the weak coupling limit of disordered copolymer models a copolymer is a chain of repetitive units monomers that are almost identical but they differ in their degree of affinity for certain solvents this difference leads to striking phenomena when the polymer fluctuates in a non homogeneous medium for example made up by two solvents separated by an interface one may observe for instance the localization of the polymer at the interface between the two solvents a discrete model of such system based on the simple symmetric random walk on z has been investigated in bolthausen and den hollander ann probab 25 1997 1334 1366 notably in the weak polymer solvent coupling limit where the convergence of the discrete model toward a continuum model based on brownian motion has been established this result is remarkable because it strongly suggests a universal feature of copolymer models in this work we prove that this is indeed the case more precisely we determine the weak coupling limit for a general class of discrete copolymer models obtaining as limits a one parameter alpha in 0 1 family of continuum models based on alpha stable regenerative sets

a series solution for the effective properties of incompressible viscoelastic media this paper presents a series solution for the homogenization problem of a linear viscoelastic periodic incompressible composite the method uses the laplace transform and the correspondence principle which are combined with the classical expansion along neumann series of the solution of the periodic elasticity problem in fourier space the terms of the neumann series appear as decoupled containing geometry dependent terms and viscoelastic properties dependent terms which are polynomial fractions whose inverse laplace transforms are provided explicitly

effective properties of linear viscoelastic heterogeneous media internal variables formulation and extension to ageing behaviours abstract the laplace carson transform classically used for homogenization of linear viscoelastic heterogeneous media yields integral formulations of effective behaviours these are far less convenient than internal variables formulations with respect to computational aspects as well as to theoretical extensions to closely related problems such as ageing viscoelasticity noticing that the collocation method is usually adopted to invert the laplace carson transforms we first remark that this approximation is equivalent to an internal variables formulation which is exact in some specific situations this result is illustrated for a two phase composite with phases obeying a compressible maxwellian behaviour next an incremental formulation allows to extend at each time step the previous general framework to ageing viscoelasticity finally with the help of a creep test of a porous viscoelastic matrix reinforced with elastic inclusions it is shown that the method yields accurate predictions comparing to reference results provided by periodic cell finite element computations

effective behavior of linear viscoelastic composites a time integration approach a new method for determining the overall behavior of composite materials comprised of linear viscoelastic constituents is presented unlike classical methods which are based on the laplace transform the present method operates directly in the time domain upon use of an implicit time discretization scheme the evolution equations describing the constitutive behavior of the phases can be reduced to the minimization of an incremental energy function this minimization problem is rigorously equivalent to a linear thermoelastic problem with a transformation strain which is a nonuniform field not even uniform within the phases the variational technique of ponte castaneda is used to approximate the nonuniform eigenstrains by piecewise uniform eigenstrains the latter problem is amenable to simpler calculations and analytical results for appropriate microstructures can be obtained the accuracy of the proposed scheme is assessed by comparison of the method with exact results obtained either by full finite element simulations in time domain or by available analytical results obtained by the laplace transform

variational second order estimates for nonlinear composites it is shown here that the approximate second order method of ponte castaneda for estimating the effective behaviour of nonlinear composite materials can be given a variational status this is accomplished by means of a slight generalization of the variational structure of talbot and willis consisting of the replacement of the infimum over the polarization field by the more general corresponding stationary operation since the relevant functions are not convex in general this replacement provides additional flexibility that can be exploited advantageously the net result is a stationary principle from which stationary variational estimates can be generated under certain hypotheses on the choice of the linear comparison composite it is shown that the resulting estimates are identical to the second order estimates thus showing that these second order estimates are indeed variational in character although only of the stationary variety the connection of these second order estimates to earlier bounds obtained from related extremum variational principles is also explored

asymptotic homogenization of viscoelastic composites with periodic microstructures a systematic way of obtaining the effective viscoelastic moduli in time and frequency domain is presented for viscoelastic composites with periodic microstructures the problem of estimating the effective moduli is formulated using the asymptotic homogenization method for theoretical aspects the memory effects due to the homogenization are shown in general form and a sufficient condition for the effects to disappear is fully discussed the computational procedure is divided into two steps the effective relaxation moduli are computed in laplace transformed domain and are numerically inverse transformed into time domain the effective complex moduli are then readily obtained by using simple formulae of the fourier transform several numerical examples are presented to illustrate and verify present approach and to discuss the memory effects

