violation of the fluctuation dissipation theorem in glassy systems basic notions and the numerical evidence this review reports on the research done during past years on violations of the fluctuation dissipation theorem fdt in glassy systems it is focused on the existence of a quasi fluctuation dissipation theorem qfdt in glassy systems and the current supporting knowledge gained from numerical simulation studies it covers a broad range of non stationary aging and stationary driven systems such as structural glasses spin glasses coarsening systems ferromagnetic models at criticality trap models models with entropy barriers kinetically constrained models sheared systems and granular media the review is divided into four main parts 1 an introductory section explaining basic notions related to the existence of the fdt in equilibrium and its possible extension to the glassy regime qfdt 2 a description of the basic analytical tools and results derived in the framework of some exactly solvable models 3 a detailed report of the current evidence in favour of the qfdt and 4 a brief digression on the experimental evidence in its favour this review is intended for inexpert readers who want to learn about the basic notions and concepts related to the existence of the qfdt as well as for the more expert readers who may be interested in more specific results

on the helicity in 3d periodic navier stokes equations i the non statistical case we study the asymptotic behavior of the statistical solutions to the navier stokes equations using the normalization map 9 it is then applied to the study of mean energy mean dissipation rate of energy and mean helicity of the spatial periodic flows driven by potential body forces the statistical distribution of the asymptotic beltrami flows are also investigated we connect our mathematical analysis with the empirical theory of decaying turbulence with appropriate mathematically defined ensemble averages the kolmogorov universal features are shown to be transient in time we provide an estimate for the time interval in which those features may still be present our collaborator and friend basil nicolaenko passed away in september of 2007 after this work was completed honoring his contribution and friendship we dedicate this article to him

discrete order preserving semigroups and stability for periodic parabolic differential equations on etudie la stabilite globale des solutions des equations paraboliques dans une forme non auto adjointe avec des coefficients periodiques dans le temps la structure mathematique est celle des semigroupes preservant l ordre a temps discret s n n e n sur un reseau de banach

absolute continuity of stable foliations for systems on banach spaces abstract we prove the absolute continuity of stable foliations for c 1 maps of banach spaces satisfying a globally defined infinitesimal invariant cones condition proofs of regularity for center and stable manifolds needed for the main theorem are included our results are applicable to dynamical systems generated by ordinary partial or functional differential equations including non autonomous differential equations that are periodic in time

asymptotic expansion for solutions of the navier stokes equations with non potential body forces abstract we study the long time behavior of spatially periodic solutions of the navier stokes equations in the three dimensional space the body force is assumed to possess an asymptotic expansion or resp finite asymptotic approximation in sobolev gevrey spaces as time tends to infinity in terms of polynomial and decaying exponential functions of time we establish an asymptotic expansion or resp finite asymptotic approximation of the same type for the leray hopf weak solutions this extends previous results that were obtained in the case of potential forces to the non potential force case where the body force may have different levels of regularity and asymptotic approximation this expansion or approximation in fact reveals precisely how the structure of the force influences the asymptotic behavior of the solutions

structural stability for time periodic one dimensional parabolic equations denote q w x 0 l x r the nonlinear term f is assumed to satisfy the following conditions f l f q r is of class c f 2 f t l x u f t x u for all te w xe o l and uer define a self adjoint operator a in x l2 0 l by a t bxx for e d a h2 0 l a ha o l for each c1 e r let a be the fractional power of a and x d a be the fractional power space equipped with the graph norm ii 11 xn 1111 11x iia i ii x throughout this section we fix a c1 1 by the above assumptions f l f 2 for any u0 e x there exists a unique local in time classical solution u t x u t x of

on the helicity in 3d periodic navier stokes equations ii the statistical case we study the asymptotic behavior of the statistical solutions to the navier stokes equations using the normalization map 9 it is then applied to the study of mean energy mean dissipation rate of energy and mean helicity of the spatial periodic flows driven by potential body forces the statistical distribution of the asymptotic beltrami flows are also investigated we connect our mathematical analysis with the empirical theory of decaying turbulence with appropriate mathematically defined ensemble averages the kolmogorov universal features are shown to be transient in time we provide an estimate for the time interval in which those features may still be present our collaborator and friend basil nicolaenko passed away in september of 2007 after this work was completed honoring his contribution and friendship we dedicate this article to him

