an explicit approach to the λ operator and the h theorem in the prigogine theory of irreversibility abstract a reformulation of the prigogine theory of irreversible dynamical systems is given in contrast to the previous works a stochastic integral with respect operator valued martingales is employed here this approach provides a mathematically rigorous and simple construction of the operator for dynamical flows it has an explicit random character we derive also the boltzmann h theorem explicitly from the properties of the semigroup only finally the form of the boltzmann kinetic equation for the irreversible dynamics is found

multi scale second order computational homogenization of multi phase materials a nested finite element solution strategy this paper presents the detailed implementation and computational aspects of a novel second order computational homogenization procedure which is suitable for a multi scale modelling of macroscopic localization and size effects the second order scheme is an extension of the classical first order computational homogenization framework and is based on a proper incorporation of the gradient of the macroscopic deformation gradient tensor into the kinematical macro micro scale transition from the microstructural analysis the macroscopic stress and higher order stress tensors are obtained thus delivering a microstructurally based constitutive response of the macroscopic second gradient continuum the higher order macroscopic constitutive tangents are derived through static condensation of the microscopic global tangent matrix for the solution of the second gradient equilibrium problem on the macrolevel a mixed finite element formulation is developed as an example the second order computational homogenization approach is applied for the multi scale analysis of simple shear of a constrained heterogeneous strip where a pronounced boundary size effect appears

the quantum state diffusion model applied to open systems a model of a quantum system interacting with its environment is proposed in which the system is represented by a state vector that satisfies a stochastic differential equation derived from a density operator equation such as the bloch equation and consistent with it the advantages of the numerical solution of these equations over the direct numerical solution of the density operator equations are described the method is applied to the nonlinear absorber cascades of quantum transitions second harmonic generation and a measurement reduction process the model provides graphic illustrations of these processes with statistical fluctuations that mimic those of experiments the stochastic differential equations originated from studies of the measurement problem in the foundations of quantum mechanics the model is compared with the quantum jump model of dalibard 1992 carmichael and others which originated among experimenters looking for intuitive pictures and rules of computation

strain driven homogenization of inelastic microstructures and composites based on an incremental variational formulation the paper investigates computational procedures for the treatment of a homogenized macro continuum with locally attached micro structures of inelastic constituents undergoing small strains the point of departure is a general internal variable formulation that determines the inelastic response of the constituents of a typical micro structure as a generalized standard medium in terms of an energy storage and a dissipation function consistent with this type of inelasticity we develop a new incremental variational formulation of the local constitutive response where a quasi hyperelastic micro stress potential is obtained from a local minimization problem with respect to the internal variables it is shown that this local minimization problem determines the internal state of the material for finite increments of time we specify the local variational formulation for a setting of smooth single surface inelasticity and discuss its numerical solution based on a time discretization of the internal variables the existence of the quasi hyperelastic stress potential allows the extension of homogenization approaches of elasticity to the incremental setting of inelasticity focusing on macro strain driven micro structures we develop a new incremental variational formulation of the global homogenization problem where a quasi hyperelastic macro stress potential is obtained from a global minimization problem with respect to the fine scale displacement fluctuation field it is shown that this global minimization problem determines the state of the micro structure for finite increments of time we consider three different settings of the global variational problem for prescribed linear displacements periodic fluctuations and constant stresses on the boundary of the micro structure and discuss their numerical solutions based on a spatial discretization of the fine scale displacement fluctuation field the performance of the proposed methods is demonstrated for the model problem of von mises type elasto visco plasticity of the constituents and applied to a comparative study of micro to macro transitions of inelastic composites copyright 2002 john wiley sons ltd

on the approach to statistical equilibrium in an infinite particle lattice dynamic model a deterministic reversible model dynamic system of infinite interacting particles on a lattice is studied the model is a modification by de haan of the kac ring model the time evolution of the statistical states can be described by a hierarchy for the reduced distribution functions it is shown that the system is isomorphic to a bernoulli shift the isomorphism is given in terms of the infinite process of observation of the states of one particle in a fixed site the approach to equilibrium is studied and time asymmetric initial states that approach equilibrium are constructed it is shown that these states contain nonvanishing correlations between infinitely distant particles the bernoulli property allows the construction of a markovian irreversible description of the model realized by a nonunitary transformation the action of this transformation on the statistical states the correlations and the rate of the approach to equilibrium is studied