the effect of disorder on polymer depinning transitions we consider a polymer with monomer locations modeled by the trajectory of a markov chain in the presence of a potential that interacts with the polymer when it visits a particular site 0 we assume that probability of an excursion of length n is given by n c varphi n for some 1 3 2 at high temperature the quenched and annealed curves differ significantly only in a very small neighborhood of the critical point the size of this neighborhood scales as beta 1 2c 3 where is the inverse temperature for c 0 for sufficiently high temperature the quenched and annealed curves are within a factor of 1 epsilon for all u near the critical point in particular the quenched and annealed critical points are equal for c 3 2 the regime depends on the slowly varying function varphi

estimates on path delocalization for copolymers at selective interfaces starting from the simple symmetric random walk s n n we introduce a new process whose path measure is weighted by a factor exp open image in new window with h 0 w n n a typical realization of an iid process and n a positive integer we are looking for results in the large n limit this factor favors s n 0 if w n h 0 and s n 0 if w n h 0 the process can be interpreted as a model for a random heterogeneous polymer in the proximity of an interface separating two selective solvents it has been shown 6 that this model undergoes a de localization transition more precisely there exists a continuous increasing function h c such that if h h c then the model is localized while it is delocalized if h h c however localization and delocalization were not given in terms of path properties but in a free energy sense later on it has been shown that free energy localization does indeed correspond to a strong form of path localization 3 on the other hand only weak results on the delocalized regime have been known so far

a numerical approach to copolymers at selective interfaces we consider a model of a random copolymer at a selective interface which undergoes a localization delocalization transition in spite of the several rigorous results available for this model the theoretical characterization of the phase transition has re mained elusive and there is still no agreement about several important issues for example the behavior of the polymer near the phase transition line from a rigorous viewpoint non coinciding upper and lower bounds on the critical line are known in this paper we combine numerical computations with rigorous arguments to get to a better understanding of the phase diagram our main results include various numerical observations that suggest that the critical line lies strictly in between the two bounds a rigorous statistical test based on concentration inequalities and super additivity for determining whether a given point of the phase diagram is in the localized phase this is applied in particular to show that with a very low level of error the lower bound does not coincide with the critical line an analysis of the precise asymptotic behavior of the partition function in the de localized phase with particular attention to the effect of rare atypical stretches in the disorder sequence and on whether or not in the delocalized regime the polymer path has a brownian scaling a new proof of the lower bound on the critical line this proof relies on a characteri zation of the localized regime which is more appealing for interpreting the numerical data

a simple computational homogenization method for structures made of linear heterogeneous viscoelastic materials in this paper a numerical multiscale method is proposed for computing the response of structures made of linearly non aging viscoelastic and highly heterogeneous materials in contrast with most of the approaches reported in the literature the present one operates directly in the time domain and avoids both defining macroscopic internal variables and concurrent computations at micro and macro scales the macroscopic constitutive law takes the form of a convolution integral containing an effective relaxation tensor to numerically identify this tensor a representative volume element rve for the microstructure is first chosen relaxation tests are then numerically performed on the rve correspondingly the components of the effective relaxation tensor are determined and stored for different snapshots in time at the macroscopic scale a continuous representation of the effective relaxation tensor is obtained in the time domain by interpolating the data with the help of spline functions the convolution integral characterizing the time dependent macroscopic stress strain relation is evaluated numerically arbitrary local linear viscoelastic laws and microstructure morphologies can be dealt with implicit algorithms are provided to compute the time dependent response of a structure at the macroscopic scale by the finite element method accuracy and efficiency of the proposed approach are demonstrated through 2d and 3d numerical examples and applied to estimate the creep of structures made of concrete

micromechanical modeling of coupled viscoelastic viscoplastic composites based on an incrementally affine formulation this study proposes a micromechanical modeling of inclusion reinforced viscoelastic viscoplastic composites based on mean field approaches for this we have generalized the so called incrementally affine linearization method which was proposed by doghri et al 2010a for elasto viscoplastic materials the proposal provides an affine relation between stress and strain increments via an algorithmic tangent operator in order to find the incrementally affine expression we start by the linearization of evolution equations at the beginning of a time step around the end time of the step next a numerical integration of the linearized equations is required using a fully implicit backward euler scheme the obtained algebraic equations lead to an incrementally affine formulation which is form similar to linear thermoelasticity therefore known homogenization models for linear thermoelastic composites can be applied the proposed method can deal with general viscoelastic viscoplastic constitutive models with an arbitrary number of internal variables the semi analytical predictions are validated against finite element simulations and experimental results

