SDE decomposition and A-type stochastic interpretation in nonequilibrium processes

An innovative theoretical framework for stochastic dynamics based on the decomposition of a stochastic differential equation (SDE) into a dissipative component, a detailed-balance-breaking component, and a dual-role potential landscape has been developed, which has fruitful applications in physics, engineering, chemistry, and biology. It introduces the A-type stochastic interpretation of the SDE beyond the traditional Ito or Stratonovich interpretation or even the α-type interpretation for multidimensional systems. The potential landscape serves as a Hamiltonian-like function in nonequilibrium processes without detailed balance, which extends this important concept from equilibrium statistical physics to the nonequilibrium region. A question on the uniqueness of the SDE decomposition was recently raised. Our review of both the mathematical and physical aspects shows that uniqueness is guaranteed. The demonstration leads to a better understanding of the robustness of the novel framework. In addition, we discuss related issues including the limitations of an approach to obtaining the potential function from a steady-state distribution.     

Stochastic differential equation derivation: Comparison of the Markov method versus the additive method

There are several methods of transforming an ordinary differential equation into a stochastic differential equation (SDE). The two most common are adding noise to a system parameter or variable and transforming to a SDE or deriving the SDE by assuming an underlying Markov process. Using simple one- and two-dimensional systems we investigate the differences in dynamics and bifurcations between SDE derived by each method from simple deterministic population models.     

Monte Carlo Simulation of Stochastic Differential Equation to Study Information Geometry

Information Geometry is a useful tool to study and compare the solutions of a Stochastic Differential Equations (SDEs) for non-equilibrium systems. As an alternative method to solving the Fokker–Planck equation, we propose a new method to calculate time-dependent probability density functions (PDFs) and to study Information Geometry using Monte Carlo (MC) simulation of SDEs. Specifically, we develop a new MC SDE method to overcome the challenges in calculating a time-dependent PDF and information geometric diagnostics and to speed up simulations by utilizing GPU computing. Using MC SDE simulations, we reproduce Information Geometric scaling relations found from the Fokker–Planck method for the case of a stochastic process with linear and cubic damping terms. We showcase the advantage of MC SDE simulation over FPE solvers by calculating unequal time joint PDFs. For the linear process with a linear damping force, joint PDF is found to be a Gaussian. In contrast, for the cubic process with a cubic damping force, joint PDF exhibits a bimodal structure, even in a stationary state. This suggests a finite memory time induced by a nonlinear force. Furthermore, several power-law scalings in the characteristics of bimodal PDFs are identified and investigated.     

A Functional Central Limit Theorem for the Becker–Döring Model

We investigate the fluctuations of the stochastic Becker–Döring model of polymerization when the initial size of the system converges to infinity. A functional central limit problem is proved for the vector of the number of polymers of a given size. It is shown that the stochastic process associated to fluctuations is converging to the strong solution of an infinite dimensional stochastic differential equation (SDE) in a Hilbert space. We also prove that, at equilibrium, the solution of this SDE is a Gaussian process. The proofs are based on a specific representation of the evolution equations, the introduction of a convenient Hilbert space and several technical estimates to control the fluctuations, especially of the first coordinate which interacts with all components of the infinite dimensional vector representing the state of the process.     

Generation of compound non-gaussian random processes with a given correlation function

A compound representation of random processes is considered. Each independent component of such a process is considered as the solution of the proper stochastic differential equation (SDE). This guarantees that the process obtained is stationary and ergodic. The analytical expressions are developed for nonlinear coefficients of the generating SDE. Theoretical results are compared with numerical simulation.     

A Stochastic Differential Equation Model with Jumps for Fractional Advection and Dispersion

The path of a tracer particle through a porous medium is typically modeled by a stochastic differential equation (SDE) driven by Brownian noise. This model may not be adequate for highly heterogeneous media. This paper extends the model to a general SDE driven by a Lévy noise. Particle paths follow a Markov process with long jumps. Their transition probability density solves a forward equation derived here via pseudo-differential operator theory and Fourier analysis. In particular, the SDE with stable driving noise has a space-fractional advection-dispersion equation (fADE) with variable coefficients as the forward equation. This result provides a stochastic solution to anomalous diffusion models, and a solid mathematical foundation for particle tracking codes already in use for fractional advection equations.     

