Differential equation stochastic

1 The stochastic differential equations and the Fokker-Planck equation 

As was shown in Section 2.1, in some cases thermal fluctuations of reactant densities affect the reaction kinetics. However, the equations of the formal chemical kinetics are not suited well enough to describe these fluctuations in fact they are introduced ad hoc through the initial conditions to equations. The role of fluctuations and different methods for incorporating them into formal kinetics equations were discussed more than once. 

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 nj t) turns out to be the average of the stochastic function Cj(f) 

Here Ci t) are solutions of the stochastic differential Ito-Stratonovich equation [26, 34, 99] 

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 / = /(ci. Cg ). The latter is nothing but solution of the Fokker-Planck equation [26, 34] 

Processes in which a certain time-dependent random variable X(t) exists are called stochastic processes. Loosely, these are the processes that evolve probabilistically in time. We can measure values jri,.j 2 3 --M etc., of X(t) at times ti,t2,t, . and we assume that a set of joint probability densities, p(xi,t 2 t2vX3,h, . exists that describes the system completely. For a treatise on stochastic processes, the reader is referred to Gardiner (2003). The applications related to chemical engineering can be followed from Rao et al. (1974a and 1974b). 

The conditional probability densities can be defined in terms of the joint probability density functions as 

The definitions in Equation (2.253) are valid independently of the ordering of the times. However, it is usual to consider only times that increase from right to left, i.e.. 

Stochastic differential equations then obviously describe the evolution of stochastic processes in terms of a dependent random variable X(t). 

Conditional probabilities can be seen as predictions of the future values of X(t) (i.e., x 2. .. at times t, t2.) given the knowledge of the past values (yi,y2 - - times Xi,X2,.. .). A stochastic process is said to be completely independent if 

We indicate a generalization of the equations for the random walk without exact proof. If a differential equation contains also nondeterministic terms, i.e., noise, then the solution can be described only statistically. The equations are often written down as system of equations, or as a vector equation. We are content here with the linear formulation  

Here n (f) is a white noise. This function obeys the following conditions  

the expectance is zero and the autocorrelation function is equal to the Dirac delta function. In addition, we postulate the initial condition x(0) = Xq. 

Finally, we observe that by calculating cr t ) and generating random increments Rn J (0, y(tn)) we have a discrete representation of the continuous time stochastic process Y(t), effectively an exact stroboscopic representation. Also it is natural to view the stochastic integral as the solution of the stochastic differential equation (SDE) with zero initial condition  

This notion of stochastic integration carries over to stochastic integration with respect to more general categories of stochastic processes. Importantly, with appropriate regularity we have the familiar relation from calculus  

In this section we briefly derive a more general stochastic differential equation as the limit of a biased random walk. The treatment here is intuitive, based on the concepts of limiting processes and stochastic integration introduced above. For a rigorous treatment of the material presented here, it is recommended to consult a reference such as the excellent book of Nelson [279] (which also contains a very lively historical discussion). 

We introduce a pair of smooth functions a(x, f) and b x, 1), where a represents a drift tendency and b the average magnitude of the random jump. In the interval [nSl, (n + l)6t], a step 3X = X +i - X of the random walk will be given by 

Take the diffusion limit St 0, Sx /St 1 with vSt = t fixed, then we obtain, under the Ito convention. 

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Hamiltonian, but in practice one often begins with a phenomenological set of equations. The set of macrovariables are chosen to include the order parameter and all otlier slow variables to which it couples. Such slow variables are typically obtained from the consideration of the conservation laws and broken synnnetries of the system. The remaining degrees of freedom are assumed to vary on a much faster timescale and enter the phenomenological description as random themial noise. The resulting coupled nonlinear stochastic differential equations for such a chosen relevant set of macrovariables are collectively referred to as the Langevin field theory description. 

Kramers solution of the barrier crossing problem [45] is discussed at length in chapter A3.8 dealing with condensed-phase reaction dynamics. As the starting point to derive its simplest version one may use the Langevin equation, a stochastic differential equation for the time evolution of a slow variable, the reaction coordinate r, subject to a rapidly statistically fluctuating force F caused by microscopic solute-solvent interactions under the influence of an external force field generated by the PES F for the reaction... 

This discussion suggests that even the reference trajectories used by symplectic integrators such as Verlet may not be sufficiently accurate in this more rigorous sense. They are quite reasonable, however, if one requires, for example, that trajectories capture the spectral densities associated with the fastest motions in accord to the governing model [13, 15]. Furthermore, other approaches, including nonsymplectic integrators and trajectories based on stochastic differential equations, can also be suitable in this case when carefully formulated. 

The stochastic differential equation and the second moment of the random force are insufficient to determine which calculus is to be preferred. The two calculus correspond to different physical models [11,12]. It is beyond the scope of the present article to describe the difference in details. We only note that the Ito calculus consider r t) to be a function of the edge of the interval while the Stratonovich calculus takes an average value. Hence, in the Ito calculus using a discrete representation rf t) becomes r] tn) i]n — y n — A i) -I- j At. Developing the determinant of the Jacobian -... 

B. Oksendal, Stochastic differential equations An introduction with applications , Springer-Verlag, Berlin, 1995... 

The exponential decay predicted by the Onsager-Machlup theory, and by the Langevin and similar stochastic differential equations, is not consistent with the conductivity data in Fig. 8. This and the earlier figures show a constant value for >.(x) at larger times, rather than an exponential decay. It may be that if the data were extended to significantly larger time scales it would exhibit exponential decay of the predicted type. 

