Stochastic function

Geostatistics is on the basis of the concept of stochastic functions, in which a particular set of measured values are considered spatially dependent random variables. A random variable is a variable which can adopt a series of different values according to a probability distribution. 

Because the velocity u contains the random component u, the concentration c is a stochastic function since, by virtue of Eq. (2.2), c is a function of u. The mean value of c, as expressed in Eq. (2.5), is an ensemble mean formed by averaging c over the entire ensemble of identical experiments. Temporal and spatial mean values, by contrast, are obtained by averaging v ues from a single member of the ensemble over a period or area, respectively. The ensemble mean, which we have denoted by the angle brackets ( ), is the easiest to deal with mathematically. Unfortunately, ensemble means are not measurable quantities, although under the conditions of the ergodic theorem they can be related to observable temporal or spatial averages. In Eq. (2.7) the mean concentration (c) represents a true ensemble mean, whereas if we decompose c as... 

In order to discuss the origin of these terms we need to allow the spins to have anisotropic shielding tensors. Molecular tumbling in solution makes the chemical shielding in the direction of the external magnetic field a stochastic function of time and acts therefore as a relaxation mechanism, called the chemical shielding anisotropy (CSA) mechanism. The Hamiltonian for each of the two spins, analogous to Eq. (5), contains therefore two... 

Classical studies of the relaxation processes, caused by translational diffusion, have been presented in the early days by Abragam (18), Torrey (136) and Pfeifer (137). Abragam (18) found, by solving the diffusion equation, the following form of the correlation function for the stochastic function Z>o under translational diffusion of two spins 1/2 ... 

To make these heuristic ideas precise we define a stochastic function 7(x) whose sample functions are the y(x). Take n different points xv in the interval (0, L) and label them in increasing order... 

As an example take a gas in a cylindrical vessel. In addition to the energy there is one other constant of the motion the angular momentum around the cylinder axis. The 6A/-dimensional phase space is thereby reduced to subshells of 6N-2 dimensions. Consider a small sub volume in the vessel and let Y(t) be the number of molecules in it. According to III.2 Y(t) is a stochastic function, with range n = 0,1,2,. .., N. Each value Y = n delineates a phase cell ) one expects that Y(t) is a Markov process if the gas is sufficiently dilute and that pi is approximately a Poisson distribution if the subvolume is much smaller than the vessel. 

The stochastic function X(t) by itself is not Markovian. This is an example of the fact discussed in IV. 1 If one has an r-component Markov process and one ignores some of the components, the remaining sstochastic process but in general not Markovian. Conversely, it is often possible to study non-Marko-vian processes by regarding them as the projection of a Markov process with more components. We return to this point in IX.7. 

The Rayleigh particle is the same particle as the Brownian particle, but studied on a finer time scale. Time differences At are regarded that are small compared to the time in which the velocity relaxes, but, of course, still large compared to the duration of single collisions with the gas molecules. Thus the stochastic function to be considered is the velocity rather than the position. It is sufficient to confine the treatment to one dimension this is sometimes emphasized by the name Rayleigh piston . 0... 

Alternatively one may postulate that all higher cumulants are zero. This specifies all stochastic properties of L(t) in terms of the single parameter F. The L(t) defined in this way is called Gaussian white noise. From the mathematical point of view it does not really exist as a stochastic function (no more than the delta function exists as a function) and in physics it never really occurs but serves as a model for any rapidly fluctuating force. 

Before embarking on this discussion one fact must be established. In many applications the autocorrelation function of L(t) is not really a delta function, but merely sharply peaked with a small rc > 0. Accordingly L(t) is a proper stochastic function, not a singular one. Then (4.5) is a well-defined stochastic differential equation (in the sense of chapter XVI) with a well-defined solution. If one now takes the limit rc —> 0 in this solution, it becomes a solution of the Stratonovich form (4.8) of the Fokker-Planck equation. This theorem has been proved officially, but the result can also be seen as follows. 

The autocorrelation function k is not a delta function, and (t) is some well-defined, non-singular stochastic function, so that (7.1) is also well-defined. Such Langevin-like equations have been amply discussed in the literature. The fact... 

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(<)... 

