Stochastic process density function

The indices k in the Ihs above denote a pair of basis operators, coupled by the element Rk. - The indices n and /i denote individual interactions (dipole-dipole, anisotropic shielding etc) the double sum over /x and /x indicates the possible occurrence of interference terms between different interactions [9]. The spectral density functions are in turn related to the time-correlation functions (TCFs), the fundamental quantities in non-equilibrium statistical mechanics. The time-correlation functions depend on the strength of the interactions involved and on their modulation by stochastic processes. The TCFs provide the fundamental link between the spin relaxation and molecular dynamics in condensed matter. In many common cases, the TCFs and the spectral density functions can, to a good approximation, be... 

Equation (64) is immediately derivable from Eq. (65) by simply taking the average of both sides of Eq. (65). The deterministic approach always assumes that E AB can be replaced by E A E B, and as Eq. (63) shows, this amounts to setting D1 C f) = 0, and this is true only for a delta function type of density function, i.e., one in which all central moments vanish. By a similar heuristic argument, it can be seen that the deterministic solution and the stochastic mean values are always the same for unimolecular processes. This was pointed out (but never really proved in general) by McQuarrie.12... 

A computational method was developed by Gillespie in the 1970s [381, 388] from premises that take explicit account of the fact that the time evolution of a spatially homogeneous process is a discrete, stochastic process instead of a continuous, deterministic process. This computational method, which is referred to as the stochastic simulation algorithm, offers an alternative to the Kolmogorov differential equations that is free of the difficulties mentioned above. The simulation algorithm is based on the reaction probability density function defined below. 

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) ... 

The limitations associated with (7) are essentially a consequence of the stochastic nature of atmospheric transport and diffusion. Because the wind velocities are random functions of space and time, the airborne pollutant concentrations are random variables in space and time. Thus, the determination of the Cj, in the sense of being a specified quantity at any time, is not possible, but we can at best derive the probability density functions satisfied by the c. The complete specification of the probability density function for a stochastic process as complex as atmospheric diffusion is almost never possible. Instead, we must adopt a less desirable but more feasible approach, the determination of certain statisical moments of Ci, notably its mean, . (The mean concentration can be... 

In the case of a single test particle B in a fluid of molecules M, the effective one-dimensional potential f (R) is — fcrln[R gBM(f )]. where 0bm( ) is th radial distribution function of the solvent molecules around the test particle. In this chapter it will be assumed that 0bm( )> equilibrium property, is a known quantity and the aim is to develop a theory of diffusion of B in which the only input is bm( )> particle masses, temp>erature, and solvent density Pm- The friction of the particles M and B will be taken to be frequency indep>endent, and this should restrict the model to the case where > Wm, although the results will be tested in Section III B for self-diffusion. Instead of using a temporal cutoff of the force correlation function as did Kirkwood, a spatial cutoff of the forces arising from pair interactions will be invoked at the transition state Rj of i (R). While this is a natural choice because the mean effective force is zero at Rj, it will preclude contributions from beyond the first solvation shell. For a stationary stochastic process Eq. (3.1) can then be... 

The power spectrum was defined here as a property of a given stochastic process. In the physics literature it is customary to consider a closely related fimction that focuses on the properties of the thennal environment that couples to the system of interest and affects the stochastic nature of its evolution. This is the spectral density that was discussed in Section 6.5.2. (see also Section 8.2.6). To see the connection between these functions we recall that in applications of the theory of stochastic processes to physical phenomena, the stochastic process x(Z) represents a physical observable A, say a coordinate or a momentum of some observed particle. Suppose that this observable can be expanded in harmonic nonnal modes uj as in Eq. (6,79)... 

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... 

Models that seek to value options or describe a yield curve also describe the dynamics of asset price changes. The same process is said to apply to changes in share prices, bond prices, interest rates and exchange rates. The process by which prices and interest rates evolve over time is known as a stochastic process, and this is a fundamental concept in finance theory. Essentially, a stochastic process is a time series of random variables. Generally, the random variables in a stochastic process are related in a non-random manner, and so therefore we can capture them in a probability density function. A good introduction is given in Neftci (1996), and following his approach we very briefly summarise the main features here. 

It should be noted that we integrate with respect to the forward variable y in (3.236). In this case, (3.236) has a very nice probabilistic interpretation. Consider the Brownian motion B t), which is a stochastic process with independent increments, such that B(t + s) - B(s) is normally distributed with zero mean and variance 2Dt. The corresponding transition probability density function p y, t x) is given by (3.237). Therefore the solution (3.236) has a probabilistic representation... 

Consider now a collection of stochastic processes X(t) that are the solutions of the SDE dX = fl(X)df + fe(X)dW. If the initial conditions are drawn from a distribution with given density p(x, 0) we would like to derive an equation for the corresponding density at time t, p(x, t). We assume that p is C°° for all t and that the density tends to zero exponentially fast as x oo, whereas a and b are C°° functions which grow at most polynomially fast in these limits, that is, there are constants pa, r]b such that... 

The simplest representation of a time varying stochastic process is an ordinary function in which a mnnher of stochastic variables are incorporated. In this case we have a simple deterministic function of the time, in which some of the variables in this function may he random, making therefore the process stochastic. In addition, if the associated probability density functions change in time (in terms of associated mean values and standard deviations increase or decrease, for instance), the random variables have different values in every realisation, and hence every realisation is different. 

The spectral density function of the fluctuation can be calculated from the autocorrelation function by the Wiener-Khintchine relation (Wiener, 1930 Khintchine, 1934). The original formulation of the theorem refers to stationary stochastic processes for a possible generalisation see, for example, Lampard, 1954. The relationship connects the autocorrelation function to the spectrum ... 

An auto-ignition process of a non-homogeneous mixture in fuel concentration was fundamentally investigated by means of a numerical calculation based on chemical kinetics and the stochastic approach. The auto-ignition process of n-heptane is calculated by means of a semi-detailed mechanism and the non-imiform state of turbulent mixing is statistically described by means of probability density functions and the stochastic method. The following conclusions are derived from the results ... 

Let the probability density for the stochastic film drainage process be represented by p h, f t hi, / ) where and are the initial values of if and F, respectively. This density function p = p h,f t hi,f ) satisfies the Fok-ker-Planck equation... 

A stochastic process is also characterized by its spectral density, the Fourier transform of its autocorrelation function. The autocorrelation function of a (stationary stochastic process) measures the correlation of the process at different time intervals while the spectral density measures the amplitudes of the component waves of different frequencies. A white noise process has a constant spectral density (i.e., the same amplitude for all frequencies) and the band-limited noise has a frequency band over which the spectral density is nearly constant. 

For an external magnetic field Bo along the z direction, the electron spins are oriented parallel or antiparallel to the z direction. Modulation of the components of the local field in the xy plane due to a stochastic process then induces stochastic electron spin transitions (spin flips) that contribute to longitudinal relaxation with time constant T. For historical reasons longitudinal relaxation is often termed spin-lattice relaxation. The relaxation rate T is proportional to the spectral density /(co) of the stochastic process at the resonance frequency Mo of the transition under consideration. This spectral density is maximum for a correlation time Tc of the stochastic process that fulfils the condition wqTc = 1. As correlation times usually are a monotonic function of temperature, there is a temperature for which the relaxation rate attains a maximum and T attains a minimum. Measurements of 7] as a function of temperature can thus be used to infer the correlation time of a dynamic process. By varying the external field Bo and thus mq, the time scale can be shifted to which EPR experiments are most sensitive. 

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