domain evolution in ferroelectric materials a continuum phase field model and finite element implementation a numerical tool for simulating the polarization distribution in ferroelectric materials is presented using the concept of a phase field a continuum physics model is established which is descretized with finite elements the main feature of the numerical implementation is an implicit time integration of the non linear evolution equation representative examples and comparison to experimental measurements from the literature show the main features of the model

time operator of markov chains and mixing times applications to financial data abstract we extend the notion of time operator from kolmogorov dynamical systems and bernoulli processes to markov processes the general methodology is presented and illustrated in the simple case of binary processes we present a method to compute the eigenfunctions of the time operator internal ages are related to other characteristic times of markov chains namely the kemeny time the convergence rate and goodman s intrinsic time we clarified the concept of mixing time by providing analytic formulas for two state markov chains explicit formulas for mixing times are presented for any two state regular markov chain the mixing time of a markov chain is determined also by the time operator of the markov chain within its age computation we illustrate these results in terms of two realistic examples a markov chain from us gnp data and a markov chain from dow jones closing prices we propose moreover a representation for the kemeny constant in terms of internal ages

a discrete invitation to quantum filtering and feedback control the engineering and control of devices at the quantum mechanical level such as those consisting of small numbers of atoms and photons is a delicate business the fundamental uncertainty that is inherently present at this scale manifests itself in the unavoidable presence of noise making this a novel field of application for stochastic estimation and control theory in this expository paper we demonstrate estimation and feedback control of quantum mechanical systems in what is essentially a noncommutative version of the binomial model that is popular in mathematical finance the model is extremely rich and allows a full development of the theory while remaining completely within the setting of finite dimensional hilbert spaces thus avoiding the technical complications of the continuous theory we introduce discretized models of an atom in interaction with the electromagnetic field obtain filtering equations for photon counting and homodyne detection and solve a stochastic control problem using dynamic programming and lyapunov function methods

quantum stochastic processes as models for state vector reduction an elementary introduction of quantum state valued markovian stochastic processes qsp for n state quantum systems is given it is pointed out that a so called master constraint must be fulfilled for a given master equation a continuous and as a new alternative possibility a discontinuous qsp are derived both are discussed as possible models for state reduction during measurement

martingale models for quantum state reduction stochastic models for quantum state reduction give rise to statistical laws that are in most respects in agreement with those of quantum measurement theory here we examine the correspondence of the two theories in detail making a systematic use of the methods of martingale theory an analysis is carried out to determine the magnitude of the fluctuations experienced by the expectation of the observable during the course of the reduction process and an upper bound is established for the ensemble average of the greatest fluctuations incurred we consider the general projection postulate of l uders applicable in the case of a possibly degenerate eigenvalue spectrum and derive this result rigorously from the underlying stochastic dynamics for state reduction in the case of both a pure and a mixed initial state we also analyse the associated lindblad equation for the evolution of the density matrix and obtain an exact time dependent solution for the state reduction that explicitly exhibits the transition from a general initial density matrix to the l uders density matrix finally we apply girsanov s theorem to derive a set of simple formulae for the dynamics of the state in terms of a family of geometric brownian motions thereby constructing an explicit unravelling of the lindblad equation

stabilizing feedback controls for quantum systems no quantum measurement can give full information on the state of a quantum system hence any quantum feedback control problem is necessarily one with partial observations and can generally be converted into a completely observed control problem for an appropriate quantum filter as in classical stochastic control theory here we study the properties of controlled quantum filtering equations as classical stochastic differential equations we then develop methods using a combination of geometric control and classical probabilistic techniques for global feedback stabilization of a class of quantum filters around a particular eigenstate of the measurement operator

a nonequilibrium entropy for dynamical systems it is proposed to define entropy for nonequilibrium ensembles using a method of coarse graining which partitions phase space into sets which typically have zero measure these are chosen by considering the totality of future possibilities for observation on the system it is shown that this entropy is necessarily a nondecreasing function of the timet there is no contradiction with the reversibility of the laws of motion because this method of coarse graining is asymmetric under time reversal under suitable conditions which are stated explicitly this entropy approaches the equilibrium entropy ast and the fine grained entropy ast in particular the conditions can always be satisfied if the system is ak system as in the sinai billiard models some theorems are given which give information about whether it is possible to generate the partition used here for coarse graining from time translates of a finite partition and at the same time elucidate the connection between our concept of entropy and the entropy invariant of kolmogorov and sinai