monotone dynamical systems with dense periodic points abstract in this paper we prove a recent conjecture by m hirsch which says that if f is a discrete time monotone dynamical system with f a homeomorphism on an open connected subset of a finite dimensional vector space and the periodic points of f are dense in then f is periodic

entropy of particle packings an illustration on a toy model a toy model of particle packing is presented which consists in arranging hexagons on a triangular lattice according to local stability rules the number of stable packings is analytically computed and found to grow exponentially with the size of the lattice which illustrates the concept of packing entropy first proposed by edwards and collaborators the analysis is carried out for both the monodispersed case and the more interesting i e more disordered bidispersed case

k dimensional invariant cones of random dynamical systems in rn with applications abstract we investigated a linear random dynamical system which strongly preserves a cone c of dimension k abbr k cone in r n under some general assumptions it is shown that such system admits a measurable family of k dimensional subspaces and a measurable family of n k dimensional subspaces which are complementary to each other and form into a tempered invariant splitting of r n we further apply the measurable bundles so obtained to study the linear random monotone cyclic feedback systems as well as the linear competitive cooperative tridiagonal systems this generalizes the floquet theory for these deterministic non autonomous or time periodic systems to the random systems

on random linear dynamical systems in a banach space i multiplicative ergodic theorem and krein rutman type theorems abstract for linear random dynamical systems in a separable banach space x we derived a series of krein rutman type theorems with respect to co invariant cone family with rank k which present a quasi equivalence relation between the measurably co invariant cone family and the measurably dominated splitting of x moreover such quasi equivalence relation turns out to be an equivalence relation whenever i k 1 or ii in the frame of the multiplicative ergodic theorem with certain lyapunov exponent being greater than the negative infinity for the second case we thoroughly investigated the relations between the lyapunov exponents the co invariant cone family and the measurably dominated splitting for linear random dynamical systems in x

characterization of dominated splittings for operator cocycles acting on banach spaces abstract versions of the oseledets multiplicative ergodic theorem for cocycles acting on infinite dimensional banach spaces have been investigated since the pioneering work of ruelle in 1982 and are a topic of continuing research interest for a cocycle to induce a continuous splitting in which the growth in one subbundle exponentially dominates the growth in a complementary subbundle requires additional assumptions a necessary and sufficient condition for the existence of such a dominated splitting also known as exponential separation was recently given by j bochi and n gourmelon for invertible finite dimensional cocycles in discrete time we extend this result to cocycles of injective bounded and possibly noncompact linear maps acting on banach spaces in both discrete and continuous time using an essentially geometric approach based on a notion of approximate singular value decomposition in banach spaces our method is constructive and in the finite dimensional case yields explicit growth estimates on the dominated splitting which may be of independent interest

edwards measure and the steady state regime of a model with kinetic constraints under tapping we study the tapping dynamics of a one dimensional ising model with symmetric kinetic constraints we define and test a variant of the edwards hypothesis that one may build a thermodynamics for the steady state by using a flat measure over the metastable states with several macroscopic quantities fixed various types of tapping are compared and the accuracy of this measure becomes quickly excellent when the number of quantities fixed on average increases independent of the way the system is excited we attribute the validity of the naive flat measure at weak tapping to the spatial separation of density defects

theory of powders starting from the observations that powders have a large number of particles and reproducible properties we show how statistical mechanics applies to powders the volume of the powder plays the role of the energy in conventional statistical mechanics with the hypothesis that all states of a specified volume are equally probable we introduce a variable x the compactivity which is analogous to the temperature in thermodynamics some simple models are considered which demonstrate how the problems involved can be tackled using the concept of compactivity there is an increasing interest in applying the methods of statistical mechanics and of transport theory to systems which are neither atomistic nor in equilibrium but which still fulfil a remaining tenet of statistical physics which is that systems can be completely defined by a very small number of parameters and can be constructed in a reproducible way powders fall into this category if a powder consists for example of uniform cubes of salt and is poured into a container falling at low density uniformly from a great height one expects a salt powder of a certain density repeating the preparation reproduces the same density a treatment such as shaking the powder by a definite routine produces a new density and the identical routine applied to another sample of the initial powder will result in the same final density clearly a maxwell demon could arrange the little cubes of nacl to make a material of different properties to that of our experiment but if such demonics are ignored and we restrict ourselves to extensive operations such as stirring shaking compressing all actions which do not act on grains individually then well defined states of the powder result in this paper we will set up a framework for describing the state of the powder basing our development on anologies with statistical mechanics some attempts have been made to apply information theory ideas directly to powders

