Fluctuation term

The average of the quantity (f Ar) has been equated to zero in obtaining Equation (24) in as much as this quantity is equally likely to be positive or negative because of the fluctuating term Ar, and the average overall possible occurrences vanishes. Equation (24) is valid in both the deformed and undeformed states. It may be written in terms of Xx, Xy, and Xz as... 

The first two terms on the right-hand side of this expression are the spatial transport terms. For homogeneous turbulence, these terms will be exactly zero. For inhomogeneous turbulence, the molecular transport term vV2e will be negligible (order Re,1). Spatial transport will thus be due to the unclosed velocity fluctuation term (u, e), and the unclosed... 

Figure 13 shows the optical birefringence measured on the (OOl)c and (llO)c faces [19]. According to the indicatrix deformation approach [25], the optical birefringence, An, for the (llO)c face can detect the antiferrodistortive transition, the square of long-range order (P>, and its fluctuation term (SP) in STO by ... 

We shall meet more general Fokker-Planck equations the special form (1.1) is also called Smoluchowski equation , generalized diffusion equation , or second Kolmogorov equation . The first term on the right-hand side has been called transport term , convection term , or drift term the second one diffusion term or fluctuation term . Of course, these names should not prejudge their physical interpretation. Some authors distinguish between Fokker-Planck equations and master equations, reserving the latter name to the jump processes considered hitherto. 

This equation relates T, the size of the fluctuating term, to the damping... 

We return to the bistable situation with clearly separated time scales and suppose that the system starts out at a site near the boundary of Da, i.e., close to the macroscopically unstable point (f)b. Then in the initial stage fluctuations across are not improbable. There is therefore a non-negligible probability that the system does not follow the macroscopic path towards (j>a but ends up in 4>c instead. Thus near a point of macroscopic instability fluctuations give rise to a macroscopic effect. It is therefore no longer possible to separate a macroscopic part from a fluctuation term as in (X.2.9) and treat the fluctuations as a small perturbation. The conclusion is that there exists no mesostate related to the stationary macrostate. Any probability distribution originally peaked near (j)b evolves in time and does not remain localized. The evolution occurs in three successive stages. 

Thus we have derived the Kramers equation (VIII.7.4) as an approximation for short tc. It becomes exact in the white noise limit (3.12). The coefficient of the fluctuation term is the integrated autocorrelation function of the fluctuating force, in agreement with (IX.3.5) and (IX.3.6).110... 

In section 2 it was mentioned that a damping term in the Schrodinger equation violates the conservation of probability. It is possible, however, to compensate this loss by adding artificially a suitable fluctuating term. The result will be an equation for if/ resembling the classical Langevin equation. 

Remark. It is easily seen that the second term of (5.2) by itself causes the norm of if/ to change. In order that this is compensated by the fluctuating term the two terms must be linked, as is done by the relation U = V V. This resembles the classical fluctuation-dissipation theorem, which links both terms by the requirement that the fluctuations compensate the energy loss so as to establish the equilibrium. The difference is that the latter requirement involves the temperature T of the environment that makes it possible to suppress the fluctuations by taking T = 0 without losing the damping. This is the reason why in classical theory deterministic equations with damping exist, see XI.5. 

The most general method to solve the set (2.2.20), (2.2.21) seems to be their transformation in terms of different-order correlations. Let us present reactant concentrations as a sum of the mean value and fluctuating term... 

In terms of the master equation for the Markov process the formal kinetics is nothing but the mean-field theory where the fluctuation terms like that on the r.h.s. of equation (2.2.43) are neglected. Strictly speaking, the macroscopic description, equation (2.1.2), were correct if the fluctuation terms vanished as V —> oo. In a general case the function P(N, t) does not satisfy the Poisson distribution [16, 27] in particular, °N (N> ... 