ISSDE: A Monte Carlo implicit simulation code based on Stratonovich SDE approach of Coulomb collision

A Monte Carlo implicit simulation program, Implicit Stratonovich Stochastic Differential Equations(ISSDE), is developed for solving stochastic differential equations(SDEs) that describe plasmas with Coulomb collision. The basic idea of the program is the stochastic equivalence between the Fokker–Planck equation and the Stratonovich SDEs. The splitting method is used to increase the numerical stability of the algorithm for dynamics of charged particles with Coulomb collision. The cases of Lorentzian plasma, Maxwellian plasma and arbitrary distribution function of background plasma have been considered. The adoption of the implicit midpoint method guarantees exactly the energy conservation for the diffusion term and thus improves the numerical stability compared with conventional Runge–Kutta methods. ISSDE is built with C++ and has standard interfaces and extensible modules. The slowing down processes of electron beams in unmagnetized plasma and relaxation process in magnetized plasma are studied using the ISSDE, which shows its correctness and reliability.     

On the interpretation of random forces derived by projection operators

We study the derivation of a Langevin equation from a microscopic basis in order to elucidate the nature of the random force. We arrive at the conclusion that the consistent interpretation of the microscopic Langevin equation in terms of a stochastic differential equation (SDE) is according to I o rules. In addition, the random force is in general not Gaussian, and it is hence not completely characterized by its second moments.     

Self-organised criticality in base-pair breathing in DNA with a defect

We analyse base-pair breathing in a DNA sequence of 12 base-pairs with a defective base at its centre. We use both all-atom molecular dynamics (MD) simulations and a system of stochastic differential equations (SDEs). In both cases, Fourier analysis of the trajectories reveals self-organised critical behaviour in the breathing of base-pairs. The Fourier Transforms (FTs) of the inter-base distances show power-law behaviour with gradients close to −1. The scale-invariant behaviour we have found provides evidence for the view that base-pair breathing corresponds to the nucleation stage of large-scale DNA opening (or ‘melting’) and that this process is a (second-order) phase transition. Although the random forces in our SDE system were introduced as white noise, FTs of the displacements exhibit pink noise, as do the displacements in the AMBER/MD simulations.     

Variational Inference for Stochastic Differential Equations

The statistical inference of the state variable and the drift function of stochastic differential equations (SDE) from sparsely sampled observations are discussed herein. A variational approach is used to approximate the distribution over the unknown path of the SDE conditioned on the observations. This approach also provides approximations for the intractable likelihood of the drift. The method is combined with a nonparametric Bayesian approach which is based on a Gaussian process prior over drift functions.     

Coarse grained approach for volume conserving models

Volume conserving surface (VCS) models without deposition and evaporation, as well as ideal molecular-beam epitaxy models, are prototypes to study the symmetries of conserved dynamics. In this work we study two similar VCS models with conserved noise, which differ from each other by the axial symmetry of their dynamic hopping rules. We use a coarse-grained approach to analyze the models and show how to determine the coefficients of their corresponding continuous stochastic differential equation (SDE) within the same universality class. The employed method makes use of small translations in a test space which contains the stationary probability density function (SPDF). In case of the symmetric model we calculate all the coarse-grained coefficients of the related conserved Kardar–Parisi–Zhang (KPZ) equation. With respect to the symmetric model, the asymmetric model adds new terms which have to be analyzed, first of all the diffusion term, whose coarse-grained coefficient can be determined by the same method. In contrast to other methods, the used formalism allows to calculate all coefficients of the SDE theoretically and within limits numerically. Above all, the used approach connects the coefficients of the SDE with the SPDF and hence gives them a precise physical meaning.     

A patch that imparts unconditional stability to explicit integrators for Langevin-like equations

This paper extends the results in [8] to stochastic differential equations (SDEs) arising in molecular dynamics. It implements a patch to explicit integrators that consists of a Metropolis–Hastings step. The ‘patched integrator’ preserves the SDE’s equilibrium distribution and is accurate on finite time intervals. As a corollary this paper proves the integrator’s accuracy in estimating finite-time dynamics along an infinitely long solution - a first in molecular dynamics. The paper also covers multiple time-steps, holonomic constraints and scalability. Finally, the paper provides numerical tests supporting the theory.     

