider a deformation of the prolongation operation defined on sets of vector fields and involving a mutual interaction in the definition of prolonged ones this maintains the invariants by differentiation property and can hence be used to reduce odes satisfying suitable invariance conditions in a fully algorithmic way similarly to what happens for standard prolongations and symmetries

local scale invariances in the bosonic contact and pair contact processes local scale invariance for ageing systems without detailed balance is tested through studying the dynamical symmetries of the critical bosonic contact process and the critical bosonic pair contact process their field theoretical actions can be split into a schrodinger invariant term and a pure noise term it is shown that the two time response and correlation functions are reducible to certain multipoint response functions which depend only on the schrodinger invariant part of the action for the bosonic contact process the representation of the schrodinger group can be derived from the free diffusion equation whereas for the bosonic pair contact process a new representation of the schrodinger group related to a nonlinear schrodinger equation with dimensionful couplings is constructed the resulting predictions of local scale invariance for the two time responses and correlators are completely consistent with the exactly known results in both models

noether theorem for μ symmetries we give a version of noether theorem adapted to the framework of symmetries this extends to such case recent work by muriel romero and olver in the framework of symmetries and connects symmetries of a lagrangian to a suitably modified conservation law in some cases this conservation law actually reduces to a standard one we also note a relation between symmetries and conditional invariants we also consider the case where the variational principle is itself formulated as requiring vanishing variation under prolonged variation fields leading to modified euler lagrange equations in this setting symmetries of the lagrangian correspond to standard conservation laws as in the standard noether theorem we finally propose some applications and examples

exact solution of the master equation for the asymmetric exclusion process using the bethe ansatz we obtain the exact solution of the master equation for the totally asymmetric exclusion process on an infinite one dimensional lattice we derive explicit expressions for the conditional probabilitiesp x1 xn t y1 yn 0 of findingn particles on lattices sitesx1 xn at timet with initial occupationy1 yn at timet 0

on non linear partial differential equations with an infinite dimensional conditional symmetry abstract the invariance of non linear partial differential equations under a certain infinite dimensional lie algebra a n z in n spatial dimensions is studied the special case a 1 2 was introduced in j stat phys 75 1994 1023 and contains the schrodinger lie algebra sch 1 as a lie subalgebra it is shown that there is no second order equation which is invariant under the massless realizations of a n z however a large class of strongly non linear partial differential equations is found which are conditionally invariant with respect to the massless realization of a n z such that the well known monge ampere equation is the required additional condition new exact solutions are found for some representatives of this class

on the geometry of lambda symmetries and pde reduction we give a geometrical characterization of prolongations of vector fields and hence of symmetries of odes this allows an extension to the case of pdes and systems of pdes in this context the central object is a horizontal 1 form and we speak of prolongations of vector fields and symmetries of pdes we show that these are as good as standard symmetries in providing symmetry reduction of pdes and systems and explicit invariant solutions

galilei invariant nonlinear systems of evolution equations all systems of n 1 dimensional quasilinear second order evolution equations invariant under chain of algebras ag 1 n contained in implied by ag1 1 n contained in implied by ag2 1 n are described the results obtained are illustrated by the examples of the nonlinear schrodinger equations hamilton jacobi type systems and reaction diffusion equations

large deviations conditioned on large deviations i markov chain and langevin equation we present a systematic analysis of stochastic processes conditioned on an empirical observable q t defined in a time interval 0 t for large t we build our analysis starting with a discrete time markov chain results for a continuous time markov process and langevin dynamics are derived as limiting cases in the large t limit we show how conditioning on a value of q t modifies the dynamics for a langevin dynamics with weak noise and conditioned on q t we introduce large deviation functions and calculate them using either a wkb method or a variational formulation this allows us in particular to calculate the typical trajectory and the fluctuations around this trajectory when conditioned on a certain value of q t for large t

a group theoretic approach to a class of second order ordinary differential equations not possessing lie point symmetries we consider the solution of a class of second order ordinary differential equations not possessing lie point symmetries by group theoretic means the method involves increasing the order of these equations using homogeneity symmetry the solution of this new third order equation is then sought in the instances that it and the new reduced second order equation possess additional symmetries as a result the number of second order equations solvable by the lie theory of extended groups is increased

kinetics of the long range spherical model the kinetic spherical model with long range interactions is studied after a quench to t tc or to t tc for the two time response and correlation functions of the order parameter as well as for composite fields such as the energy density the ageing exponents and the corresponding scaling functions are derived the results are compared to the predictions which follow from local scale invariance

evolution equations invariant under two dimensional space time schrodinger group the most general second order evolution equation t f x t x x xx xx 0 invariant under the galilei galilei similitude and schrodinger groups in two dimensions is constructed a preliminary step is a classification of all possible realizations of the corresponding lie algebras of vector fields in r2 c parametrized by x t and applications of this study include the investigation of nonlinear alternatives to quantum mechanics and nonrelativistic classical field theories among the schrodinger invariant equations in particular are found integrable equations linearizable by contact transformations

intertwining operators for l conformal galilei algebras and hierarchy of invariant equations the l conformal galilei algebra denoted by is a non semisimple lie algebra specified by a pair of parameters d l the algebra is regarded as a nonrelativistic analogue of the conformal algebra we derive hierarchies of partial differential equations which have invariance of the group generated by with a central extension as kinematical symmetry this is done by developing a representation theory such as verma modules singular vectors of and vector field representations for d 1 2

local and nonlocal solvable structures in the reduction of odes solvable structures likewise solvable algebras of local symmetries can be used to integrate scalar ordinary differential equations odes by quadratures solvable structures however are particularly suitable for the integration of odes with a lack of local symmetries in fact under regularity assumptions any given ode always admits solvable structures even though finding them in general could be a very difficult task in practice a noteworthy simplification may come by computing solvable structures which are adapted to some admitted symmetry algebra in this paper we consider solvable structures adapted to local and nonlocal symmetry algebras of any order i e classical and higher in particular we introduce the notion of nonlocal solvable structure

nonlocal aspects of λ symmetries and odes reduction a reduction method of odes not possessing lie point symmetries makes use of the so called symmetries muriel and romero 2001 ima j appl math 66 111 the notion of covering for an ode is used here to recover symmetries of as nonlocal symmetries in this framework by embedding into a suitable system determined by the function any symmetry of can be recovered by a local symmetry of as a consequence the reduction method of muriel and romero follows from the standard method of reduction by differential invariants applied to

nonlinear oscillations and boundary value problems for hamiltonian systems we prove the existence of solutions of various boundary value problems for nonautonomous hamiltonian systems with forcing terms begin gathered dot x t h p t x t p t g t hfill dot p t h x t x t p t f t hfill end gathered