While the hydrodynamic theory always predicts this near equivalence of the friction and the viscosity, microscopic theories seem to provide a rather different picture. In the mode coupling theory (MCT), the friction on a tagged molecule is expressed in terms of contributions from the binary, density, and transverse current modes. The latter can of course be expressed in terms of viscosity. However, in a neat liquid the friction coefficient is primarily determined not by the transverse current mode but rather by the binary collision and the density fluctuation terms [59]. Thus for neat liquids there is no a priori reason for such an intimate relation between the friction and viscosity to hold. 

For molecules and colloids the forces due to molecular impacts are large, but they are short in duration and random in direction, as can be surmised from the observation of Brownian motion. These forces can be described by a rapidly fluctuating term A(t) in the equation of motion. With this Brownian term added to the other forces on the right, Eq. 3.8 assumes the form... 

There are monopolar fluctuations of the net charge on the colloid and its surrounding solution there are dipolar fluctuations, the first moment of the ionic-charge distribution around the colloid as well as polarization of the colloid itself. Monopolar and dipolar fluctuations couple to create a hybrid interaction, d-m, again in the limit of the n = 0 sampling frequency at which the ions are able to fluctuate. The salt solution screens even the dipolar fluctuation the same way that the low-frequency-fluctuation term is screened in planar interactions. For dielectric spheres of radius a, ss whose incremental contribution to dielectric response is a =... 

Table C.4. "Thin" dielectric cylinders parallel and at all angles, interaxial separation z <

If ever needed, it is trivial to add magnetic-fluctuation terms to all these point-particle results. 

C.4. "Thin" dielectric cylinders, parallel and at all angles, interaxial separation z << radius R Lifshitz form retardation, magnetic, and ionic-fluctuation terms not included C.4.a. Parallel, interaxial separation z C.4.b.l. At an angle 0, minimal interaxial separation z C.4.b.2. Torque r(z, 0)... 

This is the energy equation for two-dimensional turbulent flow. Comparing it with the equation for laminar flow shows that in turbulent flow extra terms arise because of the fluctuating velocity and temperature components. These terms arise because of the enthalpy transport caused by these fluctuating terms. 

Even without the fluctuation term the plasma equations differ from the neutral particle equations in two important aspects ... 

Using these standard notations one obtains the ten equations listed now. The terms Sn[fa] in these equations due to atomic and molecular interactions with neutral particle species n will be detailed below. For simplicity here the fluctuation term will not be considered. It is common (bad) practise in plasma edge modeling to account for its effect only in an ad-hoc manner (see below) after the fluid equations have first been derived without it. A comprehensive discussion, and proper derivation with this term included from the beginning, for plasma edge models, is given in [4]. 

In contrast with formulae like Eq. (4.72), p. 90, there is no surviving factor of 1/2 with the fluctuation term of Eq. (4.79). This fluctuation contrihution comes from a quadratic cross-term and the customary factor of 2 with that cross-term has cancelled the 1/2 again see Eq. (4.15), p. 67. 

Inserting now into (191) the moment (199a) and the deviation tensor (195) without the translational fluctuation term, we have ... 

The growth law can also be written a.s the sum of mean and fluctuating terms ... 

Iwamoto and Seno (1980) proposed a three dimensional mathematical model and analyzed it under the influence of external fluctuation terms. In (1982) Iwamoto et al. gave a three dimensional model to explain small and large amplitude oscillations observed in the reactions. 

There is no detail of the derivation of the equations of the Debye-Hiickel theory that has not been criticized. Its incompleteness mathematically is evident, since only the first term of the expansion of equations (3) and (29) is used. The extensions of the theory to overcome this deficiency are, however, briefly considered below. A possibly more serious deficiency of the theory as given is that it does not take account of "fluctuation terms. This amounts to the statement that the Boltzmann equation does not yield a correct average potential this being subject to wide variations for which allowance should he made in the theory. This part of the criticism is still in active progress, and cannot be briefly summarized. In Chapters 8 and 12 an attempt will be made to show both the successes and the limitations of the theory in its present state from the experimental point of view. 

---