large time behavior and convergence rate for quantum filters under standard non demolition conditions a quantum system mathcal s undergoing continuous time measurement is usually described by a jump diffusion stochastic differential equation such an equation is called a quantum filtering equation or quantum stochastic master equation and its solution is called a quantum filter or quantum trajectory this solution describes the evolution of the state of mathcal s in the context of quantum non demolition measurement we investigate the large time behavior of this solution it is rigorously shown that for large time this solution behaves as if a direct von neumann measurement has been performed at time 0 in particular the solution converges to a random pure state which can be directly linked to the wave packet reduction postulate using the theory of girsanov transformation we obtain the explicit rate of convergence towards this random state the problem of state estimation used in experiment is also investigated

a continuous counting observation and posterior quantum dynamics a stochastic model for a continuous photon counting measurement is proposed an equation for the generating functional of the statistics of counts is found a stochastic linear equation for the unnormalised posterior state vector of atoms continuously collapsing without mixing is derived by using quantum stochastic calculus methods

computational micro to macro transitions for discretized micro structures of heterogeneous materials at finite strains based on the minimization of averaged incremental energy the paper investigates algorithms for the computation of homogenized stresses and overall tangent moduli of micro structures undergoing large strain deformations typically these micro structures define representative volumes of nonlinear heterogeneous materials such as inelastic composites polycrystalline aggregates or particle assemblies we consider a priori given discretized micro structures without focussing on details of specific discretization techniques for space and time the key contribution of the paper is the construction of a family of algorithms and matrix representations of overall properties of discretized micro structures which are motivated by a minimization of averaged incremental energy it is shown that the overall stresses and tangent moduli of a typical micro structure may exclusively be defined in terms of discrete forces and stiffness properties on the boundary we focus on deformation driven micro structures where the overall macroscopic deformation is controlled in this context three classical types of boundary conditions are investigated i linear deformation ii uniform tractions and iii periodic deformation and antiperiodic tractions incorporated by lagrangian multiplier methods these conditions generate three classes of constrained minimization problems with associated solution algorithms for the computation of equilibrium states and overall properties of micro structures the proposed algorithms and matrix representations of the overall properties are formally independent of the interior spatial structure and the local constitutive response of the micro structure and are therefore applicable to a broad class of model problems we demonstrate their performance for some representative model problems including finite elastic plastic deformations of composites texture developments in polycrystalline materials and equilibrium states of particle assemblies

wavelet transformation induced multi time scaling watmus model for coupled transient electro magnetic and structural dynamics finite element analysis abstract multi functional devices that integrate electromagnetic and mechanical fields are gaining importance in a wide variety of applications the combined mechanical and electromagnetic regimes encompassed in these structures make it necessary to develop effective multi physics analysis tools at a range of temporal scales an important consideration is that the different fields governing multi physics response may have large frequency discrepancies e g the ultra high electromagnetic frequencies and moderate vibration frequencies computational analyses of these discrepant frequency problems using conventional time integration schemes can become intractable this paper develops a framework for coupling transient electromagnetic and dynamic fields to predict the evolution of electric and magnetic fields and their fluxes in a vibrating substrate undergoing finite deformation it addresses the issue of time integration with large frequency ratios by introducing a novel wavelet transformation induced multi time scaling watmus method in the finite element framework the method significantly enhances the computational efficiency in comparison with conventional single time scale integration methods an adaptive enhancement of watmus scheme allows for the optimal wavelet bases in the transformation and integration step sizes the accuracy and efficiency of the proposed watmus scheme is verified by comparing the results with the single time scale simulations of the coupled transient electromagnetic nonlinear vibration problems

