Stochastic Kinetic Equation Research Articles

Cauchy problem of stochastic kinetic equations

Cauchy problem of stochastic kinetic equations

Asymptotic behavior of a class of multiple time scales stochastic kinetic equations

We consider a class of stochastic kinetic equations, depending on two time scale separation parameters ɛ and δ: the evolution equation contains singular terms with respect to ɛ, and is driven by a fast ergodic process which evolves at the time scale t/δ2. We prove that when (ɛ,δ)→(0,0) the spatial density converges to the solution of a linear diffusion PDE. This result is a mixture of diffusion approximation in the PDE sense (with respect to the parameter ɛ) and of averaging in the probabilistic sense (with respect to the parameter δ). The proof employs stopping times arguments and a suitable new perturbed test functions approach which is adapted to consider the general regime ɛ≠δ.

Generalized Space-Time Fractional Stochastic Kinetic Equation

In this paper, we study a class of nonlinear space-time fractional stochastic kinetic equations in Rd with Gaussian noise which is white in time and homogeneous in space. This type of equation constitutes an extension of the nonlinear stochastic heat equation involving fractional derivatives in time and fractional Laplacian in space. We firstly give a necessary condition on the spatial covariance for the existence and uniqueness of the solution. Furthermore, we also study various properties of the solution, such as Hölder regularity, the upper bound of second moment, and the stationarity with respect to the spatial variable in the case of linear additive noise.

On a class of stochastic fractional kinetic equation with fractional noise

In this article we study a class of stochastic fractional kinetic equations with fractional noise which are spatially homogeneous and are fractional in time with H>1/2. The diffusion operator involved in the equation is the composition of the Bessel and Riesz potentials with any fractional parameters. We prove the existence of the solution under some mild conditions which generalized some results obtained by Dalang (Electron. J. Probab. 4(6):1–29, 1999) and Balan and Tudor (Stoch. Process. Appl. 120:2468–2494 , 2010). We study also its Hölder continuity with respect to space and time variables with b=0. Moreover, we prove the existence for the density of the solution and establish the Gaussian-type lower and upper bounds for the density by the techniques of Malliavin calculus.

Probabilistic quantitative analysis on the contents of sulfate corrosion products in concrete

A probabilistic quantitative method is proposed in this paper to depict the random evolution process of the contents of sulfate corrosion products (CSCP) in concrete. The method includes the stochastic kinetic equation established according to sulfate corrosion reactions leading to formation of corrosion products, and probability description of the CSCP. The stochastic kinetic equation is presented based on the chemical Langevin equation for complex reactions (CLE-CR), in which internal fluctuations are regarded as white Gaussian noises. Sulfate corrosion reactions are assumed to occur in a well-stirred system without considering diffusion effects of ions. The probability distribution of the CSCP can be obtained by using the statistic approach to analyze the computation results of the stochastic kinetic equation. The model is validated by the experiments performed in the study and in the literature respectively. The results show that the CSCP in concrete at each time step should be treated as a random variable following a certain probability distribution. The discreteness of experimental results can be quantitatively described by the coefficient of variations (CVs) of random variables. The probabilities of experimental results attained by the proposed method can be employed to judge the reliability of data. Furthermore, in order to evaluate the effects of the CSCP on concrete durability, the model can also be used to predict the developing trends of the CSCP, as well as providing their probabilistic information.

Higher-order approximate solutions of fractional stochastic point kinetics equations in nuclear reactor dynamics

Stochastic point kinetics equations (SPKEs) are a system of Ito stochastic differential equations whose solution has been obtained by higher-order approximation. In this study, a fractional model of SPKEs has been analyzed. The efficiency of the proposed higher-order approximation scheme has been discussed in the results section. The solutions of SPKEs in the presence of Newtonian temperature feedback have also been provided to further discuss the physical behavior of the fractional model.

Analysis of an Asymptotic Preserving Scheme for Stochastic Linear Kinetic Equations in the Diffusion Limit

We present an asymptotic preserving scheme based on a micro-macro decomposition for stochastic linear transport equations in kinetic and diffusive regimes. We perform a mathematical analysis and pr...

Developed mathematical technique for fractional stochastic point kinetics model in nuclear reactor dynamics

Fractional stochastic kinetics equations have proven to be valuable tools for the point reactor kinetics model, where the nuclear reactions are not fully described by deterministic relations. A fractional stochastic model for the point kinetics system with multi-group of precursors, including the effect of temperature feedback, has been developed and analyzed. A major mathematical and inflexible scheme to the point kinetics model is obtained by merging the fractional and stochastic technique. A novel split-step method including mathematical tools of the Laplace transforms, Mittage–Leffler function, eigenvalues of the coefficient matrix, and its corresponding eigenvectors have been used for the fractional stochastic matrix differential equation. The validity of the proposed technique has been demonstrated via calculations of the mean and standard deviation of neutrons and precursor populations for various reactivities: step, ramp, sinusoidal, and temperature reactivity feedback. The results of the proposed method agree well with the conventional one of the deterministic point kinetics equations.