time domain formulation in computational dynamics for linear viscoelastic media with model uncertainties and stochastic excitation the paper is devoted to the computational time domain formulation of linear viscoelastic systems submitted to a nonstationary stochastic excitation and in the presence of model uncertainties which are modeled in the framework of the probability theory the objective is to introduce and to develop an adapted and complete formulation of such a problem in the context of computational mechanics a reduced order model in the time domain with stochastic excitation is derived from the computational model for the reduced order model the stochastic modeling of both computational model parameter uncertainties and modeling errors is carried out using the nonparametric probabilistic approach and the random matrix theory we present a new formulation of model uncertainties to construct the random operators for viscoelastic media we then obtained a linear stochastic integro differential equation side with random operators and with a stochastic nonhomogeneous part stochastic excitation a time discretization of this side is proposed in a first step the side is transformed to a linear ito stochastic differential equation isde with random operators then the isde is discretized using an extension of the stormer verlet scheme which is a particularly well adapted algorithm for long time good behavior of the numerical solution finally for the stochastic solver and statistical estimations of the random responses we propose to use the monte carlo simulation for gaussian and non gaussian excitations

a differential homogenization method for estimating the macroscopic response and field statistics of particulate viscoelastic composites abstract this work provides an analytical method for determining the time dependent effective constitutive response of linear viscoelastic composites with particulate microstructures by means of a differential algebraic system das of equations by making use of differential equations instead of difference equations and utilizing as primary variables the variances of the field fluctuations instead of the properties of fictitious linear comparison composites the new formulation is found to be more robust and to alleviate some of the issues associated with earlier incremental approaches remarkably the new formulation can be shown to recover the exact results of the correspondence principle for isotropic incompressible composites subjected to proportional loading as well as for isotropic compressible composites subjected to purely hydrostatic loading more generally the proposed formulation is not exact but still delivers reasonably accurate estimates in most cases lastly a generalization of the das formulation is provided for nonlinear viscoelastic composites with incompressible phases and also found to be more robust than earlier approaches

free energy and some sample path properties of a random walk with random potential we study the asymptotic behavior of the free energy for a model defined by sinai of one dimensional random walk with random potential in particular we obtain a central limit theorem and a strong law of large numbers for this free energy we use some results on the free energy to study some sample path properties of this random walk which are related respectively to its recurrence and localization some exponents describing the recurrence and localization are found

on the localization transition of random copolymers near selective interfaces in this note we consider the de localization transition for random directed 1 1 dimensional copolymers in the proximity of an interface separating selective solvents we derive a rigorous lower bound on the free energy this yields a substantial improvement on the bounds from below on the critical line that were known so far our result implies that the critical curve does not lie below the critical curve set forth by monthus eur phys j b 13 111 130 2000 on the base of a renormalization group analysis we discuss this result in the light of the rigorous and non rigorous approaches present in the literature and by making an analogy with a particular asymptotics of a disordered wetting model we propose a simplified framework in which the question of identifying the critical curve as well as understanding the nature of the transition may be approached

on the effective viscoelastic moduli of two phase media iii rigorous bounds on the complex shear modulus in two dimensions the translation and the hashin shtrikman methods are used to provide bounds on the effective complex shear modulus of a two phase two dimensional viscoelastic composite they are both given by inequalities that depend on six parameters the best bounds are obtained by optimizing these parameters over the admissible set which is larger for the translation method than for the hashin shtrikman method consequently the translation method generally leads to tighter bounds than would be obtained via the standard hashin shtrikman approach equivalence classes of two dimensional viscoelastic composites directly analogous to similar classes for the pure elastic problem are found combination of the simplified versions of the translation or the hashin shtrikman type bounds and this equivalency results in simple algorithms for computing tight bounds for any choice of phase moduli and volume fractions the bounds constrain the effective shear modulus to lie inside a region of the complex plane bounded by arcs of circles the four points which correspond to the hashin shtrikman walpole bounds on the shear modulus of an elastic composite always lie inside or on the boundary of the bounding region in many cases the bounding region tends to hug the corresponding parallelogram in the complex compliance plane having these four points as vertices

