A theorem allowing the derivation of deterministic evolution equations from stochastic evolution equations. III The Markovian-non-Markovian mix

The proof of a theorem that allows one to construct deterministic evolution equations from a set, with two subsets, containing two types of discrete stochastic evolution equation is developed. One subset evolves Markovianly and the other non-Markovianly. As an illustrative example, the deterministic evolution equations of quantum electrodynamics are derived from two sets of Markovian and non-Markovian stochastic evolution equations, of different type, after an average over realization, using the theorem. This example shows that deterministic differential equations that contain both first-order and second-order time derivatives can be derived after a Taylor series expansion of the dynamical variables. It is shown that the derivation of such deterministic differential equations can be done by solving a set of linear equations. Two explicit examples, the first containing updating rules that depend on one previous time step and the second containing updating rules that depend on two previous time steps, are given in detail in order to show step by step the linear transformations that allow one to obtain the deterministic differential equations.     

A theorem allowing to derive deterministic evolution equations from stochastic evolution equations, II: The non-Markovian extension

Deterministic evolution equations of classical as well as quantum mechanical models are derived from a set of non-Markovian stochastic evolution equations after an average over realization using a theorem. Examples are given, show that deterministic differential equations that contain derivatives with respect to time higher than or equal to two can be derived after a Taylor series expansion of the dynamical variables. It is shown that the derivation of such deterministic differential equations can be done by solving a set of linear equations that increase in number after increasing the number of previous time steps in the updating rules that define a given model. Two explicit examples, the first containing updating rules that depend on two previous time steps and the second on three, are worked in some detail in order to show some features of the linear transformation that allow one to obtain the deterministic differential equations.     

A theorem allowing to derive deterministic evolution equations from stochastic evolution equations

The deterministic evolution equations of classical as well as quantum mechanical models are derived from a set of stochastic evolution equations after taking an average over realizations using a theorem. Examples are given that show that deterministic quantum mechanical evolution equations, obtained initially by R.P. Feynman and subsequently studied by Boghosian and Taylor IV [B.M. Boghosian, W. Taylor IV, Phys. Rev. E 57 (1998) 54. See also arXiv:quant-ph/9904035] and Meyer [D.A. Meyer, Phys. Rev. E 55 (1997) 5261], among others, are derived from a set of stochastic evolution equations. In addition, a deterministic classical evolution equation for the diffusion of monomers, similar to the second Fick law, is also obtained.     

Dynamical criteria for the evolution of the stochastic dimensionality in flows with uncertainty

We estimate and study the evolution of the dominant dimensionality of dynamical systems with uncertainty governed by stochastic partial differential equations, within the context of dynamically orthogonal (DO) field equations. Transient nonlinear dynamics, irregular data and non-stationary statistics are typical in a large range of applications such as oceanic and atmospheric flow estimation. To efficiently quantify uncertainties in such systems, it is essential to vary the dimensionality of the stochastic subspace with time. An objective here is to provide criteria to do so, working directly with the original equations of the dynamical system under study and its DO representation. We first analyze the scaling of the computational cost of these DO equations with the stochastic dimensionality and show that unlike many other stochastic methods the DO equations do not suffer from the curse of dimensionality. Subsequently, we present the new adaptive criteria for the variation of the stochastic dimensionality based on instantaneous (i) stability arguments and (ii) Bayesian data updates. We then illustrate the capabilities of the derived criteria to resolve the transient dynamics of two 2D stochastic fluid flows, specifically a double-gyre wind-driven circulation and a lid-driven cavity flow in a basin. In these two applications, we focus on the growth of uncertainty due to internal instabilities in deterministic flows. We consider a range of flow conditions described by varied Reynolds numbers and we study and compare the evolution of the uncertainty estimates under these varied conditions.     

An investigation of the dynamical evolution of a class of Bianchi VI−1/9 cosmological models

We consider the evolution of a subclass of the orthogonal spatially homogeneous cosmologies of Bianchi type VI_–1/9. Expansion normalized variables are introduced to write the Einstein field equations for these models as a three-dimensional autonomous system of ordinary differential equations. This system is analyzed qualitatively using the techniques of dynamical systems, and a cosmological interpretation of the phase portraits is given.     