Solution of the stochastic point kinetics equations using the implicit Euler-Maruyama method

In this work, the implicit Euler-Maruyama method is used to solve the stochastic point kinetics equations for different forms of reactivity: constant, linear and sinusoidal. A covariance matrix is proposed considering the one-speed diffusion theory of a nuclear reactor subject to the initial condition applying to the pulsed neutron methods. The results obtained with this method are more efficient in the calculation of neutron densities and delayed neutron precursors. It was possible to reduce the standard deviation with a very good precision for the expected values.

On the Boltzmann Equation with Stochastic Kinetic Transport: Global Existence of Renormalized Martingale Solutions

This article studies the Cauchy problem for the Boltzmann equation with stochastic kinetic transport. Under a cut-off assumption on the collision kernel and a coloring hypothesis for the noise coefficients, we prove the global existence of renormalized (in the sense of DiPerna/Lions) martingale solutions to the Boltzmann equation for large initial data with finite mass, energy, and entropy. Our analysis includes a detailed study of weak martingale solutions to a class of linear stochastic kinetic equations. This study includes a criterion for renormalization, the weak closedness of the solution set, and tightness of velocity averages in $L^1$.

A Simple "Boxed Molecular Kinetics" Approach To Accelerate Rare Events in the Stochastic Kinetic Master Equation.

The chemical master equation is a powerful theoretical tool for analyzing the kinetics of complex multiwell potential energy surfaces in a wide range of different domains of chemical kinetics spanning combustion, atmospheric chemistry, gas-surface chemistry, solution phase chemistry, and biochemistry. There are two well-established methodologies for solving the chemical master equation: a stochastic "kinetic Monte Carlo" approach and a matrix-based approach. In principle, the results yielded by both approaches are identical; the decision of which approach is better suited to a particular study depends on the details of the specific system under investigation. In this Article, we present a rigorous method for accelerating stochastic approaches by several orders of magnitude, along with a method for unbiasing the accelerated results to recover the "true" value. The approach we take in this paper is inspired by the so-called "boxed molecular dynamics" (BXD) method, which has previously only been applied to accelerate rare events in molecular dynamics simulations. Here we extend BXD to design a simple algorithmic strategy for accelerating rare events in stochastic kinetic simulations. Tests on a number of systems show that the results obtained using the BXD rare event strategy are in good agreement with unbiased results. To carry out these tests, we have implemented a kinetic Monte Carlo approach in MESMER, which is a cross-platform, open-source, and freely available master equation solver.

On the comparison of two split-step methods for the numerical simulation of stochastic point kinetics equations in presence of Newtonian temperature feedback effects

In this article, a comparative study between two numerical approximation methods viz., split-step forward Euler-Maruyama method and derivative-free Milstein method have been established. The stochastic point reactor kinetics equations consist of a system of stiff nonlinear differential equations. This system has been solved for step and ramp external reactivity using split-step forward Euler-Maruyama method and derivative-free Milstein method. The obtained numerical results show that the split-step approximations are straightforward and effective methods in studying the behavior of neutron density of the stochastic point reactor kinetics equations which also have been represented graphically.

A novel method to compute the time dependence of state distributions in the stochastic kinetic description of an autocatalytic system

A novel method is introduced to estimate the solution of the stochastic kinetic master equation of an autocatalytic reaction scheme. The method first computes an accurate probability master distribution for relatively low particle numbers, then uses a time shift property of the distributions to extend this solution to the entire state space and time scale. A limited comparison with accurately calculated probability distributions suggests that the approximation works exceptionally well and is able to provide a very reliable way to obtain the solution of the stochastic master equation of a system without any limitation on its size. The method will most probably be applicable with minor modifications for the stochastic kinetic description of other irreversible reactions as well.

Regularity of stochastic kinetic equations

We consider regularity properties of stochastic kinetic equations with multiplicative noise and drift term which belongs to a space of mixed regularity (Lp-regularity in the velocity-variable and Sobolev regularity in the space-variable). We prove that, in contrast with the deterministic case, the SPDE admits a unique weakly differentiable solution which preserves a certain degree of Sobolev regularity of the initial condition without developing discontinuities. To prove the result we also study the related degenerate Kolmogorov equation in Bessel-Sobolev spaces and construct a suitable stochastic flow.