an incremental variational formulation of dissipative magnetostriction at the macroscopic continuum level this paper outlines a new variational based modeling and computational implementation of macroscopic continuum magneto mechanics involving non linear inelastic material behavior with a special focus on dissipative magnetostriction it is based on a constitutive variational principle that optimizes a generalized incremental work function with respect to the internal state variables in an incremental setting at finite time steps this variational problem defines a quasi hyper magnetoelastic potential for the stresses and the magnetic induction and incorporates energy storage as well as dissipative mechanisms the existence of this potential further allows the incremental boundary value problem of quasi static inelastic magneto mechanics to be recast into a principle of stationary incremental energy the second focus of this paper is on the careful construction of the energy storage and dissipation functions for the model problem of hysteretic magnetostriction at the macroscopic level it is then demonstrated that the proposed model is capable of predicting the ferromagnetic and field induced strain hysteresis curves characteristic of magnetostrictive material response in good agreement with experiments the numerical solution of the coupled non linear boundary value problem is based on a monolithic multi field finite element implementation as a consequence of the proposed incremental variational principle the discretization of the multi field problem appears in a compact symmetric format in this sense the proposed formulation provides a canonical framework for the simulation of boundary value problems in dissipative magnetostriction at the macro level the performance of the proposed algorithm is tested by application to relevant numerical examples

stabilization of a delayed quantum system the photon box case study a feedback scheme stabilizing an arbitrary photon number state in a microwave cavity is analyzed the quantum non demolition measurement of the cavity state allows in open loop a non deterministic preparation of photon number states by the mean of a controlled classical field injection this preparation process is made deterministic the system evolves through a discrete time markov process and the feedback law relies on lyapunov techniques this feedback design compensates an unavoidable pure delay by a stochastic version of a kalman type predictor after illustrating the efficiency of the proposed feedback law through simulations the global closed loop convergence is proved it relies on tools from stochastic stability analysis a brief study of the lyapunov exponents of the linearized system around the target state gives a strong indication of the robustness of the method

intrinsic irreversibility of kolmogorov dynamical systems we give the mathematical details and various extensions of the results stated in previous work of courbage and prigogine intrinsic random systems are deterministic and conservative dynamical systems for which we can associate two dissipative markov processes through a one to one change of representation the first leading to equilibrium for t and the second for t the microscopic formulation of the second principle of thermodynamics permits to lift the degeneracy by the exclusion of all states that do not approach equilibrium for t the set of admitted initial conditions d is then characterized by a non equilibrium entropy functional which is infinite for rejected initial states and takes finite values for admitted initial conditions thus rejected initial states correspond to an infinite amount of information to realize this selection rule we consider general probability measures on phase space that are not necessarily absolutely continuous and we extend the theory of transition to markov processes to such measures owing to the non invariance of d under the time inversion the evolution of these states in the new representation can only be given by one of the two possible markov processes

on multiscale fe analyses of heterogeneous structures from homogenization to multigrid solvers heterogeneous structures like composites often need a fine scale resolution of micro effects which influence the macroscopic overall response this is of particular relevance in the fully non linear range of large strains and inelastic material response of the constituents suitable solution methods introduce a multifield scenario of hierarchically superimposed states on different length scales for big differences of micro and macro scales the argument of scale separation induces the application of homogenization methods such types of physical multiscale approaches can be treated by nested multilevel finite element analyses that discretize both the fine scale micro structure as well as the macroscopic boundary value problem in contrast small scale differences require full resolution of the heterogeneous structure effective solution methods for the resulting large scale problems with strongly oscillating properties are suitably designed geometric multigrid techniques which may be considered as numerical multiscale approaches in both scenarios a key ingredient is the suitable formulation of scale bridging algorithms that govern the transfer between different scales the paper outlines new mesh bridging techniques in a deformation driven context for fully non linear response which exploit in a non trivial manner weak constraints on the average deformation in typical finite element patches the framework is based on an incremental variational structure of finite inelasticity the proposed new formulations provide variational based homogenization algorithms for physical multiscale scenarios and problem dependent optimal finite element grid transfers for numerical multiscale scenarios of heterogeneous materials copyright 2007 john wiley sons ltd

