Statistical/probabilistic models examples

Unlike simple random variables that have no space or time dependence, the statistics of the random velocity field in homogeneous turbulence can be described at many different levels of complexity. For example, a probabilistic theory could be formulated in terms of the set of functions U(x, t) (x, t) e R3 x R However, from a CFD modeling perspective, such a theory would be of little practical use. Thus, we will consider only one-point and two-point formulations that describe a homogeneous turbulent flow by the velocity statistics at one or two fixed points in space and/or time. 

A probabilistic or statistical model that does provide for uncertainty associated with the system is illustrated in Figure 4.3. For this example, it is assumed that the underlying response is zero and that any value of response other than zero is caused by some random process. This model might appropriately describe the vertical velocity (speed and direction) of a single gas molecule in a closed system, or white noise in an electronic amplifier - in each case, the average value is expected to be zero, and deviations are assumed to be random. The model is... 

Monte Carlo simulation can involve several methods for using a pseudo-random number generator to simulate random values from the probability distribution of each model input. The conceptually simplest method is the inverse cumulative distribution function (CDF) method, in which each pseudo-random number represents a percentile of the CDF of the model input. The corresponding numerical value of the model input, or fractile, is then sampled and entered into the model for one iteration of the model. For a given model iteration, one random number is sampled in a similar way for all probabilistic inputs to the model. For example, if there are 10 inputs with probability distributions, there will be one random sample drawn from each of the 10 and entered into the model, to produce one estimate of the model output of interest. This process is repeated perhaps hundreds or thousands of times to arrive at many estimates of the model output. These estimates are used to describe an empirical CDF of the model output. From the empirical CDF, any statistic of interest can be inferred, such as a particular fractile, the mean, the variance and so on. However, in practice, the inverse CDF method is just one of several methods used by Monte Carlo simulation software in order to generate samples from model inputs. Others include the composition and the function of random variable methods (e.g. Ang Tang, 1984). However, the details of the random number generation process are typically contained within the chosen Monte Carlo simulation software and thus are not usually chosen by the user. 

Needless to say, real world chemical explosions are multi-step phenomena involving the competition between various pathways, many of which contain autocatalytic of inhibitory effects associated with the appearance of free radicals and chain reactions. We expect that in such a complex dynamics the role of fluctuations will be even more important than in the simple models studied in the present Chapter. More generally, it seems to us that chain reactions and explosive behavior should be characteristic examples of a fluctuation chemistry [1], in which probabilistic elements are built into the system and confer to the process an essentially statistical character. 

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