Equation (2.5) is a stochastic differential equation. Some required characteristics of stochastic process may be obtained even from this equation either by cumulant analysis technique [43] or by other methods, presented in detail in Ref. 15. But the most powerful methods of obtaining the required characteristics of stochastic processes are associated with the use of the Fokker-Planck equation for the transition probability density. 

The diffusion term in this expression differs from the Fokker-Planck equation. This difference leads to a fT-dependent term in the drift term of the corresponding stochastic differential equation (6.177), p. 294. 

In terms of the equivalent stochastic differential equation (6.106), p. 279, this choice yields a diffusion matrix of the form B() = S -(i(), where the matrix G does not depend on the molecular-diffusion coefficients. 

In order to illustrate the properties of the FP model, it is easier to rewrite it in terms of an equivalent stochastic differential equation (SDE) (Arnold 1974 Risken 1984 Gardiner... 

V, ip, x, and t) in the PDF transport equation makes it intractable to solve using standard discretization methods. Instead, Lagrangian PDF methods (Pope 1994a) can be used to express the problem in terms of stochastic differential equations for so-called notional particles. In Chapter 7, we will discuss grid-based Eulerian PDF codes which also use notional particles. However, in the Eulerian context, a notional particle serves only as a discrete representation of the Eulerian PDF and not as a model for a Lagrangian fluid particle. The Lagrangian Monte-Carlo simulation methods discussed in Chapter 7 are based on Lagrangian PDF methods. 

The stochastic differential equations in (6.152)-(6.154) generate a Lagrangian PDF which is conditioned on the initial location 125... 

This is a direct result of forcing the stochastic differential equations to correspond to the Eulerian PDF. From (6.152), it follows that... 

Thus, correspondence between the notional-particle system and the Eulerian PDF of the flow requires agreement at the moment level. In particular, it requires that (U(x, /)) = (U (r) X (0 = x) and ((x, t)) = (0 (r) X (O = x). It remains then to formulate stochastic differential equations for the notional-particle system which yield the desired correspondence. 

Owing to the sensitivity of the chemical source term to the shape of the composition PDF, the application of the second approach to model molecular mixing models in Section 6.6, a successful model for desirable properties. In addition, the Lagrangian correlation functions for each pair of scalars (( (fO fe) ) should agree with available DNS data.130 Some of these requirements (e.g., desirable property (ii)) require models that control the shape of /, and for these reasons the development of stochastic differential equations for micromixing is particularly difficult. 

Note that Lagrangian correspondence does not automatically imply Eulerian correspondence, since the latter also requires that /x (x t) must be a uniform distribution for all t. Nevertheless, Lagrangian correspondence is the preferable choice, and is obtained by working directly with the stochastic differential equations. 

The model equation for particle position, (7.27), is a stochastic differential equation (SDE). The numerical solution of SDEs is discussed in detail by Kloeden and Platen (1992).28 Using a fixed time step At, the most widely used numerical scheme for advancing the particle position is the Euler approximation ... 

Arnold, L. (1974). Stochastic Differential Equations Theory and Applications. New York John Wiley. 

Kloeden, P. E. and E. Platen (1992). Numerical Solution of Stochastic Differential Equations. Berlin Springer-Verlag. 

Constraints may be introduced either into the classical mechanical equations of motion (i.e., Newton s or Hamilton s equations, or the corresponding inertial Langevin equations), which attempt to resolve the ballistic motion observed over short time scales, or into a theory of Brownian motion, which describes only the diffusive motion observed over longer time scales. We focus here on the latter case, in which constraints are introduced directly into the theory of Brownian motion, as described by either a diffusion equation or an inertialess stochastic differential equation. Although the analysis given here is phrased in quite general terms, it is motivated primarily by the use of constrained mechanical models to describe the dynamics of polymers in solution, for which the slowest internal motions are accurately described by a purely diffusive dynamical model. 

The theory of Brownian motion for a constrained system is more subtle than that for an unconstrained system of pointlike particles, and has given rise to a substantial, and sometimes confusing, literamre. Some aspects of the problem, involving equilibrium statistical mechanics and the diffusion equation, have been understood for decades [1-8]. Other aspects, particularly those involving the relationships among various possible interpretations of the corresponding stochastic differential equations [9-13], remain less thoroughly understood. This chapter attempts to provide a self-contained account of the entire theory. 

The transformation rule given in Eq. (2.166) is instead an example to the so-called Ito formula for the transformation of the drift coefficients in Ito stochastic differential equations [16]. ft is shown in Section IX that V q) and V ( ) are equal to the drift coefficients that appear in the Ito formulation of the stochastic differential equations for the generalized and Cartesian coordinates, respectively. 

In this section, we consider the description of Brownian motion by Markov diffusion processes that are the solutions of corresponding stochastic differential equations (SDEs). This section contains self-contained discussions of each of several possible interpretations of a system of nonlinear SDEs, and the relationships between different interpretations. Because most of the subtleties of this subject are generic to models with coordinate-dependent diffusivities, with or without constraints, this analysis may be more broadly useful as a review of the use of nonlinear SDEs to describe Brownian motion. Because each of the various possible interpretations of an SDE may be defined as the limit of a discrete jump process, this subject also provides a useful starting point for the discussion of numerical simulation algorithms, which are considered in the following section. 

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes (North-Holland, Amsterdam, 1981). 

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