Stochastic modeling of physical processes

More generally, while in a stationary system = P( ) contains no 

As another example consider the internal vibrational energy of a diatomic solute molecule, for example, CO, in a simple atomic solvent (e.g. Ar). This energy can be monitored by spectroscopic methods, and we can follow processes such as thermal (or optical) excitation and relaxation, energy transfer, and energy migration. The observable of interest may be the time evolution of the average 

The relevance of stochastic descriptions brings out the issue of their theoretical and numerical evaluation. Instead of solving the equations of motion for 6x102 degrees of freedom we now face the much less demanding, but still challenging need to construct and to solve stochastic equations of motion for the few relevant variables. The next section describes a particular example. 

More generally, while in a stationary system P n,t) = P n) contains no dynamical information, such information is contained in joint distributions like P n2,t2, n, t ) (the probability to observe i molecules at time and 2 molecules at time t2) and conditional distributions such as P ri2, /2 I i, ) (the probability to observe 112 molecules at time t2 given that i molecules were observed at time /i). We will expand on these issues in Section 7.4. 

---

They point out that at the heart of technical simulation there must be unreality otherwise, there would not be need for simulation. The essence of the subject linder study may be represented by a model of it that serves a certain purpose, e.g., the use of a wind tunnel to simulate conditions to which an aircraft may be subjected. One uses the Monte Carlo method to study an artificial stochastic model of a physical or mathematical process, e.g., evaluating a definite integral by probability methods (using random numbers) using the graph of the function as an aid. 

Detailed modeling study of practical sprays has a fairly short history due to the complexity of the physical processes involved. As reviewed by O Rourke and Amsden, 3l() two primary approaches have been developed and applied to modeling of physical phenomena in sprays (a) spray equation approach and (b) stochastic particle approach. The first step toward modeling sprays was taken when a statistical formulation was proposed for spray analysis. 541 Even with this simplification, however, the mathematical problem was formidable and could be analyzed only when very restrictive assumptions were made. This is because the statistical formulation required the solution of the spray equation determining the evolution of the probability distribution function of droplet locations, sizes, velocities, and temperatures. The spray equation resembles the Boltzmann equation of gas dynamics[542] but has more independent variables and more complex terms on its right-hand side representing the effects of nucleations, collisions, and breakups of droplets. 

In the past, the equivalence between the size distribution generated by the Smoluchowski equation and simple statistical methods [9, 12, 40-42] was a source of some confusion. The Spouge proof and the numerical results obtained for the kinetics models with more complex aggregation physics, e.g., with a presence of substitution effects [43,44], revealed the non-equivalence of kinetics and statistical models of polymerization processes. More elaborated statistical models, however, with the complete analysis made repeatedly at small time intervals have been shown to produce polymer size distributions equivalent to those generated kinetically [45]. Recently, Faliagas [46] has demonstrated that the kinetics and statistical models which are both the mean-field models can be considered as special cases of a general stochastic Markov process. 

An alternative view of the same physical process is to model the interaction of the reaction coordinate with the environment as a stochastic process through the generalized Langevin equation (GEE)... 

Somewhat closer to the designation of a microscopic model are those diffusion theories which model the transport processes by stochastic rate equations. In the most simple of these models an unique transition rate of penetrant molecules between smaller cells of the same energy is determined as function of gross thermodynamic properties and molecular structure characteristics of the penetrant polymer system. Unfortunately, until now the diffusion models developed on this basis also require a number of adjustable parameters without precise physical meaning. Moreover, the problem of these later models is that in order to predict the absolute value of the diffusion coefficient at least a most probable average length of the elementary diffusion jump must be known. But in the framework of this type of microscopic model, it is not possible to determine this parameter from first principles . 

By employing a very strong external field, a gedankexperiment may be set up whereby the natural thermal motion of the molecules is put in competition with the aligning effect of the field. This method reveals some properties of the molecular liquid state which are otherwise hidden. In order to explain the observable effects of the applied fields, it is necessary to use equations of motion more generally valid than those of Benoit. These equations may be incorporated within the general structure of reduced model theory " (RMT) and illustrate the use of RMT in the context of liquid-state molecular dynamics. (Elsewhere in this volume RMT is applied to problems in other fields of physics where consideration of stochastic processes is necessary.) In this chapter modifications to the standard methods are described which enable the detailed study of field-on molecular dynamics. 

This model is also used for many problems outside of physics and chemistry, for example, card shuffling, power supply, and parking lot congestion problems [Feller (SS)]. A more exhaustive survey of the varied application of this model can be found in Feller (39), Bharucha-Reid (40), Bellman 41), Chandrasekhar 42), and many other books on stochastic processes. 

---