how colored environmental noise affects population extinction environmental noise can cause an exponential reduction in the mean time to extinction mte of an isolated population we study this effect on an example of a stochastic birth death process with rates modulated by a colored that is correlated gaussian noise a path integral formulation yields a transparent way of evaluating the mte and finding the optimal realization of the environmental noise that determines the most probable path to extinction the population size dependence of the mte changes from exponential in the absence of the environmental noise to a power law for a short correlated noise and to no dependence for long correlated noise we also establish the validity domains of the white noise limit and adiabatic limit

global format for energy momentum based time integration in nonlinear dynamics summary a global format is developed for momentum and energy consistent time integration of second order dynamic systems with general nonlinear stiffness the algorithm is formulated by integrating the state space equations of motion over the time increment the internal force is first represented in fourth order form consisting of the end point mean value plus a term containing the stiffness matrix increment this form gives energy conservation for systems with internal energy as a quartic function of the displacement components this representation is then extended to general energy conservation via a discrete gradient representation the present procedure works directly with the internal force and the stiffness matrix at the time integration interval end points and in contrast to previous energy conserving algorithms it does not require any special form of the energy function nor use of mean value products at the element level or explicit use of a geometric stiffness matrix an optional monotonic algorithmic damping increasing with response frequency is developed in terms of a single damping parameter in the solution procedure the velocity is eliminated and the nonlinear iterations are based on the displacement components alone the procedure represents an energy consistent alternative to available collocation methods with an equally simple implementation copyright 2014 john wiley sons ltd

exact energy and momentum conserving algorithms for general models in nonlinear elasticity implicit time integration schemes that inherit the conservation laws of total energy linear and angular momentum are considered for initial boundary value problems in finite deformation elastodynamics conserving schemes are constructed for general hyperelastic material models both compressible and incompressible and are formulated in a way that is independent of spatial discretization three numerical examples for ogden type material models implemented using a finite element discretization in space are given to illustrate the performance of the proposed schemes these examples show that relative to the standard implicit mid point rule the conserving schemes exhibit superior numerical stability properties without a compromise in accuracy

tight frames for cyclotomic fields and other rational vector spaces abstract here we consider the construction of tight frames for rational vector spaces this is a subtle question because the inner products on q d are not all isomorphic we show that a tight frame for c d can be arbitrarily approximated by a tight frame with vectors in q i q d and hence there are many tight frames for rational inner product spaces we investigate the minimal field for which there is a tight frame with a given gramian we then consider the rational vector space given the cyclotomic field q with a primitive n th root of unity we give a simple formula for the unique inner product which makes the n th roots 1 2 n 1 into a tight frame for q from this we conclude that the associated canonical coordinates have many nice properties e g multiplication in q corresponds to convolution which makes them well suited to computation along the way we give a detailed description of the space of q linear dependencies between the n th roots which includes a cyclically invariant tight frame

internal noise stochastic resonance for intracellular calcium oscillations in a cell system by constructing a mesoscopic stochastic model for intracellular calcium oscillations in a cell system we have investigated how the internal noise would influence the calcium oscillations of such a system using stochastic simulation methods and chemical langevin method it is found that stochastic calcium oscillations appear when the internal noise is considered while the deterministic model only yields steady state the performance of such oscillations undergoes a maximum with the variation of the internal noise level indicating the occurrence of internal noise stochastic resonance interestingly we find that the optimal system size matches well with the real cell size when the control parameter is tuned near the left hopf bifurcation point and such a match is robust to the variation of the control parameters

the role of geometric stiffness in momentum and energy conserving time integration a momentum and energy conserving time integration algorithm is developed for the motion of elastic bodies described in terms of the quadratic green strain momentum conserving algorithms are formulated from an integral of the equations of motion and energy conservation has traditionally been obtained by evaluating the contribution from the internal forces by use of a combined mean value of stresses and virtual strains on the element level it is here demonstrated that momentum and energy conservation can be obtained from the classic central difference formulation by including an extra global term in the form of the increment of the geometric stiffness matrix over the current time step usually directly available in global form in existing finite element programmes the theory is derived by the use of a state space formulation where this extra term is located in the same position as the viscous damping matrix indicating that the effect of the extra incremental geometric stiffness term in the non linear algorithm is equivalent to a variable damping term depending on the change of the state of stress over a time increment in the actual numerical algorithm the new value of the velocity vector is eliminated leaving a non linear equation for the displacement increment alone followed by an explicit vector update of the velocity increment copyright 2006 john wiley sons ltd