scaling limits of equilibrium wetting models in 1 1 dimension we study the path properties for the pinning wetting model in 1 1 dimension in other terms we consider a random walk model with fairly general continuous increments conditioned to stay in the upper half plane and with a measure reward for touching zero that is the boundary of the forbidden region it is well known that such a model displays a localization delocalization transition according to the size of the reward our focus is on getting a precise pathwise description of the system in both the delocalized phase that includes the critical case and in the localized one from this we extract the brownian scaling limits of the model

replica field theory for deterministic models i binary sequences with low autocorrelation we study systems without quenched disorder with a complex landscape and we use replica symmetry theory to describe them we discuss the golay bernasconi derrida approximation of the low autocorrelation model and we reconstruct it by using replica calculations we then consider the full model its low t properties with the help of number theory and a hartree fock resummation of the high temperature series we show that replica theory allows us to solve the model in the high t phase our solution is based on one link integral techniques and is based on substituting a fourier transform with a generic unitary transformation we discuss this approach as a powerful tool to describe systems with a complex landscape in the absence of quenched disorder

a copolymer near a selective interface variational characterization of the free energy in this paper we consider a random copolymer near a selective interface separating two solvents the configurations of the copolymer are directed paths that can make i i d excursions of finite length above and below the interface the excursion length distribution is assumed to have a tail that is logarithmically equivalent to a power law with exponent 1 the monomers carry i i d real valued types whose distribution is assumed to have zero mean unit variance and a finite moment generating function the interaction hamiltonian rewards matches and penalizes mismatches of the monomer types and the solvents and depends on two parameters the interaction strength 0 and the interaction bias h 0 we are interested in the behavior of the copolymer in the limit as its length tends to infinity the quenched free energy per monomer h g que h has a phase transition along a quenched critical curve h que separating a localized phase where the copolymer stays close to the interface from a delocalized phase where the copolymer wanders away from the interface we derive variational formulas for both these quantities we compare these variational formulas with their analogues for the annealed free energy per monomer h g ann h and the annealed critical curve h ann both of which are explicitly computable this comparison leads to 1 a proof that h ann 1 and 0 2 a proof that g que h g ann h for all 1 and h in the annealed localized phase 3 an estimate of the total number of times the copolymer visits the interface in the interior of the quenched delocalized phase 4 an identification of the asymptotic frequency at which the copolymer visits the interface in the quenched localized phase the copolymer model has been studied extensively in the literature the goal of the present paper is to open up a window with a variational view and to settle a number of open problems

copolymers at selective interfaces new bounds on the phase diagram we investigate the phase diagram of disordered copolymers at the interface between two selective solvents and in particular its weak coupling behavior encoded in the slope mc of the critical line at the origin we focus on the directed walk case which has turned out to be in spite of the apparent simplicity extremely challenging in mathematical terms the partition function of such a model does not depend on all the details of the markov chain that models the polymer but only on the time elapsed between successive returns to zero and on whether the walk is in the upper or lower half plane between such returns this observation leads to a natural generalization of the model in terms of arbitrary laws of return times the most interesting case being the one of return times with power law tails with exponent 1 1 2 in the case of the symmetric random walk the main results we present here are

non periodic long range order for one dimensional pair interactions we show how non periodic quasiperiodic long range order can arise in one dimension due to long range translation invariant pair interactions with quasiperiodically alternating signs we discuss the parisi overlap distribution between the infinitely many pure states

reduced basis hybrid computational homogenization based on a mixed incremental formulation a new reduced basis method for the computer assisted homogenization of microheterogeneous materials is proposed key features of the nonuniform transformation field analysis ntfa 1 2 are taken as a point of departure a short coming of the ntfa method is its limitation to simple constitutive models on the microscale a second disadvantage is the possible loss of accuracy when the reduced basis approximating the plastic strains has increasing dimension both issues are related to the simple evolution law for the mode activity coefficients which are the new macroscopic internal variables of the homogenized material in the present contribution a generalization of the ntfa is proposed in which the evolution of the new internal variables is derived from a mixed incremental variational formulation the derivation is based on purely micro mechanical considerations the modification allows for arbitrary generalized standard materials 3 on the microscopic scale including e g crystal visco plasticity additionally the fidelity of the homogenized response is now directly linked to the reduced basis approximating the plastic strain field numerical examples for nonlinear viscous materials and single crystal plasticity outline the accuracy of the proposed multi scale method