Stieltjes Integrals of Hölder Continuous Functions with Applications to Fractional Brownian Motion

We give a new estimate on Stieltjes integrals of Hölder continuous functions and use it to prove an existence-uniqueness theorem for solutions of ordinary differential equations with Hölder continuous forcing. We construct stochastic integrals with respect to fractional Brownian motion, and establish sufficient conditions for its existence. We prove that stochastic differential equations with fractional Brownian motion have a unique solution with probability 1 in certain classes of Hölder-continuous functions. We give tail estimates of the maximum of stochastic integrals from tail estimates of the Hölder coefficient of fractional Brownian motion. In addition we apply the techniques used for ordinary Brownian motion to construct stochastic integrals of deterministic functions with respect to fractional Brownian motion and give tail estimates of its maximum.     

A class of third-order nonlinear evolution equations admitting invariant subspaces and associated reductions

With the aid of symbolic computation by Maple, a class of third-order nonlinear evolution equations admitting invariant subspaces generated by solutions of linear ordinary differential equations of order less than seven is analyzed. The presented equations are either solved exactly or reduced to finite-dimensional dynamical systems. A number of concrete examples admitting invariant subspaces generated by power, trigonometric and exponential functions are computed to illustrate the resulting theory.     

Linear integral transformations and hierarchies of integrable nonlinear evolution equations

Integrable hierarchies of nonlinear evolution equations are investigated on the basis of linear integral equations. These are (Riemann-Hilbert type of) integral transformations which leave invariant an infinite sequence of ordinary differential matrix equations of increasing order in an (indefinite) parameter k. The potential matrices in these equations obey a set of nonlinear recursion relations, leading to a heirarchy of nonlinear partial differential equations. In decreasing order the same equations give rise to a “reciprocal” hierarchy, associated with Heisenberg ferromagnet type of equations.Central in the treatment is an embedding of the hierarchy into an infinite-matrix structure, which is constructed on the basis of the integral equations. In terms of this infinite-matrix structure the equations governing the hierarchies become quite simple. Furthermore, it leads in a straightforward way to various generalizations, such as to other types of linear spectral problems, multicomponent system and lattice equations. Generalizations to equations associated with noncommuting flows follow as a direct consequence of the treatment. Finally, some results on conserved densities and the Hamiltonian structure are briefly discussed.     

Critical Stability Effects in Epidemiology

Basic properties of the epidemic and endemic version of the SIR model are described and the stability behaviour of the deterministic equilibria under random perturbations is studied. Domain restrictions of the dynamical variables lead to stochastic differential equations with boundary conditions, for which some characteristic numerical results are presented.     

A new deterministic model for chaotic reversals

We present a new chaotic system of three coupled ordinary differential equations, limited to quadratic nonlinear terms. A wide variety of dynamical regimes are reported. For some parameters, chaotic reversals of the amplitudes are produced by crisis-induced intermittency, following a mechanism different from what is generally observed in similar deterministic models. Despite its simplicity, this system therefore generates a rich dynamics, able to model more complex physical systems. In particular, a comparison with reversals of the magnetic field of the Earth shows a surprisingly good agreement, and highlights the relevance of deterministic chaos to describe geomagnetic field dynamics.     

Multispin coherences and asymptotic similarity of time correlation functions in solids

The time evolution of multispin (n-particle) correlations in solids (the growth in the number of correlated states) observed by means of multiquantum NMR spectroscopy has been investigated. The contributions from the spins of the immediate environment of each of the spins in the lattice to the time correlation functions that describe this evolution are shown to be mutually asymptotically similar. In this case, the infinite system of coupled ordinary differential equations for the time correlation functions turns out to be equivalent to a diffusion-type partial differential equation with a purely imaginary diffusion coefficient. Its analytical solution has been obtained. It is concluded that the evolution of multispin correlations is probably attributable to multiparticle processes among the spins of a “distant” (with respect to some spin) environment similar to the processes that shape the NMR absorption line wings.     

Stochastic models for lattice gauge theories

Stochastic equations are derived which describe the (Euclidean) time evolution of lattice field configurations, with and without fermions, on a three-dimensional space lattice. It is indicated how the drifts and transition functions may be obtained as asymptotic solutions of a differential equation or from a ground state ansatz. For non-Abelian gauge fields (without fermions) a ground state is constructed which is an exact eigenstate of a Hamiltonian with the same (naive) continuum limit as the Kogut-Susskind Hamiltonian. It is described how Euclidean correlations (like the Wilson loop) are obtained from the stochastic equations and how mass gaps may be obtained from the technique of exit times.     