characteristic exponents and invariant manifolds in hilbert space the multiplicative ergodic theorem and the construction almost everywhere of stable and unstable manifolds pesin theory are extended to differentiable dynamical systems on hilbert manifolds under some compactness assumptions the results apply to partial differential equations of evolution and also to non invertible maps of compact manifolds

phase translation group actions on strongly monotone skew product semiflows we establish a convergence property for pseudo bounded forward orbits of strongly monotone skew product semiflows with invariant phasetranslation group actions the results are then applied to obtain global convergence of certain chemical reaction networks whose associated systems in reaction coordinates are monotone as well as the dynamics of certain reactiondiffusion systems in time recurrent structure including periodicity almost periodicity and almost automorphy

asymptotic expansion for solutions of the navier stokes equations with potential forces abstract we derive an asymptotic expansion for smooth solutions of the navier stokes equations in weighted spaces this result removes previous restrictions on the number of terms of the asymptotics as well as on the range of the polynomial weights we also write the expansion in terms of expressions involving non linear quantities

asymptotic behavior of strongly monotone time periodic dynamical processes with symmetry given a strongly monotone discrete time dynamical system tn x x n z in an open and order convex subset x of a separable strongly ordered banach space v and a compact connected metrizable group g whose action on x is monotone and commutes with t we prove under some reasonable additional hypotheses that the limit set x of every stable point x x consists of symmetric points only i e g w w for all w x and g g moreover the set of all unstable points is contained in a union of at most countably many lipschitz manifolds of codimension one in v where each manifold is invariant under t and the action of g this result is applied to the time periodic spatially independent irreducible cooperative system of n reaction diffusion equations u t d t u f t u for t x r1 rn with spatially periodic boundary conditions in rn and with an initial distribution u0 which is continuous and satisfies the boundary conditions if u0 is stable then u0 contains spatially constant functions only and the dynamics on u0 is given by the irreducible cooperative system of n odes dudt f t u for t r1 here t is the poincare map if n 1 then u0 is a single fixed point of t if n 2 then v0 is a single fixed point of t for each v0 u0

definition of temperature in dense granular media in this paper we report the measurement of a pseudotemperature for compacting granular media on the basis of the fluctuation dissipation relations in the aging dynamics of a model system from the violation of the fluctuation dissipation theorem emerges an effective temperature a dynamical temperature t dyn whose value depends on the particle density we compare the results for t dyn at several densities with the values of edwards compactivity at the corresponding densities s f edwards in granular matter an interdisciplinary approach edited by a mehta springer verlag new york 1994 and references therein it turns out that the dynamical temperature and edwards compactivity coincide on a large range of densities opening in this way the door to experimental checks as well as theoretical constructions

convergence to equilibrium on invariant d hypersurfaces for strongly increasing discrete time semigroups abstract convergence to equilibrium for every positive semi orbit of strongly increasing discrete time semigroups on strongly ordered spaces is proved the domain of the semigroup is assumed to be slightly more general than a closed order interval in a strongly ordered banach space which is continuously imbedded into a banach lattice the semigroup is assumed to be order compact and every positive semi orbit is assumed to be order bounded the crucial hypothesis is the ljapunov stability of all equilibria it is also proved that the set of equilibria is a simply ordered arc the key tools are invariant d hypersurfaces