Mean First-Passage Time in the Stochastic Theory of Biochemical Processes. Application to Actomyosin Molecular Motor

Many studies performed in recent years indicate a rich stochastic dynamics of transitions between a multitude of conformational substates in native proteins. A slow character of this dynamics is the reason why the steady-state kinetics of biochemical processes involving protein enzymes cannot be described in terms of conventional chemical kinetics, i.e., reaction rate constants. A more sophisticated language of mean first-passage times has to be used. A technique of summing up the stochastic dynamics diagrams is developed, enabling a calculation of the steady-state fluxes for systems of enzymatic reactions controlled and gated by the arbitrary type stochastic dynamics of the enzymatic complex. For a single enzymatic reaction, it is shown that the phenomenological steady-state kinetics of Michaelis–Menten type remains essentially unaltered but the interpretation of its parameters needs substantial change. A possibility of dynamical rather then structural inhibition of enzymatic activity is supposed. Two coupled enzymatic cycles are studied in the context of the biologically important process of free energy transduction. The theoretical tools introduced are applied to elucidate the mechanism of mechanochemical coupling in actomyosin molecular motor. Relations were found between basic parameters of the flux-force dependences: the force stalling the motor, the degree of coupling between the ATPase and the mechanical cycles as well as the asymptotic turnover number, and the mean first-passage times in a random movement between the particular conformational substates of the myosin head. These times are to be determined within a definite model of conformational transition dynamics. The theory proposed, not contradicting the presently available experimental data, is capable to explain the recently demonstrated multiple stepping produced by a single myosin head during just one ATPase cycle.     

A long-range memory stochastic model of the return in financial markets

We present a nonlinear stochastic differential equation (SDE) which mimics the probability density function (PDF) of the return and the power spectrum of the absolute return in financial markets. Absolute return as a measure of market volatility is considered in the proposed model as a long-range memory stochastic variable. The SDE is obtained from the analogy with an earlier proposed model of trading activity in the financial markets and generalized within the nonextensive statistical mechanics framework. The proposed stochastic model generates time series of the return with two power law statistics, i.e., the PDF and the power spectral density, reproducing the empirical data for the one-minute trading return in the NYSE.     

A canonical dilation of the Schrödinger equation

In this paper we shall re-visit the well-known Schrödinger equation of quantum mechanics. However, this shall be realized as a marginal dynamics of a more general, underlying stochastic counting process in a complex Minkowski space. One of the interesting things about this formalism is that its derivation has very deep roots in a new understanding of the differential calculus of time. This Minkowski-Hilbert representation of quantum dynamics is called the Belavkin formalism; a beautiful, but not well understood theory of mathematical physics that understands that both deterministic and stochastic dynamics may be formally represented by a counting process in a second-quantized Minkowski space. The Minkowski space arises as a canonical quantization of the clock, and this is derived naturally from the matrix-algebra representation [1, 2] of the Newton-Leibniz differential time increment, dt. And so the unitary dynamics of a quantum object, described by the Schrödinger equation, may be obtained as the expectation of a counting process of object-clock interactions.     

Can stochastic physics be a complete theory of nature?

The prospects for a complete stochastic theory of microscopic phenomena are considered. The two traditional schools of stochastic physics, the diffusion process school and the zero-point electromagnetic field school, are reviewed. A completely relativistic theory, stochastic field theory, is proposed as an extension of the ideas of these two schools. Within the context of stochastic field theory we present the following new results: an elementary stochastization scheme which produces the zero-point electromagnetic field; a physical interpretation of the mathematical methods developed by Lukosz for calculating zero-point energies; a calculation of the first-order Lamb shift which generalizes that of Welton; a new setting for a finite-temperature theory; and comments on the bag model for quark confinement.Research financed in part by Colciencias.     