一类非线性发展方程的复合型双孤子新解

通过下列步骤,构造了一类非线性发展方程的无穷序列复合型双孤子新解: 步骤一, 给出两种函数变换,把一类非线性发展方程化为二阶非线性常微分方程; 步骤二, 再通过函数变换, 二阶非线性常微分方程转化为一阶非线性常微分方程组,并获得了该方程组的首次积分; 步骤三, 利用首次积分与两种椭圆方程的新解与Bäcklund 变换, 构造了一类非线性发展方程的无穷序列复合型双孤子新解.     

Thermostats for “Slow” Configurational Modes

Thermostats are dynamical equations used to model thermodynamic variables such as temperature and pressure in molecular simulations. For computationally intensive problems such as the simulation of biomolecules, we propose to average over fast momentum degrees of freedom and construct thermostat equations in configuration space. The equations of motion are deterministic analogues of the Smoluchowski dynamics in the method of stochastic differential equations.     

The nature of bursting noises,stochastic resonance and deterministic chaos in excitable neurons

We study the Hindmarsh-Rose model of excitable neurons and show that in the asymptotic limit this monostable model can possess some kind of dynamical bistability: small-amplitude quasiharmonic and large-amplitude relaxational oscillations can be simultaneously excited and their formation is accompanied by a narrow hysteresis. We show that bursting noises, stochastic resonance and deterministic chaos are determined by random transitions between these two dynamical states under slow and small changes of one of the model variables (z). We find that these effects take place even for such model parameters when hysteresis transforms into a step and they disappear when this step is smoothed out enough. We analyze some characteristics and conditions of formation of the deterministic chaos. We emphasize that such dynamical bistability and the effects related to it are universal phenomena and occur in a wide class of dynamical systems of different nature including brusselator.     

Algebraic dynamics solutions and algebraic dynamics algorithm for nonlinear ordinary differential equations

The problem of preserving fidelity in numerical computation of nonlinear ordinary differential equations is studied in terms of preserving local differential structure and approximating global integration structure of the dynamical system. The ordinary differential equations are lifted to the corresponding partial differential equations in the framework of algebraic dynamics, and a new algorithm—algebraic dynamics algorithm is proposed based on the exact analytical solutions of the ordinary differential equations by the algebraic dynamics method. In the new algorithm, the time evolution of the ordinary differential system is described locally by the time translation operator and globally by the time evolution operator. The exact analytical piece-like solution of the ordinary differential equations is expressed in terms of Taylor series with a local convergent radius, and its finite order truncation leads to the new numerical algorithm with a controllable precision better than Runge Kutta Algorithm and Symplectic Geometric Algorithm.     

Generalized negative flows in hierarchies of integrable evolution equations

A one-parameter generalization of the hierarchy of negative flows is introduced for integrable hierarchies of evolution equations, which yields a wider (new) class of non-evolutionary integrable nonlinear wave equations. As main results, several integrability properties of these generalized negative flow equation are established, including their symmetry structure, conservation laws, and bi-Hamiltonian formulation. (The results also apply to the hierarchy of ordinary negative flows). The first generalized negative flow equation is worked out explicitly for each of the following integrable equations: Burgers, Korteweg-de Vries, modified Korteweg-de Vries, Sawada-Kotera, Kaup-Kupershmidt, Kupershmidt.     

On the evolution of nearly circular vortex patches

Recently, the classical problem of the evolution of patches of constant vorticity was reformulated as an evolution equation for the boundary of the patch. We study this equation in the neighborhood of the circular vortex patch and introduce a hierarchy of area-preserving nonlinear approximate equations. The first of these equations is shown to have a rich rigid structure: it possesses an exhaustive increasing sequence of linear invariant manifolds of arbitrarily large finite dimensions. On each of these manifolds the equation can be written as an explicit finite system of ordinary differential equations. Solutions of these ODEs, starting from arbitrarily small neighborhoods of the circular vortex patch, are shown to blow up.