Certain Problems with the Application of Stochastic Diffusion Processes for the Description of Chemical Engineering Phenomena. Numerical Simulation of One-Dimensional Diffusion Process

1996 ◽  
Vol 61 (4) ◽  
pp. 512-535 ◽  
Author(s):  
Pavel Hasal ◽  
Vladimír Kudrna
Keyword(s):  
Numerical Simulation ◽  
Diffusion Coefficient ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Chemical Engineering ◽  
Diffusion Processes ◽  
Stochastic Integral ◽  
Random Motion ◽  
One Dimensional ◽  
Dimensional Diffusion

Some problems are analyzed arising when a numerical simulation of a random motion of a large ensemble of diffusing particles is used to approximate the solution of a one-dimensional diffusion equation. The particle motion is described by means of a stochastic differential equation. The problems emerging especially when the diffusion coefficient is a function of spatial coordinate are discussed. The possibility of simulation of various kinds of stochastic integral is demonstrated. It is shown that the application of standard numerical procedures commonly adopted for ordinary differential equations may lead to erroneous results when used for solution of stochastic differential equations. General conclusions are verified by numerical solution of three stochastic differential equations with different forms of the diffusion coefficient.

Certain problems with the application of stochastic diffusion processes for description of chemical engineering phenomena; Modelling of processes by means of stochastic differential equations

1988 ◽  
Vol 53 (7) ◽  
pp. 1500-1518 ◽  
Author(s):  
Vladimír Kudrna
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Chemical Engineering ◽  
Diffusion Processes ◽  
Transport Equations ◽  
Diffusion Equations ◽  
Stochastic Diffusion ◽  
Direct Use

The paper points at certain problems associated with direct use of stochastic differential equations for description of chemical engineering processes or with the use of corresponding diffusion equations. It is shown that on the basis of various definitions one can write down three types of stochastic differential equations which might, in principle, describe the same process. One of these types is at the same time equivalent to the classic transport equations common in chemical engineering. A method is described removing these inconsistencies.

Certain Problems with the Application of Stochastic Diffusion Processes for the Description of Chemical Engineering Phenomena. Relations between Different Types of Diffusion Equations

1992 ◽  
Vol 57 (6) ◽  
pp. 1248-1261 ◽  
Author(s):  
Vladimír Kudrna ◽  
Daniel Turzík
Keyword(s):  
Differential Equations ◽  
Diffusion Equation ◽  
Stochastic Differential Equations ◽  
Chemical Engineering ◽  
Diffusion Processes ◽  
Point Of View ◽  
Diffusion Equations ◽  
Complex Diffusion ◽  
Different Types ◽  
Individual Particles

The dependence is discussed between the "clasical" diffusion equation commonly used in chemical engineering and the stochastic differential equations which describe this diffusion from the point of view of micromotion of individual particles. The resulting equations can be useful above all for the modelling of more complex diffusion processes.

Stochastic differential equations: Langevin, Fokker–Planck (FP) equations

2021 ◽  
pp. 831-856
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Langevin Equation ◽  
Diffusion Processes ◽  
Random Noise ◽  
Path Integrals ◽  
Brst Symmetry ◽  
Random Motion ◽  
Definition Of ◽  
Fokker Planck

This chapter is devoted to the study of Langevin equations, first order in time differential equations, which depend on a random noise, and which belong to a class of stochastic differential equations that describe diffusion processes, or random motion. From a Langevin equation, a Fokker–Planck (FP) equation for the probability distribution of the solutions, at given time, of the Langevin equation can be derived. It is shown that observables averaged over the noise can also be calculated from path integrals, whose integrands define automatically positive measures. The path integrals involve dynamic actions that have automatically a Becchi–Rouet–Stora–Tyutin (BRST) symmetry and, when the driving force derives from a potential, exhibit the simplest form of supersymmetry. In some cases, like Brownian motion on Riemannian manifolds, difficulties appear in the precise definition of stochastic equations, quite similar to the quantization problem encountered in quantum mechanics (QM). Time discretization provides one possible solution to the problem.

scholarly journals The log log law for multidimensional stochastic integrals and diffusion processes

1971 ◽  
Vol 5 (3) ◽  
pp. 351-356 ◽  
Author(s):  
Ludwig Arnold
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Wiener Process ◽  
Diffusion Processes ◽  
Stochastic Integrals ◽  
Dimensional Diffusion ◽  
Vector Process ◽  
Log Law ◽  
Image Position ◽  
And Diffusion

Let for t ∈ [a, b] ⊂ [0, ∞) where Ws is an n-dimensional Wiener process, f(s) an n-vector process and G(s) an n × m matrix process. f and G are nonanticipating and sample continuous. Then the set of limit points of the net in Rn is equal, almost surely, to the random ellipsoid Et = G(t)Sm, Sm = {x ∈ Rm: |x| ≤ 1}. The analogue of Lévy's law is also given. The results apply to n-dimensional diffusion processes which are solutions of stochastic differential equations, thus extending the versions of Hinčin's and Lévy's laws proved by H.P. McKean, Jr, and W.J. Anderson.

Certain problems with the application of stochastic diffusion processes for description of chemical engineering phenomena; Kolmogorov and classic diffusion equations

1988 ◽  
Vol 53 (6) ◽  
pp. 1181-1197
Author(s):  
Vladimír Kudrna
Keyword(s):  
Mass Transfer ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Chemical Engineering ◽  
Diffusion Processes ◽  
Parabolic Type ◽  
Diffusion Equations ◽  
Stochastic Motion ◽  
Energy Component ◽  
Classic Form

The paper presents alternative forms of partial differential equations of the parabolic type used in chemical engineering for description of heat and mass transfer. It points at the substantial difference between the classic form of the equations, following from the differential balances of mass and enthalpy, and the form following from the concept of stochastic motion of particles of mass or energy component. Examples are presented of the processes that may be described by the latter method. The paper also reviews the cases when the two approaches become identical.

Certain Problems with the Application of Stochastic Diffusion Processes for the Description of Chemical Engineering Phenomena. Stochastic Model of Non-Isothermal Flow Chemical Reactor

1996 ◽  
Vol 61 (2) ◽  
pp. 242-258 ◽  
Author(s):  
Vladimír Kudrna ◽  
Libor Vejmola ◽  
Pavel Hasal
Keyword(s):  
Stochastic Model ◽  
Chemical Engineering ◽  
Diffusion Processes ◽  
Dispersion Model ◽  
Chemical Reactor ◽  
Segregation Index ◽  
One Dimensional ◽  
A Value ◽  
Isothermal Case ◽  
Flow Through

Recently developed stochastic model of a one-dimensional flow-through chemical reactor is extended in this paper also to the non-isothermal case. The model enables the evaluation of concentration and temperature profiles along the reactor. The results are compared with the commonly used one-dimensional dispersion model with Danckwerts' boundary conditions. The stochastic model also enables to evaluate a value of the segregation index.

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