three dimensional competitive lotka volterra systems with no periodic orbits the following conjecture of m l zeeman is proved if three interacting species modeled by a competitive lotka volterra system can each resist invasion at carrying capacity then there can be no coexistence of the species indeed two of the species are driven to extinction it is also proved that in the other extreme if none of the species can resist invasion from either of the others then there is stable coexistence of at least two of the species in this case if the system has a fixed point in the interior of the positive cone in r3 then that fixed point is globally asymptotically stable representing stable coexistence of all three species otherwise there is a globally asymptotically stable fixed point in one of the coordinate planes of r3 representing stable coexistence of two of the species

principal lyapunov exponents and principal floquet spaces of positive random dynamical systems ii finite dimensional systems abstract this is the second part in a series of papers concerned with principal lyapunov exponents and principal floquet subspaces of positive random dynamical systems in ordered banach spaces the current part focuses on applications of general theory developed in the authors paper j mierczy ski w shen principal lyapunov exponents and principal floquet spaces of positive random dynamical systems i general theory trans amer math soc in press http dx doi org 10 1090 s0002 9947 2013 05814 x preprint available at http arxiv org abs 1209 3475 to positive random dynamical systems on finite dimensional ordered banach spaces it is shown under some quite general assumptions that measurable linear skew product semidynamical systems generated by measurable families of positive matrices and by strongly cooperative or type k strongly monotone systems of linear ordinary differential equations admit measurable families of generalized principal floquet subspaces generalized principal lyapunov exponents and generalized exponential separations

asymptotic behaviour of comparable skew product semiflows with applications the almost 1 cover lifting property of omega limit sets is established for nonmonotone skew product semiflows which are comparable to uniformly stable eventually strongly monotone skew product semiflows these results are then applied to study the asymptotic behavior of solutions to the nonmonotone comparable systems of odes reaction diffusion systems differential systems with time delays and semilinear parabolic equations

edwards measures for powders and glasses can one construct a thermodynamics for compact slowly moving powders and grains a few years ago edwards proposed a possible step in this direction raising the fascinating perspective that such systems have a statistical mechanics of their own different from that of maxwell boltzmann and gibbs allowing us to have some information while still ignoring dynamic details recent developments in the theory of glasses have come to confirm these ideas within mean field in order to go beyond we explicitly generate edwards measure in a 3d model comparison of the results with the irreversible compaction data shows very good agreement the present framework immediately suggests new experimental checks

roughness of tempered exponential dichotomies for infinite dimensional random difference equations abstract in this paper we study the roughness of tempered exponential dichotomies for linear random dynamical systems in banach spaces such a dichotomy has a tempered bound and describes nonuniform hyperbolicity we prove the roughness without assuming their invertibility and the integrability condition of the multiplicative ergodic theorem we give an explicit bound for the linear perturbation such that the dichotomy is persistent we also obtain explicit forms for the exponent and the bound of tempered exponential dichotomy of the perturbed random system in terms of the original ones and the perturbations

remark on the rate of decay of higher order derivatives for solutions to the navier stokes equations in rn we present a new derivation of upper bounds for the decay of higher order derivatives of solutions to the unforced navier stokes equations in rn the method based on so called gevrey estimates also yields explicit bounds on the growth of the radius of analyticity of the solution in time moreover under the assumption that the navier stokes solution stays sufficiently close to a solution of the heat equation in the l2 norm a result known to be true for a large class of initial data lower bounds on the decay of higher order derivatives can be obtained

perturbative theory of the packing of mixtures and of non spherical particles abstract we present a simple theory for the packing of irregular particles and mixtures considered as a perturbation of a packing of monodisperse spheres using the perturbation theory we extract the different geometrical effects which effect the packing density and discuss why packings of irregular particles are generally less dense than packings of monodisperse spheres whereas packings of mixtures of particles of different sizes are generally denser than monodisperse sphere packings

symmetry and convergence properties for non negative solutions of nonautonomous reaction diffusion problems nonautonomous parabolic equations of the form u t u f u t on a symmetric domain are considered using the moving hyperplane method it is proved that any bounded nonnegative solution symmetrises as t this is then used to show that for nonlinearities periodic in t any non negative bounded solution approaches a periodic solution