extended state space time integration with high frequency energy dissipation the introduction of algorithmic dissipation into conservative time integration algorithms is addressed in two steps the problem is formulated in state space and it is demonstrated that the use of two balanced dissipation terms one in the equation of motion and the other in the relation between the state space variables can produce dissipation that increases monotonically with the modal frequency this formulation is only partially satisfactory because it leads to dissipation also in the low frequency range a substantially improved form of algorithmic dissipation is then developed by representing the two balanced dissipation terms via auxiliary state space variables related to the displacement and velocity increments by a first order filter the filter technique retains the high frequency behavior of the original balanced algorithm but reduces the low frequency dissipation from first order in the frequency to third order energy balance equations are derived and the behavior of the two algorithms is described by detailed spectral analysis with explicit asymptotic results directly in terms of the damping parameter of the algorithms both algorithms satisfy full symmetry of displacement and normalized velocity variables for the free algorithmically damped vibration of any single mode and therefore exhibit monotonic transients without overshoot effects it is also demonstrated how the algorithm can be arranged in single solve format where only the displacement update requires a matrix equation solution while the velocity and the auxiliary state space variables are updated subsequently in each time step by simple vector relations with scalar coefficients copyright 2007 john wiley sons ltd

path integral approach to quantum brownian motion abstract we apply the influence functional method of feynman and vernon to the study of brownian motion at arbitrary temperature by choosing a specific model for the dissipative interaction of the system of interest with its environment we are able to evaluate the influence functional in closed form and express it in terms of a few parameters such as the phenomenological viscosity coefficient we show that in the limit h 0 the results obtained from the influence functional formalism reduce to the classical fokker planck equation in the case of a simple harmonic oscillator with arbitrarily strong damping and at arbitrary temperature we obtain an explicit expression for the time evolution of the complete density matrix x x t when the system starts in a particular kind of pure state we compare our results with those of other approaches to the problem of dissipation in quantum mechanics

explicit time integration algorithms for structural dynamics with optimal numerical dissipation abstract a new predictor corrector explicit time integration algorithm is presented for solving structural dynamics problems the basis of the algorithm is the implicit generalized method developed by the authors like its implicit parent the explicit generalized method is a one parameter family of algorithms in which the parameter defines the high frequency numerical dissipation for a given value of high frequency dissipation the explicit generalized method minimizes low frequency dissipation the algorithm can be utilized effectively for structural dynamics calculations in which numerical dissipation is needed to reduce spurious oscillations parameter values are given that enable physical damping to be treated explicitly while maintaining second order accuracy without the need for an expensive second corrector pass

energy conservation in newmark based time integration algorithms abstract energy balance equations are established for the newmark time integration algorithm and for the derived algorithms with algorithmic damping introduced via averaging the so called methods the energy balance equations form a sequence applicable to newmark integration of the undamped equations of motion an extended form including structural damping and finally the generalized form including structural as well as algorithmic damping in all three cases the expression for energy appearing in the balance equation is the mechanical energy plus some additional terms generated by the discretization by the algorithm the magnitude and character of these terms as well as the associated damping terms are discussed in relation to energy conservation and stability of the algorithms it is demonstrated that the additional terms in the energy lead to periodic fluctuations of the mechanical energy and are the cause of the phenomenon of response overshoot previously observed empirically in the application of newmark based algorithms to high frequency components it is also demonstrated that the stability limit of the explicit newmark algorithm is reached when the stiffness term in the algorithmic energy vanishes and that energy fluctuations take place for integration intervals close to the stability limit

rare event statistics in reaction diffusion systems we present an efficient method to calculate probabilities of large deviations from the typical behavior rare events in reaction diffusion systems this method is based on a semiclassical treatment of an underlying quantum hamiltonian encoding the system s evolution to this end we formulate the corresponding canonical dynamical system and investigate its phase portrait this method is presented for a number of pedagogical examples

chemical langevin equation a path integral view of gillespie s derivation in 2000 gillespie rehabilitated the chemical langevin equation cle by describing two conditions that must be satisfied for it to yield a valid approximation of the chemical master equation cme in this work we construct an original path integral description of the cme and show how applying gillespie s two conditions to it directly leads to a path integral equivalent to the cle we compare this approach to the path integral equivalent of a large system size derivation and show that they are qualitatively different in particular both approaches involve converting many sums into many integrals and the difference between the two methods is essentially the difference between using the euler maclaurin formula and using riemann sums our results shed light on how path integrals can be used to conceptualize coarse graining biochemical systems and are readily generalizable

properties of time integration with first order filter damping introduction of algorithmic damping by increasing the parameter values in the newmark algorithm leads to undesirable low frequency damping and reduced order of accuracy it is demonstrated how these effects can be removed by introducing an extra damping term in the form of a first order linear filter when the linear filter is discretized in time the state variable associated with the filter can be eliminated leading to a weighted average of the equations of motion at two consecutive times the filter procedure contains the known versions of alpha weighted newmark methods as special cases but gives a different and improved weighting of the excitation terms a complete analysis of the properties of the algorithm when used on linear systems is given including the frequency response properties it is demonstrated that the effect of overshoot is the consequence of a conservation relation that operates on a modified form of the mechanical energy of the system and analytic results are presented for the magnitude of the effect copyright 2005 john wiley sons ltd

