Distribution function time-dependent

Diffusion. - Distribution of the diffusivitity of fluid in a horizontally oriented cylinder was demonstrated by NMR imaging in two papers on a granular flow system and in the earth s magnetic field. Correlation time (ic) and diffusion coefficient (D = Xc) imaging (CTDCI) was applied to a granular flow system of 2 mm oil-filled sphere rotated in a half-filled horizontal cylinder, ie. to an Omstein-Uhlenbeck process with a velocity autocorrelation function. Time dependent apparent diffusion coefficients are measured, and Tc... 

A systematic study of the relaxation of rubbing induced birefringence in PS has been conducted. Extensive and clear experimental evidence have been foimd that show the absence of the physical aging effects in the relaxation of RIB, and the relaxation of RIB involves very small length scales. The RIB relaxation is then modeled by a relaxation times distribution function that depends only on temperature but not on thermal or strain history. An individual birefringence elements model has been proposed and a systematic way has been devised to extract the parameters in the model from specifically designed experiments, namely the Temperature Lag measurements and the Continuous Curve measurements. The results predicted by the model agree well with experiments. 

The state of a gas is characterized by the distribution function / which depends on the peculiar velocity, the radius vector and the time t. For a stationary gas, the distribution function is Maxwellian and is denoted by f , which is given by... 

Distribution and Correlation Functions.—We consider a single spherical particle with position r and velocity v at time t in a concentrated dispersion of mean number density p. The distribution function measures the probability of finding a particle (the same or another particle) with position r" and velocity v" at time t". The osmotic equation of state is related to a time-averaged distribution function that depends on r alone, whereas the dynamic behaviour depends on time-dependent functions. A basic premise of statistical mechanics is that a time-average is equivalent to an ensemble average at fixed time the ensemble average is denoted by angular brackets (...). 

Wavefunctions describing time-dependent states are solutions to Schrodinger s time-dependent equation. The absolute square of such a wavefunction gives a particle distribution function that depends on time. The time evolution of this particle distribution function is the quantum-mechanical equivalent of the classical concept of a trajectory. It is often convenient to express the time-dependent wave packet as a linear combination of eigenfunctions of the time-independent hamiltonian multiplied by their time-dependent phase factors. 

The state of the surface is now best considered in terms of distribution of site energies, each of the minima of the kind indicated in Fig. 1.7 being regarded as an adsorption site. The distribution function is defined as the number of sites for which the interaction potential lies between and (rpo + d o)> various forms of this function have been proposed from time to time. One might expect the form ofto fio derivable from measurements of the change in the heat of adsorption with the amount adsorbed. In practice the situation is complicated by the interaction of the adsorbed molecules with each other to an extent depending on their mean distance of separation, and also by the fact that the exact proportion of the different crystal faces exposed is usually unknown. It is rarely possible, therefore, to formulate the distribution function for a given solid except very approximately. 

P(x, t) dx has the familiar bell shape of a normal distribution function [Eq. (1.39)], the width of which is measured by the standard deviation o. In Eq. (9.83), t takes the place of o. It makes sense that the distribution of matter depends in this way on time, with the width increasing with t. 

It is truly possible to imagine the characteristics of an ideal radiopharmaceutical only in the context of a specific disease and organ system to which it might be appHed. Apart from the physical factors related to the radioisotope used, the only general characteristic that is important in defining the efficacy of these materials is the macroscopic distribution in the body, or biodistribution. This time-dependent distribution at the organ level is a function of many parameters which may be divided into four categories factors related to deUvery of the radiopharmaceutical to a particular tissue factors related to the extraction of the compound from circulation factors related to retention of the compound by that tissue and factors deterrnined by clearance. The factors in the last set are rarely independent of the others. 

Sihcate solutions of equivalent composition may exhibit different physical properties and chemical reactivities because of differences in the distributions of polymer sihcate species. This effect is keenly observed in commercial alkah sihcate solutions with compositions that he in the metastable region near the solubihty limit of amorphous sihca. Experimental studies have shown that the precipitation boundaries of sodium sihcate solutions expand as a function of time, depending on the concentration of metal salts (29,58). Apparently, the high viscosity of concentrated alkah sihcate solutions contributes to the slow approach to equihbrium. 

If it cannot be guaranteed that the adsorbate remains in local equilibrium during its time evolution, then a set of macroscopic variables is not sufficient and an approach based on nonequihbrium statistical mechanics involving time-dependent distribution functions must be invoked. The kinetic lattice gas model is an example of such a theory [56]. It is derived from a Markovian master equation, but is not totally microscopic in that it is based on a phenomenological Hamiltonian. We demonstrate this approach... 

Note carefully that the same random variable (function) may have many different distribution functions depending on the distribution function of the underlying function X(t). We will avoid confusion on this point by adopting the convention that, in any one problem, and unless an explicit statement to the contrary is made, all random variables are to be used in conjunction with a time function X(t) whose distribution function is to be the same in all expressions in which it appears. With this convention, the notation F is just as unambiguous as the more cumbersome notation so that we are free to make use of whichever seems more appropriate in a given situation. 

This apparent time dependent cell disruption is caused because of the statistically random distribution of the orientation of the cells within a flow field and the random changes in that distribution as a function of time, the latter is caused as the cells spin in the flow field in response to the forces that act on them. In the present discussion this is referred to as apparent time dependency in order to distinguish it from true time-dependent disruption arising from anelastic behaviour of the cell walls. Anelastic behaviour, or time-dependent elasticity, is thought to arise from a restructuring of the fabric of the cell wall material at a molecular level. Anelasticity is stress induced and requires energy which is dissipated as heat, and if it is excessive it can weaken the structure and cause its breakage. 

Equations (41.15) and (41.19) for the extrapolation and update of system states form the so-called state-space model. The solution of the state-space model has been derived by Kalman and is known as the Kalman filter. Assumptions are that the measurement noise v(j) and the system noise w(/) are random and independent, normally distributed, white and uncorrelated. This leads to the general formulation of a Kalman filter given in Table 41.10. Equations (41.15) and (41.19) account for the time dependence of the system. Eq. (41.15) is the system equation which tells us how the system behaves in time (here in j units). Equation (41.16) expresses how the uncertainty in the system state grows as a function of time (here in j units) if no observations would be made. Q(j - 1) is the variance-covariance matrix of the system noise which contains the variance of w. 

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