on intrinsic randomness of dynamical systems we discuss the problem of nonunitary equivalence via positivity preserving similarity transformations between the unitary groups associated with deterministic dynamical evolution and semigroups associated with stochastic processes dynamical systems admitting such nonunitary equivalence with stochastic markov processes are said to beintrinsically random in a previous work it was found that the so called bernoulli systems discrete time are intrinsically random in this sense this result is extended here by showing that a more general class of dynamical systems the so calledk systems andk flows are intrinsically random the connection of intrinsic randomness with local instability of motion is briefly discussed we also show that markov processes associated through nonunitary equivalence tononisomorphic k flows are necessarily non isomorphic

evolution semigroups and time operators on banach spaces we present new general methods to obtain shift representation of evolution semigroups defined on banach spaces we introduce the notion of time operator associated with a generalized shift on a banach space and give some conditions under which time operators can be defined on an arbitrary banach space we also tackle the problem of scaling of time operators and obtain a general result about the existence of time operators on banach spaces satisfying some geometric conditions the last part of the paper contains some examples of explicit constructions of time operators on function spaces

feedback control of quantum state reduction feedback control of quantum mechanical systems must take into account the probabilistic nature of quantum measurement we formulate quantum feedback control as a problem of stochastic nonlinear control by considering separately a quantum filtering problem and a state feedback control problem for the filter we explore the use of stochastic lyapunov techniques for the design of feedback controllers for quantum spin systems and demonstrate the possibility of stabilizing one outcome of a quantum measurement with unit probability

nonunitary transformations and hydrodynamic limit in the one dimensional hard point gas abstract the ergodic theoretical formalism of nonunitary transformations is compared with the derivation of kinetic equations by the hydrodynamic scaling limit the one dimensional hard point system is used as an illustration both methods rest on the k mixing property of the physical systems considered we show that the specific form of the operator is important only on the microscopic time scale on the macroscopic scale it acts like a projector with broken time reversal symmetry we also discuss the connection between the kolmogorov sinai k s entropy of the model dynamical system the relative entropy production rate in the nonunitary formalism and the macroscopic entropy production rate

nonlocality of the misra prigogine courbage semigroup we show that the markov semigroups constructed by misra prigogine and courbage through nonunitary similarity transformations of kolmogorov systems are not implementable by local point transformations i e they are not the frobenius perron semigroups associated with noninvertible point transformations in contrast with the semigroups obtained by coarse graining projections our result is a straightforward generalization of the proof of the nonlocality of the similarity transformation given by goldstein misra and courbage and also of the previous illustration by misra and prigogine for the baker transformation and completes the characterization of the misra prigogine courbage semigroups

multi scale constitutive modelling of heterogeneous materials with a gradient enhanced computational homogenization scheme a gradient enhanced computational homogenization procedure that allows for the modelling of microstructural size effects is proposed within a general non linear framework in this approach the macroscopic deformation gradient tensor and its gradient are imposed on a microstructural representative volume element rve this enables us to incorporate the microstructural size and to account for non uniform macroscopic deformation fields within the microstructural cell every microstructural constituent is modelled as a classical continuum and the rve problem is formulated in terms of standard equilibrium and boundary conditions from the solution of the microstructural boundary value problem the macroscopic stress tensor and the higher order stress tensor are derived based on an extension of the hill mandel condition this automatically delivers the microstructurally based constitutive response of the higher order macro continuum and deals with the microstructural size in a natural way several examples illustrate the approach particularly the microstructural size effects

an introduction to quantum filtering this paper provides an introduction to quantum filtering theory an introduction to quantum probability theory is given focusing on the spectral theorem and the conditional expectation as a least squares estimate and culminating in the construction of wiener and poisson processes on the fock space we describe the quantum ito calculus and its use in the modeling of physical systems we use both reference probability and innovations methods to obtain quantum filtering equations for system probe models from quantum optics

homogenization and two scale simulations of granular materials for different microstructural constraints the paper presents a new method for quasi static homogenization of granular microstructures and its embedding into two scale simulations on the coarse scale a homogenized standard continuum is considered where the granular microstructures are locally attached at each point these representative volumes are defined by aggregates of discrete solid particles which may come into contact on the micromechanical side the interparticle contact mechanisms between particles is governed by a coulomb type frictional contact in particular elliptical shaped plane particles are investigated a consistent extension of classical stiff soft and periodic boundary conditions from continuous to granular microstructures induce new classes of micro to macro transitions for granular aggregates these include constraints not only for particle center displacements but also for particle rotations at a driving boundary frame the stiff and soft constraints at the driving frame of the particle aggregate induce upper and lower bounds of the particle aggregate s stiffness on the computational side we outline a unified implementation of the displacement and rotational constraints by penalty methods that proves to be convenient for straightforward integration into discrete element codes representative numerical examples with elliptical shaped particles are investigated where the macroscopic stress responses for different boundary constraints are comparatively discussed finally we embed the granular microstructures into a coarse graining discrete to finite element model where they govern the micromechanical behavior of a two scale simulation copyright 2010 john wiley sons ltd