Finite temperature field from stochastic mechanics

Stochastic mechanics of Nelson when generalized to positive temperature for a scalar field, gives rise to a stochastic field which appears to be a hybrid of euclidean and minkowskian field if the usual value of the diffusion parameter is taken. The stochastic process associated with it is a gaussian non-Markov process. The thermal expectations of this stochastic field fails to satisfy the KMS periodic condition. If the diffusion parameter is allowed to continue analytically to a purely imaginary value, the resulting field can be identified with the usual finite temperature quantum field in minkowskian space-time. The relation of this field with that of thermo field dynamics is discussed.     

New sub-grid stochastic acceleration model in LES of high-Reynolds-number flows

The experimental observations of intermittent dynamics of Lagrangian acceleration in a “free” high-Reynolds-number turbulence are shown to be consistent with the Kolmogorov-Oboukhov theory. In line with Kolmogorov-Oboukhov’s predictions, a new sub-grid scale (SGS) model is proposed and is combined with the Smagorinsky model. The new SGS model is focused on simulation of the non-resolved total acceleration vector by two stochastic processes: one for its norm, another for its direction. The norm is simulated by stochastic equation, which was derived from the log-normal stochastic process for turbulent kinetic energy dissipation rate, with the Reynolds number, as the parameter. The direction of the acceleration vector is suggested to be governed by random walk process, with correlation on the Kolmogorov’s timescale. In the framework of this model, a surrogate unfiltered velocity field is emulated by computation of the instantaneous model-equation. The coarse-grid computation of a high-Reynolds-number stationary homogeneous turbulence reproduced qualitatively the main intermittency effects, which were observed in experiment of ENS in Lyon. Contrary to the standard LES with the Smagorinsky eddy-viscosity model, the proposed model provided: (i) non-Gaussianity in the acceleration distribution with stretched tails; (ii) rapid decorrelation of acceleration vector components; (iii) “long memory” in correlation of its norm. The turbulent energy spectra of stationary and decaying homogeneous turbulence are also better predicted by the proposed model.     

Physical models of cognition

This paper presents and discusses physical models for simulating some aspects of neural intelligence, and, in particular, the process of cognition. The main departure from the classical approach here is in utilization of a terminal version of classical dynamics introduced by the author earlier. Based upon violations of the Lipschitz condition at equilibrium points, terminal dynamics attains two new fundamental properties: it is spontaneous and nondeterministic. Special attention is focused on terminal neurodynamics as a particular architecture of terminal dynamics which is suitable for modeling of information flows. Terminal neurodynamics possesses a well-organized probabilistic structure which can be analytically predicted, prescribed, and controlled, and therefore which presents a powerful tool for modeling real-life uncertainties. Two basic phenomena associated with random behavior of neurodynamic solutions are exploited. The first one is a stochastic attractor—a stable stationary stochastic process to which random solutions of a closed system converge. As a model of the cognition process, a stochastic attractor can be viewed as a universal tool for generalization and formation of classes of patterns. The concept of stochastic attractor is applied to model a collective brain paradigm explaining coordination between simple units of intelligence which perform a collective task without direct exchange of information. The second fundamental phenomenon discussed is terminal chaos which occurs in open systems. Applications of terminal chaos to information fusion as well as to explanation and modeling of coordination among neurons in biological systems are discussed. It should be emphasized that all the models of terminal neurodynamics are implementable in analog devices, which means that all the cognition processes discussed in the paper are reducible to the laws of Newtonian mechanics.     

Stochastic climate dynamics: Random attractors and time-dependent invariant measures

This article attempts a unification of the two approaches that have dominated theoretical climate dynamics since its inception in the 1960s: the nonlinear deterministic and the linear stochastic one. This unification, via the theory of random dynamical systems (RDS), allows one to consider the detailed geometric structure of the random attractors associated with nonlinear, stochastically perturbed systems. We report on high-resolution numerical studies of two idealized models of fundamental interest for climate dynamics. The first of the two is a stochastically forced version of the classical Lorenz model. The second one is a low-dimensional, nonlinear stochastic model of the El Niño-Southern Oscillation (ENSO). These studies provide a good approximation of the two models’ global random attractors, as well as of the time-dependent invariant measures supported by these attractors; the latter are shown to have an intuitive physical interpretation as random versions of Sinaï-Ruelle-Bowen (SRB) measures.