Exact mean value equations

Closed exact equations for the mean values (x) whose probability distribution is P x t), will now be derived. As they remain valid in all cases, however, they have a more complicated structure than (3.77) which makes their evaluation more difficult. 

Firstly the Fokker-Planck equation and its formal solution is written in the more abstract form  

Secondly, a so-called relevant distribution P (x A(t)) depending on time t via the C parameters A(t) = Ai (t), only is chosen. The relevant dis- 

Thirdly, a closed exact set of equations has to be set up for the - as yet -unknown mean values or, equivalently, for the parameters Ai(t),. ..,A (t). This can be achieved by evaluating the expression 

It turns out that an appropriate choice for the relevant distribution is 

---

It may be noted that using Underwood s approximation (equation 9.10), the calculated values for the mean temperature driving forces are 41.9 K and 39.3 K for counter- and co-current flow respectively, which agree exactly with the logarithmic mean values. 

Polarization equations of the type (14.35) or (14.38) contain the mean values of true current density. However, the rate-determining step is more often concentrated at just a few segments of the electrode the true working area changes continuously and an exact determination of this area is practically impossible. This gives rise to difficulties in an interpretation of polarization data. 

Equation (28) is still exact. To introduce the classical-path approximation, we assume that the nuclear dynamics of the system can be described by classical trajectories that is, the position operator x is approximated by its mean value, namely, the trajectory x t). As a consequence, the quantum-mechanical operators of the nuclear dynamics (e.g., Eh (x)) become classical functions that depend parametrically on x t). In the same way, the nuclear wave functions dk x,t) become complex-valued coefficients dk x t),t). As the electronic dynamics is evaluated along the classical path of the nuclei, the approximation thus accounts for the reaction of the quantum DoE to the dynamics of the classical DoF. 

Lowest amount of the analyte in the sample which can be detected but not necessarily quantified as an exact value. LOD is expressed as a concentration or a quantity derived from the smallest signal xLi which can be detected with a reasonable certainty for an analytical procedure. The LOD can be determined using the equation xL = xbl +ksbl with xbl and sbl as the mean value and the standard deviation of blank, respectively. LOD of an element or element species is often used as 3sbl (3cr) of blank.3... 

An exact analysis of plasticizers according to this method involves fluctuations in the Rp values, characterized by the authors as daily deviations. To eliminate these fluctuations the authors related their Rp values to tricresyl phosphate (of high meta content) with a mean Rp value equated to 0.66. 

The differences between the cluster skeletons of the three molecules of 2 are very small with the mean values of the Ru-Ru distances being similar and the mean Ru-C(carbide) distances being identical. The most notable differences between the structures arise from the orientation of the tricarbonyl units attached to the apical Ru atoms above and below the molecular equator of the octahedral cluster (the molecular equator is defined as the Ru4 plane in which the bridging carbonyl ligand is present). The two tricarbonyl units are almost exactly staggered in the crystal obtained from benzene, whereas they approach an eclipsed conformation in the other polymorph. Although the 13C-NMR spectrum of 2 has not been recorded in solution (or in the solid state), it is not unreasonable to anticipate that... 

By means of equations (30) and (36) it is possible to calculate the number of aromatic rings per average molecule of mineral oil fractions with Ra < 1 and Ra > 1, respectively, from the molecular weights and the percentages of carbon atoms in aromatic structure. The calculated values are not exact values since it is assumed that the multiple ring systems are kata-condensed however, this often seems to be the case. 

We used p instead of = in Equation 5.37 because the exact numerical value depends on the definition of the uncertainties—you will see different values in different books. If we define At in Figure 5.13 as the full width at half maximum or the root-mean-squared deviation from the mean, the numerical value in Equation 5.37 changes. It also changes a little if the distribution of frequencies is not Gaussian. Equation 5.37 represents the best possible case more generally we write... 