conservation properties of a time fe method part i time stepping schemes forn body problems in the present paper one step implicit integration algorithms for the n body problem are developed the time stepping schemes are based on a petrov galerkin finite element method applied to the hamiltonian formulation of the n body problem the approach furnishes algorithmic energy conservation in a natural way the proposed time finite element method facilitates a systematic implementation of a family of time stepping schemes a particular algorithm is specified by the associated quadrature rule employed for the evaluation of time integrals the influence of various standard as well as non standard quadrature formulas on algorithmic energy conservation and conservation of angular momentum is examined in detail for linear and quadratic time elements copyright 2000 john wiley sons ltd

a time integration algorithm for linear transient analysis based on the reproducing kernel method in this paper a new algorithm based on the reproducing kernel method is presented for linear transient analysis using very large time steps in the conventional time integration methods the required accuracy is always achieved by either decreasing the size of time steps or increasing the order of approximations the proposed method however does not require a restriction on the size of the time step the required approximating functions for integration of governing equations are determined using the concept of reproduction of the solution within each time step a recurrence relationship is obtained through solving the governing equation by the point collocation based reproducing kernel method within the time step in collocation points the values of external excitation can be obtained either through the approximation between the values at the beginning and the end of each time step or assigning the exact values of the external force hence one advantage of this new method is the independency of the sampling rates of external excitation from the size of the time step the proposed method is applied for solving first and second order differential equations and advantages of the method are illustrated through a number of numerical examples

asymptotic euler maclaurin formula over lattice polytopes abstract formulas for the riemann sums over lattice polytopes determined by the lattice points in the polytopes are often called euler maclaurin formulas an asymptotic euler maclaurin formula by which we mean an asymptotic expansion formula for riemann sums over lattice polytopes was first obtained by guillemin and sternberg 2007 11 then the problem is to find a concrete formula for each term of the expansion in this paper an asymptotic euler maclaurin formula of the riemann sums over general lattice polytopes is given the formula given here is an asymptotic form of the so called local euler maclaurin formula of berline and vergne 2007 3 for delzant polytopes our proof given here is independent of the local euler maclaurin formula furthermore a concrete description of differential operators which appear in each term of the asymptotic expansion for delzant lattice polytopes is given by using this description when the polytopes are delzant lattice a concrete formula for each term of the expansion in two dimension and a formula for the third term of the expansion in arbitrary dimension are given

state space time integration with energy control and fourth order accuracy for linear dynamic systems a fourth order accurate time integration algorithm with exact energy conservation for linear structural dynamics is presented it is derived by integrating the phase space representation and evaluating the resulting displacement and velocity integrals via integration by parts substituting the time derivatives from the original differential equations the resulting algorithm has an exact energy equation in which the change of energy is equal to the work of the external forces minus a quadratic form of the damping matrix this implies unconditional stability of the algorithm and the relative phase error is of fourth order an optional high frequency algorithmic damping is constructed by optimal combination of three different damping matrices each proportional to either the mass or the stiffness matrix this leads to a modified form of the undamped algorithm with scalar weights on some of the matrices introducing damping of fourth order in the frequency thus the low frequency response is virtually undamped and the algorithm remains third order accurate even when algorithmic damping is included the accuracy of the algorithm is illustrated by an application to pulse propagation in an elastic medium where the algorithmic damping is used to reduce dispersion due to the spatial discretization leading to a smooth solution with a clearly defined wave front copyright 2005 john wiley sons ltd

state dependent diffusion thermodynamic consistency and its path integral formulation the friction coefficient of a particle can depend on its position as it does when the particle is near a wall we formulate the dynamics of particles with such state dependent friction coefficients in terms of a general langevin equation with multiplicative noise whose evaluation requires the introduction of specific rules two common conventions the ito and the stratonovich provide alternative rules for evaluation of the noise but other conventions are possible we show that the requirement that a particle s distribution function approach the boltzmann distribution at long times dictates that a drift term must be added to the langevin equation this drift term is proportional to the derivative of the diffusion coefficient times a factor that depends on the convention used to define the multiplicative noise we explore the consequences of this result in a number of examples with spatially varying diffusion coefficients we also derive a path integral representation for arbitrary interpretation of the noise and use it in a perturbative study of correlations in a simple system

the euler maclaurin formula for simple integral polytopes we give a euler maclaurin formula with remainder for the sum of a smooth function on the integral points in a simple integral lattice polytope our proof uses elementary methods