Due to the contagious distribution of the toxicant and the stochastic function describing exposure and effects, multiple discrete outcomes often are possible from the same initial conditions (e.g., ranging from extinction to the reaching of the carrying capacity for a patch). 

With turbulent combustion viewed as a random (or stochastic) process, mathematical bases are available for addressing the subject. A number of textbooks provide introductions to stochastic processes (for example, [55]). In turbulence, any stochastic variable, such as a component of velocity, temperature, or the concentration of a chemical species, which we might call v, is a function of the continuous variables of space x and time t and is, therefore, a stochastic function. A complete statistical description of a stochastic function would be provided by a probability-density functional, tf, defined by stating that the probability of finding the function in a small range i (x, t) about a particular function v(x, t) is [t (x, t)]<3t (x, t) ... 

Although it is essential conceptually to make a distinction between a stochastic function and a particular function i (x, t) that appears as the argument of we choose not to do so notationally for the sake of simplicity. Symbols like v always will refer to arguments of probabilities unless there is an indication to the contrary. 

It would be most useful to apply this to the field population in general. In addition, models are currently bdng studied which allow for the formation of stars of different masses by using different reaction channels in the Langevin systems of the following sections in order to see whether the IMF is a stable stochastic function of time. If it changes, the star formation rate, the metallicity evolution of the disk, and the IMF variations become inexorably linked and impossible to separate. ... 

For an inhomogeneous medium with a chaotic structure, the permittivity e(r, co) and the conductivity c> (r, co) are random (stochastic) functions of the coordinates r. 

The arrangement can be still more complex. For example, in a class of neural networks called thermodynamic models, each unit can output 0 or 1, and a stochastic function operating on the unit s inputs determines the probability that its output will be 1 (2). 

The internal spin interaction Hamiltonian Hmt can be decomposed into spatial Tm[ ua(l) ] and spin Sm degrees of freedom Hin (t) = 2mTm[ ua(t) ]Sm. The spatial contribution, hereafter an NMR interaction rank-2 tensor T, is a stochastic function of time Tm[ ua(t) because it depends on generalized coordinates < ( ) of the system (atomic and molecular positions, electronic or ionic charge density, etc.) that are themselves stochastic variables. To clarify the role of these coordinates in the NMR features, a simple model is developed below.19,20 At least one physical quantity should distinguish the parent and the descendant phase after a phase transition. For simplicity, we suppose that the components of the interaction tensor only depend on one scalar variable u(t) whose averaged value is modified from m to m + ( at a phase transition. To take into account the time fluctuations, this variable is written as the sum of three terms, i.e. u(t) — m I I 8us(t). The last term is a stationary stochastic process such that — 0, where <.) denotes a... 

Conditioning often is needed in turbulent combustion. For example, often there is interest in flows (for example, jets and wakes) that are partly turbulent and partly nonturbulent (in that the flow is partly irrotational), and the conditioning may be that rotational fluid be present. These flows are said to exhibit intermittency of the turbulence an intermittency function may be defined as a stochastic function having a value /(x, t) = 0 in irrotational flow and /(x, t) = 1 in rotational flow, and conditioning on / = 1 may be desirable for a number of purposes. We may decompose a probability-density function as... 

We now come back to the important example of two spin 1/2 nuclei with the dipole-dipole interaction discussed above. In simple physical terms, we can say that one of the spins senses a fluctuating local magnetic field originating from the other one. In terms of the Hamiltonian of equation Bl.13.8. the stochastic function... 

P and Q are stochastic functions of X, both with an expected value of zero ... 

Table 2.4 shows the power spectra of some input functions. Impulse and stochastic functions have high limit frequencies. Therefore impulse or stochastic methods should be prefered for tracer experiments in soil columns. 

To apply the Schrodinger form of the continuum model, the steps involved are (1) calculate the potential, V y) (2) solve the Schrodinger equation for t/r and co and (3) transform back to the stochastic function, W x,f). To show these steps, we consider the example of PVAc undergoing a downward step from40°C to 30°C. This is one of Kovac s experiments shown in Figure 4.2. 

Event A During performance of a mission of mileage L a vehicle failure occurs that is repaired with the defined maintenance resource U. In this case the mission is completed and it is possible to express by way of two-dimensional stochastic function (1) the probability of the event occurring ... 

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