V-clcctron state T, correlation energy can be defined for any stationary state by Ec = E — / o, where Eo = ( //1) and E = ( // 4 ). Conventional normalization ) = ( ) = 1 is assumed. A formally exact functional Fc[4>] exists for stationary states, for which a mapping — F is established by the Schrodinger equation [292], Because both and p are defined by the occupied orbital functions occupation numbers nt, /i 4>, E[p and E[ (p, ] are equivalent functionals. Since E0 is an explicit orbital functional, any approximation to Ec as an orbital functional defines a TOFT theory. Because a formally exact functional Ec exists for stationary states, linear response of such a state can also be described by a formally exact TOFT theory. In nonperturbative time-dependent theory, total energy is defined only as a mean value E(t), which lies outside the range of definition of the exact orbital functional Ec [ ] for stationary states. Although this may preclude a formally exact TOFT theory, the formalism remains valid for any model based on an approximate functional Ec. 

This is the continuity equation for turbulent flow when the mean motion is two-dimensional. It will be noted that this equation has exactly the same form as the continuity equation for two-dimensional steady laminar flow with the mean values of the velocity components substituted in place of the steady values that apply in laminar flow. This result can, in fact, be deduced by intuitive reasoning and simply states that if an elemental control volume through which the fluid flows is considered, then over a sufficiently long period of time, the fluctuating components contribute nothing to the mass transfer through this control volume. 

The simplest way of employing this relationship is to perform a calibration experiment with a standard solution that has a known concentration [A]q. After a carefully measured reaction time t, [A], is determined and is used to evaluate the constant e by means of Equation 29-27. Unknowns are then analyzed by measuring fA], after exactly the same reaction time and employing the calculated value for e to compute the analyte concentrations. 

It is important to recognize that the criteria of instability for Gk> 0 and stability for Gk - Ois strictly valid only for Gk independent of /. Indeed, if Gk depends on t, the equation (4 312) is nonautonomous, and it is well known that stability can generally be established only by exact integration of the equation. If Gk is slowly varying, a local analysis based on an instantaneous value of Gk will be qualitatively correct for a finite time interval, but for more general time-dependent forms for Gk, we should not be surprised if the situation turns out to be more complex. To see an example of this, we can consider the special case in which the bubble volume changes periodically with time about some mean value, as may occur in response to an oscillatory variation in p at a frequency different from the resonant frequency co0 (given just prior to Eq. (4-229)). Thus, we let... 

Equation 35-34 is closely related to the radial equation 18-37 of the hydrogen atom and may be solved in exactly the same manner. If this is done, it is found that it is necessary to restrict v to the values 0, 1, 2, in order to obtain a polynomial solution.1 If we solve for W by means of Equations 35-35 and the definitions of Equations 35-33, 35-32, and 35-28, we obtain the equation... 

This equation is only approximately true, because we have multiplied the probability that X is in the subinterval (x, x, - - Ax) by x,-, which is only one of the values of X in the subinterval. However, Eq. (5.68) can be made more and more nearly exact by making n larger and larger and Ax smaller and smaller in such a way that n Ax is constant. In this limit, f becomes independent of Ax. We replace the symbol fi by f(Xi) and assume that /(x,) is an integrable function of x,. It must be at least piecewise continuous. Our formula for the mean value of x now becomes an integral as defined in Eq. (5.21) ... 

We take a variational approach so that there is no question of requiring an exact solution of the Schrodinger equation for reference. Let J be a variational trial function for the valence electrons of a many-electron system and let h be the valence many-electron Hamiltonian. We seek a minimum in the mean value of H with respect to such (normalised) trial functions together with the constraint that be orthogonal to the wavefunction of a subset of the electrons (the core). We will then recast the equation into a pseudopotential form and examine this form with a view to modelling the pseudopotential. 

Supposing that u is constant, by virtue of the mean value theorem, the exact solution of equation (1.79) at time step n+1, can be written as follows ... 

In fliis case, the potential well has a sharp and deep minimum. This means that the vicinity of this minimum determines flic value of the integral in the more exact Muller (76) equation. For examination of this assumption, this integral was calculated numerically and according to the ap-... 

We are interested in properties of the ammonia molecule in its ground and excited states e.g.. we would like to know the mean value of the nitrogen-hydrogen distance. Only quantum mechanics gives a method for calculation this value (p. 26) we have to calculate the mean value of an operator with the ground-state wave function. But where could this function be taken from Could it be a solution of the Schrddinger equation Impossible unfortunately, this equation is too difficult to solve (14 particles cf. problems with exact solutions in Chapter 4). 

---