convergence and stability for essentially strongly order preserving semiflows abstract this paper is concerned with a class of essentially strongly order preserving semiflows which are defined on an ordered metric space and are generalizations of strongly order preserving semiflows for essentially strongly order preserving semiflows we prove several principles which are analogues of the nonordering principle for limit sets the limit set dichtomy and the sequential limit set trichotomy for strongly order preserving semiflows then under certain compactness hypotheses we obtain some results on convergence quasiconvergence and stability in essentially strongly order preserving semiflows finally some applications are made to quasimonotone systems of delay differential equations and reaction diffusion equations with delay and the main advantages of our results over the classical ones are that we do not require the delicate choice of state space and the technical ignition assumption

asymptotic integration of navier stokes equations with potential forces ii an explicit poincare dulac normal form abstract we study the incompressible navier stokes equations with potential body forces on the three dimensional torus we show that the normalization introduced in the paper c foias j c saut linearization and normal form of the navier stokes equations with potential forces ann inst h poincare anal non lineaire 4 1 1987 1 47 produces a poincare dulac normal form which is obtained by an explicit change of variable this change is the formal power series expansion of the inverse of the normalization map each homogeneous term of a finite degree in the series is proved to be well defined in appropriate sobolev spaces and is estimated recursively by using a family of homogeneous gauges which is suitable for estimating homogeneous polynomials in infinite dimensional spaces

an extension of the fuglede commutativity theorem modulo the hilbert schmidt class to operators of the form _ _ we study the operators a x y m xn and a x m xn which map the algebra of all bounded linear operators on a separable hilbert space to itself where m j and n are separately commuting sequences of normal operators we prove that 1 when m 2 the hilbert schmidt norms of a x and a x are equal finite or infinite 2 for m 2 3 if a x and a x are hilbert schmidt operators then their hilbert schmidt norms are equal 3 if a a have the property that for each x a x 0 implies a x 0 then for each x if a x is a hilbert schmidt operator then a 2 x is also and the latter has the same hilbert schmidt norm as a2 x note that fuglede s theorem is immediate from 1 in the case mi 2 m1 n2 and n1 i m2 the proofs employ the duality between the trace class and the class of all bounded linear operators and unlike the early proofs of fuglede s theorem they are free of complex function theory let h denote a separable finite dimensional complex hilbert space let l h d k h d c d f h c2 c1 11 ilp 11 11 denote respectively the class of all bounded linear operators acting on h the compact operators the schatten p class the finite rank operators the hilbert schmidt class the trace class the c norm and the operator norm definition let k mn n 1 and k nn m 1 be two separately commuting sequences of normal operators define a a l h l h by

principal lyapunov exponents and principal floquet spaces of positive random dynamical systems i general theory this series of papers is concerned with principal lyapunov exponents and principal floquet subspaces of positive random dynamical systems in ordered banach spaces the current part of the series focuses on the development of general theory first the notions of generalized principal floquet subspaces generalized principal lyapunov exponents and generalized exponential separations for general positive random dynamical systems in ordered banach spaces are introduced which extend the classical notions of principal floquet subspaces principal lyapunov exponents and exponential separations for strongly positive deterministic systems in strongly ordered banach to general positive random dynamical systems in ordered banach spaces under some quite general assumptions it is then shown that a positive random dynamical system in an ordered banach space admits a family of generalized principal floquet subspaces a generalized principal lyapunov exponent and a generalized exponential separation we will consider in the forthcoming part s the applications of the general theory developed in this part to positive random dynamical systems arising from a variety of random mappings and differential equations including random leslie matrix models random cooperative systems of ordinary differential equations and random parabolic equations institute of mathematics and computer science wroc law university of technology wybrzeze wyspianskiego 27 pl 50 370 wroc law poland url http www im pwr wroc pl mierczyn index html e mail address mierczyn pwr wroc pl department of mathematics and statistics auburn university auburn university al 36849 usa e mail address wenxish auburn edu 2010 mathematics subject classification primary 37h15 37l55 37a30 secondary 15b52 34f05 35r60