homogenization and optimal bounds in linear elasticity in their celebrated paper of 1963 hashin shtrikman 1 addressed the problem of determining optimal bounds for the bulk and shear moduli of a statistically isotropic elastic composite material with arbitrary isotropic phase geometry they derived a set of bounds for these moduli in the physically meaningful case of three dimensional elasticity in the case of a two phase composite let k 1 1 and k 2 2 respectively denote the bulk and shear moduli for the first and the second phase let k denote their analogues for the composite and let 0 stand for the volume fraction of the first phase in the composite under the ordering restriction that both

sturmian ground states in classical lattice gas models we construct for the first time examples of non frustrated two body infinite range one dimensional classical lattice gas models without periodic ground state configurations ground state configurations of our models are sturmian sequences defined by irrational rotations on the circle we present minimal sets of forbidden patterns which define sturmian sequences in a unique way our interactions assign positive energies to forbidden patterns and are equal to zero otherwise we illustrate our construction by the well known example of the fibonacci sequences

overlap distributions for deterministic systems with many pure states we discuss the parisi overlap distribution function for various deterministic systems with uncountably many pure ground states we show examples of trivial countably discrete and continuous distributions

breaking of periodicity at positive temperatures we discuss a classical lattice gas model without periodic or quasiperiodic ground states the only ground state configurations of our model are nonperiodic thue morse sequences we show that low temperature phases of such models can be ordered in fact we prove the existence of an ordered nonmixing low temperature translation invariant equilibrium state which has nonperiodic gibbs states in its extremal decomposition

classical lattice gas models of quasicrystals one of the fundamental problems of quasicrystals is to understand their occurrence in microscopic models of interacting particles we review here recent attempts to construct stable quasicrystalline phases in particular we compare two recently constructed classical lattice gas models with translation invariant interactions and without periodic ground state configurations the models are based on nonperiodic tilings of the plane by square like tiles in the first model all interactions can be minimized simultaneously the second model is frustrated its nonperiodic ground state can arise only by the minimization of the energy of competing interactions we put forward some hypotheses concerning stabilities of nonperiodic ground states in particular we introduce two criteria the so called strict boundary conditions and prove their equivalence to the zero temperature stability of ground states against small perturbations of potentials of interacting particles we discuss the relevance of these conditions for the low temperature stability i e for the existence of thermodynamically stable nonperiodic equilibrium states

regularity properties and pathologies of position space renormalization group transformations scope and limitations of gibbsian theory we reconsider the conceptual foundations of the renormalization group rg formalism and prove some rigorous theorems on the regularity properties and possible pathologies of the rg map our main results apply to local in position space rg maps acting on systems of bounded spins compact single spin space regarding regularity we show that the rg map defined on a suitable space of interactions formal hamiltonians is always single valued and lipschitz continuous on its domain of definition this rules out a recently proposed scenario for the rg description of first order phase transitions on the pathological side we make rigorous some arguments of griffiths pearce and israel and prove in several cases that the renormalized measure is not a gibbs measure for any reasonable interaction this means that the rg map is ill defined and that the conventional rg description of first order phase transitions is not universally valid for decimation or kadanoff transformations applied to the ising model in dimension d 3 these pathologies occur in a full neighborhood 0 h e of the low temperature part of the first order phase transition surface for block averaging transformations applied to the ising model in dimension d 2 the pathologies occur at low temperatures for arbitrary magnetic field strength pathologies may also occur in the critical region for ising models in dimension d 4 we discuss the heuristic and numerical evidence on rg pathologies in the light of our rigorous theorems in addition we discuss critically the concept of gibbs measure which is at the heart of present day classical statistical mechanics we provide a careful and we hope pedagogical overview of the theory of gibbsian measures as well as the less familiar non gibbsian measures emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations we give a rather complete catalogue of the known examples of such occurrences the main message of this paper is that despite a well established tradition gibbsiannessshould not be taken for granted

