Stochastic differential equation systems

Setting W(t) = (2kBT )l/2W(t), the Langevin relation—describing the motion of the particle in the phase space—can be written as a two-variable stochastic differential equation system of the form of... 

Chapter 4 is devoted to the description of stochastic mathematical modelling and the methods used to solve these models such as analytical, asymptotic or numerical methods. The evolution of processes is then analyzed by using different concepts, theories and methods. The concept of Markov chains or of complete connected chains, probability balance, the similarity between the Fokker-Plank-Kolmogorov equation and the property transport equation, and the stochastic differential equation systems are presented as the basic elements of stochastic process modelling. Mathematical models of the application of continuous and discrete polystochastic processes to chemical engineering processes are discussed. They include liquid and gas flow in a column with a mobile packed bed, mechanical stirring of a liquid in a tank, solid motion in a liquid fluidized bed, species movement and transfer in a porous media. Deep bed filtration and heat exchanger dynamics are also analyzed. 

At its most basic level, molecular dynamics is about mapping out complicated point sets using trajectories of a system of ordinary differential equations (or, in Chaps. 6-8, a stochastic-differential equation system). The sets are typically defined as the collection of probable states for a certain system. In the case of Hamiltonian dynamics, they are directly associated to a region of the energy landscape. The trajectories are the means by which we efficiently explore the energy surface. In this chapter we address the design of numerical methods to calculate trajectories. 

Imagine a distribution po(X) which we may take to be an initial macroscopic state of a stochastic differential equation system. This might be a smooth probability density such as a Gaussian, the indicator function for a small disk D in the phase space, or, in the extreme case a Dirac delta distribution (indicating that all initial conditions are clustered at a single point in phase space). The density evolves according to the partial differential equation... 

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. 

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. 

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. 

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. 

Other approaches to genetic networks include study of small circuits with either differential equations or stochastic differential equations. The use of stochastic equations emphasizes the point that noise is a central factor in the dynamics. This is of conceptual importance as well as practical importance. In all the families of models studied, the non-linear dynamical systems typically exhibit a number of dynamical attractors. These are subregions of the system s state space to which the system flows and in which it thereafter remains. A plausible interpretation is that these attractors correspond to the cell types of the organism. However, in the presence of noise, attractors can be destabilized. 

A stochastic differential equation is a differential equation whose coefficients are random numbers or random functions of the independent variable (or variables). Just as in normal differential equations, the coefficients are supposed to be given, independently of the solution that has to be found. Hence stochastic differential equations are the appropriate tool for describing systems with external noise (see IX.5). 

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(<)... 

The stochastic differential equation (2.2.15) could be formally compared with the Fokker-Planck equation. Unlike the complete mixing of particles when a system is characterized by s stochastic variables (concentrations the local concentrations in the spatially-extended systems, C(r,t), depend also on the continuous coordinate r, thus the distribution function f(Ci,..., Cs]t) turns to be a functional, that is real application of these equations is rather complicated. (See [26, 34] for more details about presentation of the Fokker-Planck equation in terms of the functional derivatives and problems of normalization.)... 

Since a number of particles involved in any reaction event are small, a change in concentration is of the order of 1 /V. Therefore, we can use for the system with complete particle mixing the asymptotic expansion in this small parameter 1 /V. The corresponding van Kampen [73, 74] procedure (see also [27, 75]) permits us to formulate simple rules for deriving the Fokker-Planck or stochastic differential equations, asymptotically equivalent to the initial master equation (2.2.37). It allows us also to obtain coefficients Gij in the stochastic differential equation (2.2.2) thus liquidating their uncertainty and strengthening the relation between the deterministic description of motion and density fluctuations. 

Polystochastic models are used to characterize processes with numerous elementary states. The examples mentioned in the previous section have already shown that, in the establishment of a stochastic model, the strategy starts with identifying the random chains (Markov chains) or the systems with complete connections which provide the necessary basis for the process to evolve. The mathematical description can be made in different forms such as (i) a probability balance, (ii) by modelling the random evolution, (iii) by using models based on the stochastic differential equations, (iv) by deterministic models of the process where the parameters also come from a stochastic base because the random chains are present in the process evolution. 

The stochastic model accepts a Markov type connection between both elementary states. So, with ai2Ar, we define the transition probability from type I to type II, whereas the transition probability from type II to a type I is a2iAr. By Pi(x,t) and P2(x, t) we note the probability of locating the microparticle at position x and time T with a type I or respectively a type II evolution. With these introductions and notations, the general stochastic model (4.71) gives the particularization written here by the following differential equation system ... 

To shed further light on this aspect, let us consider the system of stochastic differential equations... 

In order to relate the system of Eqs. (77) to a time-independent Fokker-Planck formalism, we replace that set of stochastic differential equations with the equivalent one. 

In order to understand how the algorithm actually works and to construct an explicit expression for the error it is not convenient to work with the metadynamics equations (12) in their full generality. Instead, we notice that the finite temperature dynamics of the collective variables satisfies, under rather general conditions, a stochastic differential equation [54,55]. Furthermore, in real systems the quantitative behavior of metadynamics is perfectly reproduced by the Langevin equation in its strong friction limit [56]. This is due to the fact that all the relaxation times are usually much smaller than the typical diffusion time in the CV space. Hence, we model the CVs evolution with a Langevin t3rpe dynamics ... 

In a previous work [56] we deduced an alternative expression for the error (11) by performing extensive numerical simulations of the stochastic differential equation (19). The metadynamics parameters, w/tq, Sa, and the system-dependent parameters, f3, D and S were systematically varied, and for each choice of the parameters the error (11) was computed by repeating several metadynamics reconstructions. Fitting the results, we obtained that the data were reproduced within an accuracy of 20% by... 

We add to A another matrix, S, whose elements are drawn from a normal probability distribution N(0, with zero mean and a variance denoted by cr. A fraction /of the elements of the matrix A will be populated in this way the rest will be zero, so 0 0, the matrix S is very sparse, and different variables, x, are relatively uncoupled by the perturbations. Likewise, as n increases, so the system becomes larger, with more and more components of the vectors coming into play. Finally, as the variance of the stochastic perturbations, cr, increases, so the broader is the influence of random effects from one interaction pathway to others. Thus, the complex system is now turned into the stochastic differential equation... 

The Langevin approach has been used by many authors in order to treat nonlinear systems. This is of importance to us since the equations of rotational motion are intrinsically nonlinear. The concept of a nonlinear Langevin equation is also subject to a number of criticisms. These have been discussed extensively by van Kampen [58] (Chapters 8 and 14). In our calculations, we shall encounter stochastic differential equations of the form... 

There are many systems that can fluctuate randomly in space and time and cannot be described by deterministic equations. For example. Brownian motion of small particles occurs randomly because of random collisions with molecules of the medium in which the particles are suspended. It is useful to model such systems with what are known as stochastic differential equations. Stochastic differential equations feature noise terms representing the behavior of random elements in the system. Other examples of stochastic behavior arise in chemical reaction systems involving a small number of molecules, such as in a living cell or in the formation of particles in emulsion drops, and so on. A useful reference on stochastic methods is Gardiner (2003). 

The inclusion of fluctuations in the description of nonlinear systems is done by two approaches [50]. On the one hand side one adds fluctuating sources in the nonlinear dynamics, transforming thus the differential equations into stochastic differential equations. The second way is the consideration of probability densities for the considered variables and the formulation of their evolution laws. Both concepts are introduced shortly in the next two subsections. 

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