simulation of the multi scale convergence in computational homogenization approaches although the asymptotic homogenization is known to explicitly predict the thermo mechanical behaviors of an overall structure as well as the microstructures the current developments in engineering fields introduce some kinds of approximation about the microstructural geometry in order for the homogenization method for periodic media to apply for general heterogeneous ones the problems arising from mathematical modeling are examined in the framework of representative volume element rve analyses here the notion of homogenization convergence allows us to eliminate the geometrical periodicity requirement when the size of rve is sufficiently large then the numerical studies in this paper realize the multi scale nature of the convergence of overall material properties as the unit cell size is increased in addition to such dependency of the macroscopic field variables on the selected size of unit cells the convergence nature of microscopic stress values is also studied quantitatively via the computational homogenization method similar discussions are made for the elastoplastic mechanical responses in both macro and microscopic levels in these multi scale numerical analyses the specific effects of the microstructural morphology are reflected by using the digital image based dib finite element fe modeling technique which enables the construction of accurate microstructural models

feedback stabilization of discrete time quantum systems subject to non demolition measurements with imperfections and delays we consider a controlled quantum system whose finite dimensional state is governed by a discrete time nonlinear markov process in open loop the measurements are assumed to be quantum non demolition qnd the eigenstates of the measured observable are thus the open loop stationary states they are used to construct a closed loop supermartingale playing the role of a strict control lyapunov function the parameters of this supermartingale are calculated by inverting a metzler matrix that characterizes the impact of the control input on the kraus operators defining the markov process the resulting state feedback scheme taking into account a known constant delay provides the almost sure convergence to the target state this convergence is ensured even in the case where the filter equation results from imperfect measurements corrupted by random errors with conditional probabilities given as a left stochastic matrix closed loop simulations corroborated by experimental data illustrate the interest of such nonlinear feedback scheme for the photon box a cavity quantum electrodynamics system

fidelity is a sub martingale for discrete time quantum filters fidelity is known to increase through any kraus map the fidelity between two density matrices is less than the fidelity between their images via a kraus map we prove here that in average fidelity is also increasing for discrete time quantum filters attached to an arbitrary kraus map fidelity between the density matrix of the underlying markov chain and the density matrix of the associated quantum filter is a sub martingale this result is not restricted to pure states it also holds true for mixed states

computational homogenization in dissipative electro mechanics of functional materials abstract the homogenization of the complex microscopic response of functional materials with electro mechanical coupling is a highly demanding research topic and of utmost importance for technological design smart materials are characterized by microstructural properties which can be changed by external stress and electric field stimuli and hence find use as the active components in sensors and actuators in this context an important challenge is to combine phase field descriptions of microscopic electric domain evolution with homogenization techniques in order to define the macroscopic overall response this paper offers a new approach to computational homogenization in dissipative electro mechanics based on a rigorous exploitation of rate type and incremental variational principles governing the scale bridging between micro and macroscales it provides a canonical insight in the structure of the coupled electro mechanical problems where all field equations appear as the euler lagrange equations of variational principles a challenge is to link the gradient type continuum description of the microstructure governed by a phase field of micro electric polarization to a local electro mechanical macro continuum this is achieved by the definition of boundary conditions on the surface of the microstructure which include the phase field to this end dirichlet neumann type and periodic boundary constraints are derived from a generalization of the hill mandel macro homogeneity condition towards the coupled electro mechanical problem we outline in detail techniques for the numerical implementation of these boundary conditions by means of lagrange multiplier and penalty methods two and three dimensional numerical simulations treat electric domain wall motions in microstructures and their hysteretic homogenized overall response for macroscopic strain and electric field driven volume elements