the statistical mechanics of random copolymers random copolymers are polymers with two or more types of monomer where the monomer sequence is determined by some random process once determined the sequence is fixed so random copolymers are an example of a system with quenched randomness we review the statistical mechanics of random copolymers focusing on self avoiding walk models where there are two types of monomers a and b which are randomly distributed along the polymer chain theoretical approximate and numerical results are reviewed for models of the random copolymer adsorption localization and collapse phase transitions we concentrate on what is known about the existence of phase transitions the morita approximation and results about self averaging we also discuss in less detail the replica trick and numerical methods including monte carlo methods exact enumeration and transfer matrix methods important open problems are identified throughout and highlighted in the conclusions

localization transition for a polymer near an interface consider the directed process i s i where the second component is simple random walk on mathbb z s 0 0 define a transformed path measure by weighting each n step path with a factor exp lambda sum 1 leq i leq n omega i h sign s i here omega i i geq 1 is an i i d sequence of random variables taking values pm 1 with probability 1 2 acting as a random medium while lambda in 0 infty and h in 0 1 are parameters the weight factor has a tendency to pull the path towards the horizontal because it favors the combinations s i 0 omega i 1 and s i 0 omega i 1 the transformed path measure describes a heteropolymer consisting of hydrophylic and hydrophobic monomers near an oil water interface we study the free energy of this model as n to infty and show that there is a critical curve lambda to h c lambda where a phase transition occurs between localized and delocalized behavior in the vertical direction we derive several properties of this curve in particular its behavior for lambda downarrow 0 to obtain this behavior we prove that as lambda h downarrow 0 the free energy scales to its brownian motion analogue

individual energy level distributions for one dimensional diagonal and off diagonal disorder we study the distribution of the n th energy level for two different one dimensional random potentials this distribution is shown to be related to the distribution of the distance between two consecutive nodes of the wave function we first consider the case of a white noise potential and study the distributions of energy level both in the positive and the negative part of the spectrum it is demonstrated that in the limit of a large system l the distribution of the n th energy level is given by a scaling law which is shown to be related to the extreme value statistics of a set of independent variables in the second part we consider the case of a supersymmetric random hamiltonian potential v x x 2 x we study first the case of x being a white noise with zero mean it is in particular shown that the ground state energy which behaves on average like exp l 1 3 in agreement with previous work is not a self averaging quantity in the limit l as is seen in the case of diagonal disorder then we consider the case when x has a non zero mean value

random walkers in one dimensional random environments exact renormalization group analysis sinai s model of diffusion in one dimension with random local bias is studied by a real space renormalization group which yields exact results at long times the effects of an additional small uniform bias force are also studied we obtain analytically the scaling form of the distribution of the position x t of a particle the probability of it not returning to the origin and the distributions of first passage times in an infinite sample as well as in the presence of a boundary and in a finite but large sample we compute the distribution of the meeting time of two particles in the same environment we also obtain a detailed analytic description of the thermally averaged trajectories by computing quantities such as the joint distribution of the number of returns and of the number of jumps forward these quantities obey multifractal scaling characterized by generalized persistence exponents ensuremath theta g which we compute in the presence of a small bias the number of returns to the origin becomes finite characterized by a universal scaling function which we obtain the full statistics of the distribution of successive times of return of thermally averaged trajectories is obtained as well as detailed analytical information about correlations between directions and times of successive jumps the two time distribution of the positions of a particle x t and x t ensuremath with tg t ensuremath is also computed exactly it is found to exhibit aging with several time regimes characterized by different behaviors in the unbiased case for t ensuremath t ensuremath ensuremath sim t ensuremath ensuremath alpha with ensuremath alpha g1 it exhibits a mathrm ln t mathrm ln t ensuremath scaling with a singularity at coinciding rescaled positions x t x t ensuremath this singularity is a novel feature and corresponds to particles that remain in a renormalized valley for closer times ensuremath alpha l1 the two time diffusion front exhibits a quasiequilibrium regime with a mathrm ln t ensuremath t ensuremath mathrm ln t ensuremath behavior which we compute the crossover to a t t ensuremath aging form in the presence of a small bias is also obtained analytically rare events corresponding to intermittent splitting of the thermal packet between separated wells which dominate some averaged observables are also characterized in detail connections with the green function of a one dimensional schr odinger problem and quantum spin chains are discussed