Analytical exponential model for stochastic point kinetics equations via eigenvalues and eigenvectors

The stochastic point kinetics equations with a multi-group of delayed neutrons, which are the system of a couple of stiff stochastic differential equations, are presented. The analytical exponential model is used to solve the stochastic point kinetics equations in the dynamical system of the nuclear reactor. This method is based on the eigenvalues and corresponding eigenvectors of the coefficient matrix. The analytical exponential model calculates the mean and standard deviations of neutrons and precursor populations for the stochastic point kinetics equations with step, ramp, and sinusoidal reactivities. The results of the analytical exponential model are compared with published methods and the results of the deterministic point kinetics model. This comparison confirms that the analytical exponential model is an efficient method for solving stochastic stiff point kinetics equations.

Numerical solution of simplified stochastic point kinetics equation involving step and pulse reactivity

In the present paper, the numerical solutions of simplified stochastic point kinetic equations have been established in the field of nuclear reactor system. The stochastic behaviour of a point reactor is modelled by means of a system of Ito stochastic differential equations. The resulting system of Ito stochastic differential equations is solved over each time-step size. The obtained numerical results are verified by considering different experimental data with step and pulse reactivities. Here, the computational results explicitly show that the numerical methods are efficient and suitable for solving simplified stochastic point kinetic equations.

Growth and division in a dynamic protocell model.

In this paper a new model of growing and dividing protocells is described, whose main features are (i) a lipid container that grows according to the composition of the molecular milieu (ii) a set of “genetic memory molecules” (GMMs) that undergo catalytic reactions in the internal aqueous phase and (iii) a set of stochastic kinetic equations for the GMMs. The mass exchange between the external environment and the internal phase is described by simulating a semipermeable membrane and a flow driven by the differences in chemical potentials, thereby avoiding to resort to sometimes misleading simplifications, e.g., that of a flow reactor. Under simple assumptions, it is shown that synchronization takes place between the rate of replication of the GMMs and that of the container, provided that the set of reactions hosts a so-called RAF (Reflexive Autocatalytic, Food-generated) set whose influence on synchronization is hereafter discussed. It is also shown that a slight modification of the basic model that takes into account a rate-limiting term, makes possible the growth of novelties, allowing in such a way suitable evolution: so the model represents an effective basis for understanding the main abstract properties of populations of protocells.

Stochastic averaging lemmas for kinetic equations

We develop a class of averaging lemmas for stochastic kinetic equations. The velocity is multiplied by a white noise which produces a remarkable change in time scale. Compared to the deterministic case and as far as we work in $L^2$, the nature of regularity on averages is not changed in this stochastic kinetic equation and stays in the range of fractional Sobolev spaces at the price of an additional expectation. However all the exponents are changed; either time decay rates are slower (when the right hand side belongs to $L^2$), or regularity is better when the right hand side contains derivatives. These changes originate from a different space/time scaling in the deterministic and stochastic cases. Our motivation comes from scalar conservation laws with stochastic fluxes where the structure under consideration arises naturally through the kinetic formulation of scalar conservation laws.

The effect of pulse reactivity for stochastic neutron point kinetic equation in nuclear reactor dynamics

In this present analysis, the numerical simulation methods are applied to calculate the solution for Stochastic Neutron Point Kinetic Equations (SNPKE) with pulse reactivity in dynamical system of a nuclear reactor. The resulting systems of differential equations are solved for each time step-size. Using experimental data, the methods are investigated with pulse reactivity. The computational results designate that these numerical approximation methods are straightforward, effective and easy for solving stochastic point kinetic equations.

Les manifestations respiratoires des vascularites

The essential properties of a spatially homogeneous gas consisting of quasiparticles (phonons, magnons, rotons, etc.) are usually described in terms of the one-point distribution function f. This is the last in a series of papers (Physica A 180 (1992) 309, 336), the overall objective of which is the construction of the Einstein-Smoluchowski promeasure μɛ in order to formulate and develop a theory dealing with the equilibrium fluctuations of f. The basic idea was the inversion of Boltzman's relation connecting entropy and probability. As a consequence, the derivation of μɛ started by considering an infinite-dimensional analog of Einstein's thermodynamic theory of equilibrium fluctuations. Now, investigating a situation in which the time evolution of f is governed by the linear stochastic differential Boltzman-Peierls equation, the object here is to find the conditions under which the stationary measure for the aforesaid equation can be obtained from the Einstein-Smoluchowski promeasure μɛ. In agreement with such a general program, the fluctuation-dissipation theorem is analysed and some evidence of the necessity for an algebraic approach to the stochastic kinetic equations is also presented. And finally, the paper discusses the time-correlation function formalism and shows that this formalism is consistent with the universal ideas of Einstein. To summarize, the present paper demonstrates how the Einstein-Smoluchowski promeasure μɛ can be used to study some problems in the extended context of the time-dependent theory of equilibrium fluctuations.