Kinetic stochastic differential equations

One of the simplest methods to generalize formal kinetics is to treat reactant concentrations as continuous stochastic functions of time, which results in a transformation of deterministic equations (2.1.1), (2.1.40) into stochastic differential equations. In a system with completely mixed particles the macroscopic concentration n (t) turns out to be the average of the stochastic function Cj(<)... 

Use of the stochastic differential equation (2.2.2) as the equation of motion instead of equation (2.1.1) results in the treatment of the reaction kinetics as a continuous Markov process. Calculations of stochastic differentials, perfectly presented by Gardiner [26], allow us to solve equation (2.2.2). On the other hand, an averaged concentration given by this equation could be obtained making use of the distribution function / = f(c, ..., cs t). The latter is nothing but solution of the Fokker-Planck equation [26, 34]... 

Numerical integration (sometimes referred to as solving or simulation) of differential equations, ordinary or partial, involves using a computer to obtain an approximate and discrete (in time and/or space) solution. In chemical kinetics, these differential equations are typically the rate laws that describe the time evolution of the system. One obtains results for the mean concentrations, without any information about the (typically very small) fluctuations that are inevitably present. Continuation and sensitivity analysis techniques enable one to extrapolate from a numerically obtained solution at one set of parameters (e.g., rate constants or initial concentrations) to the behavior of the system at other parameter values, without having to carry out a full numerical integration each time the parameters are changed. Other approaches, sometimes referred to collectively as stochastic methods (Gardiner, 1990), can provide data about fluctuations, but these require considerably more computational labor and are often impractical for models that include more than a few variables. 

As far as the present work is concerned, the relevance of numerical stochastic methods for polymer dynamics in micro/macro calculations resides in their ability to yield (within error bars) exact numerical solutions to dynamic models which are insoluble in the framework of polymer kinetic theory. In addition, and mainly as a consequence of the correspondence between Fokker Planck and stochastic differential equations, complex polymer dynamics can be mapped onto extremely efficient computational schemes. Another reason for the efficiency of stochastic dynamic models for polymer melts stems from the reduction of a many-chain problem to a single-chain or two-chain representation, i.e., to linear computational complexity in the number of particles. This circumstance permits the treatment of global ensembles consisting of several tens of millions of particles on current hardware, corresponding to local ensemble sizes of O(IO ) particles per element. 

The CLE is a multivariate Ito stochastic differential equation with multiple, multiplicative noise. We define the CLE again and present methods to solve it in Chapter 18, where we discuss numerical simulations of stochastic reaction kinetics. 

The authors applied this model to the situation of dissolving and deposited interfaces, involving chemically interacting species, and included rate kinetics to model mass transfer as a result of chemical reactions [60]. The use of a stochastic weighting function, based on solutions of differential equations for particle motion, may be a useful method to model stochastic processes at solid-liquid interfaces, especially where chemical interactions between the surface and the liquid are involved. 

After the kinetic model for the network is defined, a simulation method needs to be chosen, given the systemic phenomenon of interest. The phenomenon might be spatial. Then it has to be decided whether in addition stochasticity plays a role or not. In the former case the kinetic model should be described with a reaction-diffusion master equation [81], whereas in the latter case partial differential equations should suffice. If the phenomenon does not involve a spatial organization, the dynamics can be simulated either using ordinary differential equations [47] or master equations [82-84]. In the latter case but not in the former, stochasticity is considered of importance. A first-order estimate of the magnitude of stochastic fluctuations can be obtained using the linear noise approximation, given only the ordinary differential equation description of the kinetic model [83-85, 87]. 

If one chooses to use partial derivatives to describe drug kinetics in the body, then expressions for each of 3/3, 3/33C, 3/3y, and 3/3z must be written. That is, a system of partial differential equations must be specified. Writing these equations involves a knowledge of physical chemistry, irreversible thermodynamics, and circulatory dynamics. Such equations will incorporate parameters that can be either deterministic (known) or stochastic (contain statistical uncertainties). Although such equations can be written for specific systems, defining and then estimating the unknown parameters... 

It is often stated that MC methods lack real time and results are usually reported in MC events or steps. While this is immaterial as far as equilibrium is concerned, following real dynamics is essential for comparison to solutions of partial differential equations and/or experimental data. It turns out that MC simulations follow the stochastic dynamics of a master equation, and with appropriate parameterization of the transition probabilities per unit time, they provide continuous time information as well. For example, Gillespie has laid down the time foundations of MC for chemical reactions in a spatially homogeneous system.f His approach is easily extendable to arbitrarily complex computational systems when individual events have a prescribed transition probability per unit time, and is often referred to as the kinetic Monte Carlo or dynamic Monte Carlo (DMC) method. The microscopic processes along with their corresponding transition probabilities per unit time can be obtained via either experiments such as field emission or fast scanning tunneling microscopy or shorter time scale DFT/MD simulations discussed earlier. The creation of a database/lookup table of transition... 

A major criticism of the stochastic probability approach is that relatively slow secondary reactions, for which the near-equilibrium assumption does not apply, cannot be accommodated. In this situation, it is necessary to derive and solve simultaneous partial differential equations for mass conservation and obtain expressions for the first and second moments of the elution profile and the concomitant plate height arising from slow kinetics of secondary equilibrium. If, once again, the process can be represented as involving the reversible binding of two forms, the resolution of the interconverting species can be given by [59]... 

Mass-action-type kinetic differential equations can be identified with the CCD model, while the more often used stochastic model is the CDS model. These two models will be the object of detailed investigations in Chapters 4 and 5. 

A direct problem is attacked if, for example, the solution of a kinetic differential equation is determined either analytically or numerically, or when the qualitative properties of the solutions are investigated (in this case a subset of. (R X /I) is determined in which stochastic model of a reaction is simulated. 

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