exponential separation and principal lyapunov exponent spectrum for random nonautonomous parabolic equations abstract the purpose of the paper is to extend the principal eigenvalue and principal eigenfunction theory for time independent and periodic parabolic equations to random and general nonautonomous ones in the random case a notion of principal lyapunov exponent serving as an analog of principal eigenvalue is introduced it is shown that the principal lyapunov exponent is deterministic and of simple multiplicity it is also shown that there is a one dimensional invariant random subbundle corresponding to the solutions that are globally defined and of the same sign which serves as an analog of principal eigenfunction in addition monotonicity of the principal lyapunov exponent with respect to the zero order terms both in the equation and in the boundary condition is proved when the second and first order terms are deterministic it is proved that the principal lyapunov exponent is greater than or equal to the principal eigenvalue of the associated time averaged equation in the general nonautonomous case the concepts of principal spectrum which serves as an analog of principal eigenvalue and principal lyapunov exponents are introduced as is known the principal spectrum is a compact interval it is proved in the paper that the principal spectrum contains all the principal lyapunov exponents when the second and first order terms are time independent a lower estimate of the infimum of the principal spectrum is given in terms of an associated time averaged equation

stability and convergence in strongly monotone dynamical systems soit x un systeme dynamique x est un espace topologique et le flot t t r est une famille d applications continues de sous ensembles ouverts de x dans x on decrit qualitativement le comportement asymptotique quand t des trajectoires t t x pour les etats initiaux x dans le cas de systemes ayant un principe de comparaison fort

floquet bundles for scalar parabolic equations for linear scalar parabolic equations such as u t u xx a t x u x b t x u on a finite interval 0 x with various boundary conditions we obtain canonical floquet solutions u n t x these solutions are characterized by the property that z u n t x n for all te where z denotes the zero crossing lap number of matano the coefficients a t x and b t x are not assumed to be periodic in t but if they are the solutions u n t x reduce to the standard floquet solutions our results may naturally be expressed in the language of linear skew product flows in this context we obtain for each n 1 an exponential dichotomy between the bundles span u n 1 and overline span u n cdot cdot n n 1 infty

uniqueness and attractivity of the carrying simplex for discrete time competitive dynamical systems abstract for the discrete time dissipative competitive dynamical system on r n it is proved that generically there is a unique n 1 dim invariant carrying simplex attracting all nontrivial orbits which solved the conjecture raised by h l smith in 1986

on the finite dimensional dynamical systems with limited competition the asymptotic behavior of dynamical systems with limited competition is investigated we study index theory for fixed points permanence global stability convergence everywhere and coexistence it is shown that the system has a globally asymptotically stable fixed point if every fixed point is hyperbolic and locally asymptotically stable relative to the face it belongs to a nice result is the necessary and sufficient conditions for the system to have a globally asymptotically stable positive fixed point it can be used to establish the sufficient conditions for the system to persist uniformly and the convergence result for all orbits applications are made to time periodic ordinary differential equations and reaction diffusion equations

tapping thermodynamics of the one dimensional ising model we analyse the steady state regime of a one dimensional ising model under a tapping dynamics recently introduced by analogy with the dynamics of mechanically perturbed granular media the idea that the steady state regime may be described by a flat measure over metastable states of fixed energy is tested by comparing various steady state time averaged quantities in extensive numerical simulations with the corresponding ensemble averages computed analytically with this flat measure the agreement between the two averages is excellent in all the cases examined showing that a static approach is capable of predicting certain measurable properties of the steady state regime

stationary probability distribution in granular media abstract we discuss recent developments in the formulation of a statistical mechanics approach to non thermal systems such as granular media we review a few important numerical results on the assessment of edwards theory and in particular we apply these ideas to study a mean field model of a hard sphere binary mixture under gravity which can be fully analytically investigated as a consequence we derive the rich phase diagram and predict the features of segregation patterns of the mixture

convergence in smooth strongly monotone flows defined by semilinear parabolic equations on considere des equations paraboliques semi lineaires dy dt ay f y ou a est un operateur sectoriel sur un banach x ayant une resolvante compacte et f est une c 2 application de l espace des puissances fractionnaires x 0 1 dans x soit y t une solution qui est quasi convergente mais non convergente alors il existe un voisinage u de y 0 y 0 avec la propriete suivante si z u y 0 a une trajectoire bornee et soit z y 0 soit z y 0 alors la solution partant de z converge vers un equilibre