two feti based heterogeneous time step coupling methods for newmark and α schemes derived from the energy method abstract the purpose of this paper is to describe a general methodology to obtain multi time step coupling methods for different time integration schemes newmark hht wbz ch on the basis of energetic considerations the general framework of the feti method is considered for building the heterogeneous time step coupling methods by ensuring the gluing of subdomains with lagrange multipliers the proposed methods for coupling subdomains with different time integration schemes and time steps fine and large time steps are based on the pseudo energy measure as introduced by hughes in the so called energy method employed for proving the stability of implicit explicit algorithms assuming a given variation of the lagrange multipliers over the large time step kinematic conditions are derived by ensuring the zero pseudo energy at the interface two methods will be derived in this paper the first new method for which the interface problem involving the lagrange multipliers is solved at the large time scale can handle the popular dissipative schemes hht wbz ch in the particular case of the newmark schemes the new method matches with the ph method proposed by prakash and hjelmstad the second new method is based at the micro time scale it can be viewed as an extension of the method proposed by gravouil and combescure gc method for the coupling of the schemes nonetheless like the gc method the new micro time based method turns out to be dissipative when non dissipative second order accurate schemes are involved and looses one order of accuracy

energy conserving and decaying algorithms in non linear structural dynamics a generalized formulation of the energy momentum methodwill be developed within the framework of the generalizedmethodwhich allows at the same time guaranteed conservation or decay of total energy and controllable numerical dissipation of unwanted high frequency response furthermore the latter algorithm will be extended by the consistently integrated constraints of energy and momentum conservation originally derived for the constraint energy momentum algorithm the goal of this general approach of implicit energyconserving and decaying time integration schemes is to compare these algorithms on the basis of an equivalent notation by the means of an overall algorithmic design and hence to investigate their numerical properties numerical stability and controllable numerical dissipation of high frequencies will be studied in application to non linear structural dynamics among the methods considered will be the newmark method the classical methods the energy momentum methodwith and without numerical dissipation the constraint energymomentum algorithm and the constraint energy method copyright 1999 john wiley sons ltd

multi time step domain coupling method with energy control a multi time step integration method is proposed for solving structural dynamics problems on multiple domains the method generalizes earlier state space integration algorithms by introducing displacement constraints via lagrange multipliers representing the time integrated constraint forces over the individual time step it is demonstrated that displacement continuity between the subdomains leads to cancelation of the interface contributions to the energy balance equation and thus stability and algorithmic damping properties of the original algorithms are retained the various subdomains can be integrated in time using different time steps and or different state space time integration schemes the solution of the constrained system equations is obtained using a dual schur formulation allowing for maximum independence of the calculation of the subdomains stability and accuracy are illustrated by a numerical example using a refined mesh around concentrated forces copyright 2010 john wiley sons ltd

conservation properties of a time fe method part ii time stepping schemes for non linear elastodynamics in the present paper one step implicit integration algorithms for non linear elastodynamics are developed the discretization process rests on galerkin methods in space and time in particular the continuous galerkin method applied to the hamiltonian formulation of semidiscrete non linear elastodynamics lies at the heart of the time stepping schemes algorithmic conservation of energy and angular momentum are shown to be closely related to quadrature formulas that are required for the calculation of time integrals we newly introduce the assumed strain method in time which enables the design of energy momentum conserving schemes and which can be interpreted as temporal counterpart of the well established assumed strain method for finite elements in space the numerical examples deal with quasi rigid motion as well as large strain motion copyright 2001 john wiley sons ltd

noise induced multistability in chemical systems discrete versus continuum modeling the noisy dynamics of chemical systems is commonly studied using either the chemical master equation cme or the chemical fokker planck equation cfpe the latter is a continuum approximation of the discrete cme approach it has recently been shown that for a particular system the cfpe captures noise induced multistability predicted by the cme this phenomenon involves the cme s marginal probability distribution changing from unimodal to multimodal as the system size decreases below a critical value we here show that the cfpe does not always capture noise induced multistability in particular we find simple chemical systems for which the cme predicts noise induced multistability whereas the cfpe predicts monostability for all system sizes

unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum we show that for rigid body dynamics the mid point rule formulated in body co ordinates exactly conserves energy and the norm of the angular momentum for incremental force free motions but fails to conserve the direction of the angular momentum vector further we show that the mid point rule formulated in the spatial representation is in general physically and geometrically meaningless an alternative algorithm is developed which exactly preserves energy and the total spatial angular momentum in incremental force free motions the implicit version of this algorithm is unconditionally stable and second order accurate the explicit version conserves exactly angular momentum in incremental force free motions numerical simulations are presented which illustrate the excellent performance of the proposed procedure even for incremental rotations over 65 degrees the procedure is directly applicable to transient dynamic calculations of geometrically exact rods and shells